perspectives in mathematical logic - 01 - admissible sets and structures. an approach to...

Perspectives in Mathematical Logic

Jon Barwise

Admissible Setsand Structures

Springer-VerlagBerlin Heidelberg NewYork

PerspectivesinMathematical Logic

Ω-Group:R.O.Gandy H.Hermes A.Levy G.H.MϋllerG.E.Sacks D.S.Scott

Jon Barwise

Admissible Sets andStructuresAn Approach to Definability Theory

Springer-VerlagBerlin Heidelberg New York 1975

JON BARWISEDepartment of Mathematics, University of WisconsinMadison, WI 53706 / USAandU.C.L.A., Los Angeles, CA 90024 / USA

With 22 Figures

AMS Subject Classification (1970): 02 F 27, 02 B 25, 02 H10, 02 K 35

ISBN 3-540-07451-1 Springer-Verlag Berlin Heidelberg New YorkISBN 0-387-07451-1 Springer-Verlag New York Heidelberg Berlin

Library of Congress Cataloging in Publication Data. Barwise, Jon. Admissible sets andstructures. (Perspectives in mathematical logic). Bibliography: p. Includes index.1. Admissible sets. 2. Definability theory. I. Title. QA9.B29. 511'.3. 75-33102.

This work is subject to copyright. All rights are reserved, whether the whole or part of thematerial is concerned, specifically those of translation, reprinting, re-use of illustrations,broadcasting, reproduction by photocopying machine or similar means, and storage indata banks. Under § 54 of the German Copyright Law where copies are made for otherthan private use, a fee is payable to the publisher, the amount of the fee to be determinedby agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1975.Printed in Germany.

Typesetting and printing: Zechnersche Buchdruckerei, Speyer. Bookbinding:Konrad Triltsch, Wurzburg.

To my motherand the memory of my father

Preface to the Series

On Perspectives. Mathematical logic arose from a concern with the nature andthe limits of rational or mathematical thought, and from a desire to systematisethe modes of its expression. The pioneering investigations were diverse and largelyautonomous. As time passed, and more particularly in the last two decades, inter-connections between different lines of research and links with other branches ofmathematics proliferated. The subject is now both rich and varied. It is the aimof the series to provide, as it were, maps or guides to this complex terrain. Weshall not aim at encyclopaedic coverage; nor do we wish to prescribe, like Euclid,a definitive version of the elements of the subject. We are not committed to anyparticular philosophical programme. Nevertheless we have tried by critical discussionto ensure that each book represents a coherent line of thought; and that, bydeveloping certain themes, it will be of greater interest than a mere assemblageof results and techniques.

The books in the series differ in level: some are introductory some highlyspecialised. They also differ in scope: some offer a wide view of an area, otherspresent a single line of thought. Each book is, at its own level, reasonably self-contained.Although no book depends on another as prerequisite, we have encouraged authorsto fit their book in with other planned volumes, sometimes deliberately seekingcoverage of the same material from different points of view. We have tried toattain a reasonable degree of uniformity of notation and arrangement. However,the books in the series are written by individual authors, not by the group. Plansfor books are discussed and argued about at length. Later, encouragement is givenand revisions suggested. But it is the authors who do the work; if, as we hope,the series proves of value, the credit will be theirs.

History of the Ω-Group. During 1968 the idea of an integrated series of monographson mathematical logic was first mooted. Various discussions led to a meeting atOberwolfach in the spring of 1969. Here the founding members of the group (R.0. Gandy, A. Levy, G. H. Mύller, G. Sacks, D. S. Scott) discussed the projectin earnest and decided to go ahead with it. Professor F. K. Schmidt and ProfessorHans Hermes gave us encouragement and support. Later Hans Hermes joinedthe group. To begin with all was fluid. How ambitious should we be? Shouldwe write the books ourselves? How long would it take? Plans for author less bookswere promoted, savaged and scrapped. Gradually there emerged a form and amethod. At the end of an infinite discussion we found our name, and that of

VIII Preface to the Series

the series. We established our centre in Heidelberg. We agreed to meet twicea year together with authors, consultants and assistants, generally in Oberwolfach.We soon found the value of collaboration: on the one hand the permanence ofthe founding group gave coherence to the over-all plans; on the other hand thestimulus of new contributors kept the project alive and flexible. Above all, wefound how intensive discussion could modify the authors ideas and our own. Oftenthe battle ended with a detailed plan for a better book which the author waskeen to write and which would indeed contribute a perspective.

Acknowledgements. The confidence and support of Professor Martin Earner ofthe Mathematisches Forschungsinstitut at Oberwolfach and of Dr. Klaus Petersof Springer- Verlag made possible the first meeting and the preparation of a provisionalplan. Encouraged by the Deutsche Forschungsgemeinschaft and the HeidelbergerAkademie der Wissenschaften we submitted this plan to the Stiftung Volkswagenwerkwhere Dipl. Ing. Penschuck vetted our proposal; after careful investigation hebecame our adviser and advocate. We thank the Stiftung Volkswagenwerk for agenerous grant (1970-73) which made our existence and our meetings possible.

Since 1974 the work of the group has been supported by funds from the HeidelbergAcademy; this was made possible by a special grant from the Kultusministeriumvon Baden-Wurttemberg (where Regierungsdirektor R. Goll was our counsellor).The success of the negotiations for this was largely due to the enthusiastic supportof the former President of the Academy, Professor Wilhelm Doerr. We thank allthose concerned.

Finally we thank the Oberwolfach Institute, which provides just the right atmos-phere for our meetings, Drs. Ulrich Feigner and Klaus Gloede for all their help,and our indefatigable secretary Elfriede Ihrig.

Oberwolfach R. O. Gandy H. HermesSeptember 1975 A. Levy G. H. Mutter

G. Sacks D. S. Scott

Author's Preface

It is only before or after a book is written that it makes sense to talk about thereason for writing it. In between, reasons are as numerous as the days. Lookingback, though, I can see some motives that remained more or less constant in thewriting of this book and that may not be completely obvious.

I wanted to write a book that would fill what I see as an artificial gap betweenmodel theory and recursion theory.

I wanted to write a companion volume to books by two friends, H. J. Keisler'sModel Theory for Infίnίtary Logic and Y.N. Moschovakis' Elementary Inductionon Abstract Structures, without assuming material from either.

I wanted to set forth the basic facts about admissible sets and admissibleordinals in a way that would, at long last, make them available to the logic studentand specialist alike. I am convinced that the tools provided by admissible setshave an important role to play in the future of mathematical logic in general anddefinability theory in particular. This book contains much of what I wish everylogician knew about admissible sets. It also contains some material that everylogician ought to know about admissible sets.

Several courses have grown out of my desire to write this book. I thank thestudents of these courses for their interest, suggestions and corrections. A roughfirst draft was written at Stanford during the unforgettable winter and spring of1973. The book was completed at Heatherton, Freeland, Oxfordshire during theacademic year 1973—74 while I held a research grant from the University ofWisconsin and an SRC Fellowship at Oxford. I wish to thank colleagues at thesethree institutions who helped to make it possible for me to write this book,particularly Professors Feferman, Gandy, Keisler and Scott. I also appreciatethe continued interest expressed in these topics over the past years by ProfessorG. Kreisel, and the support of the Ω-Group during the preparation of this book.I would like to thank Martha Kirtley and Judy Brickner for typing and JohnSchlipf, Matt Kaufmann and Azriel Levy for valuable comments on an earlierversion of the manuscript. I owe a lot to Dana Scott for hours spent helpingprepare the final manuscript. I would also like to thank Mrs. Nora Day and theother residents of Freeland for making our visit in England such a pleasant one.

X Author's Preface

A final but large measure of thanks goes to my family: to Melanie for allowingme to use her room as a study during the coal strike; to Jon Russell for help withthe corrections but most of all to Mary Ellen for her encouragement and patience.To Mary Ellen, on this our eleventh anniversary, I promise to write at most onebook every eleven years.

September 19, 1975 K.J.B.Santa Monica

Table of Contents

Introduction 1

Part A. The Basic Theory 5

Chapter I. Admissible Set Theory 7

1. The Role of Urelements 72. The Axioms of KPU 93. Elementary Parts of Set Theory in KPU 114. Some Derivable Forms of Separation and Replacement 145. Adding Defined Symbols to KPU 186. Definition by Σ Recursion 247. The Collapsing Lemma 308. Persistent and Absolute Predicates 339. Additional Axioms 38

Chapter II. Some Admissible Sets 42

1. The Definition of Admissible Set and Admissible Ordinal 422. Hereditarily Finite Sets 463. Sets of Hereditary Cardinality Less Than a Cardinal K 524. Inner Models: the Method of Interpretations 545. Constructible Sets with Urelements HYP^ Defined 576. Operations for Generating the Constructible Sets 627. First Order Definability and Substitutable Functions 698. The Truncation Lemma 729. The Lόvy Absoluteness Principle 76

Chapter III. Countable Fragments of L^ 78

1. Formalizing Syntax and Semantics in KPU 782. Consistency Properties 843. SDΐ-Logic and the Omitting Types Theorem 874. A Weak Completeness Theorem for Countable Fragments 925. Completeness and Compactness for Countable Admissible Fragments 95

XII Table of Contents

6. The Interpolation Theorem 1037. Definable Well-Orderings 1058. Another Look at Consistency Properties 109

Chapter IV. Elementary Results on HYP^ 113

1. On Set Existence 1132. Defining Πj and Σ} Predicates 1163. Π} and Δ} on Countable Structures 1224. Perfect Set Results 1275. Recursively Saturated Structures 1376. Countable $R-Admissible Ordinals 1447. Representability in 501-Logic 146

PartB. The Absolute Theory 151

Chapter V. The Recursion Theory of Σ! Predicates on Admissible Sets . . 1531. Satisfaction and Parametrization 1532. The Second Recursion Theorem for KPU 1563. Recursion Along Well-founded Relations 1584. Recursively Listed Admissible Sets 1645. Notation Systems and Projections of Recursion Theory 1686. Ordinal Recursion Theory: Projectible and Recursively Inaccessible

Ordinals 1737. Ordinal Recursion Theory: Stability 1778. Shoenfield's Absoluteness Lemma and the First Stable Ordinal . . 189

Chapter VI. Inductive Definitions 197

1. Inductive Definitions as Monotonic Operators 1972. Σ Inductive Definitions on Admissible Sets 2053. First Order Positive Inductive Definitions and HYPan 2114. Coding HF^ on Wl 2205. Inductive Relations on Structures with Pairing 2306. Recursive Open Games 242

Part C. Towards a General Theory 255

Chapter VII. More about L^ω 257

1. Some Definitions and Examples 2572. A Weak Completeness Theorem for Arbitrary Fragments 2623. Pinning Down Ordinals: the General Case 2704. Indiscernibles and upward Lόwenheim-Skolem Theorems 2765. Partially Isomorphic Structures 2926. Scott Sentences and their Approximations 2977. Scott Sentences and Admissible Sets 303

Table of Contents XIII

Chapter VIII. Strict Π} Predicates and Konig Principles 311

1. The Konig Infinity Lemma 3112. Strict Π} predicates: Preliminaries 3153. Konig Principles on Countable Admissible Sets 3214. Konig Principles K! and K2 on Arbitrary Admissible Sets 3265. Kόnig's Lemma and Nerode's Theorem: a Digression 3346. Implicit Ordinals on Arbitrary Admissible Sets 3397. Trees and Σi Compact Sets of Cofinalityω 3438. Σ t Compact Sets of Cofinality Greater than ω 3529. Weakly Compact Cardinals 356

Appendix. Nonstandard Compactness Arguments and the Admissible Cover. 3651. Compactness Arguments over Standard Models of Set Theory . . . 3652. The Admissible Cover and its Properties 3663. An Interpretation of KPU in KP 3724. Compactness Arguments over Nonstandard Models of Set Theory . 378

References 380

Index of Notation 386

Subject Index 388

Major Dependencies

Appendix

(Va denotes the first three §§ of Chapter V, similarly for Vila.)

Introduction

Since its beginnings in the early sixties, admissible set theory has become amajor source of interaction between model theory, resursion theory and settheory. In fact, for the student of admissible sets the old boundaries betweenfields disappear as notions merge, techniques complement one another, analogiesbecome equivalences, and results in one field lead to results in another. This isthe view of admissible sets we hope to share with the reader of this book.

Model theory, recursion theory and set theory all deal, in part, with problemsof definability and set existence. Definability theory is (by definition) that partof mathematical logic which deals with such problems. The Craig InterpolationTheorem, Kleene's analysis of Δj sets by means of the hyperarithmetic sets,GodeΓs universe L of constructible sets and Shoenfΐeld's Absoluteness Lemmaare all major contributions to definability theory. The theory of admissiblesets takes such apparently divergent results and makes them converge in a singlecoherent body of thought, one with ramifications for all parts of logic.

This book is written for the student who has taken a good first space yeargraduate course in logic. The specific material we presuppose can be summarizedas follows. The student should understand the completeness, compactness andLόwenheim-Skolem theorems as well as the notion of elementary submodel.He should be familiar with the basic properties of recursive functions and re-cursively enumerable (hereinafter r.e.) sets. The student should have seen thedevelopment of intuitive set theory in some formal theory like ZF (Zermelo-Fraenkel set theory). His life will be more pleasant if he has some familiaritywith the constructible sets before reading §§ II.5,6 or V.4—8, but our treatmentof constructible sets is self-contained.

A logical presentation of a reasonably advanced part of mathematics (whichthis book attempts to be) bears little relation to the historical development ofthat subject. This is particularly true of the theory of admissible sets with itscomplicated and rather sensitive history. On the other hand, a student is handi-capped if he has no idea of the forces that figured in the development of his sub-ject. Since the history of admissible sets is impossible to present here, we com-promise by discussing how some of the older material fits into the currenttheory. We concentrate on those topics that are particularly'relevant to thisbook. The prerequisites for understanding the introduction are rather greaterthan those for understanding the book itself.

2 Introduction

Recursive ordinals and hyperarithmetic sets. In retrospect, the study of ad-missible ordinals began with the work of Church and Kleene on notation systemsand recursive ordinals (Church-Kleene [1937], Church [1938], Kleene [1938].)This study began as a recursive counterpart to the classical theory of ordinals;the least nonrecursive ordinal ω{ is the recursive analogue of ω l 9 the first un-countable ordinal. (Similarly for ωc

2 and ω2, etc.) The theory of recursive ordinalshad its most important application when Kleene [1955] used it in his studyof the class of hyperarithmetic sets, the smallest reasonably closed class of setsof natural numbers which can be considered as given by the structure,yΓ = <ω, + , x > of natural numbers. Kleene's theorem that

hyperarithmetic = Δ}

provided a construction process for the class of Δ} sets and constituted the firstreal breakthrough into (applied) second order logic. One of our aims is toprovide a similar analysis for any structure 501. Given 50Ϊ we construct the smallestadmissible set HYPW above 50ί (in § II.5) and use it in the study of definabilityproblems over 50ί (in Chapters IV and VI).

The study of hyperarithmetic sets generated a lot of discussion of the analogybetween, on the one hand, the Π} and hyperarithmetic sets, and the r.e. and re-cursive sets on the other. These analogies became particularly striking whenexpressed in terms of representability in ω-logic and first order logic, by Grze-gorczyk, Mostowski and Ryll-Nardzewski [1959]. The analogy had some defects,though, as the workers realized at the time. For example, the image of a hyper-arithmetic function is hyperarithmetic, not just Π} as the analogy would suggest.

Kreisel [1961] analyzed this situation and discovered that the correct an-alogy is between Π} and hyperarithmetic on the one hand and r.e. and finite(not recursive) on the other. He went on to develop a recursion theory on thehyperarithmetic sets via a notation system. (He also proved the Kreisel Com-pactness Theorem for ω-logic: If a Π} theory T of second order arithmetic isinconsistent in ω-logic, then some hyperarithmetic subset T0^T is inconsistentin ω-logic.) This theory was expanded in the metarecursion theory of Kreisel-Sacks [1965]. Here one sees how to develop, by means of an ordinal notationsystem, an attractive recursion theory on ω{ such that for X^ω:

X is Πj iff X is ωί-r.e,

X is Δ} iff X is ωί-finite.

In § IV.3 we generalize this, by means of HYP^, to show that for any countablestructure 501 and any relation R on 501:

R is Π} on 50Ϊ iff R is HYP^-r.e.,

R is Δ} on 50Ϊ iff R is HYP^-finite,

thus providing a construction process for the Δ} relations over any countablestructure 50Ϊ whatsoever. The use of notation systems then allows us to transferresults from HYP^ to 501 itself (see §§ V.5 and VI.5).

Introduction 3

Constructible sets. The other single most important line of development leadingto admissible sets also goes back to the late thirties. It began with the intro-duction by Gόdel [1939] of the class L of constructible sets, in order to providea model of set theory satisfying the axiom of choice and generalized continuumhypothesis (GCH).

Takeuti [1960, 1961] discovered that one could develop L by means of arecursion theory on the class Ord of all ordinals. He showed that GόdeΓs proofof the GCH in L corresponds to the following recursion theoretic stability:If K is an uncountable cardinal and if F: Ord -» Ord is ordinal-recursive thenF(β)<κ for all β<κ. In modern terminology, every uncountable cardinal isstable. Takeuti' s definition of the ordinal-recursive functions was by means ofschemata, Tague [1964] provided an equivalent definition by means of an equa-tion calculus obtained by adjoining an infinitistic rule to Kleene's equationcalculus for ordinary recursion theory.

Admissible ordinals and admissible sets. The notion of admissible ordinal can beviewed as a common generalization of metarecursion theory and Takeuti' s re-cursion theory on Ord. Kripke [1964] introduced admissible ordinals by meansof an equation calculus. Platek [1965] gave an independent equivalent definitionusing schemata and another by means of machines as follows. Let α be an or-dinal. Imagine an idealized computer capable of performing computationsinvolving less than α steps. A function F computed by such a machine is calledvi-recursiυe. The ordinal α is said to be admissible if, for every α-recursive func-tion F, whenever β<α and F(β) is defined then F(β)<α, that is, the initialsegment determined by α is closed under F.

The first admissible ordinal is ω. An ordinal like ω + ω cannot be admissiblesince, for α > ω, the equation

defines an α-recursive function and F(ω} = ω + ω. The second admissible or-dinal is, in fact, ω{ and the ω{ -recursion theory of Kripke and Platek agreeswith that from metarecursion theory (see §§IV.3 and V.5). The theorem ofTakeuti mentioned above implies that every uncountable cardinal is admissible.The important advance made possible by the definition of admissible ordinalis that it allows one to study recursion theory on important ordinals (like ω\)which are not cardinals.

Takeuti' s work had shown that recursion theory on Ord amounts to de-finability theory on L. Analogously, the Kripke- Platek theory on an admissibleordinal α has a definability version on L(α), the sets constructible before stage α.It is this second approach which is most useful and is the one followed here.It leads us to consider admissible sets, sets A which, like L(α) for α admissible,satisfy closure conditions which insure a reasonable definability theory on A.These principles are formalized in a first order set theory KP. In order to studygeneral definability, though, not just definability theory in transitive sets, wemust strengthen the general theory weakening KP to a new theory KPU. Butthis is taken up in detail at once in Chapter I.

4 Introduction

Infinίtary Logic. There is just one other idea that needs to be introduced here,that of infinitary logic. The model theory of Lωω, the usual first order predicatecalculus, consists largely of global results, results which have to do with all modelsof some first order theory. These results have little to say about any one partic-ular structure since only finite structures can be characterized up to isomorphismby a theory of Lωω. Recursion theory, on the other hand, is a local theory aboutthe single structure ΛΛ If we are to have a global theory with non-trivial localconsequences, we must extend the model theory of Lωω to stronger logics, logicswhich can characterize structures and properties not characterizable in Lωω.For the study of admissible sets the appropriate logics turn out to be admissiblefragments LA of L^, as developed in Barwise [1967, 1969 a, b]. The countablecase is studied in Chapter III; the uncountable case, in Chapters VII and VIII.

Some material not covered in this book. This book is a perspective on admissiblesets, not a definitive treatment. It is far bigger and contains somewhat less mate-rial than we foresaw when we began writing. In particular, the following topics,all highly relevant to definability theory, are either omitted or slighted:

recursion theory in higher types,Spector classes,non-monotonic inductive definitions,relative recursion theory on admissible ordinals,forcing on admissible sets,forcing and infinitary compactness arguments.

It is planned that some of these topics will be treated in other books in this series.

A note to the casual reader. There is one bit of notation that might be confusingto the casual reader of this book. We use 21 for arbitrary models of the theoryKPU or, more generally, for arbitrary structures for the language L* = L(e,...)in which KPU is formulated. We switch to the notation A when our structureis well founded. In 99.44% of the uses A will denote an admissible set.

Part A

The Basic Theory

"Logic is logic. That's all I say."

Oliver Wendel HolmesThe Deacon's Masterpiece

Chapter I

Admissible Set Theory

Admissible sets are the intended models of a certain first order theory. In thischapter we discuss the theory itself and show how to develop a significant partof intuitive set theory within it.

1. The Role of Urelements

Our approach to admissible sets is unorthodox in several respects, the mostobvious being that we allow admissible sets to contain urelements. Bluntly put,we consider admissible sets which are built up out of the stuff of mathematics,not just the sets built up from the empty set. To make this a little clearer, and tosee why it is an obvious step to take, we begin by reviewing the developmentof ZF, Zermelo-Fraenkel set theory, as it is correctly presented (in, for example,§9.1 of Shoenfield [1967]).

The fundamental tenet of set theory is that, given a collection M of mathe-matical objects, subcollections are themselves perfectly reasonable mathematicalobjects, as are collections of these new objects, and so on. We begin with a col-lection M of objects called urelements (sometimes called points, atoms or in-dividuals, depending principally on our subject), which we think of as beinggiven outright. The objects in M might be real numbers, elements of some groupor even physical objects. We construct sets out of the objects of M in stages.At each stage α we are allowed to form sets out of urelements and the sets formedat earlier stages. An object is a set on M just in case it is formed at some stagein this construction; the collection of all sets on M is denoted by VM.

Now it turns out, and it must have been a surprising discovery, that if weallow strong enough principles of construction at each stage α, and if we assumethat there are enough stages, then urelements become superfluous. All ordinarymathematical objects occur, up to isomorphism, in V, i. e. in ¥M where M is theempty collection. It is consistent with this that the extensionality axiom of ZFexplicitly rule out the existence of objects which are not sets; the combinationof the power set and replacement axioms is so strong as to make urelementsunnecessary.

Set theory, as formalized in ZF, provides an extremely powerful and elegantway to organize existing mathematics. It is not without its drawbacks, never-

8 I. Admissible Set Theory

theless. While it is too weak to decide some questions (like the continuum hypo-thesis) which seem meaningful (even important), it is in some ways too strong.Some examples:

(1) The most obvious advantage of the axiomatic method is lost since ZFhas so few recognizable models in which to interpret its theorems.

(2) Important distinctions on the nature of the sets asserted to exist arecompletely lost.

(3) The principle of parsimony, of established value throughout the mathe-matical ages, is violated at every turn.

(4) Large parts of mathematical practice are distorted by the demand thatall mathematical objects be realized as sets (as opposed to being isomorphic tosets). If these objections are not too clear, they should becomes so as we inves-tigate the theory of admissible sets. At any rate these considerations, and othersfamiliar to anyone versed in generalized recursion theory, eventually dictatethe study of set theories weaker than ZF, weaker in the principles of set existencewhich they attempt to formalize. The theory we have in mind here, of course,is the Kripke-Platek theory KP for admissible sets.

It is at this point that one is tempted to make a simplifying mistake. We havefirst thrown out urelements from ZF because ZF is so strong. When we thenweaken ZF to KP we must remember to reexamine the justification for banningthe urelements. Doing so, we discover that the justification has completely dis-appeared. In this book we readmit urelements by "weakening" KP to a theoryKPU. The original KP will be equivalent to the theory

KPU + "there are no urelements".

This approach has many advantages. The chief is that it allows us to form,for any structure Wl = (M,Rl... Rky a particularly important admissible setHYPgn above 9JΪ, one which is of great use in the study of definability over 9M.The approach has no disadvantages since we can always restrict attention tothe special case where there are no urelements.

1.1—1.4 Examples

1.1. The point made in (1) above becomes clearer when we recall that if ZF isconsistent, so is

ZF + "There is no transitive model of ZF".

(Prove this without using GodeΓs Incompleteness Theorems!)

1.2. The observation in (2) is illustrated by considering, for example, an arbitraryabelian group (5 = <G, +>. Consider the following subgroups of G:

pG = {px\xeG},

T = {χ\nx = Q for some natural number n>0}

= the torsion subgroup of G,

D = \J {HIH is a divisible subgroup of G}

= the divisible part of G.

2. The Axioms of KPU 9

While these definitions are clearly increasing in logical complexity, there is nodistinction to be made between them from ZF's point of view. We will returnto this example in Chapter IV.

1.3. As an example of the way one is tempted to violate the principle of parsi-mony when working in ZF, one need only look in the average text on set theory.There you will find the power set axiom (a very strong axiom from our pointof view) used to verify a simple fact like the existence of a x b.

1.4. The point made in (4) above is illustrated by considering the real line. Whilewe know how to construct something isomorphic to the real line in ZF (eitherby Cauchy sequences or by Dedekind cuts), in practise the mathematician is notinterested in the details of this construction. For example, he would never thinkof worrying about what the elements of |/2 happen to be.

1.5 Notes. The notes at the end of sections are used to collect historical remarks,credit for theorems (when possible) and various remarks which might otherwisehave gone into footnotes.

In the early days of set theory, certainly in the work of Zermelo, urelementswere an integral part of the subject. The rehabilitation of urelements in thecontext of admissible set theory is such a simple idea that it would be silly toassign credit for it to any one person. Probably everyone who has thought atall about infinitary logic and admissible sets has had a similar idea.

Karp [1968] suggests the study of nontransitive admissible sets. Kreisel [1971]points out that "the principal gap in the existing model theoretic [generalizedrecursion theory] ... is its preoccupation with sets (that is sets built up fromthe empty set by some cumulative operation...); not even sets of individualsare treated." Barwise [1974] contains the first published treatment of admissiblesets with urelements. This book grew out of that paper, to some extent. It is worthremembering that the defense of urelements given in § 1 would have been un-necessary not too long ago. Perhaps it will be equally pointless sometime inthe future.

2. The Axioms of KPU

Let L be a first order language with equality, some relation, function and con-stant symbols and let 9JΪ = <M,---> be a structure for this language L. We wishto form admissible sets which have M as a collection of urelements; these ad-missible sets are the intended models of a theory KPU which we begin to developin this section.

The theory KPU is formulated in a language L* = L(e,...) which extends Lby adding a membership symbol e and, possibly other function, relation andconstant symbols. Rather than describe L* precisely, we describe its class ofstructures, leaving it to the reader to formalize L* in a way that suits his tastes.


2.1 Definition. A structure 91 = (9K A, £,...) for L* consists of(i) a structure 3R = <M,— > for the language L, where M = 0 is kept open

as a possibility (the members of M are the ur elements of 21 );(ii) a nonempty set ,4 disjoint from M (the members of A are the seίs of 91 );

(iii) a relation £^(Mu,4)x,4 (which interprets the membership symbol e);(iv) other functions, relations and constants on M u A to interpret any other

symbols in L(e, . . .) (that is the symbols in the list indicated by the three dots).The equality symbol of L* is always interpreted as the usual equality relation.We use variables of L* subject to the following conventions: Given a structure

819, = (SUM, £,...) for L*,

p,g,p1? ... range over M (urelements),

a,b9c9d9f,r9al9... range over A (sets),

x,y,z, ... range over MuA.

This notation gives us an easy way to assert that something holds of sets, or ofurelements. For example, VpBαVx (xea<^>x = p) asserts that {p} exists for anyurelement p, whereas VplaVq (qea*-*q = p) asserts that there is a set a whoseintersection with the class of all urelements is {p}.

We sometimes use (e.g. in 2.2 (iii)) w,t;,w to denote any kind of variable.

The axioms of KPU are of three kinds. The axioms of extensionality andfoundation concern the basic nature of sets. The axioms of pair, union and Δ0

separation deal with the principles of set construction available to us. The mostimportant axiom, Δ0 collection, guarantees that there are enough stages in ourconstruction process. In order to state the latter two axioms we need to definethe notion of Δ0 formula of L(e, ...), of Levy [1965].

2.2 Definition. The collection of Δ0 formulas of a language L(e, ...) is the smal-lest collection Y containing the atomic formulas of L(e, ...) closed under:

(i) if φ is in Y, then so is — iφ;(ii) if φ,ψ are in Y, so are (φ Λ φ) and (φvψ);

(iii) if φ is in Y, then so are Vuevφ and Ίuevφ for all variables u and υ.

The importance of Δ0 formulas rests in the metamathematical fact that anypredicate defined by a Δ0 formula is absolute (see 7.3), and the empirical fact(which we will verify) that many predicates occuring in nature can be definedby Δ0 formulas (see Table 1).

2.3 Definition. The theory KPU (relative to a language L(e, ...)) consists of theuniversal closures of the following formulas:

Extensionality: Vx (x£a<^>xeb)-+a = b;

Foundation: 3xφ(x)-+3x\_φ(x) Λ Vyeχ—\φ(y)'] for all formulas φ(x) in whichy does not occur free

3. Elementary Parts of Set Theory in KPU 11

Pair: 30(xe0Λye0);

Union: ΊbVyeaVxGy (xeb);

Δ0 Separation: 3fo Vx (xefo<->oce0Λ φ(x)) for all Δ0 formulas in which b doesnot occur free;

Δ0 Collection: Vxea3yφ(x,y)^>3bVxea:lyebφ(x,y) for all Δ0 formulas inwhich b does not occur free.

Note that the formulas φ(x\ φ(x,y) used above may have other free variables.

2.4 Definition. KPU+ is KPU plus the axiom:

30 Vx [xe0<-»3p (x =pγ\ ,

which asserts that there is a set of all urelements.

2.5 Definition. KP is KPU plus the axiom:

Vx 3α (x = a) ,

which asserts that every object is a set, i.e. that there are no urelements.

2.6 A word of caution. There are some axioms built into our definition of struc-ture for L(e, . . .). The following sentences make these conditions explicit andshould be considered part of the axioms of KPU:

VpVa(p*a) (cf. 2.1 (ii));

30 (0 = 0) (expresses A^ψ in 2.1(ii));

VpVx(xφp) (cf. 2.1 (iii)).

2.7 Notes. The notions of Δ0 and Σ! are due to Levy [1965]. The axioms ofKP go back to Platek (the P in KP), in particular, to Platek [1966]. He definedan admissible set A to be a transitive, nonempty set closed under TC satisfyingΔ0 separation and Σ reflection. Kripke [1964] (the K in KP) had, independently,a similar notion with Σ reflection replaced by Σ replacement. (For the modelsKripke had in mind (Lα's) they are equivalent; but in general it is Σ reflectionwhich matters.) Both of these men were influenced by Kreisel [1959] and Kreisel[1965]. See, e.g. Kreisel [1965, p. 199(b)]. (For the notion of a Σ formula, see4.1 below.)

3. Elementary Parts of Set Theory in KPU

In this section we show how to define some of the elementary concepts of intuitiveset theory in KPU. We thus want to show that certain sentences of L(e,...) arelogical consequences of KPU. We do this here by translating these sentences


into English and then giving their proofs in English, being careful to use onlyaxioms from KPU. For example, rather than state:

we state:

Given x9y9 there is a unique set a = {x,y} with only x9y as members;

and then we give an informal proof of the latter. (Given x,y, there is a b withx,yeb, by Pair. By Δ0 separation there is an a with zeα<-»zeb Λ [z = x or z = y],the part in brackets being a Δ0 formula. By Extensionality, there can be at mostone such a.) Thus all results in this section are proved in KPU.

3.1 Proposition, (i) There is a unique set 0 with no elements.(ii) Given a, there is a unique set b = (Ja such that xeb iff "Bye a (xey).

(iii) Given a,b there is a unique set c = a\jb such that xec iff xea or xeb.(iv) Given a,b there is a unique set c = ar\b such that xec iff xea and xeb.

Proof. These are all routine. By 2.1 (ii) there is a set b. For (i) we apply Δ0 sep-aration to b and the formula x φ x. For (ii) use the union axiom to get a b' suchthat VyeaVxey (xeb'\ and then form

b = {xeb'\'Byea (xey)}

by Δ0 separation. For (iii), form \J{a,b}. To prove (iv), let c = {xea\xeb}9

which exists by Δ0 separation. In each case uniqueness follows from the axiomof extensionality. D

We define, as ususal, the ordered pair of x9y by

and prove that <x,y> = <z,w> iff x = z and y = w.

3.2 Proposition. For all a9b there is a set c — axb, the Cartesian product of aand fe, such that

and yeb} .

Proof. By Table 1 the predicate of a9b9u:

u is an ordered pair <x,y> with xea and yeb

is Δ0 so we can use Δ0 separation once we know that there is a set c with <x,y>ecfor all xea, yeb. This follows from Δ0 collection as follows. Given any xeawe first show that there is a wx such that <X,J>EW X for all yeb. Why? Well,given yeb there is a set d = <x,y>. So, by Δ0 collection there is a set vvx such

3. Elementary Parts of Set Theory in KPU 13

that <x,y>ewx for all yeb. Now, apply Δ0 collection again. We have

3w Vyebldew (d = <x,y»

so there is a cv such that for all xeα, yeb, <x,y>ew for some wec^ Thus, ifc = \Jci, then <x,y>ec for all xea, yeb. D

The above is a good example of the principle of parsimony. In ZF, whereone has the power set axiom, the set c needed in the proof can be taken to bejust P(P(αuί>)), but this proof does not carry over to KPU.

We can define ordered n-tuples, for n>2, as follows, by induction on n:

and, similarly,

a± x ••• xan = aί x(a2x '" xan) .

Thus a± x ••• x an is the set of n-tuples ^xί9...9xny with x^e^ for i = l,...,n.Now that we have ordered pairs, we can give the usual definitions of intuitive

notions like relation, function, etc., all by Δ0 formulas as in Table 1.A set a is transitive, written Tran(α), iff

Vyex Vzey (zed) ,

so that Tran(α) is a Δ0 formula. Urelements are not considered transitive. Everyset of urelements is transitive. The empty set 0 is transitive.

3.3 Definition. Let «5^(fl) = αu{fl} .

3.4 Exercise. Prove (by induction) that for each n,

3.5 Exercise. Show that if a is a set of transitive sets, then (Jα is transitive.Show that if a is transitive and b<^a, then α u j f c } is transitive. In particular,if a is transitive, so is ^(d).

3.6 Definition. An ordinal is a transitive set a such that every member x of ais also a transitive set. Thus, we may write this definition as:

Ord(α) <->> Tran(α) Λ Vxeα Tran(x) .

We use α,jS,y, ... to range over ordinals. We write α<jS for αeβ. An ordinal αis a natural number if for all jS^α, if β/0 then β = (y) for some y. We usevariables n,m, ... over natural numbers.


3.7 Exercise. We assume that the reader has some familiarity with ordinalnumbers. He should verify that all the usual things are provable in KPU:

(i) 0 is an ordinal;(ii) If α is an ordinal so is (α), usually written α + 1.

(iii) If α/β then α<β or jδ<α. (This uses the axiom of foundation!)(iv) For all α, α^α.(v) If a is a set of ordinals, then \Ja is an ordinal β with α^β whenever

αeα, and Ξαeα(y^α) whenever y<β. (Thus β is the supremum of α, and wewrite β = sup(ά).)

(vi) If α<jS then α + l^β.(vii) Every nonempty set of ordinals has a smallest element.

3.8 Definition. A set a is finite if there is a one-one function / with άom(f) = aand range some natural number n. A set a is countable if there is a one-onefunction / with domain a such that f(x) is a natural number for every xeα.

3.9 Exercise, (i) Show that every member of an ordinal is an ordinal.(ii) Show that a set is an ordinal iff it is transitive and its elements are

linearly ordered by e.(iii) Show that an ordinal is finite iff it is a natural number.

Table 1. Some Δ0 Predicates

Predicate Abbreviation Δ0 Definition

x^y Vzex (zey)yeaΛzectΛVxeα (x =3beα3ceα (b = ί y } Λ c =

a = (x,yy for some ya = (x,yy for some xa = (x,yy for some x,ya is a relationf is a functionr is a relation with

domain ar is a relation with

range ar is a relation with

field a

y=f(χ)a=\Jb

1st (a) = χ2nd(a)=y"a is an ordered pair'Reln(α)Fun(/)

dom(r) = α

rng(r) = α

fιeld(r) = fl

3c6fl3yec (α = <x,y»3ceα3xec (a = <(x,y»3ceα 3xec 3yec (α = <(x,_y))Vxeα "x is an ordered pair"Reln(/) Λ VaεfVbef (isia = lsib->2nda = 2

R e l n ( r ) Λ V b ε r ( l s t b e α ) Λ V x G α 3 b e r ( l s t b =

Reln(r) Λ Vb e r (2ndb e a) Λ Vx e a 36 6 r (2nd/?

α = dom(r)urng(r)Fun(/)Λ<x,y>e/Vxefr Vyex (yea) A Vyef l 3xefe (yex)

-dfe)= χ)

4. Some Derivable Forms of Separationand Replacement

Our development of set theory progressed smoothly as long as the predicatesinvolved were definable by Δ0 formulas. With the notions of finite and count-able in 3.8 we hit the first examples of predicates which cannot be so expressed.

4. Some Derivable Forms of Separation and Replacement 15

For example, if we write either of these out they take the form

where φ is Δ0. A formula of the form 3u φ(u\ where φ is Δ0, is called a Σί formula.It turns out that a wide class of formulas are equivalent to Σ1 formulas and thatwe can use these formulas in various forms of separation, collection and replace-ment.

4.1 Definition. The class of Σ formulas is the smallest class Y containing theΔ0 formulas and closed under conjunction and disjunction (2.2 (ii)), boundedquantification (2.2 (iii)) and satisfying:

(i) if φ is in Y so is 3u φ for all variables u.

The class of Π formulas, on the other hand, is the smallest class Y' containingthe Δ0 formulas closed under conjunction, disjunction, bounded quantificationand satisfying:

(ii) if φ is in Y' so is Vw φ, for all variables u.

For example, the two formulas:

Vfoeα [b is countable] and Vxeα 3fo [Tran(b) Λ xεb~] ,

are Σ but not Σ^. Clearly the negation of any Σ formula is logically equivalentto a Π formula and vice versa. As a corollary to Theorem 4.3 we will see thatfor every Σ formula φ, there is a Σx formula φ' such that

Given a formula φ and a variable w not appearing in φ, we write φ(w) for theresult of replacing each unbounded quantifier in φ by a bounded quantifier; thatis we replace:

3w by 3wew, and

Vw by Vw e w ,

for all variables u. Thus φ(w) is a Δ0 formula. If φ is Δ0 then φ(w) = φ, since thereare no unbounded quantifiers in φ. We always assume that w does not alreadyappear in φ.

4.2 Lemma. For each Σ formula φ the following are logically valid (I e., true inall structures 21^):

(i) φ(u)

(ϋ) φ(u}


where u^v abbreviates the formula V.x [xeu^xev]. (Actually it is the universalclosures of these formulas which are true in all 21 since φ may have other freevariables. We will not bother with this comment in the future.)

Proof. Both facts are proved by induction following the inductive definition 4.1of Σ formula. Let us just prove the first, the second being similar. Fix a structure^ = (9 ; A,E, ...) and x,yeAvM so that x^y is true in 21 . For Δ0 formulasφ, we have, obviously, φ = φ(x} = φ(y\ Assume first that ( φ / \ ψ ) ( x } (i.e., assumeit's true in 21 ). Hence, φ(x) and ψ(x\ By induction φ(y) and φ(y\ so (φ/\φ)(y).Similarly for (φ v ψ)(x)-^(φ vψ)(y} and bounded quantifiers.

Now assume (3wφ(w))(x\ so there is a wex such that φ(w)(x\ By inductionφ(w)(y}; and, since x^y, 3wey(φ(w)()0); i.e., (3wφ(w))(y). D

4.3 Theorem. (The Σ Reflection Principle). For all Σ formulas φ we have thefollowing:

(Here a is any set variable not occurring in φ\ we will not continue to make theseannoying conditions on variables explicit.) In particular, every Σ formula isequivalent to a Σx formula in KPU.

Proof. We know from the previous lemma that 3α φ(a)-+φ is valid, so the axiomsof KPU come in only in showing φ-+3aφ(a\ The proof is by induction on φ,the case for Δ0 formulas being trivial. We take the three most interesting cases,leaving the other two to the reader.

Case 1. φ is \jι Λ θ. Assume that

, and

as induction hypothesis, and prove that

Let us work in KPU, assuming ψ Λ θ and proving 3α \ψ(a} Λ θ(α)]. Now thereare al9a2 such that ψ(a*\θ(a2\ so let a = aίva2. Then φ(α) and \l/(a} hold by theprevious lemma.

Case 2. φ is Vuevψ(u). Assume that

Again, working in KPU, assume Vuevψ(u) and prove 3aVuEvψ(u)(a\ Foreach uev there is a b such that ψ(u)(b\ so by Δ0 collection there is an α0 such that

4. Some Derivable Forms of Separation and Replacement 17

Vuev3bea0\ls(u)(b\ Let a=\Ja0. Now, for every uεv, we haveso Vw e v ψ(ύ)(a\ by the previous lemma.

Case 3. φ is 3uψ(u). Assume ψ(u)<^>3bψ(u)(b) proved and supposetrue. We need an a such that 3uεaψ(u)(a\ If ψ(u) holds, pick b so that ι/^(w)(fe)

and let a = bu{u}. Then weα and ψ(u)(a\ by the previous lemma. D

In Platek's original definition of admissible set he took the Σ reflectionprinciple as basic. It is very powerful, as we'll see below. The Δ0 collection axiomis easier to verify in particular structures, however, and is also more like thereplacement axioms with which one is familiar from ZF.

4.4 Theorem. (The Σ Collection Principle). For every Σ formula φ the followingis a theorem of KPU: // Vxeα3y φ(x,y) then there is a set b such thatVxea1yebφ(x,y) and \/yeb'Bxeaφ(x,y).

Proof. Assume that

By Σ reflection there is a set c such that

(1)

Let

(2)

by Δ0 separation. Now since φ(c}(x,y)^κp(x,y) by 4.2, (1) gives us:

whereas (2) gives us:

) . D

4.5 Theorem. (Δ Separation). For any Σ formula φ(x) and Π formula ψ(x\ thefollowing is a theorem of KPU: If for all xea, φ(x)+-+\jj(x\ then there is a setb = {xea\φ(x}}.

Proof. Assume Vxea (φ(x)++ψ(x)). Then Vxεa [φ(x) v ~ι (x)], which is equiv-alent to a Σ formula, so there is a c such that Vxea [φ(c)(x) v ~Ί^(C)(X)]. Let, byΔ0 separation, b = {xea\φ(c)(x)}. Clearly every xeb satisfies φ(x). If xeα andφ(x) then i/φc), so ^(c)(x) (since ψ(x)-*ψ(c\x))', so φ(c\x). Thus xeb. D

4.6 Theorem. (Σ Replacement). For each Σ formula φ(x,y) the following is atheorem of KPU: If VxEa3lyφ(x,y) then there is a function /, with dom(/) = α,such that Vxeaφ(x,f(x)).


Proof. By Σ Collection there is a set b such that Vxealyeb φ(x,y). UsingΔ Separation there is an / such that

}. D

The above is sometimes unsuable because of the uniqueness requirement 3 !in the hypothesis. In these situations it is usually 4.7 which comes to the rescue.

4.7 Theorem. (Strong Σ Replacement). For each Σ formula φ(x,y) the followingis a theorem of K PU : // Vx e a 3y φ(x, y) then there is a function fwith dom(/) = asuch that

(i)

(ii) VxeaVyef(x)φ(x,y).

Proof. By Σ Collection there is a b such that Vxeα Ίyeb φ(x,y) andV y e b 3xeα φ(x,y). Hence there is a w, by 4.3, such that

(w\x,y), and V y e b 3xeα φ(

For any fixed xeα there is a unique set cx such that

by Δ0 Separation and Extensionality; so, by Σ Replacement, there is a function/ with domain a such that f(χ) = cx for each xea. D

4.8—4.9 Exercises. There are a number of minor variations on the above.

4.8. For example, prove that, for each Σ formula φ,

4.9. Given a Σ formula φ let φ*a denote the result of replacing some, but notnecessarily all, existential quantifiers 3u by 3t/eα for some new set variable a.Show that:

5. Adding Defined Symbols to KPU

The introduction of defined relation and function symbols is a common practice,but it must be used with just a little care in KPU. In a theory like ZF one is ableto take any formula φ(x1 ?..., xn\ define a new relation symbol by

(R) Vx 1...Vxn[R(x 1,...,xπ)^φ(x 1,...,xJ],

5. Adding Defined Symbols to KPU 19

and then use R as an atomic formula in other formulas — even in the axiom ofreplacement. After all, one could always go back and replace R by φ. For KPU,however, where we must pay attention to the syntactic form our axioms take,a definition like (R) would work, at first glance, only if the φ in (R) were Δ0. Wehave tacitly used this form of introducing new relation symbols repeatedlyin § 3. Using the principles of § 4 we may allow ourselves a bit more freedom.

5.1 Definition. Let φ(xl,...,xn) be a Σ formula of L* and ^(X I ? . . . ,X M ) be aΠ formula of L* such that

Let R be a new rc-ary relation symbol and define R by (R) above. R is then calleda Δ relation symbol of KPU.

To be really precise it would be the triple R,φ,ψ such that the above holdwhich constitute a Δ definition of the relation symbol R, but we do not need tobe this careful. The next lemma shows that we can treat Δ relation symbols asthough they were atomic formulas of L*. Here, and elsewhere, we abbreviatex1? ...,xk by x.

5.2 Lemma. Let KPU be formulated in L* and let R be a Δ relation symbol ofKPU. Let KPU' be KPU as formulated in L*(R), plus the defining axiom (R)above.

(i) For every formula θ(xl, ..., xk, R) of L*(R), there is a formula Θ0(xί,...,xk)of L* such that KPU + (R) implies

Moreover, if θ is a Σ formula of L*(R) then Θ0 is a Σ formula of L*.(ii) For every Δ0 formula Θ(x1? ..., xk, R) of L*(R) there are Σ and Π formulas

00(xι ,...,**), 0ι(xι,. ,Xfc) of L* such that KPU + (R) implies

θ(x,R)<->00(x), and θ(x,R)^θ1(x) .

(iii) KPU' is a conservative extension of KPU. That is, for any sentence θ of L*,

KPU'KΘ iff KPUh-0.

Proof. Let us suppose that R is defined by

R(x1,...,xJ^->φ(x1,...,xn),

where φ is a Σ formula, and that


where ψ is a Π formula. The first sentence in (i) is obvious since we may replaceR by its definition. It is to make the second sentence of (i) true that we need Rto be a Δ relation symbol of L*. Using de Morgan's laws, push all negations inθ inside as far as possible so that they only apply to atomic formulas. Now re-place each positive (i.e., unnegated) occurrence of R in θ by φ, each occurrence-|R by the Σ formula equivalent to —\ψ. The result is called Θ0. Since

KPUVR^φ and

it is clear that

It is also clear that this transformation takes Σ formulas into Σ formulas. Note,however, that the transformation does not take Δ0 formulas into Δ0 formulas,but only into Σ formulas. However, since the Δ0 formulas are closed undernegation, (ii) immediately follows from (i). To prove (iii) it suffices to show thatevery axiom of KPU' is turned into a theorem of KPU when R is replaced asabove. For example, Δ0 Separation of KPU' becomes Δ Separation in KPUand Δ0 Collection for KPU' becomes a consequence of Σ Collection for KPU. D

Using this lemma we can clear up a point which may have been botheringthe reader. One way of formalizing L* = L(e, . . .) is to make it a single sortedlanguage with predicate symbols U for urelements and S for sets. In this wayVp(...p...) would stand for Vx(U(x)->(...x...)), and Vα(...) would stand forVx(S(x)->(...x...)), and "x is an urelement" would be a Δ0 formula, U(x). Theother way of formalizing L* is to have a many-sorted language with the threesorts of variables

α, fo, . . . , and

The predicate "x is an urelement" is no longer Δ0, but it is Δ. Our definition ofL* insures that

x is an urelement <-> 3p (x = p) , and

x is not an urelement <-> 30 (x = a)

so the predicate and its negation are Σ1. The lemma assures us that we canintroduce a new symbol by:

U(x)<-»x is an urelement,


and use it in Δ0 formulas without fear. Similarly we can introduce

for "x is a set" and treat it as a Δ0 formula.A predicate of intuitive set theory is said to be a Δ predicate of KPU if it

can be defined by a Δ relation symbol. Using the above lemma we see that wemay treat Δ predicates as though they were defined by atomic formulas of L*.Furthermore, the Δ predicates are closed under Λ , v, Vwer, IUEV. Usingthese observations, we see that all the predicates listed in Table 2 are indeedΔ predicates.

The introduction of defined relation symbols is a convenience, but the intro-duction of defined function symbols is a practical necessity (though theoreticallya luxury). The conditions necessary for us to be able to do this are given in thefollowing definition.

5.3 Definition. Let φ(xl9 ..., xn,y) be a Σ formula of L* such that

Let F be a new n-ary function symbol and define F by:

(F) Vxι,...,

F is then called a Σ function symbol of KPU.The next lemma lets us treat Σ function symbols as though they were atomic

symbols of the basic language L*.

5.4 Lemma. Let KPU be formulated in L* and let F be a Σ function symbol ofKPU. Let KPU' be KPU as formulated in L*(F), plus the defining axiom (F) above.

(i) For every formula θ ( x ί , . . . , x k , f ) of L*(F) there is a formula 00(x l 9...,xk)of L* such that KPU + (F) implies

0(x,F)<->00(Jc).

Moreover, if θ is a Σ formula of L*(F) then 00 is a Σ formula of L*.

(ii) For every Δ0 formula Θ(xί9 ..., xfc, F) of L*(F) there are Σ and Π formulas00(*i,...,xJ, 0ι(xι,...,*k) of L* such that KPU + (F) implies

0(x,F)<e-»θ0(*)> and

(iii) KPU' is a conservative extension of KPU.

22

Table 2.

I. Admissible Set Theory

Some Δ predicates

Predicate

x is an urelementx is a setx is transitivex is an ordinalx is a limit ordinalx is a natural numberless than for ordinalsless than or equal

Abbreviation

U(x)S(x)Tran(x)Ord(x)Lim(x)Nat No(x)α<β

*<β

Definition

3p(x = p) (or Vtf(x^a))3α (x = a) (or Vp (x p))S ( x ) Λ V y e x V z e y ( z e x )Tran(x) Λ Vyex Tran(y)Ord(x) Λ x 0 Λ Vyex 3zex (z = yu{y})Ord(x) Λ Vy e x— i Lim(y) Λ — i Lim(x)Ord(α)Λθrd(β)Λαeβa<βvx = β.

Proof. Note that if φ(xx, ...,xn,y) is a Σ formula and if F(x1,...,xn) = -y iffφ(x1,...,xπ,y), then we can get a Σ definition for F(x l5 ...9xn)^y by

(1) iff

Thus the graph of F is a Δ predicate. The only complication, then, that can occurhere but not in the previous lemma, is that F may occur in θ in complicatedcontexts like:

F(G(x))=HGO and

Call a formula simple if F only appears in simple contexts like:

F(x1,...,xB) = 3; and

Repeated uses of the equivalences below allow us to transform every formulainto an equivalent simple formula in such a way that Σ formulas transform intoΣ formulas:

F(G(x),x2,...,xB)) = ; [G(x) = zΛ F(z,x2, ..., xj =

φ(F(x), ,..) [z= F(x) Λ φ(z, ...)] (φ quantifier free).

The proof now proceeds as in 5.2, replacing occurrences of F(xl9...9xJ = yby φ(xι,...,xπ,;y), occurrences of F(x l 9 ..., xn)ϊy by the Σ formula in (1). D

When we use 5.1 (or 5.3) to introduce a Δ relation symbol R (or Σ functionsymbol F) we often abuse notation by using KPU to denote the new theoryKPU' of 5.2 (or 5.4). The lemmas insure us that we can't get into trouble withthis abuse of notation.


Table 3. Some Σ operations

Operation

domain of frange of fthe first coordinate of xthe second coordinate of xthe restriction of f to athe image of f restricted to asuccessorordinal successorsupremum

Domain

all functions /all functions /all ordered pairs xall ordered pairs xall functions / and sets aall functions / and sets aall sets xall ordinals αsets of ordinals

Abbre-viation

dom(/)mg(/)ls tx2ndxf\af"a

y(χ)α + 1sup(α)

Σ Definition (the unique zsuch that)

see Table 1see Table 1see Table 1see Table 1z = {xef\isixea]

z = {xerng(/)|3yeα(/(x) = }>)}z = xu{x}z = (α)

* = !>

An operation of intuitive set theory is a Σ operation of KPU if it can bedefined by a Σ function symbol of KPU. The following exercises summarizesome of the ways, in addition to 5.3, we have of defining Σ operations. The mostimportant method, though, must wait for the next section.

5.5—5.7 Exercises

5.5. Every function symbol of L* is a Σ function symbol.

5.6. The Σ operations are closed under composition.

5.7. The Σ operations are closed under definition by cases. That is, if G 1 ? . . . ,G k

are rc-ary Σ operations and φί(xl,...,xn),...,φk(x1,...,xn) are Σ formulas suchthat

indicates exclusive or), then we may define a Σ operation F by:

.,*„)=<if

if φ f c(x 1 ?...,xn).

Frequently we are interested in the value of a function symbol only for cer-tain kinds of objects. For example, we want to define lstα to be the first coordi-nate of a if a is an ordered pair, but we don't really care what lstα means other-wise. To introduce lstα as a function symbol then, we should, to be completelyrigorous, first do something like prove: Mx1\y φ(x,y), where φ(x,y) is:

x is an ordered pair with first coordinate y, or

x is not an ordered pair and y is the empty set,

and then define:

isix = y iff φ(x,y).


Similarly, we are interested in \Jx only when x is a set. We will not bother withsuch details in the future, as long as it is clear that the intended domain of ournew function symbol is Δ definable.

6. Definition by Σ Recursion

Definition by recursion is a powerful tool. It will allow us to introduce, in ac-cordance with 5.3, operations such as ordinal addition, ordinal multiplicationand the support function sp:

which gives the set of urelements which go into the construction of a set a. Beforeshowing how to justify such recursions we must first prove outright what is ineffect a special case.

6.1 Theorem (Existence of Transitive Closure). We can introduce a Σ functionsymbol TC into KPU so that the following becomes a theorem of KPU: For everyx, TC(x) is a transitive set such that x^TC(x); and for any other transitive set α,if x^a, then

The axiom of foundation will be used in the proof of 6.1, in the form ofProof by Induction over e. If one takes the contrapositive of foundation onegets the following scheme. For every formula φ the following is a theorem ofKPU:

Vx(Vyex φ(y)

Thus in proving Vxφ(x), we pick an arbitrary x and prove φ(x) using, in theproof, φ(y) for any yex. (Of coure if x is an urelement then there are no suchyex.)

Proof 0/6.1. If we had the ordinal ω at our disposal (we cannot prove it existsin KPU) we could use it to define

This definition should be kept in mind to understand the following proof. Defineβ(x,α) to be:

x c a Λ Tran(α) Λ Vb (x c b Λ Tran(b)->α c b) .

Thus Q is defined by a Π formula and Q(x,a) iff a is the smallest transitive setcontaining x. It is clear that Q(x9a)^Q(x9a')->a = a'.

6. Definition by Σ Recursion 25

Now let P(x,a) be the following Σ predicate:

x is an urelement Λ a = Q, or

x is a set, x^α, Tran(α)Λ Vzeα3/[Fun(/)Λdom(/)

is a natural number n + \ = {0, ..., n} Λ z = /(0)e/(l)e e/(n)ex.]

(This can be easily formalized without writing "•••"; so there is no hidden re-cursion.) A simple induction on natural numbers n shows that P(x,α)->Q(x,α).In particular, P(x,a) Λ P(x,a')-*a = d .

If we can prove that for every x there is an a such that P(x,a) then we willbe able to define a Σ function symbol TC by

ΎC(x) = a iff P(x,a)

and TC(x) will have the desired property of the transitive closure of x. We stillneed to show that Vx la P(x,a). If x is an urelement, take a = ΰ. Thus, we needonly prove Vb 3α P(b, a), which we do by induction on e. Given b, in proving3α P(b,a) we may assume

and hence, by the above,

Vxe/?3!cP(x,c).

By Σ replacement there is a function g with dom(^) = fo, such that P(x,g(x))holds for all xeb. Let

It is clear that b^a and it is not difficult to check that a is transitive. Let usverify the last clause oϊ P(b,a). Thus, let zea. If zeb then take /={<0,z>}. Nowassume zeUrng(#), i.e. ze#(x) for some xeb. But then there is an h such thatdom(/z) is an integer π + 1, /z(0) = z, Λ(ί)e/ι(ΐ + l) and Λ(n)ex since P(x,0(x)).Let / = Λ u { < w + l,x>}. Then/(0) = ze/(l)e/(2)e e/(n + l) = xefe so P(a,b\ D

6.2 Exercise. Verify

(i) TC(p) = 0, and

(ii) ΊC(a) = av\J{ΊC(x)\xea}.

Once we have Theorem 6.4 we could use the equations in 6.2 to define TC;unfortunately we need 6.1 and 6.3 to state and prove 6.4. The following is astrengthening of the method of proof by induction over e.


6.3 Theorem (Proof by Induction over TC). For any formula φ(x) the followingis a theorem of KPU: //, for each x, (VyeTC(x) φ(y)) implies φ(x), then Vx φ(x).

Proof. We show, under the hypothesis, that V x V y e Ύ C ( x ) φ ( y ) . This impliesVx φ(x), since xeTC({x}). We may assume, by induction on e, that for all zex

(1) Vy€ τC(z)φ(y)

in showing VyeTC(x) φ(y\ But by the hypothesis, (1) implies φ(z) so we have<p00, for all 3;exuQ{TC(z)|zex}=TC(x). D

The following theorem is of central importance to all that follows.

6.4 Theorem (Definition by Σ Recursion). Let G be an n + 2-ary Σ functionsymbol, n^Q. It is possible to define a new Σ function symbol F so that the fol-lowing is a theorem of KPU (+ the defining axiom (F)): for all x1 ? ..., xπ,y,

(i) F(x1,...,xn,y) = G(x1,...,xn,};, «z,F(x1,...,xπ,z)>|zeTC(y)}).

Before turning to the rather tedious proof of 6.4, let us make some remarkson variations which follow from it. For example, we could replace 6.4 (i) by:

(Let G'(x,y,/) = G(x,y,/f)0, and apply 6.4 to G'.) We could also start out withtwo functions G,H and define

F(x 1,...,xπ,α) = G(x1,...,xn,α, «z, F(x1,...,xJ>:zeTC(α)}).

This is the form we usually use. (Let G'(x,y,/) be H(x,}0, if y is an urelement,otherwise G(x,y,/) if y is a set. Then apply 6.4 to G'.)

Proof of 6.4. To be a little more formal, what we really want to prove about F,once we find a way of defining it, is that for all x l 5 . . . , xn,y there is an / such that

(1) / is a function Λ dom(/) - ΊC(y) ,

(2) Vwedom(/)(/(w)=F(x1,...,x l l,w)), and

(3) F(x1,...,xn,j;) = G(x1,...,xII,);,/).

This suggests the correct defining formula for F, Let n = i to simplify notation.Let P(x,y,z,/) be the Σ predicate given by:


/ is a function Λ dom(/) = TC(y)

We will prove:

(4) VxVy3!z3/P(x,j;,z,/);

and so we can introduce a Σ function symbol F by:

(5) F(x,y) = z

where it is clear that the right-hand side of (5) is a Σ formula. In order to prove(4) it suffices to prove;

(6) P(x,y,z,f)*P(x,y,z',f')^z = z'*f=fr

9 and

(7)

We prove both (6) and (7) by induction on TC(y). We use, in these proofs, lines(8), (9) below which are obtained by inspecting the definition of P:

(8) P(x9y9z9f)^z = G ( x 9 y 9 f ) ' 9

(9) P(x,>;,z,/)ΛweTC(y)^P(x,w,/(w),/rTC(w)).

We now prove (6) by induction on ΎC(y). Thus, we may assume that forweTC(y) there is at most one u and g with P(x9 w,u,g) and prove thatP(x9y9z9f)ΛP(X9y9z'9f')^z = z'Λf = f'. Since z = G ( x 9 y 9 f ) and zf = G ( x 9 y 9 f t ) 9

it suffices to prove / = /'. But / and /' are functions with common domainTCGO so it suffices to show that /(w) = /'(w) for all weTC(y). But by (9),P(x,w,/(w),/fTC(w)) and P(x,w,/'(w),/7TC(w)); so /(w) = /'(w) by the in-duction hypothesis. It remains to prove (7), and this is where Δ0 Collectionenters in the guise of Σ Replacement. We prove 3z3/P(x,y,z,/) assuming, byinduction on TC, that VweTC(y)3w3#P(x,w,w,#); and hence, by (6), there isa unique uw,gw such that P(x,w,ww,#J. By Σ Replacement the function

exists. To prove (7) it suffices to prove P(X,y,G(x,y,f),f) and this will followfrom VzeTC(x)(/(z) = G(x,z,/fTC(z))). Since we have P(x,z,wz,#J we havef(z) = uz = G(x,y,gz). Thus, all we have to show is f\ΎC(z)=gz. For wedom(#z)= TC(z), (9) implies P(x,w,^z(w),^zfTC(w)). Thus by (6) we have fifz(w) = ww=/(w);so gz = f\ΎC(w) as desired. This proves (7). Now let us introduce F by line (5)


and go back to prove 6.4 (i). By (5) we have

F(x,y) = G ( x 9 y 9 f ) where P ( x 9 y 9 G ( x 9 y 9 f ) 9 f ) 9

so we need only show that

/ = {<z,F(x,z)>|zeTCGO}.

For zeTC(y) we have, by (9), P(x,z,/(z),/fTC(z)) so, by (5), F(x,z) = /(z)as desired. D

6.5 Exercise. Prove that if two operations F1?F2 both satisfy 6.4 (i) in place ofF for all xί9...9xn9y then F^x^ ...,xn,y) = F2(x1, ..., xn9y)9 for all xί9...9xn9y.

In applications of 6.4 one does not usually bother to introduce the explicitfunction symbols G,H first.

6.6 Corollary (Δ Predicates Defined by Recursion). Let P,Q be Δ predicates ofn + 1, n + 2 arguments respectively, π^O. We can introduce a Δ predicate R bydefinition so that the following are provable in the resulting KPU:

(i)

(ii)

Proof. Introduce the characteristic functions G,H of P,Q respectively. Use ΣRecursion to define the characteristic function F of R and then note that

so that R is shown to be Δ. D

In Table 4 we give some examples of operations defined by recursion. Thereader not familiar with this type of thing should work through the followingexercises.

6.7—6.9 Exercises

6.7. (The rank function), (i) Show how to make the definition of rk given inTable 4 fit into the form of Theorem 6.4.

(ii) Prove that rk(x) is an ordinal rk(α) = α for ordinals α, and rk(y)<rk(x)whenever yeTC(x).

(iii) Prove that rk(α)={rk(y)|yeTC(α)}. (This could be used to give a dif-ferent recursive definition of rk.)


6.8. (The support function), (i) Show how to make the definition of sp given inTable 4 fit into the form demanded by 6.4.

(ii) Prove that sp(a) = {xEΎC(a)\x is an urelement}.

6.9. (Ordinal addition), (i) Show how to make ordinal addition fit into theform of 6.4.

(ii) Prove:

and

, if

(iii) Prove:

To conclude this section we point out that, like much of axiomatic mathe-matics, the development of set theory in KPU is largely a matter of refiningproofs from ZF. Among its rewards is the Σ recursion theorem (1.6.4). Since weend with a Σ operation symbol, the operation defined by recursion is absolute.The usual development in ZF completely looses track of this vital information.(This is relevant to the point we made in § 1, line (2).)

Table 4. Some Σ Operations Defined by Recursion

Operation

rank function

support function

Domain

everything

everything

Abbreviation Recursive definitions

rk(x) rk(p) = 0rk(α) = sup (rk(x) + 1 1 xe a}

sp(x) sp(p) = {p}

ordinal additionordinal multiplicationcollapsing function(cf. 1.7)constructive sets(cf. II.5)

pairs of ordinals α, β ot + βpairs of ordinals α, β aβpairs a,x Cα(*)

pairs a, a L(α,α)

α β = sup{αy + α| y < /?}

L(a,λ) = (Jα<λL(α,α), for limit λ.


7. The Collapsing Lemma

We return to the development of set theory in KPU to discuss an importantoperation C of two arguments; we write Cx(y) instead of C(x,y). The operationis defined in KPU using Σ Recursion by the equations:

C*(p)=p;

Cx(a) = {Cx(y)\yeaπx}.

(This falls under the second variation on Theorem 6.4.) C will be calledMostowski's collapsing function. We shall compute Cx(y) for some specific valuesof x and y after we have a lemma to aid us. In this section we will only be inter-ested Cx(y) for yex.

7.1 Lemma, (i) Cp(a) = Q, for all p,a.(ii) If a^b and a is transitive, then Cb(x) = x for all xea.

(iii) For any b the set {Cb(x)\xeb} = Cb(b) is transitive.

Proof, (i) is obvious. We prove (ii) by e-induction. Thus, given xea we supposethat Cb(y) = y for all yex. But since a is transitive, x^a^b, so we have

Cb(x) = {Cb(y)\yεxnb}

= {Cb(y)\yex}

= {y\yex}

= x .

To prove (iii), let a = {Cb(x)\xeb}. We must show that a is transitive. Letzeyeα. Thus y = Cb(x) for some xεb, hence ze{Cb(x')\xΈxnb}; so z = Cb(x')for some x'efc. Hence zeα. D

7.2 Example. Let fc = {0,l,2,4,{l,3,4},{!,4}}. If we let α = 3 = {0,l,2} then7.1 (ii) applies to give:

Let us compute Cb(4):

Cb(4) = {Cb(x)\xεb,xε4}

= {C>(x)|x = 0,l,2}

= {0,1,2}

__ α

7. The Collapsing Lemma 31

Thus Q "collapses" 4 to 3 since 3 wasn't in b. Now let us compute Cfc({l,3 4})and €,({1,4}):

= {€„(!), Cfc(4)}

= {1,3}.

Thus both the sets {1,3,4} and {1,4} are collapsed to {1,3}, all because 3 wasleft out of b. Note that

which is a transitive set, just as 7.1 (iii) foretold.

7.3 Definition. For any set b let cb denote the restriction of Cb( ) to fc; i. e.cb = {(x,Cb(x)y:xeb}9 and let

clpse(fc) = mg(cb) = {Cb(x) :xeb} = Cb(b) .

Note that the function cb exists (as a set) by Σ replacement and that clpse(b)is a transitive set by 7.1 (iii).

A set b is extensional if for every two distinct sets a1,a2eb there is an xebsuch that x is in one of aί9a2 but not both; in other symbols,

Vxefo

We would like to say that b is extensional if

<b, € n b2y \= "Extensionality" ,

but we cannot do this because we have not yet defined syntax and semantics(say 1=) in KPU. So, what we have done is simply to write this out in full.

In Example 7.2, b was not extensional because of the two sets

Any transitive set is extensional, as is any set of ordinals. The next lemma showsthat any extensional set is isomorphic to a transitive set.


7.4 Theorem (The Collapsing Lemma). // a is extensional then ca maps a one-one onto the transitive set clpse(α). Furthermore, for all x,yea

(i) xey iff ca(x)eca(y).

In other words, ca is an isomorphism of <#, enα 2) onto <clpse(α), enclpse(0)2>.

Proof. We need to show ca is one-one and that ca(x)eca(y) implies xey. Weprove both of these by proving VxVyP(x,y) where P(x,y) is the conjunction of:

x,yea/\ca(x) eca(y)-+x ey , and

x,yea/\ca(y) eca(x)-*y ex.

Given an x0 we can assume, by induction on e,

(1) Vxεx0VyP(x,y)

in our proof of VyP(x0,y). Given an arbitrary y0 we can assume

(2) . Vyey0P(x0,y)

in our proof of P(x0,y0), again using e-induction. Thus, suppose x0,j;0eα.

Case 1. ca(xQ) = ca(y0). Suppose Xo^y0. We see that both x 0>yo must besets since ca(p)=p. But then, since a is extensional there is a zea withze(x 0uy 0) — (*on.Vo) Suppose zex0 — y0, the other possibility being similar.Then ca(z)Eca(xo) = ca(y0) but, by (1), P(z,y0) so zey0, a contradiction.

Case 2. ca(x0) e ca(y0). But then cα(x0) = cα(z), for some zey0> but P(x0,z)by (2), so x0 = z and

Case 3. cα(>;)ecα(x0). Similar to Case 2. D

We hint at some of the types of applications of the collapsing lemma in theexercises.

7.5—7.8 Exercises

7.5. Show that if a is finite so is Cb(a). [Hint: Use induction on natural numbers.]

7.6. Show in KPU that a set a of ordinals is finite iff clpse(α) is a natural num-ber. This shows that the predicate "α is a finite set of ordinals" is a Δ predicatein KPU. (For contrast see the remarks in 9.1.)

8. Persistent and Absolute Predicates 33

7.7. Assuming intuitive set theory, or ZF, use the collapsing lemma and theLόwenheim-Skolem theorem to show that for every transitive A there is a count-able transitive set B such that <,4,e>ΞΞ<£,e>. (= denotes elementary equiv-alence; we use <v4,e> for <,4,en,42> when A is transitive.) Show that = can-not in general be replaced by •< (elementary substructure).

7.8. Let a,b be transitive sets, / an isomorphism of <α,e> and <b,e>. Showthat if f ( p ) = p for all urelements pea then f(x) = x for all xea and hence a = b.

7.9 Notes. The collapsing lemma is due to Mostowski [1949] and is one of thestandard tools of the set-theorist. (See also the notes to § 9.) The value Cb(x) ofthe collapsing function is of interest even when xφb. For example if b is count-able one can use Cb(x) as a kind of countable approximation to x. Using a notionof "almost all" due to Kueker and Jech, one can prove that if P is a Σ predicateand P(x) holds, then P(Cb(x)) holds for almost all countable sets b. For moreon this see Kueker [1972], Jech [1973] and Barwise [1974].

8. Persistent and Absolute Predicates

In this section we discuss the reason for the restriction to Δ0 formulas in theaxioms of separation and collection. The rationale behind this restriction restsin one of the basic notions of the subject, that of absoluteness.

Recall the discussion of VM from § 1. The sets in VM come in stages andseparation tells us what principles are allowed in forming the sets at each stage.The content of Δ0 Separation is that we allow ourselves to form the setb = {xea\φ(x,y)} at stage α if we already have formed a and y, but only if themeaning of φ(x9y) is completely (or absolutely) determined solely on the basisof the sets formed before stage α. In other words, when we come to a later stageβ and form (xea\φ(x,y)} we want to get the same set b, even though there arenow more sets around which might conceivably affect the truth of φ(x9y) byaltering the range of any unbounded quantifiers in φ.

Similar considerations apply to collection. Suppose that, in the process ofbuilding VM, we suddenly notice that Vxea3yφ(x,y) is true. We want to beable to form at the next stage a set b for which Vxea'Byeb φ(x,y) is true, andremains true. But what if the introduction of this very set b destroyed the truthof φ(x,y) for some xeaΊ This can happen if φ has unbounded universal quan-tifiers in it. If we want this stability, we must apply collection only if φ(x9y) can-not become false when we add new sets to our universe of set theory. That is,φ(x9y) should persist.

The aim of this section is to extract formal consequences from these ideas.

8.1 Definition. Let 2ί8M = (STO; A,E,...) be a structure for L*. For aeA we define

A\yEa].


Note that the value of aE in 8.1 depends on 91 , and a. The import of 8.1 is clear.Speaking very loosely, the set aE is "the set that a believes itself to be". In thenatural intended structures aE will just be a itself.

The usual notion of substructures has an obvious generalization to L*. Wesay that 93^ is an extension of 91 , and write 91^93^ (where 9ΪSW = (9W;^, £,...)and 93w = (SR;β,F,...)) if 501 c 91 (as L-structures), if A^B, and if the inter-pretations £, . . . are just the restrictions to M u A of the interpretations £',... .

A moment's reflection shows that 91 93 is not really the natural notionof extension when one is thinking of models of set theory. For suppose aεA.The trouble is that a may be "schizophrenic" in its role as a set in 91 and asa set in 93 . The relation 91^93^ guarantees that aEâE> but it does notrule out the possibility that for some xεB-A, xε(aE,-aE). This is clearly achaotic situation (since a set is supposed to be determined by its members), sowe introduce a stronger notion of extension suitable for the study of set theory.

8.2 Definition. Given structures 91 = (SR; A, £,...) and 95W = (5R;B,F,...) forL*, we say that 93^ is an end extension of 91 , written either as:

if 9ΪS0J c 93<R and if for each aεA, aE = aE, . One sometimes reads, aloud, 91 end®<nas "9X351 is an initial substructure of 93^".

8.3 Example. If A is a transitive set, BÂ, E = εr\A2, and E'=εr\B2, then

for in both structures any aeA has aE = aE>=a. If A were not transitive, how-ever, this could fail.

8.4 Lemma. Let 9X^,93^ be structures for L*, 939?3end^laR. // φ is a Σ formulaof L* then for any xί9 ..., xêSl^, 2Iarι^φ[>ι, ..., *„

Proof. This just repeats the proof of Lemma 4.2 proceeding by induction on φ.The end extension hypothesis is used to assure that Vxeα has the same mean-ing in 9IOT and 93^. D

8.5 Definition. A formula φ(uί9...,uj of L* is said to be persistent relative toa theory T of L* if for all models 91 , 93^ of T with 93^^91 , and all xί9...,xn

in 91 ,:<Άm\=φ\_xl,...,xn] implies ®9lϊ=φ[xl,...9xn].

The formula φ is absolute relative to T if for all 9Isw,Sw,x1,...,x l l as above:

> •••> i f f 9 3 < N ( ? , . . . , * .


The significance of 8.5 should be clear enough. Absolute formulas don'tshift their meaning on us as we move from 21 to its end extension 23^ and backagain. Absoluteness is a precious attribute.

8.6 Corollary. All Σ formulas are persistent and all Δ0 formulas are absolute(relative to any theory Ύ).

Proof. By 8.4 all Σ formulas are persistent, hence all Δ0 formulas are persistent.But the Δ0 formulas are closed under negation and φ is absolute iff φ and —\φare both persistent. D

8.7 Example. Let Sl and 23^ be models of KPU, 21^^93 . We can inter-pret all the definitions and theorems of KPU in these two models. For example,let αeSlan. Since Ord(x) is a Δ0 formula,

iff 93

Now let us return to consider the rationale behind the Δ0 in Δ0 Separationand Δ0 Collection. We see from Corollary 8.6 that we have asserted separationand collection for absolute formulas, at least some of them. For example, ifwe form the set b = {xεa\φ(x)} in 21OT, a model of KPU, (with φ a Δ0 formula),then in any ®<n — end%[R> tne equation for b will remain true.

Have we asserted separation and collection for all absolute formulas? Yes,but not explicitly. There are formulas φ(x,y) which are absolute relative to KPUwhich are not Δ0; separation for such φ is not an axiom of KPU. It is a theoremof KPU, though, as we see from the following result of Feferman-Kreisel [1966].

8.8 Theorem. For any theory T of L*, if φ(xl,...,xn) is persistent relative toT then there is a Σ formula ψ(xΐ9 ..., xn) such that

Hence, if φ is absolute relative to T, there are Σ and Π formulas ψ(xί9 ..., xn),Θ(xί9 ...,xπ) such that

From the results in § 4 it follows that we can prove separation in KPU forall formulas absolute relative to KPU and collection for all formulas persistentrelative to KPU. Furthermore, if we later extend KPU to a stronger theory T(and we will from time to time) then we'll still have separation for all φ absoluterelative to T and collection for all φ persistent relative to T. (If T is strongerthen it has fewer models so, in general, it is easier for a formula to be persistentor absolute.) These results are not used in the actual study of KPU but theyare reassuring.

We conclude this section with a lemma which will prove useful later on.We include it here so that the student can become familiar with the conceptof absoluteness. First some remarks.


If 95^t=KPU and we use a phrase like "b is an ordinal of 9V, what wemean, of course, is that be 33^ and Swl= Ord(b). The object b need not be areal ordinal at all. If %(κênd®<n anc^ êân then, as we saw in Example 8.7,a is an ordinal of 21 iff a is an ordinal of 33 . Furthermore, since Slsoϊênd®^the ordinals of 21 form an initial segment of the ordinals of 33 . (Why?) Thisinitial segment may or may not exhaust the ordinals of 33 , even though 21 33 .In the case where it is a proper initial segment there need not be any ordinal bof 33gj which is the least upper bound of this segment.

8.9 Lemma. Let 2ίαnênd®sR where 33^1= KPU. Suppose that whenever33 N rk (a) = a we have aeA iff OLE A. Suppose further that there is no ordinal βof 33^ which is the least upper bound of the ordinals of 51 . Then, with the possibleexception of foundation, all the axioms of KPU hold in

Proof. We check three axioms and trust the student to verify the other two,Extensionality and Union.

Pair: Suppose, xêSί^, let a,βeA be such that

Then, if 33αrιt=:y = (αH-l)u(/? + l), we have ye A since otherwise γ would be the

least upper bound of the ordinals of 21 . Thus if we choose be 33 with33^^ b = {x, y} so that 33ant=rk(b) = y, then be A and $lmt=b = {x,y} by ab-soluteness of the formula b = {x,y}, from Table 1.

Δ0 Separation: Suppose a,yeWm. Let φ(x,y) be Δ0. We want to findί^beSl^ such that

(i)b = {xea\φ(x,y)}

holds in $1 . Let be932R be such that (1) holds in 33 , using Δ0 Separation in 93 .But since

the set b is in <&m. It still satisfies (1) in 21 by absoluteness.

Δ0 Collection: Suppose that αe^l^, the formula φ(x,y) is Δ0 with parametersfrom SIsK and that Vxeα3y φ(x,y) holds in 91 . Then we have:

(2) for each xea there is a ye A and aeA such that (ΆyJι^=φ(x,y) and) = a, and hence, by absoluteness ^Bm

Thus in 93^ we have Vxeα3α3j;[rk(y) = α Λ φ(x,y)\. So, by Σ Reflectionin 33 , there is an ordinal βe33aϊl such that


and hence:

(3)

holds in 93 . In 93^ pick the least ordinal β satisfying (3): it exists by foundation.By (2), β is a sup of ordinals αe^ϊl^, so βe9lan. Apply Δ0 Collection in 93^ to(3) to find a set be^m such that

, y) Λ rk(y) < β~]

holds in 93 . Since 93αrίt=rk(ί?)^β, be^. But then the formula

is Δ0, it holds in 93 , and it has all its parameters in 91 . Hence by absoluteness,it holds in 91 . D

8.10—8.12 Exercises

8.10. Given 91 93 we write Mm ^ if for all Σ1 formulas φ(xί9...,xj ofL* and all xl9...,xn in *Άm:

9IOTl=<p [>!,..., xn] iff 93«l= φlX,... ,xπ].

Show that if 2lsw£end®«» ®m<ι®9i> and 939lt=KPU then, with the possibleexception of foundation, all the axioms of KPU hold in 91 . (The end extensionhypothesis is used to insure that Σ formulas persist from 91 to 93 . Withoutthis, the exercise is false.)

8.11. Given 91^^93^, show that 91 ^ 93^ iff for every Δ0 formulaφ(v1,...,vn,vn+ΐ) and all x1,...,xIIe2ϊsw,

ι<p[xι> >Xn] implies 3xn+ίe<Ά9Λ(Ά9t\=φ[xί9...9xn + l ] ) .

(We are not assuming ^l^^e^®^!)

8.12 (Schlipf). Find an example of two structures 91 and $}m satisfying thehypotheses of Lemma 8.9 but where 91 fails to satisfy the axiom of foundation.[Let 5&m be a proper elementary extension of HF^. (Cf. § II.2.)]

8.13 Notes. The considerations involved in the choice of Δ0 Separation andΔ0 Collection are suggested by the informal notion of "predicative". Kripke, infact, called his axioms for admissible sets PZF, for predicative ZF. As an expli-cation of the intuitive idea of predicativity, however, KP has certain debatablefeatures. See, for example, Feferman [1975] for a discussion and examples of settheories which are predicative in a more stringent sense.


9. Additional Axioms

There are certain extensions of KPU which surface from time to time. We havealready defined KPU+ and KP in §2. We catalogue some of the others here.

9.1 Definition. The axiom of infinity, or Infinity, is the axiom:

3αLim(α)

where Lim(α) is defined in Table 2. We use ω as a symbol for the first limit ordinal.

Note that ~ι (Infinity) asserts only that all ordinals are finite, not that allsets are finite.

The axiom of infinity is often used to form sets by taking a=\Jn<ωbn wherebn is defined by recursion on n. We saw one example where this would havebeen convenient in the proof of 6.1. For another example, define (in KPU)

by Σ recursion. We find that F(a,n) is the set of rc-element subsets of a. InKPU + (Infinity) we can introduce a new Σ operation symbol Pω by

as the student should verify. We can use Pω to convert quantifiers over finitesubsets of a to bounded quantifiers :

Vfc [b^a^b finite -+(...£>...)]

becomes

in KPU 4- (Infinity). Since a is finite iff aePω(a\ we see that "α is finite" isΔx in KPU 4- (Infinity) (whereas it is only Σί in KPU).

The remaining axioms will be of secondary importance for our study.

9.2 Definition. Σί Separation is the set of axioms of the form

(i) 3b Vx(xeb<r-+xea/\φ(x)),

where φ is a Σί formula of L*.

9.3 Definition. Full separation asserts 9.2 (i) for all formulas φ of L*.

9. Additional Axioms 39

9.4 Definition. Full collection asserts the collection scheme

Vxeα3yφ(x,3;)-» 3b VxeaByeb φ(x,y)

for all formulas φ of L*.

9.5 Definition. The axiom Beta. A relation r is well founded on a if

If râxa and r is well founded on α then we say that r is well founded. (If ris well founded on a, then rn(α x α) is well founded, but r itself may have somefunny things going on outside the set a.) The axiom Beta asserts: for every well-founded relation râxa on a set a there is a function /, dom (f) = a, satisfying:

for all xea. The function / is said to be collapsing for r.

The axiom Beta has the effect of making the E^ predicate "r is well foundedon α" a Δj predicate since it becomes equivalent to:

3/ [dom(/) = <2 Λ / is collapsing for r] .

(See 9.8(ii)(b).) Beta is not provable in KPU but it is provable if we addΣ! Separation.

9.6 Theorem. Beta is provable in KPU+ (Σx Separation).

Sketch of proof. Let us work in KPU+î Separation). Let r be well foundedon α, and write x<y for <x,y>er. Define a operation F on the ordinals byΣ recursion:

= the set of all xea such that (yea\y-<x}^\Jβ<OLF(β).

Note that oί^β implies F(a)^F(β). Let us show that every xea is in some). If not, then the set b = a-b0 is non empty, where

this being the place where we need Σ! Separation. Let xeb be such that for allye b, we have y^x (using the well-foundedness of r). Then

Vyea[y<x-+lβ(yeF(β))']


so by Σ Reflection there is an α such that

Vyeα[)Kx-»3jβ<α(.yeF(β))], and hence

So xeF(α) by the definition of F(α), which contradicts xφb0. Now sincea = \JaF((x) there is, by Σ Reflection, a y such that a = \JΛ<yF(oί).

The rest of the proof is easy. Define /α, for α^y, by recursion on α: fΛ isthe function with domain F(α) such that

for all xeF(α). These /α are increasing (β^α implies fβ^fΛ9 by induction on α),and / = fy = \JΛ<yfΛ is the desired function satisfying

f(χ)={f(y) ye<**y<χ}

for all xea. D

9.7 Definition. T/ze power set axiom. We think of the power set operation asa primitive operation. When we use the power set axiom we will assumeL* = L(e, P, . . .) where P is a 1 -place operation symbol. The power set axiom asserts

where, as in Table 2, S(x) means "x is a set".


9.8. Prove in KP (not KPU) that every set of finite rank is finite. HenceKP + —ι(Infinity) implies that every set is finite. This greatly limits KP as opposedto KPU, as we'll see in later chapters.

9.9. Let r^axa and let / be a function with dom(/) = fl which is collapsingfor r. Prove the following in KPU:

(i) r is well founded;(ii) rng(/) is transitive and has no urelements in it;

(iii) If g is a function with dom(g) = a and g is collapsing for r, then f=g.[Hint: Prove VbVxEa(f(x) = b++g(x) = b) by ε-induction on fo]

(iv) If for all x,yξa, x^y, there is a zea with -i«z,x>er<-Xz,y>er), then/ is one-one and hence is an isomorphism of <α, r> with a transitive set </?,enb2>.

9.10. Show that in KPU + (Beta) we can introduce a Σ operation symbol Bsuch that

B(r,α) = 0 iff r is not well founded on α,

9. Additional Axioms 41

but if r is well founded on a, then B(r,a) is a (the) function / with dom(f) = asuch that f(x) = {f(y): y e α Λ <y,x>er} for all xeα. [Use 9.9.]

9.11. A relation r^axa well orders a if it orders α linearly and is well founded.Show in KPU + (Beta), that if r well orders a, then there is a (unique) ordinal αsuch that <α,r>

9.12. Show that adding Σ1 Separation to KPU has the same effect (i.e. sametheorems) as adding all the following axioms, where φ is Δ0:

3bVxeα [By φ(x,y)-+1yeb φ(x,y)].

9.13 Notes. In a theory like ZF containing Σ1 Separation, Beta becomes a theo-rem and the collapsing lemma of § 7 is a consequence of it. In such theories Betaitself is often called The Collapsing Lemma. It is due to Mostowski [1949]. InKPU we must separate the two aspects since one is provable and the other isnot. Beta is so named because Mostowski [1961] used the terminology "β-modeΓ(with "/Γ for bon ordre) for models where well-orderings were absolute.

Chapter II

Some Admissible Sets

Having gained some feeling for the theory KPU we turn to its intended models,admissible sets. Admissible sets come in many sizes and shapes. In this chapterthe student is introduced to some of the more attractive ones in a cursory fashion.We will delve into their structure and properties later.

1. The Definition of Admissible Setand Admissible Ordinal

It facilitates matters if we fix a largest possible universe of sets over an arbitrarycollection M of urelements once and for all. We define by recursion:

VM(α + l) = Power set of (MuVM(α));VMW = (Jα<Λ VΛί(α)> if ^ is a limit; and

Fig. 1 A. The universe VM of sets on M

1. The Definition of Admissible Set and Admissible Ordinal 43

where the union in the last equation is taken over all ordinals α. (The reasonfor letting VM(0) = 0, rather than VM(0) = M, is that VM is to be a collectionof sets on M.) We use eM for the membership relation on YM, dropping thesubscript if there is little room for confusion. If SDt = <M,---> we write Wm

for VM. If M is the empty collection we write V(α) for VM(α) and V for VM.

1.1 Definition. Let L* = L(e,...) and a structure $01 for L be given. An admissibleset over SOΐ is a model 91 of KPU of the form

where MuA is transitive in VM, and e is the restriction of eM to M^jA. Theadmissible set 91 is admissible above $R if Me A, i.e., if Sl^KPU"1". We usespecial Roman A, IB, <C to range over admissible sets. When we need to exhibitthe underlying structure 9M we write A^.

Fig. 1B. A typical admissible set over $01

In other words, admissible sets are models of KPU which are transitive hunksof VM with the intended interpretation eM of the membership symbol. Warning:While the interpretation of the membership symbol must be the natural one,Definition 1.1 makes no such demands on the interpretations of any other symbolsin the list ... of L(e, . . .). They must fend for themselves. For example, if L* = L(e, P)and the admissible set AM = (SOt; A,e,P) is a model of Power, then there isnothing to guarantee that P(ά) is the real power set of α; it may very well beonly a small subset of the real power set of a.

1.2 Lemma. Suppose 9ίsw = (aK;Λ,eM,...) and 93w = (5R;B,ew,...) and// MuA is transitive in VM, then ytm—end'&M

Proof. Recall the definition given in 1.8.1. If aeA then aeM=a = a€N sinceis transitive in VM. D

This lemma also holds for 33^ = %,, except that V^ is not a proper structure.This trivial lemma is of real importance. With the results of 1.8 it insures that Δpredicates and Σ operations of KPU have the same meaning in all admissiblesets that they have in V^.

44 II. Some Admissible Sets

1.3 A Comparison. Consider the two operations TC and P (Power set) and anadmissible set A = (9K,,4,e,P) satisfying the Power set axiom. Given aeA theexpressions

TC(α), P(fl)

each have ίwo possible interpretations. For TC there are the sets b0, b± such that

At=TC(α) = b0 and VaRN=TC(α) = fe1,

where A = (9K;/l,e). For P there are the sets c0, cl such that

(α) = c0 and VOT \=P(a) = cί.

Since A^^V^ and TC is a Σ operation, we have Vmt=TC(a) = bQ:, and sobQ = b^. Thus ί?0 is the real transitive closure of α, so that

ô = Π {^ I transitive, a^b] .

For P, however, this fails. Since x^y is Δ0 we get cQ^cί but that's all. Typically,c0 will be a proper subset of the real power set q of a.

1.4 Definitions. A pwre set in VM is a set a with empty support; i. e., one withTC(α)nM = 0. (For example, ordinals are pure sets.) A pure admissible set is anadmissible set which is a model of KP; i. e., one without urelements. Pureadmissibles can be written A = (,4,e,...). If L* = {e} then we write A for

Fig. 1C. A pure admissible set A

1.5 Theorem. // A^^ΪR; A,ε) is admissible and A0 = {aEA\a is a pure set},then A0 is a pure admissible set, called the pure part of Am. (See Fig. ID.)

Proof. By 1.2 we have A0êndAmêndWm. By absoluteness, sp(α) has the samemeaning in A^ and V^. Let us check Δ0 Collection leaving the easier axiomsas exercises to help the student master absoluteness arguments. Suppose α, beAQ

and suppose A0 satisfies Vxealy φ(x,y,b), where φ is Δ0. If φ(x,y,b) is true

1. The Definition of Admissible Set and Admissible Ordinal 45

Fig. 1 D. The pure part A0 of an admissible set

in A0 it is also true in A^ by absoluteness so A^ satisfies:

Vx e a ly [sp(y) = 0 Λ φ(x, y, b)~] .

Applying Σ collection in AOT, we get a ceA^ such that

(1) Vxef l ίByec [sp(y) = Q Λφ(x,y,b)~], and

(2) Vy 6 c 3x e a [sp(j ) = 0 Λ φ(x, y, b)~\ .

From (2) we get sp(c) = 0, since sp(c) = \J{sp(y)\yec}; so ceA0. But then (1)is a Δ0 formula with parameters from A0, true in A^, hence true in AQ. D

1.6 Exercise. Verify Pair, Union and Δ0 Separation for the proof of 1.5. Noticethat Extensionality is trivial from the transitivity of A0, and that Foundation istrivial by the well-foundedness of A^.

1.7 Definitions. The ordinal of an admissible set A^, denoted by ^(A^), is theleast ordinal not in A^; equivalently, it is the order type of the ordinals in AOT.An ordinal α is admissible if α = o(Aarί) for some SOΪ and some admissible set A^.An ordinal α is ^-admissible if a = o(A,m) f°r some A^ which is admissibleabove 5DΪ (in the sense of 1.1).

1.8 Corollary. The ordinal α is admissible iff oί = o(A) for some pure admissible set.

Proof. If α = o(AOT) and A^ is admissible, then a = o(A0), where A0 is the purepart of ASR. D

What kinds of ordinals are admissible? In the next section we will see that ωis admissible. From our development of ordinal arithmetic in Chapter I we seethat if α is admissible then α is closed under ordinal successor, addition, multi-plication, exponentiation and similar functions of ordinal arithmetic. Thus theleast admissible α > ω is bigger that

ω + ω, ω ω, ωω, ωωω,...,ε0,...,

46 11. Some Admissible Sets

where the operations are from ordinal (not cardinal) arithmetic. In 8 3 we will prove that every infinite cardinal K is admissible and that for any P < K , there are K admissible ordinals a between 0 and u. (Thus, u is a limit of admissibles.)

1.9 Definitions. Let A =A, = (m; A,€, . . .). We often use the following notation and terminology. An object x is in A if X E MuA, and we write X E A . A relation on A is a relation on MuA. An n-ary relation S on A is C, on A if there is a C, formula cp, possibly having constants y,, . . . , y, from A, such that

S(x,,. .. , x,) iff Al=q[x,,. .. , x,]

for all x,,. . . ,x,EA. The relation S is FI, on A if (1) holds for some FI, formula cp, and S is A, on A if S is both C, and H, on A. A function F on A is a function with domain a subset of (MuA)" for some n and range a subset of MuA. We say F is C, on A if its graph is C, on A .

1.10 Proposition. Let A be admissible. (i) If ~ E A then a is A, on A.

(ii) If' x c A then {x) is A, on A. (iii) The C, relations of A are closed under A , v , 3 x ~ a , Vxca, 3x.

Proof. (i) xga iff A ! = x ~ a ; so a is A, as a subset of A . Part (ii) follows from (i). Part (iii) is immediate from the fact that every C formula is equivalent, over A, to a C, formula and the X formulas are closed under the operations mentioned. O

1.11 Exercise. Let A =A, be admissible and let G be an operation defined on all triples in A, whose restriction to A, is C, definable on A,. Define, in V,,

by C recursion. Show that XEA, implies F(x)EA,, and that FrA, x A, is C, on A,. (This should be easy if the student has understood what has come before.)

2. Hereditarily Finite Sets

A set acV, is hereditarily finite if TC(a) is finite. IHF, is the set of hereditarily finite sets of V,. It can also be defined by:

HF,w(0) = 0;

HF,,,(n + 1) =set of all finite subsets of (MuHF,(n));

JHFH = U, <, HF,w(n).

2. Hereditarily Finite Sets 47

Fig. 2A.

2.1 Theorem. HF^ is the smallest admissible set over SOΪ. More precisely, letL* = L(e,...) and let JHFm = (3Λ', HFM,e,...) bean ^-structure.

(i) HFggj is admissible.(ii) // ASH = (501; A, e,...) is admissible, then

There is a difference between HF^ as a set and as an L*-structure, but it isusually clear which we have in mind.

Proof of 2.1. (ii) is trivial since A must be closed under pair and union so thatH¥M(n)^A for all π, by induction on n. Let us prove that HF^ is admissible.Since HF^ is transitive in Wm we get extensionality and foundation for free.Note that each H¥m(n) is also transitive. If x, yGHFm(n) then {x,y}eHF^n +1)so we have Pair. If aeHFm(n) then [_)a is a finite subset of H¥m(n) so is anelement of HF^rc + l), and we have Union. If a^beH¥m(n) then αeHF^rc)since a subset of a finite set is finite, so we have full separation, hence Δ0 Sepa-ration. Similarly, we have full collection for if αeHF^ has say k elementsyί9...,yk and for each of these yt there is an xt such that φ(xhyί) holds, thenall x l 5...,x f c occur in some H¥m(n)9 hence {x1,...,xk}eHFaR(rc + l). D

2.2 Corollary. The smallest admissible set is

HF = {αeV|β is a pure hereditarily finite set).

The smallst admissible ordinal is ω.

Proof. HF is the pure part of any HF^, and o(HF) = ω. D

HF is really where the study of admissible sets began. It was in attemptingto generalize recursion theory on the integers that admissible sets developed (bya rather tortuous route) and, as we now show, recursion theory on the integersamounts to the study of Σ1 and Δ! on HF.

2.3 Theorem. Let S be a relation on natural numbers.(i) S is r.e. iff S is Σt on HF.

(ii) S is recursive iff S is Δ: on HF.

There are relativized versions of 2.3 that are just as easy to prove. For ex-ample, S is recursive in f iff S is Δ! on <HF,e,/>, which by 2.1 is admissible.


For the proof of 2.3 we assume familiarity with the elements of ordinary re-cursion theory.

Proof of 2.3 (=>). Note that (i) implies (ii) since 5 is recursive iff S and —\S are r.e.Nevertheless, first we prove the (=>) part of (ii) to help us prove the correspondinghalf of (i). It clearly suffices to show that every recursive total function on theintegers / can be extended to a Σl function / on HF by the definition:

for xeω

= 0, for xφω.

To prove this we take a definition of recursive function where one startswith basic total functions and closes under some operations which take onefrom total functions to total functions. We choose the one given in Shoenfield[1967], though any other will go through just as easily. Thus, the (total) re-cursive functions are the smallest class containing + , , K< (the characteristicfunction of <), F(xl5...,xn) = xί (the projection functions), closed under com-position and closed under the μ-operation (if G is a recursive function such thatVπ3w [G(n,w) = 0] and for all π,

F(ή) = μm [G(n, m) = 0] , the least m such that G(π, m) = 0 ,

then F is recursive).We have already defined Σί operations + and in § 1.6 and the Δ0 relation

α<β in Table 2. The composition of total Σ^ functions is total and Σ^ so weneed only verify that the class of / with Σ1 f are closed unter the μ-operator.Suppose Vn3w(G(rc,w) = 0), that G is recursive, that G is Σί on HF by theinductive hypothesis and that F(n) = μw[G(rc,w) = 0].

Then F(x) = y iff

Some xt is not a natural number Λ y = 0; or all xt and y arenatural numbers and G(x,j;) = 0 and Vz<y 3π [n^Q /\G(x,z)=n].

This is Σ (since G is Σx), and hence it is Σl by 1.4.3. Thus every recursive functionand predicate on ω is Δx on HF. But every r.e. predicate S(x) can be writtenin the form 3nR(x,n), where R is recursive by a standard result of recursiontheory; so every r.e. predicate is Σi on HF. D

To prove the other half of 2.3 we need the following lemma.

2.4 Lemma. There is a function e:ω-+ HF with the following properties:(i) e is a bίjection (e is one-one and onto);

(ii) e is Σx on <HF,e>;(iii) n = e(m) is a recursive relation of m, n; and

(iv) for any Δ0 formula φ(x1?...,xk) the relation <HF,e>l=φ(e(n1),...χnk))of n1 ?...,n f c is recursive.


Proof. Let us define:

{0} (1=2°)

e(2nι + 2"2 + + 2"*) = {φj, . . . , φk)} K > n2 >

We are using the binary expansion of integers, so e(ή) is defined for all n byΣ recursion. Hence e is Σ1 by 1.6.4. and 2.16. An easy induction shows that eis one-one and onto. To prove (iii), note that if e(k) is an integer n, thene(fc + 2k) = n + l. To prove (iv), note that e(n)ee(m) iff n is an exponent in thebinary expansion 2kl + + 2kl of m. Other Δ0 formulas follow by induction onΔ0 formulas using familiar closure properties of the recursive predicates. D

Proof of 2.3 (<=). Now suppose S in Σl on HF, say S(n) iff HF N 3y φ(n, y),where φ is Δ0, the case where S has more than one argument being similar.Then S(ri) iff 3fc3m [e(k) = n/\φ(e(k),e(m))~\. The part within brackets is recursiveby 2.4 (iii) and 2.4 (iv), so S is r.e. D

There is another way one might want to consider ordinary recursion theory.Suppose we think of the natural numbers not as finite ordinals but as primitiveobjects (urelements) given to us with some structure, say

where we use 0,1,2,... for these natural numbers, JV = {0,1,2,...}, and (x), ©for addition and multiplication in 9ϊ.

2.5 Theorem. Let S be a relation on 91 = <N,®,©>. Then(i) S is r.e. iff S is Σt on HF^;

(ii) S is recursive iff S is Δx on HF^.

The proof is similar to 2.3. For a different proof one can use Theorem VI.4.12.We include 2.5 because it suggests that one might consider Δx and Σ1 on HF^as definitions of recursive and r.e. on 501, for an arbitrary structure 9K. This is,in effect, what Montague suggested in Montague [1968] for the case of whathe calls K0-recursion theory.

Another definition of a recursion theory over an arbitrary structure 9W waspresented in Moschovakis [1969 a], the generalizations of recursive and r.e. beingcalled search computable and semi-search computable. What Moschovakis didwas this. He started with SDl = <M,R1,...,Rfc>, chose a new object 0<£M andclosed Mu{0} under an ordered pair function, calling the result M*. Thenin M* he introduced, via an inductive definition similar to Kleene's for highertype recursion theory, the class of search computable functions. Theorem 2.6


below, due to Gordon [1970] shows that these two approaches coincide. Thisresult will not be used in this book. The reader unfamiliar with search com-putability should consider 2.6 as a definition. A proof is sketched in the notesfor those familiar with the notions involved.

2.6 Theorem. Let ΪR = <M,JR1,...,^k>, and let S be a relation on 9JI.(i) S is semi-search computable on 90Ϊ iff S is Σt on

(ii) S is search computable on 50Ϊ iff S is Δt on

In the context of recursion theory one often works with HF^ as opposedto 9JI itself since the relations on 9JI which are semi-search computable are notalways definable at all over 9Ή itself. The trouble with your average structure SRis that it lacks coding ability. This lack is what rests behind the need for thefollowing class of formulas. We will not use them until Chapters IV and VI.

2.7 Definition. The extended first order formulas of L* = L(e,...) form thesmallest collection containing:

(i) all formulas of L,(ii) all Δ0 formulas of L*,

and closed under:(iii) Λ , v , Vuev, Ίuev (u, v any kind of variables), Vp, 3p,(iv) 3α.

The coextended first order formulas of L* form the smallest collection con-taining (i) and (ii) and closed under (iii) and under:

(v) Vα.

The extended first order formulas do not allow unbounded universal quan-tifiers over sets. The coextended formulas form the dual collection. That thesecollections are more natural than they seem at first is shown by the next resultand the fact that its converse also holds. The converse is a theorem of Feferman[1968] and will not be needed here.

2.8 Proposition. Let φ(v1,...,vn) be extended first order. For any structures

%0ί^end®sri> and any *!,..., e^:(i) ^NφExi,...,^] implies ®TO C*!*-- •>*«]•

Proof. The difference between this and Lemma J.8.4 rests in the fact that thesestructures have the same urelement base 9W. The proof is a trivial proof by in-duction. D

2.9 Example. Let L be the language of number theory with 0, 1, (x), 0. In amodel 91 of arithmetic the set of standard finite integers is defined in ΉF^ bythe extended first-order formula ψ(x) shown here:

3α [xeαΛ Vzeα [z^O


This formula is Σ l5 in fact, so that the set of finite integers is semi-search com-putable over $1. The sentence Vp ψ(p) is extended first order, and HF^N Vp ψ(p)iff 91 is the standard model of arithmetic.

The extended and coextended first order formulas of L(e), when interpretedover HF^, form a very small fragment of so called weak second-order logic.Weak second-order logic over $R just consists of the language L(e) interpretedin HF^. At least that is one way of describing it.

2.10—216 Exercises

2.10. Prove that HFOT c V^αί), and that HF^^V^ω) iff SR is finite.

2.11. If A is a pure admissible set, A^HF, then ωeA.

2.12. If A^ is admissible and ^(A^^ω then the pure part of Aw is HF.

2.13. Prove that ΉF is a Δλ subset of any admissible set.

2.14. Let X be Σt on IHF. Prove that X is Σl on every admissible set.

2.15. Prove that VM(ω) is admissible iff M is finite.

2.16. Prove that H(l) = {nί9...,nk}, where l = 2nι + ~ + 2nk, nv>- >nk, is a Σx

operation of /.

2.17 Notes. Theorem 2.3 is a standard result of recursion theory, as is 2.5.Theorem 2.6 is due to Gordon [1970]. The class of extended first order formulas,introduced in 2.7, will be quite important in Chapters IV and VI when dealingwith structures without much coding machinery built into them.

We conclude the notes to this section with a sketch of a proof of Theorem 2.6.The proof uses results from later chapters. We first show that every semi-searchcomputable relation on 9K is Σί on HF^. The basic relation of the theory is

and it is defined by means of a first order positive Σ inductive definition and so,by the main result of § VI.2, is Σ1 on MF^.

To prove the other half, it suffices to show that some complete Σt relationon HR0J is semi-search computable. Let T be the diagram of $R plus the axiomsKPU coded up on M* by means of the pairing function and let S(x) iff "x codesa sentence provable from T\

It is implicit in Chapter V (and explicit in Chapter VIII) that S is a com-plete Σ! prediciate. But the relation "p is a proof of x from axioms in T" mustbe search computable (if the notion is to make any sense).

Hence the relation 3p ("p is a proof of x from axioms in T') is semi-searchcomputable, since the semi-search computable relations are closed under 3.Note that this gives another proof of 2.3 and 2.5.


3. Sets of Hereditary Cardinality Less Thana Cardinal K

The next admissible set we come across is a simple generalization of HF^. Let Kbe any infinite cardinal and define

H(κ)M = {aeWM\ΊC(a) has cardinality less than K;} .

In particular H(ω)M = IHFΛί. If M is empty then we write H(κ) for H(κ)M. If Kis regular then we can also characterize H(κ)M as follows:

G(0)=0;

G(α + 1 ) = {α £ M u G(α) | card (a) <κ}

) = \JΛ<λ G(a), if λ is a limit ordinal;

For singular K this characterization fails: a bad set sneaks into G(κ + l), if notbefore (see Exercise 3.7). We use the axiom of choice in this section.

3.1 Theorem. For all infinite cardinals K, the set H(κ)m = (yjl;H(κ)M,e) is ad-missible. It is admissible above 9W iff κ>card(M).

The proof of this is not as simple as one might expect in the case when K isa singular cardinal. For K regular, though, it is a trivial result. We will returnto the proof of 3.1 after Theorem 3.3.

3.2 Theorem. Let K be regular. If (3Jl;H(κ)M,e,...) is a structure for L(e,...),then it is admissible.

Proof. Just like for the case κ = ω. In fact, we get full separation and full col-lection. D

The next result, besides giving us a lot of new examples of admissible sets,also allows us to prove Theorem 3.1 for singular K. By card(L*) we mean thecardinality of the set of symbols of L*.

3.3 Theorem (A Lowenheim-Skolem Lemma). Let L* = L(e,...) and letAan = (9W;>4,e,...) be admissible. Let AQ<^M\jA be transitive and let K be acardinal with K card (L*)u card (A0). There is an admissible set 1BK = (ΪI; B,e,...)with the following properties:

(i) 9KΪR (91 is an elementary submodel of^Jl);(ii) c

(iii) A^(iv) For any φ of L* and any xί,...,xneA0, JR9ί\=φ\_xί9...9xn\ iff

Aawt=φ[x1,...,xπ]; and(v) In particular, Ag^^B^ (ΞΞ indicates elementary equivalence).

3. Sets of Hereditary Cardinality Less Than a Cardinal K 53

Proof. (Note that it is not asserted that BÂÎ) Think of A^ as a singlesorted structure

91 =

where 9K = <M,— >. Find 9I1<9I with ^0^^ι and card^X* by the usualLόwenheim-Skolem-Tarski Theorem. 91 ! has the form:

9IX =<NuXl9 AT,

Since there are no urelements in 4l5 clpseiNu^)^^. Let 5 =(i. e. B is the set of sets in clpseίTVu^)) and note that the set of urelements inclpseC/Vu^) is just N. Let f = cNuAl in the notation of 1.7. Since N^>A1 isextensional, / establishes an isomorphism between 9^ and a structureS3 = <ΛΓuB;]V,£,e, ---,- 'X by the collapsing lemma. The isomorphism/ is theidentity for xεA0 by Lemma 1.7.1. Let 91 = <JV,— > and B^ = (^;5,e,...), andall the properties of the theorem are clear. D

Proof of 3.1. It remains to show that if /c is singular then H(κ)yJl = (9Jlι H(κ)M,e)is admissible over 9K. Let κ+ be the next cardinal >κ. The only axiom whichis not immediate is Δ0 Collection. (H(κ)M still satisfies full separation sincea^beH(κ)M=>aeH(κ)M.) Suppose

(1) Vxea3yφ(x,y,z)

is true in //(K;)^, where zeH(κ)m. Now φ has only a finite number of symbolsof L* in it, so we may ignore the rest of L* in what follows. Thus we assumecard(L*)^K0</c. Let a,zeX, X transitive, card(X)<fc; say X = ΊC({a,z}).Since (1) is true in H(κ)^ it persists to H(κ+)m, which is admissible by 3.2. Using3.3 we can get an admissible A^, with ϊl^gjί, so that XÂ^, card(A9ί)<κ:and (1) holds in A^. By Δ0 Collection in A^ there is a feeA^ so that

(2)

holds in A^. But A9lcend//(/c)g[rϊ, so (2) holds in H(κ)m by persistence. D

3.4 Corollary. Every infinite cardinal is an admissible ordinal. For every un-countable cardinal K and β<κ, there is an admissible α where

Proof. κ = o(H(κ}} proves the first assertion in view of 3.1. The second assertionfollows from 3.3 by setting Am = H(κ), A0 = β + l and τc = card(/?). D

We could have also proved 3.1 by using the following result of Levy [1965](proved there for M = 0).

3.5 Theorem. For all uncountable cardinals κ<λ we have H(κ)m-<1 H(λ)m. Thatis, any Σx sentence with constants from H(κ)^ true in H(λ)^ is already true in H^)^.


Proof. This is really just like the proof of 3.1. Suppose the formula 3y φ(x,y)holds in H(λ)m, where xe//(κ:)aR. As in 3.1 we find an admissible set A^, with

such that the formula holds in A^ and card(Ayι)<κ. But thenand so tne formula holds in H(κ)m by persistence. D

One of the earliest generalizations of ordinary recursion theory on the in-tegers goes back to papers of Takeuti where he defines recursive functions onordinals less than some cardinal K. When one looks for the analogue of ΉF forordinal recursion theory on K, the proper structure turns out to be L(JC), the setof sets constructible before K, rather than H(κ). The reason is that one needsto be able to code up the sets by ordinals in some way analogous to Lemma 2.4,if one is to prove a result like Theorem 2.3. We will study the constructible setsin § 5 and again in Chapter V.

3.6 — 3.7 Exercises

3.6. Let κ<λ be infinite cardinals and let X be a transitive subset of H(λ) withcard(X) = κ. Prove that there is an admissible set A of cardinality K with X^Asuch that A^H(λ}, where ^ is explained in 1.8.10 and 1.8.11. [Iterate 3.2.]

3.7. Let K: be a singular cardinal, let M be a set of urelements of cardinality Kand define G as above. Show that already in G(2) there is a set not in H(κ)M.[G is defined just before 3.1.]

3.8 Notes. The technique of following an application of the Downward Lowen-heim-Skolem Theorem with an application of the Collapsing Lemma (as in 3.3)is extremely important. In some sense, it goes back to GόdeΓs original proofthat the GCH holds in L, the constructible universe. It was later used implicitlyby Takeuti when proving, in our terminology, that uncountable cardinals arestable. Theorem 3.5 is due to Levy [1965]. Theorem 3.1 is due to Kripke andPlatek.

4. Inner Models: The Method of Interpretations

We assume that the reader understands the notion of an interpretation, say /,of one theory 7J (formulated in a language LJ in another theory T2 (formulatedin a possibly different language L2). Readable accounts of this can be found inEnderton [1972] and Shoenfeld [1967]. We use φ1 for the interpretation of φgiven by /. Thus φ is in L1? φ1 is in L2; and if φ is an axiom of 7J, then φ1 is atheorem of T2. If 501 is a model of T2, then we use SR~7 for the Lrstructure givenby $R and /; W1 is a model of T^ Note that Enderton uses π$R for our 9JT7;while Shoenfield doesn't make explicit the model theoretic counterpart of thesyntactic transformation /.

4. Inner Models: the Method of Interpretations 55

We give a simple example. We can interpret Peano arithmetic in KPU byhaving / define

"natural number"

"addition"

"multiplication"

"x < /'

by "finite ordinal",

by "ordinal addition",

by "ordinal multiplication",

by »xεy\

Then every axiom φ of Peano arithmetic (in +, , <) goes over to a theorem φ1

of KPU (formulated in L(e,...)). If 21OTI=KPU then 9Iϋ/ = <W, +,', <> is themodel of Peano arithmetic whose domain ΛΓ is the set of finite ordinals of 21 ,and where +, , < are the restrictions of the corresponding functions and re-lations of Sί to ΛΓ. Rather than launch into a discussion of just how we useinterpretations to construct admissible sets, we give a straight-forward illustra-tion. The following result is a generalization of Theorem 1.5.

4.1 Theorem. Let Jk^ = ΰl', A,ε) be admissible and let 9W0c9Jί be a sub-structure of yjl whose universe M0 is Σl definable on A^. // JBmo = (9Jl0; £,e) isdefined by B = {aeA\sp(a)^M0}, then BαRo is admissible over 5R0.

|M

Fig. 4 A. B , the left half of Aa

Proof. B is transitive so extensionality and foundation come for free. Pair, Unionand Δ0 Separation are routine. We prove Δ0 Collection. Suppose IBαίio satisfiesVxea1yφ(x,y), where a and any other parameters in φ are in B . For fixed

, we find in A^ that Vpεsp(y) θ(p)9 where θ(p) is the Σ! formula definingin A^. Hence A^ satisfies the formula:

Vxeα 3y \_φ(x,y) Λ Vpesp(y) θ(p)~] .

By Σ Collection in A^, there is a b in A^ so that

(1) Vxεa3yebφ(x,y) and Vyefo Vpesp(^) θ(p).


But then sp(b)cM0. So beB, and (1) holds in BSEWo by absoluteness. D

Properly viewed, Theorem 4.1 is a trivial application of an interpretation /.If θ(p) defines 50Ϊ0 then /, in effect, simply redefines:

"x is an urelement" by "x is an urelement Λ 0(x)",

"x is a set" by "x is a set Λ Vpesp(x) θ(pj\

and leaves 6 and the symbols of L unchanged. The proof that every axiom φof KPU becomes a theorem φ1 of KPU' is just like the proof of 4.1 (where KPU'is KPU with axioms asserting θ is closed under any function symbols of L).Hence, for every model 21 of KPU', the structure SI 1 is also a model of KPU.In Theorem 4.1 we have BaRo = Sls^

I

5 In this example we don't gain much bylooking at it from the point of view of interpretation, but we will in more com-plicated situations.

The interpretation we just used has some important features in common withmost of the interpretations we use. They are what Shoenfield [1967, § 9.5] callstransitive e-interpretations.

4.2 Definition. Let L* = L(e) and let / be an interpretation of L* into KPU (asformulated in L*). / is a transitive ^-interpretation if / leaves the symbols of Land e unchanged and merely "cuts down on the urelements and sets" so thatthe following are provable in KPU:

(i) if (x is an urelement)7 then x is an urelement;(ii) if (x is a set)7, then x is a set and for all yex, (y is an urelement)7 or

(y is a set)7.If / is a transitive e-interpretation and SI^^KPU then 9I ,7 is called the

inner submodel of A^ given by /.

Fig. 4 B. A model W.w and an inner submodel

The conditions in 4.2 guarantee that Sla/^end^m Fig. 4B indicates the ideabehind transitive e-interpretations and inner submodels.

The following lemma is useful to keep in mind.

5. Constructible Sets with Urelements; HYP^ Defined 57

4.3 Lemma. Let I be a transitive e-interpretation.(i) KPUh-(Extensionality)J;

(ii) For each instance of foundation φ, we have KPUl-φ1;(iii) For each Σ formula φ(x):

KPU h- Urelementίx)7 v Set(x)' -> [φ(xY-+φ(x)],

(iv) For each Δ0 formula φ:

KPU h- Urelement(x)7 v Set(x)7 -* [Xx)'«-><p(x)].

Proo/. (i) (Extensionality/ can be written as:

Set7(α) Λ Set7(fc) Λ α^b -* 3x [(SetJ(x) v Urelement^x)) Λ -ι(xeβ<->xe&)] .

This follows immediately from property 4.2 (ii). To prove (ii) let φ bela\l/(ά)-^3a [^(α)Λ-ι3feeα^(fe)]. Then φ7 states: If 3α [Set'fc) A ψ'fa)'], thenthere is an a such that Set'fa) and ^J(α); but there is no b with SetJ(fe) suchthat be a and ι/^(b). This follows immediately by applying foundation to theformula: Se1/(α)Λ ι/^7(α). Part (iii) follows model theoretically by the commentabove about Slsw^end^sw* fc>r all ^MI=KPU. It can also be proved directlyby induction on φ. Part (iv) follows from (iii). D

4.4 Exercise. Verify that the specific / defined on p. 56 is a transitive e-inter-pretation.

5. Constructible Sets with Urelements;Defined

In this section we construct most of the more important admissible sets in onefell swoop by means of GodeΓs .hierarchy of constructive sets. For reasons whichwill become apparent, we restrict ourselves to the case where the language Lhas only a finite number of nonlogical symbols and where L* = L(e). For sim-plicity we assume the symbols of L are relation symbols: a simple modificationwill extend the results to languages with function and constant symbols.

5.1 Apologia. There are two well known ways of defining the Constructible setsin a theory without urelements, both developed by Gόdel. The most intuitive isby iterating definability through the ordinals; the other uses some form of GδdeΓsJ^,..., Jζ. We have always preferred the former method but find ourselves forcedto use the latter here. The reason is simple enough, but is one that doesn't arisein ZF. Many admissible sets AOT have ordinal 0(AOT) = ω, i.e., are models of— i Infinity, whereas natural ways of iterating first order definability need ω.


Even though we give up the iteration of full first order definability, we modifythe usual approach (along lines used by Gandy [1975] and Jensen [1972]) viathe ^s to make it as similar to the definability approach as possible.

5.2 Assumption. For the rest of § 5 we assume that J^,...,^ are Σ1 operations(of two arguments each) introduced into KPU so that the following hold, wherewe define

(i)(ii)

(iii) KPUh-sp(J*.(x,.y))csp(χ)usp(;μ), for all(iv) KPUH[Tran(&HTran(0(fe))];(v) For each Δ0 formula φ(x1,...,xn) with free variables among x l 5...,xπ and

each variable xi9 i^n, there is a term 2F of n arguments built fromJ . . , J so that:

There are many ways of fulfilling the assumptions. We will return to give aspecific solution in § 6. Next, with 5.2 firmly in mind, we return to the develop-ment of set theory in KPU begun in Chapter I. First note that 2 is a Σ operationsince J^,...,^ are. Define, in KPU, a Σ operation L( , •) by recursion overthe second argument:

L(α, α + 1) = ®(y (L(α, α))) = 0 (L(α, α) u {L(α, α)}) ,

L(α,A) = |Jα<λL(α,α) if Lim(λ).

5.3 Definition. An object x is constructive from a, written xeL(α), if3α[xeL(α,α)]. If x is constructible from 0, we say x is constructible and writexeL.

5.4 Lemma (of KPU). For all sets a and ordinals α:(i) 0eL(α,l) if a is transitive;

(ii) L(α,α) is transitive;(iii) %<β implies L(0,α)cL(α,jβ);(iv) L(α,α)eL(0,α + l);(v) x,>;eL(α,α) implies ^.(x,^)eL(α,α-f 1), Ki^JV;

(vi) αeL(α,jβ) /or some β;(vii) An urelement p is in L(a) iff pesp(α).

Proof, (i), (iii), (iv), (v) follow from the definition of L(α,α) directly. Part (ii) is byinduction on α using Assumption 5.2 (iv). Part (vii) is proved by showing thatpeL(α,α) iff pesp(α) by induction on α (using 5.2 (iii)). This leaves (vi) whichis also proved by induction on α. By the induction hypothesis we haveVy<α3(5 [yeL(α,δ)]. So, by Σ Reflection, there is an ordinal λ such that

5. Constructible Sets with Urelements; IHYP^ Defined 59

Vy<α3<5<Λ, [yeL(α,<))]. But then by (iii), every y<α is in L(α,A); that is,α^ L(α,Λ,). Now, applying Assumption 5.2 (v) for the first time, we see that the setb = {xeL(a,λ)\Ord(x)} is in L(α,Λ/) for some λ"^λ. Since L(a,λ) is transitive,b is an ordinal β and a^β. Again, since L(α, /Γ) is transitive, αeL(α,/l'), becauseeither a = β or αeβ. D

We now define a transitive e-interpretation φL(α) by the following:

(x is a urelement)L(a} is (xesp(α)),

(x ί's α set)L(a) is (x is a set ΛxeL(α)),

leaving e and all symbols of the original language L unchanged. (We apologizefor the two L's, but note that one is sanserif.) Note that this is indeed a tran-sitive e-interpretation in the sense of § 4.

5.5 Theorem. For every axiom φ of KPU + , we have KPU\-φL(a\

Proof. We run through the axioms of KPU + . Extensionality and Foundationfollows from 4.3. Pair and Union follow from 5.2 (i), (ii), and 4.3 (iv). Δ0 sepa-ration follows from 5.2 (v) and 4.3 (iv).

Δ0 Collection: Suppose that φ(x, y,z) is Δ0. Working in KPU assume(fl) and Vxea0 3yeL(α) [φ(x,y,z)L(fl)].

We suppress mention of z. Writing out yeL(α) and using 4.3 (iv) on φ(x,y)we get Vxeα 0 3α [3yeL(α,α) φ(x,y)\. By Σ collection there is a β such that

So, by 5.4 (iii), Vxeα 0 3yeL(α,jβ)φ(x,y). Using 4.3 (iv) again, setting b = L(a,β),we find:

Thus, the interpretation of Δ0 Collection is provable.Finally, we need to prove [3b Vx(xeb ^->3p(x = p)Y\ L(fl). By Δ Separation it

suffices to prove [3f? Vp (pe^)]L(fl). Let fc = TC(a) = L(a,0). By definition, feeL(a,l)and (x is an urelement)L(a) is just xesp(α); but sp(α)^fo. D

5.6 Definition. L(α)sw = (aR;L(M,α)nVAf,e).

L(oί)m is a structure (for the language L* = L(e)) which may or may not beadmissible. We use the intersection with ¥M in 5.6 is just to take out theurelements in strict accord with our definition of structure for L*.


5.7 Theorem. // there is an admissible set A^A^ above 2R withthen LίαJjR is the smallest such. In other words L(α)aίl is admissible, L(α)aϊίcAarl,MeL(α)<m and

Proof. For /?<α, L(M,β) has the same meaning in A^ and Vm by absoluteness.Thus L(α)m is the inner model of A^ given by the interpretation defined above.Thus, in particular, I^α^cA^. By Theorem 5.5, L(α)αrι is admissible, andMeL(α)<m. We see that o(L(a)m) = u from 5.4 (vi). D

If we had the option, the following definition would be printed in red. Itintroduces one of the principal objects of our study. Recall that A^ is admissibleabove 9K if ATO1=KPU + , the " + " being the part that gives "above".

5.8 Definition (The Next Admissible).(i) HYPTO = (2»; A,E), where A = f] {B|(2R; B,e) is admissible above 2R}.

(ii)

5.9 Theorem, (i) HYP^ is the smallest admissible set above 9Jt.(ii) HYP^Lία)^ for α =

Proof. We need only see that HYP^ is admissible over 501, since it is certainlycontained in all other admissibles over $R with M an element. There is an ad-missible Agjj with MeAgjt by 3.1. Let α be the least ordinal of the form o(Aαϊl),where A^ is admissible above $R. Apply 5.7 to α and A^. D

We will study the structure of HYP^ off and on in Chapters IV, VI, VII, VIII.For now we will simply state without proof, for the reader who understands thenotions involved, that if Jf = (N, +,•> is the usual structure of the naturalnumbers, then for any relation R on Jf,R is hyper arithmetic iff ReHYP^, andR is Π\ on N iff R is Σ^ on HYP^. Furthermore, O(Jf ) = ω[ = the least non-recursive ordinal Proofs will appear later.

For the next result recall that L(α) = L(0,α); so L(α) is a pure set. The proofis immediate from 1.8 and 5.7 with M = 0.

5.10 Corollary. An ordinal α is admissible iff L(α) is a pure admissible set.

An urelement free version of 5.9 is given below; the proof is similar.

5.11 Theorem. Let a be a pure transitive set, A = f\ {B: B admissible, αeB}. ThenA is admissible; it is of the form L(α,α) for some admissible α; and it is the smallestadmissible set with a as an element.

5.12 Corollary. If a is admissible and αeL(α), then L(α,α) = L(α).

Proof. Both L(α,α) and L(α) are the smallest admissible sets with a an elementand ordinal α. D

5. Constructible Sets with Urelements; HYPOT Defined 61

The final results of this section will appear rather technical at present, butthey are extremely important for much that is to follow.

5.13 Definition. Let A^ be admissible. Let φ(v) be a Σx formula with one freevariable but with parameters from some set XÂm.

(i) If Am\=1lvφ(v) and Am\=φ[a], then φ(v) is a Σx definition of a withparameters from X.

(ii) If, in addition to (i), for every ^Byjιênd^m which is a model of KPUwe have 33 μ= 3 ! v φ(v), then φ(v) is a good Σl definition of a withparameters from X.

5.14 Theorem. Let M = sp(a) where a is transitive and let α be the least ordinalsuch that A = ($ft; L(α,α)nVM,e) is admissible. Every xeA has a good Σ^ defi-nition on A with parameters from a\j{a}.

Proof. Let B be the set of xeA which have good Σ1 definitions on A withparameters from £f(ά). Note the following:

(1) y(a)^B; and(2) x,yeB implies î(x,y)eB for

For (2) we need the fact that is Σ1 definable in KPU without parameters,which was implicit in 5.2.

(3) If b^B, then

This follows from the fact that 9(b) = bv{^(x,y) \x,yeb,lιζi^N} and from (2).Next since L( , ) is a Σl operation of KPU we find:

(4) // βeB, then L(a,β)eB.

We now prove, by induction on β<a, that βeB and L(a,β)<^B.Case 1. β=Q. 0 has a good Σx definition and L(a,Q) = a^B by (1).Case 2. β = γ + i. By induction hypothesis γeB and L(a,γ)^B. But if γeB

so is 7 + 1. L(ά,y)eB by (4). Thus

so L(fl,y + l) = ®(^(L(fl,y)))cβ, by (3).C«5^ J. jβ is a limit ordinal. By the induction hypothesis we have β^B and

'L(a,β)^B, since j8 = {7|y.<j8} and L(a9β) = \Jv<βL(a9γ). Thus we need onlyprove βeB. This, however, is the main point of the proof. By our choice of αand β<tt we have L(α,j?)aR = (50l; L(α,β)nVM,e) is not admissible so there isa Δ0 formula φ(x,j;,z) and there are objects z,ί?eL(α,β)aR so that

(5) L(a,β)mϊ=Vx€b3yφ(x,y,z), and

(6) L(a,


(Since Lim(β) holds, Δ0 Collection is the only way for L(a,β)m to fail to beadmissible by Exercise 5.16). Now b,ze£, so they have good Σt definitionsσ(u\ ^ί(w) with parameters from <f(ά). Consider the following Σ formula θ(β):

Ord(jS) Λ 3b 3z [σ(b) Λ ψ(z) ΛVxeb3yeL(a, β) φ(x, y, z)

Now clearly A\=θ(β) so every end extension 95TON0()8). If SOTI=KPU thenno "ordinal" of 33 greater than β can satisfy θ by (5), (6). Similarly, no ordinalsmaller can satisfy θ. Thus θ(β) defines β in every end extension of A^ satisfyingKPU, so βeB. D

5.15 Corollary. Every αelHYP^ has a good Σl definition on ΉYP^ with noparameters other than M and some pl9...,pkeM.


5.16. Let Mesp(α) and let λ be a limit ordinal. Show that (9JI; L(α,/ί)n¥M,e)satisfies all the axioms of KPU except, possibly, Δ0 Collection.

5.17. If K is a cardinal, κ>card(M), then L(κ:)αϊl is admissible above 9JΪ.

5.18. If K, λ are uncountable cardinals, κ<λ then L(/c)aϊl-<1L(/l)an.

5.19. L(/c) is admissible for all cardinals κ^ω.

5.20. Improve 5.4(vi) by proving that αeL(M,α + ω), assuming ω exists.

5.21 Notes. The constructible sets were first used by Godel [1939] in his famousproof of the consistency of the generalized continuum hypothesis. In this paper,Godel used iterated first-order definability. In the proof of Godel [1940] thefundamental operations were introduced and used to generate the constructiblesets. The approach to the constructible sets taken here borrows some ideas fromJensen [1972], but it is a little more complicated due to the presence of urelementsand relations on them. We shall see that the complications only come up infulfilling Assumption 5.2 in the next section.

6. Operations for Generating the Constructible Sets

We now turn to the task of finding J^,...,^ satisfying Assumption 5.2. Wewill see that we can get by with especially simple functions (substitutable func-tions). This will prove useful in understanding the sets constructible in ω steps.

6. Operations for Generating the Constructible Sets 63

The real strength of 5.2 resides in the requirement that for each Δ0 formulaφ(xl9...,xn), there is a term built from the symbols J^,...,^ so that

We take care of this condition first.We already have by 5.2:

and

From these we obtain by various simple compositions the following:

The function J^ corresponds (in Lemma 6.1) to v in Δ0 formulas. To handlenegations we need to define:

From this we get, by composition,The need to treat quantifiers leads us to the following more complicated

functions :

(SF'S) ^5(x,y) = dom(x) = { ί s t ( z ) \ z G x , z an ordered pair},

, z an ordered pair},

The functions J^, J^ are annoying. They arise from the peculiar nature of theordered n-tuple. We tend to think of (xl,x2,x3,x4) as a rather symmetricobject but it is, in fact, far from it. We can form it from xl and <x 2,X3,X4>(since it is just <x l5<x2,x3,x4») but we cannot form it from, say x4 and<Xι,x2,x3> or from x3 and <x l 5x 2,X4> using J^,...,J^. This accounts for theappearance of J^ and J^.


It now remains only to add the functions which correspond to atomic formulas :

^u(x,.y) = {zex|zisanurelement},

and for each relation symbol R(x1?...,xn) of L an operation:

In order to prove the desired result we prove something a little more general.It gives us a better inductive hypothesis in our proof which uses induction onΔ0 formulas. For technical reasons, we have inverted the order of the variablesin 6.1. For the same reason, there is an inversion taking place in lines (^ 10),

6.1 Lemma. For every Δ0 formula φ(x l5...,xπ) with free variables among xί9...9xn9

there is a term 2Fφ built up from the symbols «^"1,...,«^κ so that

KPU

Proof. We treat L* = L(e) as a single sorted language with symbols U (for urelement)and S (for set), e, =, R 1 ? ..., R / ? and variables x1,x2,x3, ... . Whenever we write aformula φ as φ(xl9...9xn) we mean that all the free variables of are amongx !,..., xw, but not all of these variables need actually appear as free variables in φ.For the purpose of this proof we need two special definitions. We call a formula ofL* an orderly formula if it satisfies the following condition: whenever a quantifierΊxj or Vx7 occurs in φ, the index j is the largest index of all the free variables in thescope of the quantifier. By simply renaming bound variables systematically, wehave:

(a) Every Δ0 formula of L* is logically equivalent to an orderly Δ0 formula with thesame free variables.

We call a formula φ(xl9...9xn) a termed-formula, or t-formula, if there is aterm J^ such that the conclusion of 6.1 holds. Note that there is a possibleambiguity here since a formula with free variables among x l 5x 2 is also a formulawith free variables among x l 5x 2,x 3 and so could be written as φ(x l 5x2) or asφ(x lsx2,x3). To be completely precise, we should say that φ with free variablesamong x1? ...,xπ is a ί-formula. Line (e) below will show us that we don't haveto be this careful.

Our goal is to prove that every Δ0 formula is a ί-formula. We want to provethis by induction on Δ0 formulas, but we must dispose of certain logical trivialitiesbefore we can treat even the atomic formulas. These trivialities are handled in(b)-O') below.


(b) // KPU h- φ(x1? ..., xn)<-+ψ(xί9 ..., xn) and φ is a t-formula then so is φ.

This last is clear. Combining (a) and (b) allows us to restrict attention to orderlyΔ0 formulas, so Lemma 6.1 follows finally from (z) below.

(c) // φ(xl9...9xn) is ψ(xί9...9xn_ί) and ψ is a t-formula then so is φ.

Define &φ(al9...9an) = anx&r

ψ(aί9...9an-ί). This proves (c).

(d) // φ(xί9 ..., xn) is ι/φc1? ..., xn+ί) and ψ is a t-formula then so is φ.

Note that {0} = {^(aί9aί)}=^ί(^3(aί9aί)9^3(al9aί))9 so we may use {0}inside terms. Define next :

= rng({<0,xn,...,x1>|x ίeα f and ^(x1,...,

= {<xπ,...,x 1>eαBx xα 1 |φ(x 1,...,xπ)}.This proves (d).

(e) // φ(xl5 ..., xn) is ι/φc1? ..., xm) and ψ is a t-formula, then so is φ.

For n>m this follows by induction on n using (c). For m>n this follows byinduction on m — n using (d). For m = n there is nothing to prove.

(f) // φ(xί9 ..., xn) is a t-formula, so is —\φ.

Define #ίΊφ(aί9...9aJ = anX' xaί-Fφ(aί9...9an). This proves (f).

(g) // φ(xί9 ..., xn) and ψ(xΐ9...9 rj are t-formulas so is φ^φ.

Define ^φ^(aί9 ..., an) = #r

φ(aί9 ...,αn)n J^(α1? . ..,ΛΠ). This proves (g).

(h) The t-formulas are closed under propositional connectives.

This follows by (b), (e), (f) and (g). In the following we use φ(x/y) to denote theresult of replacing all free occurrences of y by x.

(i) // ψ(xί9...9xn) is a t-formula and φ(x1,...,xn+1) is ^(x1,...,xπ_1,xπ + 1/xπ),then φ is a t-formula.

If n = l, define ^φ(a1,a2) = ll/(a2)xaΐ . If n>l, define:

and


(j) // ^(x l5x2) is a t-formula and φ(x1? ..., xn) is ψ(xn_1/x1,xn/x2), then φ is at-formula.

This makes sense only if n^2 and is non-trivial only if n>2. To prove (j) define:

In (k) — (v) we prove that atomic formulas are ί-formulas.

(k) For all n, if φ(xΐ, ...,xπ) is U(xJ then φ is a t-formula.

For (k) define &φ(aί9 ...,flπ) = "u(flII,αII)x an.γ x ••• x^.

(1) (xj = x2) is α t-formula by (J^IO).

(m) (xn = xn + i) is α t-formula by (1) and (j).

(n) (xπ = xm) is a t-formula for all m>n.

This follows by induction on m using (m) for the base and (i) for the induction step.

(p) (xn

= χ

m) is a t-formula for all n,m.

For n<m, this is (n). For n = m, take ^φ(aί9 ..., an) = anx ••• x aγ. For n>m,note that (xπ^xm) iff (xm = xn), so the result follows from (b) and (n).

(q) (x1ex2) is a 't-formula by (2F 11).

(r) (xn + ί 6 xn + 2) is a t-formula by (q) and (j).

(s) // φ(xι, ..., xn) is (Xj eXj), then φ is a t-formula.

Let ψ(xl9...9xn+2) be (^i = +ι)Λ(xj = xπ + 2 )A(x I I + 1 ex I I + 2), so that ψ is a

ί-formula by (p), (r), (e), (q). Hence we define:

We now use J 6 to obtain the proof of (s) :

(t) // <P(XI, ...,x f c + m) is R(xk +ι, ...,xk+m), where R is an m-ary relation symbol ofL and /c>l, ί/ien φ is a t-formula.


Define J%(αl9 ...,ak+m) = #:

R(ak+mx ••• xα k + 1 ,α k x ••• xαj . This proves (t).

(u) // R is an m-ary relation symbol of L and φ(xl, ..., xπ) is R(x t l, ..., x, J, ί/zβπ φis a t-formula.

Let ^(x1?...,xn,xn+1,...,xn+m) be R^+iv . + jΛ^ = XΠ + I ) Λ - Λ(x ί m = xw+m).Thus ψ is a ί-formula by (t), (p), (e) and (g). Define

where we apply rng w-times. This proves (u).

(v) All atomic formulas are t-formulas.

The only ones not covered by earlier cases are those of the form S(x,), butS(Xf)<-»"~ιU(xI ) so this follows from (b), (f) and (k). We have not only shown thatevery atomic formula is a ί-formula, but also that the ί-formulas are closed underpropositional connectives. We now turn to bounded quantifiers.

(w) // ψ(xί9...9xn+ί) is a t-formula and φ(x1?...,xπ) is 3x l l+1ex</^(x1,...,xn+1),then φ is a t-formula.

Let Θ(x1? ..., X Λ + I ) be (xw + 1ex7 ) so ψ /\θ is a ί-formula σ(x1? ...,xπ+1). Notethat

,^

So we may define &φ by &φ(al9..., αf l) = rng(^(α1, ..., an,\Jaj)). This proves (w).

(x) If ψ(x 19...9 xk) is a t-formula and φ(xί9...,xn) is 3xkexJ (x1, ..., xfc), wherek>n, then φ is a t-formula.

The proof of (x) is just like that for (w) except we must apply rng k — n times.

(y) // ^(x1? ..., xj is a t-formula and φ(xl9...9xn) is Vx fcex7-^, where k>n,then φ(xl9 ..., xπ) is a t-formula.

This follows from (b), (f) and (x) since

(z) All orderly Δ0 formulas are t-formulas by (v), (h), (x) and (y). D

6.2 Corollary. 3?^, ..., &*κ satisfy Assumption 5.2(v).

Proof. Let φ(xί9 ..., xn) be a Δ0 formula. We need a term 2? so that


But we can form this set from J^({xJ, ..., {*<_ J,α,{xί+1}, ..., {xn}) by usingZF^ (rng) n — i times and then ^^ (dom). D

It may seem discouraging, but we are not through yet because 2F^...,2FK

do not give us the transitivity condition demanded by 5.2 (iv). Recall that wewant to show that Tran(f?) implies Tran(^(fc)), where:

This reduces to showing that for 1< z < N we have :

(*) b transitive and x,yeb implies TC ( (x, y)) c

The only functions among ^, ..., κ for which condition (*) could fail are thoseinvolving rc-tuples. To satisfy (*) for these functions define, for each n 2, functions91&Ϊ, and jel9jr2,Jtr3 by:

yt(χ>y)=<χn> ',χι>yy > if χ = <χB,...,χι>= 0 . , otherwise;

&ϊ(χ,y) = {*πXχ,ι-ι •••*!»)'>}> if χ = <χ π . . -χ ι>— 0 , otherwise

= 0 , otherwise

^3(x,y) = {W,<y,ι;>} , if X = <M,I;>

= 0 , otherwise.

6.3 Definition. Let J be the largest number of places of a symbol of L Thefunctions ^...^N use to generate L consist of J^,...,^x together with , ,for all n^J, plus J^1?^f2,^f3.

6.4 Theorem. The functions ^\,...,J^y satisfy Assumption 5.2.

Proof. We need to see that condition (*) holds for those functions involvingπ-tuples. Let us check 2FΊ in some detail.

Suppose x,y are in the transitive set b. Let us list the members of ΎC(^Ί(x,y))which are not in b, together with the reason they are in @(b). Recallthat Jr

7(x,y) = «w,i;,w>|<w,i;>ex, wey}

7. First Order Definability and Substitutable Functions

Members ofΎC(έFΊ(x,y)) Excuse for appearing in<M,I;,W> with <w,ι;>6x,{u}

Anything else in ΎC(έFΊ(x,y)) is in b, since b is transitive. J^ and the ^R aresimilar. The others are simpler. D


6.5. Show that each of J^ + 1, ..., J*^ can be written as a term in J^,...,^.[Hint: This is fairly easy using 6.1.]

6.6. Define L'(α,/l) using only ^Ί...^. Show that for limit ordinalsλ, L'(a,λ) = L(a,λ). The only point of using J^ + 1, ..., J^y was to make eachL(α,α) transitive.

6.7. Verify condition (*) in the proof of 6.2 for J*8.

6.8 Notes. The proof of 6.1 is one of the few places where the addition of urelementsand relations on them causes extra work. Neither space nor memory permit us tolist all the people who have found gaps in earlier proofs of this lemma.

When used in a class or seminar, section 6 should be supplemented withcoffee (not decaffeinated) and a light refreshment. We suggest Heatherton Rock'Cakes. (Recipe: Combine 2 cups of self-rising flour with 1 ί. allspice and a pinchof salt. Use a pastry blender or two cold knives to cut in 6 T butter. Add ^ cupeach of sugar and raisins (or other urelements). Combine this with 1 egg andenough milk to make a stiff batter (3 or 4 T milk). Divide this into 12 heaps,sprinkle with sugar, and bake at 400 °F. for 10 — 15 minutes. They taste betterthan they sound.)

7. First Order Definabilityand Substitutable Functions

The functions J*^,...,^ defined in 6.3 are actually quite simple comparedwith some Σ operations we might have used to satisfy Assumption 5.2. We willexploit this to prove the following theorem; the first corollary is of special im-portance.


7.1 Theorem. Let Wl=(M,Rΐ,...,Rιy, let a be transitive in VM with M^a.Let A = ar\VM and let A9R = (9R;X,ε). Then a relation S on Am is first-orderdefinable using parameters from a iff

7.2 Corollary. // O(9Jί) = ω, then the relations on 9JI in HYP^ are just the first-order relations.

Proof. If o(HYPsw) = α then HYP^L^. D

7.3 Corollary. The relations on L(α,α) in L(α,α + ω) are the relations first orderdefinable over (9K0; L(0,α)nKWo,e), where 9W0

ίs tne substructure of 9Jί withdomain Sp(α).

Proof. Apply 7.1, reading L(α,α) for a and $R0 for 9JI. D

We begin the proof of 7.1 by studying substitutable functions.

7.4 Definition. A Σ operation symbol F of π-arguments is substitutable if theΔ0 formulas are closed under substitution by F; that is, if for each Δ0 formulaφ(w,vl9 . . ., vk), there is a Δ0 formula \l/(ul9 ...,un,vl9...,vk) not involving F so thatKPυ\-φ(P(u),ΰ)<-*\l/(u9ΰ).

7.5 Lemma, (i) The substitutable operations are closed under composition.(ii) // KPU I- Viί(F(w) is a set), then F is substitutable iff for each Δ0 formula φ,

the formula 3xeF(iί)φ(x,i;) is equivalent (in KPU) to a Δ0 formula \l/(ύ,v).(iii) If F is substitutable, so is G defined by G(x,y) — {F(z,y)|zex} .

Proof: (i) is more or less obvious. For example, if φ(F(w))<-M/φ) andthen φ(F(G(x)))^(G(x))^θ(x).

The necessity in (ii) is a special case of Definition 7.4. To prove the otherhalf note that

yeF(Jc) iff 3zeF(3c)(y = z) ,

a=F(x) iff Vzefl(zeF(x))ΛVzeF(x)(z6fl),

F(5c)eα iff

p=F(x) iff

.) iff

So all atomic formulas involving F are Δ0. A simple induction on Δ0 formulas,using the hypothesis of (ii), shows that F is substitutable in each of them.

To prove (iii) note that 3weG(x, y)(p(w)<-»3zex<p(F(z,y)); so G is substitutableby (ii). D

7.6 Lemma. Each of the operations 2F^ ..., ^ is substitutable.

7. First Order Definability and Substitutable Function 71

Proof. We run through a few cases, using 7.5 (ii) quite heavily.

V lu e {x, y } <p(w) <-> φ(x) v

2\ 3we(Jx φ(ι/)<->3zex 3w

tfΊ: 3ze<x?<y> φ(z)«-xp({x}) v(/>({x,y}), whichis Δ0 since J^ issubstitutable;

Thus we see that J 4 is substitutable, since J 2

and i are> by compositionand 7.5 (iii).

JV 3M6dom(x)φ(M)^->3M,ι;6UUx[<M,t;>6XΛ<jθ(M)], so J 5 follows from Jf7^

The other J^ are just as routine. D

For the remainder of the section fix 9W,0 and A^ as in the statement of thetheorem to be proved, Theorem 7.1.

7.7 Lemma. For every element xeL(α,ω) there is a term 3F in the symbolsJ^,...,^,^ and yί9...,ymεav{a} such that x = &(yl9...9yj.

Proof. Note that L(α, n) = @^ (&&(. . .(a). . .)) for n repetitions of ^o^ ψ> is aterm in J^, J 2 as we saw in § 6) so each L(α,π) is of the appropriate form. We nowshow that each xeL(α,π) is of the appropriate form by induction on n. SinceL(α,ω) = \Jn<ωL(a,n) the result follows.

For n = Q we have L(β,0) — α, since a is transitive, so the result is trivial.If x 6 L(α, n + 1) — L(α, n), then x = L(α, π) or x = J*^(z, y) for some y,ze L(α,«)u {L(α,n)}.The first case is taken care of by the first part of the proof. If x = #Γ

i(y,z) withy,zeL(α,π)u (L(α,π)}, then y,z are of the appropriate form. Hence, x is also of thecorrect form. D

7.8 Lemma. // φ(xl5 ..., xn,y) is Δ0 without parameters, then the relation

is first-order definable over A^.

Proof. A trivial induction on Δ0 formulas; just replace Vxeα by Vx, etc. D

Proof of Theorem 7.1. Suppose S^cf, SeL(a,ω). Then, by 7.7, there is a termJ* in J^,...,^,^ such that S = (x1,...,xk,α) for some x1,...,x f ceα. Butthen %!,...,>? j iff <3Ί,. . ^G^X!, ...,xfe,α).

The right hand side is equivalent to a Δ0 formula (p(y1? ...,yπ, x1? ...,x fc,α)by the substitutability of & (using 7.6 and 7.5 (i)) and < >. The relationφ(y^ ...,yπ,x !,..., xfc,fl) is definable on A^, and hence S is definable using theparameters x 1 ?...,x k. The converse is trivial since every definable relation S onASH is ΔO on L(α,l) and so is in L(α,ω) by, say, Exercise 5.16. D



7.9. F is effectively substitutable if the ψ of 7.4 can be found effectively from φ.Show that each J^,..., J^,^ is effectively substitutable. [Use Church's Thesis.]

7.10. Verify that the effective version of 7.8 holds.

7.11 Notes. It seems to be an open problem whether the converse of 7.2 is true ingeneral. The study of substitutable functions goes back to Levy [1965]. Hecalled them "admissible terms", terminology clearly inadmissible in our context.They were used by Gandy [1975] and Jensen [1972] (written later than Gandy[1975]) to prove the urelementless version of Corollary 7.3. Gandy called them"substitutable", Jensen called them "simple".

8. The Truncation Lemma

Recall (from 1.9.5) that a binary relation £ on a set X is well founded iff for allnonempty Y^X there is an x e Y such that for all ye Y we have —\(yEx).The notion is what we have tried to capture in the axiom of foundation, but ofcourse we fail since it is just not expressible in the first-order language of set theory.A nonstandard model of KPU is one of the form 21 = (ΪR; A, E, . . .), where E is notwell founded; the other models are the standard, or intended models since, by thenext result, they are isomorphic to admissible sets. The proof is essentially thesame as that of 1.9.6.

8.1 Proposition. // ^Xaκ = (9W;>l, £,...) is a well-founded model of extensionalitythen, it is isomorphic to a structure of the form ]BM = (SOt;.B,e, ...) withtranstitive. Both B^ and the isomorphism f are unique, and f satisfies

P, far

f ( a ) = { f ( b ) \ b E a } 9 for aεA.

Now let M<m = (Wl;A,E) be any structure and let if = {®OT£ endânlôRis well founded}. Assume if^Q9 which is the case iff 91 N 3x Vx (yφx).

8.2 Lemma. There is a largest %$mEif (one which is an end extension of allother members of if) .

Proof. Let 33 be the union of all structures in if. It is easy to check that®aRê«ΛR and (£^^93^ for all GêTT. To see that SOT is well founded, letX be a non empty subset of Mu£. We must find an xεX such that yεXimplies —\yEx. Since 93^ is the union of if, there is a Hmeif such thatX' = Xπ(MuC) is nonempty. Since (t is well founded there is an xeX' suchthat yeX' implies —lyEx. But yEx implies jeMuC for all yeMvA (byân — end^αn)? so we have ~ΊyEx for all yzX. D

8. The Truncation Lemma 73

8.3 Definition. The largest well-founded 93OT such that Sswênd^sw is calledthe well-founded part of 21 and is denoted by

Note that this makes sense whether or not $1 is not well founded. If $1is well founded, then i^/(tyίm) = 91 . If Wm is a model of extensionality, so isWYCΆml since /^7(2iari)êndân- In this case we often identity i^/(^) with theUnique transitive structure isomorphic to it, as given by 8.1. We make this identifi-cation in the next result, for example, which is an example of one way in which(CPU is better behaved than stronger theories like ZF. It gives us a new method ofConstructing admissible sets, which accounts for its occurrence in this chapter.

8.4 Truncation Lemma. Let 9IOT = (SDl;^,£,...) and BOT = (2Fl;B,e, ...) be L*-ktructures with SI^KPU and JBmênd<Άm, where (W;£,eH^(9Jt;,4,£).Then Ban is admissible over 9JΪ.

Proof. We need to show that the hypotheses of Lemma 1.8.9 are satisfied, for thenwe get all the axioms of KPU except Foundation true in B . But B^ is wellfounded, so it certainly satisfies Foundation. First note:

(1) If aeA and aE^B, then aeB.

This follows from the maximality of BêY/^.

(2) If aεB and Mm N rk(α) = α, then aeB.

This follows by e induction on a, using (1), since SIaRt=α = sup{rk(x) +

(3) If ae£ and <&&*= rk(ά) = a, then

This follows by induction on α using (1). Thus we see that if 2Xaίϊl=rk(α) = α,then αeJ5 iff

(4) There is no sup in 21 for the ordinals of ]Bm.

This follows from (1). Thus, we have what we need to apply 1.8.9. D

We have worded 8.4 in a roundabout way because of the functions whichmight appear in the list.... The universe of i^/(3R;A9E) might not be closedunder them. Perhaps it is worth stating a special case of 8.4, the one we usuallyapply. It follows at once from 8.4.

8.5 Corollary. // 91OT = (2R;>4,E) is a model of KPU then its wellfounded part is anadmissible set over 501.

ί j

8.6 Theorem. Let Wi = (M,Rl ....,#,>. The admissible set MYPm is the inter sectionof all models 21 , well-founded or not, of KPU + . More accurately, given any model%CT of KPU+, there is a unique embedding of ΉΎPm onto an initial substructure of

74 11. Some Admissible Sets

Proof: By 8.5, ( ) is admissible above 'JJZ and hence IHYP,s W/('U,)c,,,'U,, the first inclusion being correct up to the unique embedding discussed above. O

Recall that O(%n) is, by definition, o(IHYP,). Structures %n such that O(YJl) = w are going to play an interesting role in our study of admissible sets and structures. We call such structures recursively suturuted. This terminology will be justified in Chapter IV (cf. Definition IV.5.1 and Theorem IV.5.3). In the next theorem we use the truncation lemma to prove that there are lots of recursively saturated structures; that is, structures YJl with O(YJl) = o.

8.7 Theorem. For every structure YJ2= (M,R,, . .., R,) there is u recursively saturated elen~entary extension % of %I of the same cardinality.

Proof: Consider IHYP, as a single-sorted structure of the form:

and let B = (NUB, N, B, R',, . . ., R;, E) be an elementary extension with nonstandard natural numbers. This exists by the ordinary Compactness Theorem. Let %=(N,R;, ..., R;), and let %,=(%;B,E), which is a model of KPU'. The well-founded part of 23, is an admissible set IB; with N E %', since rk(N) = 1. Also o(IB;)=o, since 23, has non-standard integers. Thus o(IHYP,) =w by Theorem 5.9. The cardinality considerations are routine. 0

This shows that we cannot expect YJl<% and O(%)=w together to imply O(YJl)= w.

Finally, we use 8.5 to get a rather technical looking results. The real content of 8.8 will emerge gradually throughout the book.

8.8. Proposition. Let S be an n-ary relation on a structure YJl= (M, R,, . . ., R,). If S is C, on IHYP, then there is a C, formula cp(x,, . . ., x,, p,, . . ., p,, YJl), with only constants p,, . . ., p , ~ M such that for all q,, . . ., q , ~ M the following are equivalent:

6) S(q1, ..., q");

(ii) HYP, I= cp(G, $2 MI;

(iii) For all models of KPU' of the form 'U, = ())32; A,E) we have 'U, cp(G,fi> MI.

ProoJ: By 6.4 every ~EIHYP, can be defined by a C, formula with constants from M u .( MI. Thus we may replace any of these a's by its definition to get a 9 of the appropriate kind such that ( i ) o ( i i ) . Since IHYP,k KPU', we see that (iii)*(ii). To see that (ii)*(iii) note that any such 21, is (isomorphic to) an end extension of IHYP,, by 8.6. Hence if cp(q,p,M) holds in IHYP,, it holds in a,, since it is &.Of course, we need to know that the isomorphism is the identity on M u j M ) , but thisfollowsfrom8.1. 0

8. The Truncation Lemma 75


8.9. Let aR = <M,R1,...,R l> be such that 2RX5R, and card (2R) = card (51)implies SR^Ή (equivalently, Th(9W) is card(9JΪ)-categorical). Show thato(HYPsw) = ίo, and hence the relations S on 9JΪ in HYP^ are just the ones first-order definable over 9JΪ.

8.10. Let 9K - <M, = > be infinite. Show that a subset X ^M is in HYPOT iff X orM — X is finite.

8.11. (F. Ville) Suppose α is not admissible and <L(α),e>^end<A,£>, where<^,£>I=KP. Show that, up to a unique isomorphism, <L(/?),e>^end<,4?£>?

where jS is the least admissible ordinal greater than α.

8.12. Use the notation of 8.11. Let S be a relation on L(α), 5 Σ! on L(β). Find aΣ! formula <p(xls ...,xπ,0) with αeL(α) and no other constants such that thefollowing are equivalent :

(i) S(x);

(ii) L(β)\=φ(X9a)'9

(in) For all models 21 = <v4,E> of KP if <L(α),e> c end9I, then 21 N <p(x,3).

[Hint: Find a good Σx definition of α to get rid of L(α) in φ.]

8.13. If 9ISM = (M;A,£,P) is a model for KPU -f Power and(an;β,6) = 7(ϊR;v4,£), then BαR = (ϊR,JB,e,Pίβ) is admissible and a model ofPower.

8.14. Show that the well-founded part of a model Sί of KPU + Beta need notsatisfy Beta. (Not for the beginner.) The well-founded part of a model <^4,£>of all of ZF need not satisfy Beta.

8.15. (For those familiar with Π}.) Let 91 = <7V, +,•> and let S be a relation on 91.Show that if S is Σ! on HYP^, then S is Π}.

8.16 Notes. The history of the Truncation Lemma is more complicated than thelemma itself. Starting from the fact that every ω-model of second-order arithmeticcontains all hyperarithmetic sets of natural numbers, Mile. F. Ville generalizedthis by proving Exercise 8.11. This was in 1966 and her proof remains unpublished.Barwise [1969] generalized this to obtain a V = L or V = L(x) version of theTruncation Lemma. It is not clear to the present author who first thought of thetrick (used back in Lemma 1.8.9) that allows the full result to go through.


9. The Levy Absoluteness Principle

We have been rather free wheeling with our metatheory, for example in § 1 and § 3of this chapter. We used the power set axiom, results on cardinal numbers andeven the axiom of choice (in the guise of the Downward Lowenheim-Skolemtheorem) in § 3. It should be clear, though, that everything we have done could beformalized within ZFC, Zermelo-Fraenkel set theory with choice. (Given astructure 9Jί = <M, ...> for example, with MeV, we can define V^ as a class in Vwithout difficulty as long as we remember that eM is distinct from e.) Weakertheories would suffice; but, because it is familiar to almost everyone, we fix ZFCas our metatheory for this book, unless some other theory like KPU is specifiedthe way it was in Chapter I.

The following version of the Lowenheim-Skolem Theorem, implicit in 3.4,will be of considerable use to us in what follows, though we usually use the simpleparameter-free version given in 9.2.

9.1 Theorem. Let φ(xι, ..., xn, JΊ, ..., ym) be a Π formula in the language of ZFC(with only e and = ) with the free variables only as shown. The following sentenceis a theorem of ZFC:

9.2 Corollary. Let φ(x) be a Π formula in the language of ZFC with only the onefree variable x. Then ZFC I— Vxe/f (KJ φ(x) -> Vx φ(x).

Proof of 9.1. Since KP^ZFC we may assume φ(x,y) is Hί9 that is, of the formVzι^(x, y,z), where ψ is Δ0 by 1.4.3. We work within ZFC and prove the sentence inquestion by contraposition. Let yl9 ..., y^H^^ and suppose there are x l9...,xn,such that —\φ(x9y)9 i.e. there is a z such that —]φ(x9y9z). Pick /C^K! so largethat xl9...9xn9zεH(κ). Then, since —\ψ is Δ0 we have <//(/c),e> t= ~ \ψ(x,y,z)by absoluteness. By 3.4, ^H(^l\ey^il <//(κ),e>, so we find

i), e> \= 3x1,...,xnz—i^(x,j),z). Pick xί9...,xn€H(ttί) so thate>N3z-ιι//(x,y,z). Then by Lemma 1.4.2, 3z~ \ψ(x9y9z) is true, which

means that — \φ(x,y). Since x1,...,xπeH(K1), this proves our result. D

We conclude this section with a simple example of the use of the AbsolutenessPrinciple.

9.3 Proposition. Let 9M = <M> be a structure with no relations. If X^M isconstructible from 9JΪ, XeL(9!Ji), then X or M-X is finite.

Proof. The statement to be proved has the form:

)-+X is finite v M-X is finite'].

9. The Lέvy Absoluteness Principle 77

In ZF, or even in KPU + Infinity, this is a Π statement (by use of Pω from 1.9)so it suffices to prove it for countable M and α. We may assume M is infinite sinceotherwise the result is trivial.

Let σ be any one-one map of M onto M. We can extend σ to an automorphismjσ of ¥M onto VM by recursion on e :

(a) = {σ(x)\xea} .

Note that σ(J^(x,j;)) = J^(σ(x),σ(y)), whenever l^ίX^Γ, by inspection. Asimple proof by induction shows that σ(L(M, α)) = L(M,α) for all α.

Now suppose that M and α are countable but that there is an JfeL(M,α) withX^M such that X and M-X are both infinite. Then, for any Y^M with Yand M — Y infinite, there is a one-one map σ mapping X onto Y so that σ(X) = Y.But then XeL(M,oc) implies σ(L(M,α)); so YeL(M,α). But there are 2X° such X,whereas L(M,α) is countable. D

9.4—9.7 Exercises

9.4. Let ZFU+ be KPU+ plus full separation, full collection, Power and Infinity.Prove that for each φeZFU + , we have ZFU h- φL(M).

9.5. Show that if M is as in 9.3 then L(M) is a model of ZFU+ plus "all subsets ofM are finite or cofinite". This shows that choice fails very badly in this particularL(M).

9.6. Let $ft = <M, #!,..., #j> and let σ be an automorphism of $R. Extend σto a σ Ψ^^ Ψ^ as in 9.3. Show that σ: L(α)ari-^L(α)aίl, one-one and onto,for all α.

9.7 Notes. The Levy Absoluteness Principle was first proved by Levy [1965]. Seethe notes from § 3 for more details on the general argument. One of the mainfeatures of this book (at least from our point of view) is the systematic use of theLevy Absoluteness Principle to simplify results by reducing them to the countablecase. This is particularly true of Part B of the book.

We will see, as a by product of § V.8, that the axiom of choice is not neededin the proof of 9.1. See the proof of V.8.10, in particular.

Chapter III

Countable Fragments of L00 CO

In this chapter the student is introduced to the infinitary logic Laoω and itscountable fragments. The reason for treating infinitary logic so early in the bookis two-fold. In the first place it offers a nice application of the very notion ofadmissible set, since the fragments of Laoω most like ordinary logic are thosegiven by countable admissible sets. More important, however, is the powerfultool that infinitary logic gives us in our study of admissible sets. The resultsfrom model theory presented in this chapter are all chosen because of theirapplicability to the theory of admissible sets and generalized recursion theory.

1. Formalizing Syntax and Semantics in KPU

In § 1.3 we formalized informal notions of mathematics in KPU, notions like"function", "natural number", and "ordinal". In this section we do the samething for informal notions of logic, notions like "language", "structure", "formula".

In this section we work in KPU but we suppose that among the atomicpredicates of our metalanguage L* are the following:

Relation-symbol (x),

Function-symbol (x),

Constant-symbol (x),

Variable (x)

and among the operation symbols of our metalanguage are two unary ones:

v and # .

We use r , r 1 ? . . . to vary over objects x satisfying Relation-symboΓ(x). Sin>ilarly h , h l 9 . . . for function symbols and c,c1 ?d,... for constant symbols. Wealso assume that among the constant symbols of our metalanguage L* are

-i, Λ> V' V, 3, = .

1. Formalizing Syntax and Semantics in KPU 79

These twelve symbols may be part of our original metalanguage L* or theymay be defined symbols introduced into KPU as in §1.5. In applications, thelatter is almost always the case.

We assume the following axioms on syntax:

(1) An axiom asserting that the classes of variables, function symbols, relationsymbols, contant symbols are all disjoint, and that none of the six constantsdisplayed above are in any of these classes.

(2) An axiom on variables which asserts, writing VΛ for v(α),

Var iab le (x) <=> 3α (x = va)

(3) An axiom on # , which tells us the "arity" of relation and function symbols :

if x is a relation or function symbol then # (x) is a positive naturalnumber.

A set L is a language if L is a set of relation, function, and constant symbols.The predicates "t is a term" and "ί is a term of L" are defined by recursion

on TC(ί):

1.1 Definition, t is a term (of !_)<-> ί is a variable, or ί is a constant symbol (in L),or ί = <h,y> where h is a function symbol (in L), ^ = <yι,...,^#(Λ )> and eachyt is a term (of L).

These two definitions ("ί is a term", and "ί is a term of L") are of the typepermitted by 1.6.6 so they define Δ predicates. (The only sticky point comes inchecking that the predicates P(y,n) iff "y is a sequence of length n" and Q(y9n9x9i)iff "P(y9 n) and i^i^n and x is the ith term in the sequence y" are Δ predicates.This also follows from 1.6.6 by recursion on n. For example, P(y,n) iff n is a na-tural number ^1 and, if n>i then there exist zl9z2eΎC(y) such that y = <z1?z2>and P(z2,rc-l).)

1.2 Definition. An atomic formula (of L) is a set of one of the following forms:

(i) < = , f l 5 r 2 > where tί9t2 are terms (of L); we write (t1 = t2) or even (tl=t2).(ii) <r,ί l 5 ..., O where r is a relation symbol (in L), n=#(ή and tί9...9tn

are terms (of L); we write r( f 1 ? ..., ίj for <r,ί l s ..., ίn>.

1.3 Definition. A set φ is a finite formula (of L) iff

φ is an atomic formula (of L), orφ is <~ι,^> and ψ is a finite formula (of L), or

φ is </\»{^'0}) OΓ <V'{^'^}) wnere *A>0 are finite formulas (of L), orφ is <3,y,^> or <V,ι;,<p> where y is a variable and φ is a finite formula (of L).

80 III. Countable Fragments of LQoω

We write ~Ί\// for <—ι,^>, ij/Aθ for /\{ψ,θ} and 3vψ for <3,ι;,ι/^>; similarlyfor v,V. We use the usual abbreviations like φ-+ψ for ((—\φ)vψ). All of theabove predicates are Δ predicates, the last again by 1.6.6.

1.4 Proposition. // Infinity is true then for any language L there is a set

Lωω = {φ\φ a finite formula of L with only variables of the form vn

occurring in φ,n<ω}.

Proof. We first show that

Terms = { t \ t a term of L with only variables of the form vn in t}

is a set. Define

7erras(0) = {ce l_ |c a constant symbol} u {v ( w ) |n<ω},

Terms(n + l)={(h,tl9...,tky\heL, h a constant symbol,fc= #(h), tl9..., tkeΓerms(n)} u Γerms(n).

by induction on n. This makes sense if ω exists, by replacement for Terms (0),as does

Terms = \Jn<(0 Terms(n).

A similar proof shows that Lωω is a set. D

For the past twenty years, and more, logicians have been working to findmanageable strengthenings of Lωω. It has turned out that languages with ex-pressions of infinite length are one of the best lines of attack. These languagesallow us to form expressions like the following

which says that every element is of the form c,h(c\ h(h(c)\ etc.; or

which says that every x is definable by some finite formula; or φΛ(x) defined byrecursion on α by:

φQυQ s

φ>α) is Vy(y<vΛ++\/β<Λφβ(y/vβ)).

Then φα(x) is going to be true in a linearly ordered structure iff the predecessorsof x have order type exactly α.


1.5 Definition. A set φ is an infinitary formula if one of the following hold:

φ is a finite formula,φis—\ψ where ψ is an infinitary formula,φ is Ίvψ or Vvψ where v is a variable and ψ is an infinitary formula,φ is </\,Φ> or φ is <\/»Φ) where Φ is a nonempty set of infinitary formulas.

Again this definition is justified by 1.6.6. We write

for <Λ,ΦX

for <V>φ>;

/\Φ is called the conjunction of the formulas in Φ, \/Φ the disjunction. The notionof infinitary formula of a language L is defined in a parallel way.

We assume that the reader can carry out all the syntactic definitions (freeand bound variable, substitution of a term t for a free variable in φ, for example)only noting that substitution must be defined by recursion over ΊC(φ). Wedenote the result of substituting ί for v in φ by φ(t/v). A sentence is a formulawith no free variables.

We define the set sub(φ) of subformulas of φ by recursion over TC as follows:

sub(φ) = {φ} if φ is atomic

= {<p} u sub(^) if φ is — \ψ, 3vψ or

if <? is φ or

1.6 Lemma. // φ /ιαs a finite number of free variables so does any ^esub(φ).In particular, if ψ is a subformula of some sentence then ψ has a finite numberof free variables.

Lemma 1.6 is proved by a routine induction on formulas, and motivates thefollowing definition.

A proper infinitary formula is one with only a finite number of free variables.The notion of "proper infinitary formula" is a Δ notion, since

φ is proper iff {v\v a free variable in φ} is finite

iff (α|rα is free in φ} is finite,

and the notion "α is a finite set of ordinals" is Δ by Exercise 1.7.6. Since we willonly be discussing proper infinitary formulas, we might just as well drop theadjective "proper" once and for all. We use the symbol Looω to denote the classof all (proper) infinitary formulas of L.

1.7 Definition. A structure SOΐ for a language L is a pair 9W = <M,/> such that,writing x*®1 for f(x) we have :


(i) M is a nonempty set,(ii) / is a function with dom(/)= L,

(iii) r e l _ implies r*01 is a subset of M#(r),(iv) h e l _ implies h9^ is a function with domain M#(h) and range c=M,(v) if ceL then cmeM.

This too is a Δ predicate of 951 and LAn assignment in 9JI is a function s with dom(s) a finite set of variables and

rng(s)^M. Given a structure 9JΪ for L, a term t of L and an assignment s in $Rwith the variables of t contained in dom(s), we let tm(s) be the value of t in 9JIat s. This is defined by recursion:

1**(s) = c** if ί is the constant symbol c

= s(v) if t is the variable v

(s),...,ί?(s)) if ί is h(ί 1 ?...,ί k).

Our next goal is to formalize the notion of satisfaction:

where 9Jί is a structure for L, φ is a formula of L and s is an assignment to thefree variables of φ. In order to make the definition fit into the form of definitionby recursion available to us, we have to be a little awkward. Since there is noset of all variables, there can't be a set of all assignments to 9JI. There is, how-ever, a Σ operation G such that for all L, all L structures $R and all φeLaoω

G(5ϋl, φ) = {s I s an assignment in 501 with dom(s) = free variables of φ} .

We outline the definition of G in Exercise 1.11. It is then a routine matter todefine

for languages L, structures $R for L and formulas φ of L^, by recursion overTC(φ). We give some of the clauses of this recursive definition:

SatL(9W, -iφ) = {SE G(9Jί, -\φ) \ sφ SatL(9K, φ)} ,

l for all φeΦ, stFree-Var(φ)eSatL(aW,φ)},

rsome xeM,su{<ι?,x>}eSatL(SDl,φ)}

if v is free in φ

= SatL(ΪR,φ) if v is not free in φ.


We can now define, for L-structures $R, formulas φ of L, and assignments s,the predicate 9Jtt=φ[s] by

iff

For sentences φ we have Wl\=φ if the empty function 0 is in SatL(9W,(p). Since•SatL(SDΪ,φ) is a Σ operation, by 1.6.4, we see that $RNφ[s] is a Δ predicate of9JΪ, φ, s and the suppressed L (Note that L can be recovered from 50ΪL = dom(2nd$R).) If the free variables of φ are among vί9...,vn we write

for 3Ά\=φ[s] where s = {(v1,aly,...,(vn,any}.

Given structures 9Jl, $1 for a language L we write

m = yi (LωJ if, for all finite sentences φ, Wl\=φ iff 9lNφ; and

9W = 9l (L^J if, for all sentences φ of L^, 9W^φ iff

As written both of these are Π1 predicates of 501,91 (and L). By Proposition 1.1,Lωω is a set if Infinity holds; in fact the operation which takes L to the set Lωω

is a Σ operation on L. Thus, in the presence of the axiom of infinity we canrewrite $RΞ$ft(Lωω) to see that it is an absolute predicate of $R,9l and L Inthe presence of Σί Separation ^Jl = yi(L^ω) also becomes absolute, but forentirely different reasons. More on that in Chapter VII.

1.8—1.12 Exercises. Work in KPU

1.8. Show that an ( = a x a x ••• x α, M-times) is a Σ operation of α and n.

1.9. Prove that the predicate Q, defined in the parenthetical remark followingDefinition 1.1, is indeed a Δ predicate.

1.10. Show that the operation α ->card(α), defined on finite sets a of variables,is a Σ operation. [Use the collapsing lemma.]

1.11. Using 1.10 and the Σ operation

S(α, n) = {b c a \ card(fc) = n} ,

show that the operation G used above is a Σ operation.

1.12. Write out the few remaining details needed for the definition of SatL(9JΪ,φ).


2. Consistency Properties

There is a very general method for constructing models which has evolved intowhat Keisler [1971] calls the "Model Existence Theorem". We will prove thistheorem here in KPU + Infinity. To prove it in this weak metatheory we mustbe a little more careful than usual. Among notions which are equivalent in ZFwe must choose those which avoid unnecessary uses of Power and Choice. Thisexplains why our presentation must diverge in minor ways from Keisler s.

The collection of infinitary formulas of a language L never forms a set butwe must usually deal with a set of formulas. Hence the next definition. Werepeat, for emphasis, that we work in KPU + Infinity in this section.

2.1 Definition. Let L be a language. A fragment of L^ is a set LA of infinitaryformulas and variables such that

(i) every finite formula of Lωω is in L^ ,(ii) if φe LA then every subformula and variable of φ is in LA,

(iii) if φ(v)E\-A and t is a term of L all of whose variables lie in LA thenφ(t/v) is in LA, and

(iv) if φ, ψ,v are in LA so are

~~ιφ, ~ φ, 3ι; φ, Vv φ, φ v ψ, φ Λ ψ .

At this stage, the subscript A serves merely as an index. It will serve a moreuseful purpose later.

We have used an undefined notion in 2.1, a silly technical device ~φ. It isdefined by:

~φ is -\φ if φ is atomic,

~(~ιφ) is φ,

is

is

is

is

We see that ~ has an explicit Σ definition, not a recursive one, since ~ doesnot occur on the right hand side of the above. Note that ~φ is logically equiv-alent to —\φ. (Keisler uses φ— \ for our ~φ.)

Let K be a language and C = {cn: n<ω] a countable set of constant symbolsnot in K. We keep K, C and L = K u C fixed for the rest of this section.

If K^ is a fragment of K,aoω then there is a natural fragment L^= K^(C) ofL^ associated with it; namely the set of all formulas of the form φ(ciί9..., cln)ιwhich result by replacing a finite number of free variables by constants from C.Fix these fragments KΛ and LA = K^(C) for the rest of this section. A term t ofLA is basic if it is in Cor if it's of the form h(c ί l ? ..., cfn) for h e K and the c/s in C.

2. Consistency Properties 85

Next comes the cumbersome but crucial definition.

2.2 Definition. A consistency property for L^ is a set S of sets s such that eachseS is a set of sentences of LA and such that all the following hold for every seS:

(CO) (Triviality rule) OeS; if s<=s'eS then su{φ}eS for each φεs'.(Cl) (Consistency rule) If φes then ~Ίφφs.(C2) (—\-rule) If ~~ιφes then sul^φjeS.(C3) (Λ-™k) If Λφ e s then for all φeΦ, su{φ}eS.(C4) (V-rule) If (Vι;<p(ι;))es then for each ceC, su {<p(c/ι?)}eS.(C5) (\/-rule) If \/Φes then for some φeΦ, su{(p}eS.(C6) (B-rule) If (3uφ(z;))es then for some ceC, 5U {<p(c/ι?)}eS.(C7) (Equality rules). Let ί be a basic term of L^ and c,deC.

i) If (c = d)es then su {(dii) If φ(t), (c = t)es then su {φ(c)}εS.

iii) For some eeC, sv {e = t}eS.

The rule (CO) was not included in Keisler's definition. It really is a trivialitythough.

2.3 Lemma. If S satisfies all of 2.2 except (CO) then there is a smallest consistencyproperty S'^S.

Proof. Define

This is easily seen to be a consistency property. If S c S" and S" is a consistencyproperty then f(n)^S" by induction on n. D

2.4 Lemma. Lef S be a consistency property, seS.

i) φ,(φ^nl/)es implies sii) ceC implies su {(c = c)}eS.

iii) c,d,θEC, (c^d)65, (d^e)es implies s^>{c = e}eS.

Proof. These are all similar. Assume φ, (φ^φ)es. Since (φ^φ) is really (— \φ v ι/^)we see by (C5) that either su{~ \φ}eS or sv{ψ}eS. The first possibility isruled out by (Cl). Next assume the hypothesis of (iii). By (C7i) we have5' = 5u{eΞd}65. By (C 7 ii) we have s'v{c = e}eS so, by (CO), su{c = e}eS.To prove (ii) let ceC. By (C7iii) there is an eeC with su (c = e}e£. By (C7i),su (CΞΘ, Q = C}ES. Applying (iii) we have su {CΞΘ, e = c, c = c}eS, so,by (CO), 5u{cΞC}e5. D

8 6 111. Countable Fragments of L,,,

The point of Definition 2.2 is that it exactly isolates the principles needed to carry out the "Henkin argument". To be more specific, it allows us to prove the Model Existence Theorem. A structure $332 for L is a canonical structure if every element of 93 is of the form cm for some C E C.

2.5 Model Existence Theorem. Let LA he a countuble fragment and let S he a consistency property for LA. For ecery S E S there is a canonical structure YJI ,for L such that YJI is a model of s, i. e., fbr every y E s, YJ1 k y .

Proof. We can't be quite as free wheeling as Keisler [1971, p. 131 since we have a rather limited metatheory. We have already taken care of most of the diffi- culties, though, by careful choice of definitions and by the wording of the theorem. Since LA is countable we can enumerate its sentences:

and the terms occurring in LA:

We shall construct a sequence

of elements of S as follows. We take s, to be the s of the theorem. Given s, we define s,, , by adding on one, two, or three sentences of LA.

Step 1. Find the first constant symbol c of C, in the list of terms, such that S,U { c - - t , ) ~ S and let s: ,=s,u{c-t ,} .

Step 2. If sLu jy,) $ S let s,, , =s i . If sAu ( ~ , ) E S then let s i = s h u (9,).

There now are three distinct cases to consider, depending on the principal connective in y,.

Step 3. If y , does not begin with 3 or V let s,, , =s;.

Step 4. If y , is 3c$ then find the first C E C in the list (2) , by (C6), such that si u { $ ( c / v ) ) E S and let s,, , be this element of S.

Step 5. If y , is V@ then use (C5) to find the least $ E @ , least in the list ( I ) , such that s : u { $ ) E S and let s,, , be this element of S.

Now let s, = U,,, s, The rest of the proof is exactly as in Keisler [1971]. We define an equivalence relation on C by

c = d iff ( c - - ~ ) E s ,

3. ΊDΐ-Logic and the Omitting Types Theorem 87

and let M={c/^:ceC}. By (C7), if φ(cί9 ..., cjesω and c^d, thenφ(d1 ?...,dn)esω. This tells us how to interpret the relation and function sym-bols of L:

<c1/^,...,c / J/^>erα n iff r(c l 9 ..., cn)εsω,

h^C!/^, ...,<:„/«) is d/w for that d such that (h(c1?...,cπ)Ξd)e5ω.

A simple proof by induction on formulas of L^ shows that y)l\=φ for everyφεsω. One uses the properties (CO)— (C7) of course. D

2.6 Extended Model Existence Theorem. Let \-A and S be as in the model existencetheorem. If T is a set of sentences of LA such that

seS, φeT implies s

then for any seS, Tus has a canonical model.

Proof. Let S' = {Tus:seS}. While S' is not a consistency property, it almostis one. It satisfies (Cl) — (C7) so we can apply Lemma 2.3 to get a consistencyproperty S"=>Sf. Apply the Model Existence Theorem to (Tus)eS". D

Note, in passing, that canonical structures for countable fragments arecountable structures.

2.7—2.8 Exercises

2.7. Prove, in KPU + Infinity, that if φ is an infinitary sentence with sub(φ)countable then there is a countable fragment LA with φ e LA .

2.8. Use the Model Existence Theorem to show that if LA is a countable frag-ment and if φe\-A has a model then it has a countable model.

2.9 Notes. A history of the Model Existence Theorem can be found in the Pre-face and Lecture 3 of Keisler [1971].

The completeness and compactness theorems of §5 cannot be proved inKPU + Infinity. The main reason for working in KPU + Infinity in this sectionis that it allows us to pinpoint the exact place where stronger principles areneeded in these other theorems.

3. Wl-Logic and the Omitting Types Theorem

As a first, and important, application of the Model Existence Theorem, we provea general version of the Henkin-Orey ω-Completeness Theorem. We then usethe same proof to obtain the Omitting Types Theorem for countable fragments.

88 III. Countable Fragments of Lαoω

Let $R = <M,/> be a structure for a countable language L and let L+ besome language containing L, a unary relation symbol M, and for each weMa constant symbol m, and possibly other symbols.

3.1 Definition. An ^-structure for L+ is a structure 9t = <Af,#> satisfying

i) The interpretation of M in 91 is Mii) The interpretation of m in 91 is m, for all raeM; and

iii) 9JΪ is a substructure of <W,gfl_).

3.2 Examples, i) ω-logic: Let M = ω={0,l,...} and 9Jl = <M>. In this case$0ϊ-logic is usually called ω-logic. Thus, in ω-logic one adds a symbol ω(x), andconstant symbols 0,1,2,... An ω-model is a structure 91 with αr^ω and ~ns{ = n.To study ω-logic one adds the ω-rule to the usual rules of proof:

// you can prove φ(n/v0) for each n<ω then conclude

Vu0[ω(i70)-χp(ι;o)] -

The ΪR-rule below is the natural generalization of this.ii) Let $R = <M, ---> be a fixed structure for L and let L* = L(e,...) be as

usual. Treat L* as a single sorted language with a symbol U for the collectionof urelements (as we have done from time to time). Among structures 21 forL* we want to single out those with 9l = 90ϊ. Let M be U and add a constantsymbol p for each peM. Let L+= L*u {p|peM}. An 9JΪ-structure for L+ has,by 3.1, the form

(with p interpreted by p for peM), for some A9E,.... If it is a model of thatpart of KPU contained in the definition of L*-structure (cf. 1.2.6) then we canwrite it as (9JI; A9£,...). In particular, any Wl-structure for L+ which is a modelof KPU has the form ^^^(W; A,E,...), with p interpreted by p for each peM.

These two examples are the most important ones but what we do in thissection is entirely general.

Thus let 9JΪ be a structure for L and let L+ be as described prior to Defini-tion 3.1. We wish to find a set of axioms and rules which generate the finite sen-tences of L+ which hold in all ΪR-structures or, more generally, in all models ofsome theory T which are 9Jΐ-structures. We can do this only if 501 is countable.Given a set of sentences T we write T\=mφ if φ holds in all $R-structures whichare models of T and \=mφ if φ is true in all 9JΪ-structures.

The results of this section must be carried out in a set theory a little morepowerful than KPU. Our metatheory is discussed in the notes for this section.

3.3 Axioms for SR-logic. Let L and L+ be as above,i) M(m) is an axiom of $Uΐ-logic, for all m in $R.

ii) Every atomic or negated atomic sentence of Lu {m: we9Jl} true in $Ris an axiom of ΪR-logic.

iii) The usual axioms for L^ω are all axioms of ΪR-logic.

3. 9Ji-Logic and the Omitting Types Theorem 89

3.4 Definition. Let T be a set of finitary sentences of L+. A finite formula φ isa consequence of T by the 9JΪ-rule, written

if φ is in the smallest set of formulas containing T and the axioms of 90ΐ-logicand closed under the following rules:

i) (Modus ponens) If TV^φ and T[-m(φ-+ψ) then T\-mψ.ii) (Generalization) If Tt— wi^^ΨΨn)) and ty, not free in φ then

iii) (9W-rM/e) If TK^m/ro) for every we9Jl then TI-TOVι;0(M(ι;0)->φ(ι;0)).A sentence is provable by the SOΪ-rule, written \-mφ, if T\-mφ for 7=0.

Notice that we have made no mention of the phrase "proof by the 9Jί-rule".Instead we gave an inductive definition of T\—^φ directly. A straightforwardproof by induction shows that

T\-mφ(vl,...,vn) implies Tt=9ΛVvl9...,Vvnφ(v1,...,υn).

If 9JΪ and L+ are countable, then the converse holds. It is known as the ω-Com-pleteness, or 9Jl-Completeness, Theorem.

A set of sentences T is consistent in $R-logic if

is false for all formulas φ of L+. (Note the 1- as opposed to 1=^.)

3.5 SDΪ-Completeness Theorem. Let L+ and SOΪ be countable and let T be a setof finitary sentences of L+. If φ is a finite sentence of L+ then

T\-mV tff T^mΨ-

Proof. We can assume L+ has a countable set {cn: n<ω} of constant symbolsnot in Lu (m: we^UΪ} and not mentioned in Γ for otherwise we simply enlargeL+ a little more. This enlargement would not enlarge the set of theorems of9W-logic of the original L+. Suppose φ is not a theorem of T in 901-logic. Ourgoal is to construct, via the Model Existence Theorem, an 9W-structure 91 whichis a model of T and ~\φ.

Let L^ be a countable fragment of Lr^ω with the sentence

V^o VmeM [-IM (l>0) V VQ = ffi]

in L^. Let S consist of all sets s of the form

90 III. Countable Fragments of Looω

where s0 is a set of finitary sentences of L+ such that Tus0 is consistent in501-logic, and s1 is a finite set of infinitary sentences of the form

(1) V«eM[-πffl(cJvcB = ffi].

Note that s = {-ιφ}eS by hypothesis. The only nontrivial step in showing thatS is a consistency property is to show that S satisfies the \/-rule, (C 5). Suppose\/ΦeseS where s = s0^>sί is partitioned as above. If \/Φ is in s0 then it isjust a binary disjunction ψ v θ. If neither ψ nor θ were consistent in SJMogicwith Tu s0 then —iψ Λ —\θ would be a consequence of Tu s0 in 501-logic, henceT0us0 would not be consistent, since ^vθes 0 . If \/Φes1 then it is of theform (1) for some n<ω. We need to show that either

(2)

or one of

(3)m

for some meM, is consistent in ΪR-logic so that one of

is in S. Suppose that none of the (3)m are consistent in 9Jί-logic. Write ψ(cn) forthe conjunction of s0. It follows that

for each meM, so, by the ΪR-rule

and hence

Taking the contrapositive we get

so — ιM(cn) is a consequence of Tus0 in 9)l-logic, so (2) is consistent in ΪR-logic.This completes the proof of (C5). Let T/ = T+all sentences of the form (1).Then if seS and ψeT' then su {ψ}eS. Thus by the Extended Model ExistenceTheorem, T'u {s} has a canonical model whenever seS. But a canonical modelof all of (1) is isomorphic to an W-structure so every seS is true in some^-structure, in particular s = {— ιφ}. D

3.6 Corollary. // 9JΪ and L+ are countable then a theory T of L+ has an ^-struc-

ture for a model iff T is consistent in Wl-logic. D

3. 9Ή-Logic and the Omitting Types Theorem 91

3.7 Corollary. // 9JI and L+ are countable and φ is a sentence of L+ then

iff

In applications in this book, L+ will usually be as given in Example 3.2(ii)and T will usually be KPU or KPU + .

The fraternal twin of the ω-Completeness Theorem is the so-called OmittingTypes Theorem, a result which helps us construct models which "omit" ele-ments not satisfying certain infinite disjunctions.

3.8 Omitting Types Theorem. Let LA be a countable fragment of Looω, and letT be a set of sentences of LA which has a model For each n let Φn be a set offormulas of LA with free variables among vί9 ..., vkn. Assume that for each n andeach formula ψ(vίy . .., vkn) of LA: if

? 1 ? . . . , vknφ

has a model, so does

T+3vί9...9vkn(\l/Λ<ρ)

for some <p(υί9 ..., vkr)eΦn. Given this hypothesis, there is a model 501 of T suchthat for each n<ω,

Proof. A simple modification of the proof of the Sϋl-Completeness Theoremsuffices. We first expand L to a language L'= Lu (cn: n<ω}. Let LB be a count-able fragment containing L^ and each of the sentences

Vvί9...9vknVφeΦnφ(vί9...9vkn)

and let L'A, L'B be the natural fragment of L/

ooω associated with LA and Lβ as in§ 2. Let S consist of all finite sets s of the form

where s0 is a finite set of sentences of L'A with Tu s0 having a model and whereSi is a finite set of sentences of the form

(1) \/φeΦnφ(Cil/Vί9...9CiJvkn).

The proof that S is a consistency property is just like the proof in 3.5 exceptwe use "has a model" for "consistent in 9JΪ-logic". If φεT or φ is of the form (1)then for each seS, su {φ}eS so there is a canonical model 501 of T and eachof (1). The student who has trouble filling in the details is referred to Lecture 11of Keisler [1971]. D

92 III. Countable Fragments of L^

3.9 Exercise. Show how the ΪR-completeness theorem can be derived from theOmitting Types Theorem.

3.10 Notes. The ω-Completeness Theorem goes back to Henkin [1954], [1957]and Orey [1956]. The extension of the Omitting Types Theorem to arbitrarycountable fragments is due to Keisler [1971].

We cannot carry out the $R-Completeness Theorem in KPU or KPU + In-finity. The problems arises from the definition of "the set of consequences / ofT by the ΪR-rule". This inductive definition would need something like Σ^ Sep-aration to justify it by proving that there is a smallest set of the kind describedin 3.4. In Chapter VI we will step back and look at such inductive definitions.

4. A Weak Completeness Theoremfor Countable Fragments

Let LA be a fragment of L^. A sentence φ of LA is valid, written Nφ, if

ΪRt-φ

for every structure ΪR for L. We would like to prove a generalization of theordinary completeness theorem for L by showing that

1= φ iff 3P [P is a proof of φ]

for some notion of "proof. For such a result to be of any use there must besomething "effective" about the notion of proof (otherwise we could take asproofs all valid sentences) and there should be a relation between LA and the"size" of proofs of sentences in LA.

We approach the notion of "proof in a tentative fashion so that we can seeexactly what it is that forces us to consider admissible fragments for the eventualresult, Theorem 5.5.

After the brief respite of § 3 we return to use KPU +Infinity as our meta-theory in this section.

4.1 Definition. Let L^ be a fragment of L. A set Γ of formulas of L^ is a validityproperty for LA if Γ contains (Al)—(A7) below, is closed under (Rl)—(R3),and does not contain φ/\—\φ, for any φeLA.

(Al) Any instance of a tautology of finitary propositional logic.(A2) (-up)<->(~φ).(A3) /\Φ^φ, if φeΦ.(A4) VΛ = VΛ.(A5) Vz = vβ->vβ = v0ί.(A 6) Vι? φ(v) -> φ(t/v), t any term free for v in φ(v).(A7) φ(v)Λ v = t-+φ(t/v), t any term free for v in φ(v).

4. A Weak Completeness Theorem for Countable Fragments 93

(Rl) (Modus Ponens). If φ and (φ->ψ) are in Γ so is ^.(R2) (Generalization). If ((/>-> (t;)) is in Γ and t; is not free in φ then (φ-ΛΛ ^(r))

is in Γ.(R3) (Conjunction). If /\ΦeL A and (ψ^xp) is in Γ for each (/>eΦ, then

OA->/\Φ) is in Γ.

All formulas in the above are assumed to be elements of LA.

4.2 Example, (i) Let $JΪ be a structure for L and let ΓOT be the set of allφ(vl9...9vn)e\-A such that

is a set by \ Separation. It is clearly a validity property.(ii) If 9C is a set of validity properties then

is a validity property.

As one might guess from the way the definition and examples were given,we cannot prove (in our current metatheory KPU + Infinity) that there is asmallest validity property, though it is very instructive to try. It is also usefulto think of the members of Γ as the "provable" formulas in the next lemma.

Let us fix for the rest of this section, a fragment LA, a set C = {cn: n<ω} ofnew constant symbols, and let K = L u C and K^=LA(C) be the natural frag-ment of Kooω associated with LA:φeKA iff there is a \j/e LA such that φ resultsfrom φ by replacing some free variables by constant symbols throughout,

We say that φ is a free substitution instance of ψ. We have purposely interchangedthe roles of K and L from § 2.

The following proposition allows us to apply the Extended Model ExistenceTheorem.

4.3 Proposition. Let Γ0 be a validity property for LA and let Γ be the set of allfree substitution instances of formulas in Γ0.Define

S = { s \ s a finite set of ^-sentences, (~]/\s)φΓ}

Then:(i) S is a consistency property for K^ .

(ii) if φeΓ, seS, then sv{φ}eS.

Proof. We first observe that Γ is a validity property for KA. Two examples shouldsuffice.

94 III. Countable Fragments of Lαoω

(A 7) Consider a sentence of K^ of the form

(1) φ(cί,...,cn9v)Λv = t-+φ(cί,...,cn,t/v)9

where t also may contain c^...,^. Now this is a free substitution of someformula of the form

(2) φ(w1? . . . , wπ, υ) Λ v = t' -> φ(wl5 . . . , W Λ , ί'/i?)

where f = f'(c1/w1,...,c l l/w l l). By (A 7) for Γ0, (2) is in Γ0 so (1) is in Γ.

(R3) Suppose (ψ^>/\Φ)εKA and that (ψ^κp)eΓ for each φeΦ. We needto see that (^->/\Φ)eΓ. Now (ι/^->/\Φ) is a free substitution instance ofsome formula in LA, say it is of the form

ι)e Φ0} .

Since (^(c1)-^φ(c1))eΓ it is a free substitution instance of, say,

where ψ(vβ}-+φ(vβ)<EΓ0. But then using (Rl), (R2) and (A 6) for Γ0 we getOWtfι)-»<Pfaι))eΓ0, for each φfyJeΦQ. By(R3)forΓ0, (A^J^Λί^i)!^)6^}is in Γ0, and hence ψ^>/\Φ is in Γ. We leave the other clauses to the student,and assume Γ is a validity property for K^. The verification of (ii) is entirelyroutine, for suppose φeΓ, sεS but su{φ}φS. Then —\(/\s/\φ)eΓ, so by(Al) and (Rl), (φ->-ι/\s). But then, by (Rl), -|/\S6Γ, so sφS, which is acontradiction. The various cases in the verification that S is a consistency prop-erty are similar, with one slight twist for (C6). Suppose 3vφ(v)eseS but thatfor each ceC, su{φ(c/ι;)}^S. Hence f\s—>~\φ(c/v) is in Γ for each ceC and,in particular, for some c not appearing in S. Since c does not appear in S,

/\s-+-\φ(v)

is also a free substitution instance of something in Γ0, so it is in Γ, and hence,so are all the following:

(by(R2)),

which contradicts seS. The other clauses are left to the student. D

4.4 Definition. A sentence φ of L^ is a theorem of LA if φ is in every validityproperty Γ for LA.

5. Completeness and Compactness for Countable Admissible Fragments 95

A word of warning: The predicate

φ is a theorem of LA

is Πi in KPU but not in general Δx in KPU. Thus we cannot assert (in KPU)that there is a set of all theorems of LA.

4.5 Weak Completeness Theorem for Countable Fragments. Let LA be a countablefragment. A sentence φ of LA is valid iff it is a theorem of LA.

Proof. Assume φ is a theorem of L . Let 9W be any model and let Γ^ be as inExample 4.2(i). Γ^ is a validity property for LA so φeΓOT, i.e. Sf f lNφ.

Now assume φ is not a theorem of LA. Hence there is a validity property Γ0

with φφΓ0. Let Γ0^Γ and S be as in 4.3. Then {—iφjeS by 4.3 (ii) and S isa consistency property so ~\φ has a model, by the Model Existence Theorem. D

The word "weak" in Theorem 4.5 is there because we still have no notions ofproof compatable with LA such that

(3) 1= φ iff 3P [P is a proof of φ] .

All we have managed to do so far is replace one Γ^ notion (\=φ) with anotherH! notion (φ is a theorem of L^). We want a Aί notion of proof so that line (3)gives a Σ: form for \=φ, and we want a proof of φ to be essentially the same"size" as the members of LA.

4.6 Exercise. Define: φ is a theorem of T iff φ is in every validity property con-taining T as a subset. Show that if TC \_A where LA is countable, then φ is atheorem of T iff every model of T is a model of φ.

4.7 Note. The weak completeness theorem is one form of the Karp CompletenessTheorem for Lωιω.

5. Completeness and Compactnessfor Countable Admissible Fragments

In this section we prove the completeness theorem alluded to in the previoussection. We have reduced the task to finding a suitable notion of LA-proof togo along with the notion of theorem of LA introduced in Definition 4.4.

The first notion of proof one thinks of in this setting is: an L^-proof is awell-ordered sequence


such that each φα, for α^β, is either an axiom (Al) — (A 7) of LA or is a con-sequence of earlier φy's (y<α) by one of the rules (Rl), (R2) or (R3). We canof course prove (given a strong enough metatheory) that φ is a theorem of L^iff there is such a sequence with φβ = φ. This notion of proof is too restrictive,however, and does not have the nice properties we need for applications. Weneed a notion which does not have the axiom of choice built into its very de-finition. There are many ways of doing this. We simply choose one.

5.1 Definition. An ordered pair P is an ίnfίnίtary proof iff one of the followingholds:

(Al) — (A 7)

(Rl)

P = <Xφ> where i^n^l and φ is an axiom of L^ by Arc ofDefinition 4.1.

P = (f^y where f is a function, dom(/) = {0,l}, /(O) is an in-finitary proof P0 with 2nd P0 of the form (φ-*\l/), and /(I) is aninfinitary proof P! with 2ndPί=φ.P = <(P0,(φ— > Vι;α^(ι;α))> where P0 is an infinitary proof with 2nd P0

of the form (φ-*ψ(vΛ)) where VΛ is not free in φ.^> = <Λ(1A-^Λ^)) where / is a function with domain Φ such thatfor each φeΦ, f ( φ ) is a nonempty set of infinitary proofs, and foreach P

(R2)

(R3)

If P is an infinitary proof and φ = 2ndP then P is said to be a proof of φ. SeeFig. 5 A, B, C.

Fig. 5 A. A proof P ending with an application of Rl

(υΛ not free in φ)

Fig. 5B. A proof P ending with an application of R2


(for all iel)

Fig. 5C. A proof P ending with an application of R3

Definition 5.1 can be given in KPU by recursion over TC(P) and conse-quently results in a Δt predicate in KPU. Consequently,

3P [P is an inβnitary proof of φ]

is a Σ! predicate of φ.

We now leave our weak metatheory and step out into the universe VM ofall sets on M (which is of course a "model" of KPU). It is the interplay betweenthis universe and admissible sets which is of interest.

Let A = Am be an admissible set with constants /\, \J, —ι, 3, V, =, andthe predicates and functions (v, # ) mentioned in § 1 satisfying the axioms onsyntax found there. We can interpret the results of § 1 and Definition 5.1 ineither AOT or Vg^. As long as we are dealing with Δj notions, we know that theresults are the same, by absoluteness. For example, we have, for φeA^

"φ is an infinitary formula"

true in A^ iff it is true (in V^). If, moreover, 91 eA^ then 91 ϊ=φ holds in A^iff it is true (in V^) and, if PeA^ then

"P is an infinitary proof of φ"

is true interpreted in A^ iff it is true (in V^).

5.2 Definition, (i) If A=A a R = ($R; A,e, ...) is admissible and L is a languagewhich is Δj on A then

is an infinitary formula of Looω}

= {φeA| A 1= "φ is an infinitary formula of L^J'}

is called the admissible fragment of L x ω given by A.(ii) If PeA and P is an infinitary proof then P is said to be an L^-proof.


It is a trivial matter to check that an admissible fragment LA really is afragment in the sense of Definition 2.1.

5.3 Theorem. Let LA be an admissible fragment of Looω and let φ be a sentenceof LA. The following are equivalent:

(i) 3P [P is an LA-proof of φ],(ii) 3P [P is an infinitary proof of φ],

(iii) φ is a theorem of LA, L e., φ is in every validity property for LA.

(Warning: One cannot in general add a (iv) asserting that φ is in every validityproperty which is an element of A. This (iv) is usually much weaker than (iii).)

Proof, (i) => (ii) is trivial.(ii) => (iii). Let Γ be any validity property for LA. A routine proof by induc-

tion on TC(P) shows that if

P is a proof of φ

then φeΓ since Γ contains (Al) — (A 7) and is closed under (Rl) — (R3).(iii) => (i). We need to show that the set

Γ={^eL A |3PeA (P a proof of φ}}

is a validity property for LA, for then φeΓ since φ is in all validity properties.We need to see, then, that Γ contains (Al) — (A 7) and is closed under (Rl) — (R3).The first part is obvious so let us check (Rl) and (R3), (R2) being similar to (Rl).(Rl) Suppose φ, (φ-n^)eΓ. There are P0, P teA with

2ndP0 = (φ^), 2nάPί=φ.

Let /(0) = P05/(1) = Λ τhen ^ = </»eA and P is a proof of (A, hence ^eΓ.

(R3) This is where admissibility and our careful choice of the notion of proofcome into play. Suppose (ι//-»/\Φ)el_A and that (ψ->φ)eΓ for all φeΦ.Thus, for each φeΦ there is a PeA such that P is a proof of (ψ-+φ). Applystrong Σ Replacement in A to get a function /eA, dom(/) = Φ so that foreach φeΦ:

and

if Pe/(φ) then P is a proof of (ψ^κρ).

Then </,(^-*/\Φ)>eA and it is a proof of (^-*/\Φ). Thus (\I/-+/\Φ) isin Γ. D

5.4 Corollary. // LA is an admissible fragment then the set of theorems of LA isa Σ! subset of A. Moreover, the Σ1 definition has no parameters in it and is in-dependent of A.


Proof. The Σ^ formula is

3P [P is a proof of φ] . D

Let us write

for the Σ! formula

3P [P is a proof of φ] .

By combining 5.3 with the Weak Completeness Theorem of 4.5, we obtain thedesired result.

5.5 Barwise Completeness Theorem. Let LA be a countable admissible fragment.Then for all φeL A , the following are equivalent:

(i) Nφ,(ii) h-φ,

(iii) A satisfies \-φ.Thus the set of valid sentences of LA is Σί on A. D

It is this completeness theorem which accounts for the tractable nature ofcountable, admissible fragments. It is used to prove many of the results in thisbook. Before going on though, we pause to point out one thing that the theoremmost emphatically does not say, but which is sometimes mistaken for the con-clusion of the theorem. It does not say that the sentence

\=φ iff \-φ

is true in the countable admissible set A. This, together with 5.5, would implythat if φeL A and φ has a model then φ has a model 9Ϊ, 9leA. This is falsefor most A. See Exercises 5.11 — 5.14.

Now we turn to our first application of the completeness theorem.

5.6 Barwise Compactness Theorem. Let LA be a countable admissible fragment°f Looω Let T be a set of sentences of LA which is Σx on A. If every T0^T whichis an element of A has a model, then T has a model

Proof. Expand L to K=Lu{c w | n<ω} as usual but do it so that K remains Δjon A. Let KA = LA(C) be the usual fragment of LA associated with K. Thus KA

is the set of all sentences of Kooω which are elements of A and have only a finitenumber of c's in them. We use the Model Existence Theorem for KA.

Let S be the set of all finite sets s of sentences of KA such that for all TG^Twith T^eA,

7^ us has a model.

100 III. Countable Fragments of LQOω

Note that if seS and φeT then su{φ}eS so we are all set to apply theExtended Model Existence Theorem to get a model of T once we show that Sis a consistence property. As usual it is (C 5) that causes the problems. So supposeyΦeseS but that for each φeΦ, sv{φ}φS. Thus, for each φeΦ there is aT0cT, T0eA such that

has no model.

Let θ(x) be the Σλ definition of T on A. The following Σ sentence is true in Aby the Completeness Theorem for K A :

(1)

By Σ Reflection there is a set 06 A such that (1) holds relativized to a. We canassume a is transitive by 1.4.2. Let

by Δ0 Separation. Then T^eA, T^T by 1.4.2 and, for each φeΦ,

since there is some 7^ c 7J with

But then suTJ can have no model since \/Φes. This contradicts the assump-tion that seS. D

Combining Completeness and Compactness we obtain the following extensionof the Completeness Theorem. We use T\=φ to indicate that every model of Tis a model of φ.

5.7 Extended Completeness Theorem. Let LA be a countable admissible fragment.Let T be a set of sentences of LA which is Σ! on A. The set

is Σ! on A.

Proof. If φ is a sentence of LA then

T\=φ iff 3T

by the Barwise Compactness Theorem (applied to Tu{— \φ}) so T\=φ iff thefollowing is true in A, where θ(x) defines T,


which gives a Σ definition of T\=φ. Note that it depends only on the defini-tion θ of T, not on A, and that is has only the same parameters occuring in itthat occur in θ. Π

One peculiar instance of the Compactness Theorem deserves special mentionbecause it comes up frequently. It applies, for example, to A = HYP<tri when 9Wis recursively saturated.

5.8 Theorem. Let A = ASOΪ be a countable admissible set with o(Am) = ω. LetT, T be theories of LA which are Σί on Am such that every φeT is a pure set.(Hence T is a set of finitary sentences.) If for each finite T0<^T,

T0uT' has a model,

then TuT' has a model.

Proof. If Tu T' has no model then, by the Compactness Theorem, N~ ι/\Φ,for some ΦeA, Φc ΓuT. Now, if we write Θl9 Θ2 for the Σ! definitions of Tand T', respectively, then we have VxeΦ[θ1(x) v Θ2M]> so by Σ Reflection thereis an aeA such that VxeΦ[θ< f l)(x) v $>fl)(x)].

Thus, if we use Δ0 Separation to form

then Φ = Φ1vΦ2, Φ^T, Φ2^T' and Φ^uΦ2 has no model. But Φx is a setof pure sets, hence a pure set, hence finite since o(A) = ω. D

There is a question that often comes up. Let Aαrί = ΉF2R. To what extentdoes the Compactness Theorem 5.6 give us the full compactness theorem forLA = Lωω? In other words, how does the requirement that T be Σ: on A^ affectus. If HFjOT is countable (i. e. if 9JI is countable) then 5.6 gives us the full com-pactness for Lωω. For let T be any theory of Lωω. T is Σ: on (9Jί; HFM,e,Γ)which is admissible by Theorem Π.2.1 so we can apply 5.6 to this admissible set.

5.9- 5.14 Exercises

5.9. Define "P is a proof from axioms in T' parallel to Definition 5.1. You mustbuild in a Σ! definition of T. Use this to prove the Extended CompletenessTheorem and Compactness Theorem in one fell swoop, without using the Com-pleteness Theorem. [You will need to use Exercise 4.6.]

5.10. Let LA be an admissible fragment. Show that if P is an LA-proof then allformulas in the proof P are LA-formulas.


5.11. Let A, IB be admissible sets with AeB and

"A is countable"

true in B. Let LA be an admissible fragment given by A. Show that

Bt= > is valid"iff

A 1= 3P [P is a proof of φ\ .

Conclude that if φel_ A has a model then it has a model in B.

5.12. Let A be admissible and satisfy the following:(i) (locally countable). A\=Va (a is countable)

(ii) (recursively inaccessible). VαeA there is an admissible Be A with αeB.Show that for any sentence φ of LA

holds in A. Show that HC^ is locally countable and recursively inaccessible.(Of course, for HC^ the conclusion is trivial since every countable structure fora countable language is isomorphic to a structure in HC^.)

5.13. Let A be a countable transitive model of ZFC (or enough of it to insurethat A is admissible and prove that K t exists). Let α be the ordinal of A whichsatisfies

A 1= "α is the first uncountable ordinal".

Write a sentence φ of LA which asserts that α is countable. Thus φ has a modelbut does not have a model 9leA. In other words, —\φ is valid in the sense of Abut it is not provable since φ does indeed have a model.

5.14. Let α be the first admissible ordinal >ω. Let A = L(α). Unlike the Ain 5.13, this A is a model of

Vα [a is countable].

Find a sentence φ of LA which has a model but none in A.

5.15 Notes. The completeness and compactness theorems of this section are dueto Barwise [1967] and appeared in Barwise [1969], The terminology "BarwiseCompleteness Theorem" and "Barwise Compactness Theorem" have become sostandard that it would be false modesty (and confusing) to give them some othername here.

The observation that these theorems go through unchanged in the presenceof urelements was first made in Barwise [1973], though it was really clear allalong. The odd 5.8 first appears in Barwise [1973].

6. The Interpolation Theorem 103

6. The Interpolation Theorem

The Interpolation Theorem is one of the results which holds for countable ad-missible fragments but not for arbitrary countable fragments; the proof requiresthe Completeness Theorem of § 5.

6.1 Theorem. Let LA be a countable admissible fragment, φ, ψ sentences of LA

such that\=φ-*\l/.

There is a sentence θ of LA whose relation, function and constant symbols arecommon to those of both φ and ψ such that

\=φ-+θ and \=θ^ψ.

Note. Equality is not treated as a relation symbol. It may appear in θ whileappearing in only one of φ, ψ.

Proof. Let L° be the set of those symbols occurring in φ and let L1 be the setof those symbols occurring in φ. Let C be a countable set of new constant sym-bols coded as a Δί subset of A and let LA(C) be the set of free substitution in-stances of formulas of LA by a finite number of symbols from C. Define LA(C)and LA(C) similarly. We define the consistency property S to be the set of allfinite sets s of LA(C) which can be written as a union S^SQUSJ satisfying thefollowing conditions:

(1) s0 is a set of sentences of LA(C), and similarly for sίι(2) If 00, ^eL^QnL^C) are such that SQ^O and s^θ^ then the sen-

tence ΘQ Λ θl has a model.

The verification that S is indeed a consistency property is routine (indeed, itis just like the Lωιω case in Keisler [1971]) except for the \/-rule, (C5). It is inthe verification of this rule that we need the Completeness Theorem for LA(C).So suppose s = s0usl is as above and that \/Φes. Since the two cases are sym-metric, we may assume that \/Φes0. We want to prove that for some σeΦ,

Suppose this is not the case. Then for every σeΦ, there is a pair Θ0, θ± such that

(3) t=/\s 0Λσ->θ 0, ^=/\s1-^θ1, and l=— I^QΛ^), and the constants fromC in Θ0 and Θ1 are in s0 u s1 .

Let us indulge in a little wishful thinking and suppose that there are functions/, g which are elements of our admissible set A with dom(/) = dom(g) = Φ suchthat, for each σeΦ, </(σ),^(σ)> is a pair <00>0i> satisfying (3). Then we can let

θ'0 = \ / { f ( σ ) \ σ ε Φ } , θ\= /\{g(σ)\σεΦ} .


By Σ Replacement, θ'0 and θ\ are elements of A so they are both sentences ofthe languages LA(C) and LA(C). Furthermore, s0l=θ'0, s^θi and l= — ι(θ'0Λ #i),which contradicts s0us1eS. But what about our bit of wishful thinking? Atfirst it seems exactly that, since we have no choice principle holding in A. Onceagain is it Strong Σ Replacement which which comes to the rescue. By the Com-pleteness Theorem for LA(C), line (3) can be expressed by a Σ^ formula. By StrongΣ Replacement there is a function he A with dom(/z) = Φ such that, for eachσeΦ, h(σ) is a nonempty set of pairs <00>0i> satisfying (3). Define / and g by

g(σ) = /\{2«dh(σ)\σeΦ}.

Then /, geA and our wish has come true since Θ0 = f(σ) and θl=g(σ) alsosatisfy (3). Thus S is a consistency property.

The conclusion of the theorem now follows easily from the observation that{φ,~ \ψ}φS. Just quantify out the finite number of new constant symbols in thesentence Θ(cί9...9cj ( = ΘQ in the notation used above):

..9ΌHθ(υl9...9Όj^ψ. D

We could use the interpolation theorem for LA to prove Beth' s Theorem forLA, but we will not be needing this result.

6.2-6.6 Exercises

6.2 (Hard). Let /19 12 be interpretations of a language L° in consistent infinitarytheories, 7^, T2 formulated in languages L1, L2 respectively, and suppose thatthere are no two models 3Λί9 $0Ϊ2 of 7J, T2 respectively such that SDlf / 1 =9K2 / 2

If L°, L1, L2 are Δt on the countable admissible A, Tl9 T2 are Σi theories ofLi and LA and the interpretation I ί 9 12 are Σx functions on A then there is asentence φ of LA such that

implies ϊR-^Nφ, for all L1 -structures SR^

m2\=T2 implies Wl2

l2\=-ιφ, for all L2 -structures 50Ϊ2.

6.3. Let LA, KA be countable admissible fragments and let / be an interpretationof L into a theory T of KA, T and / being Σx on A. Let φ be a sentence of KA

such that for all models 911? 912 of T with SRΓ^Ϊl^, ^i^^ iff ^2^^, Thenthere is a ιAeL A such that

6.4. Let / be an interpretation of a complete theory T of Lωω in an incompletetheory 7J of Kωω. Show that if φeKω ω is not decided by 7Ϊ then there aremodels 9tR, 91 of Γ with 9K^φ, 9l^-πφ and arr'^ΪΓ'.

7. Definable Well-Orderings 105

6.5. Show that there are models <3ll = (N,+,x1y9 9i2 = <AΓ, +,x2> of Peano

arithmetic, with the same integers and addition, but 9

6.6. Show that there are models 9W = <M,£>, SR = <JV,F> of ZF + V = L withthe same ordinals, <Ord9CR,£fOrd9[ίί>-<Ord9i,FίOrd9ί>, but with different setsof hereditarily finite sets. [Use GδdeΓs Incompleteness Theorem, 6.4 and the factthat every true sentence about the ordinals (with <) is provable in ZF.]

6.7 Notes. The interpolation theorem for Lωω is due to Craig [1957]. For thefull Lωιω it is due to Lopez-Escobar [1965]. Theorem 6.1 is due to Barwise [1969].Exercise 6.2, a generalization of 6.1, is useful in abstract logic and is due toBarwise [1973]. References for the other exercises can be found there.

7. Definable Well-Orderings

In this section we prove a model theoretic result which will have applicationsto HYP^ in Chapter IV. The basic question is: What ordinals can be define inan admissible fragment? We solve this problem here for countable fragments.The uncountable case is taken up in Chapters VII and VIII.

7.1 Example. Define 0α(x) by recursion over α as follows:

00(x) is Vy-\(y<x),

ΘJix) is Vy(y<x~\/β<Λθβ(y/x)).

Let 9W = <M, <> be a linear ordering. A simple proof by induction shows thatif 9JΪNθα[x] then {ye$R\y<x} is well ordered and has order type α. HenceWl\=Vx\/ β < Λ θ β ( x ) iff 9ΪI is well ordered and has order type ^α. Thus, if we setσα equal to

V* V0 < α θβ(x) Λ /\p < α 3X θβ(x)

then 5Ht=σα iff $0ΐ has order type exactly α. D

The formulas from Example 7.1 were defined by recursion on α so the de-finition can be phrased as a Σ recursion in KPU. Hence, if α is in an admissibleset A then φα(x), σ α el_ A (as long as the symbol < is in L and A).

7.2 Definition. Let L have a binary symbol < and possible other symbols. Asentence φ( < ) of an admissible fragment LA pins down the ordinal α if

(i) yi\=φ implies < w is a well-ordering of its field.(ii) φ has a model $1 with <* of order type α.

A theory T of LA pins down α if /\T pins down α.

106 III. Countable Fragments of Looω

The example above shows that every ordinal in the admissible set A can bepinned down by a sentence of LA; in fact,

("< is a linear ordering") Λ σα

has only models of order type α. For countable admissible fragments, no otherordinal can be pinned down, as we show below.

For uncountable admissible fragments one can often pin down an ordinalα > o(A) (though one cannot give an explicit definition like σα above). The leastordinal which cannot be pinned down plays a key role in the model theory ofuncountable fragments. We will go into this further in Chapter VII.

Note that if φ(<i) pins down α then ^(<1? <,f) defined by

φ(<1) Λ "f maps < into <q in an order preserving fashion",

as a sentence about <, pins down all ordinals ^α. Thus we can always workwith sentences which pin down an initial segment of the ordinals. We will usethis implicitly in the proof of the next theorem and several times in Chapter VII.

7.3 Theorem. Let I_Δ be a countable admissible fragment, φ(<) a sentence whichpins down ordinals. There is an ordinal α in the admissible set A such that everyordinal pinned down by φ is less than α.

Proof. Suppose, to prove the contrapositive, that for every αeA, φ«) has amodel ϊt with -<** of order type α. We prove that φ(<) has a model 91 where-<** is not well ordered. It is instructive to split into cases, though not reallynecessary.

Case 1. o(A) = ω. If A^ΉF^ then φ(<) is just a finitary sentence and theresult is well known to follow from the compactness theorem. But even ifA^HF^, the proof goes through. Let <, C0,c1 ?... be new symbols in the purepart of A and let T be the theory

Let ψ be

Λ "f maps < order preserving into -<".

We need only show T + ψ has a model. Since T is a set of pure sets ando(A) = ω we need (by 5.8) only see that ψ is consistent with every finite subsetof T, which is obvious.

Case 2 o(A)>ω. The basic idea is the same, but we must work a littleharder. Let a0=ΎC(φ). We may assume that A is the smallest admissible setwith α0 as an element. Introduce the following new symbols into L: e, unaryU, S, N for urelement, set and member of N respectively, a function symbol f,a constant c and, for each xeA a constant x. Let T be the following set ofsentences of LA :

7. Definable Well-Orderings 107

(0) "U, S, N are disjoint and their union is everything",(1) φN, the relativization of φ«) to N,(2) KPU formulated in terms of U, S, e,(3) diagram (A),(4) Vx[xea->\/ b e α.χΞΞb], for each αeA,(5) " β e C Λ C is an ordinaΐ\ for each β<o(A),

(6) Vx^cV^κPiP<AL(α°'x)>(7) "f maps the e predecessors of c mίo -< so ί/iαί

This theory is Σ! (in fact Δ0) on A so we can apply the Compactness Theorem.If T0^T, T0eA then T0 will have a model of the form

where βeA, / maps the ordinals <β into -<*, and 9lt=φ. Let

be a model of the whole theory T. By (1), ϊlNφ. By (2)— (5), A^end93, but,by (6), c is not in the well-founded part of 23. Hence, by (7), -<** cannot be awell-ordering. D

7.4 Corollary. Lei LA be α countable admissible fragment, T a Σί theory of LA

which pins down ordinals. There is an α<o(A) such that every ordinal pinneddown by T is less than α.

Proof. If not, then every T0^T, T0eA would be consistent with

by 7.3 and hence T would also be consistent with this set by the Barwise Com-pactness Theorem. D

We can use Theorem 7.4 to prove a general version of a theorem of Friedmanon models of set theories T^KP. Note that if < is a linear ordering then, bythe definition in II.8.3, i^/(<) is the largest well-ordered initial segment of <.We can identify Wf(<] with an ordinal without confusion. In Theorem 7.5, L isa language containing a binary symbol < among its symbols.

7.5 Theorem. Let LA be a countable admissible fragment of LX)ω and let α = o(A).Let T be a Σ{ theory of LA such that:

i) T N " < is a linear ordering"ii) for each β<α, T has a model 9JΪ with

/*7(<w)>0.

Then T has a model M with


Proof. Notice that T is consistent with the set

where θβ is as in 7.1, by the Barwise Compactness Theorem, so we may as wellassume that the sentences are actually in T. This insures that any model 9JIof T has ιT/(<αίl)^α. By Theorem 7.4, T also has a model 9ϊί where <m isnot well ordered. Let T' be the theory

where {dn\n<ω} is a new set of constant symbols. Thus T is consistent. LetK=Lu{dJ«<ω} and let KA be the corresponding admissible fragment. Let K£be the set of formulas in which at most a finite member of dπ's occur. Thus T'is a theory of K£. We are going to use the Omitting Types Theorem for K£ tofind a model (StR,^,...,^,...) of T' and the sentence

where "vφ Field (<)" stands for \/x(v^x Λ X ^ V ) . Such a model 9K must have^y(<9W) = α. Suppose there were no such model. Then by the Omitting TypesTheorem there is a σ(v,d^...,dn)e\(% such that T' + 3vσ(v,d^...,dn) is con-sistent, but such that all the following are theorems of T'\

(8) \fv [<J(Ό, d)^υe Field (<)],(9) VϋCφ.dH-iθXi;)],

(10) Vu[σ(ι?,d)-»ι;<dm], for all m<ω.

Note that, by (10),

is consistent. Let c be a new constant symbol and let T" be

Γu{σ(c,d 1,...,dn)ΛC<dn< <d1}

which is consistent since T^T'. We claim that

(11) For every model 9Jί of T", the <αrί predecessors of c = c5" are well ordered.

If not, then there is a descending sequence below c, so we can name itsmembers dπ+1,..., giving us

which makes (9W,J1 ?...,rfn,...)^T', where the element c violates (10).

8. Another Look at Consistency Properties 109

From (11) and Theorem 7.4 we obtain

(12) there is a y<α such that

(To see this consider τf" = T" + "~< = <\ the predecessors of c" and applyTheorem 7.4 to T'" as a theory about -<.)

Thus we have

and hence

contradicting (9). D


7.6 (Friedman). Assume ZF has an uncountable transitive model. Show that theorder types of the ordinals in countable nonstandard ω-models of ZF are exactlythe order types α(l+τ/) for α a countable admissible ordinal, α>ω, and η theorder type of the rationals. [Use Theorem 7.5.]

7.7. Let α be nonadmissible. Let T = KP + "< is ef ordinals". Show that therecan be no model 9JΪ of T with α = i

7.8 Notes. Theorem 7.3 is from Barwise [1969]. It refines older results of Lopez-Escobar [1966] and Morley [1965]. Theorem 7.5 is new here. It generalizes aresult in Friedman [1973].

8. Another Look at Consistency Properties

There is room for a lot of creativity inside the proof of the Model ExistenceTheorem. Let S be a consistency property, s0eS, and recall the way we con-structed a model of s0. As we built our sequence s0^sl^" ^sn^'"^sω (andin so doing built a canonical model) there was freedom in defining sπ+1 that wedidn't use. At the nth stage, after defining sn we could first enlarge sn to someother 5*^5Π before going on to get sπ + 1^s*, as long as s*eS. The resulting

with

Sω ~ \Jn <ωsn = \Jn <

110 III. Countable Fragments of Laoω

would give rise to a canonical model $R of each sn and each s*. We give a modestillustration of this technique here, just to illustrate the general method. We willreturn to it in § IV.4.

Let l_4 c Kβ be fragments. A theory T of Kβ is complete for LA if for eachsentence φ of LA, Tϊ=φ or T\=~Ίφ, but not both. Given a structure 9Jί forK we define

) = (φε LA \ φ is a sentence true in 9W} .

Note that this is a complete L^-theory.

8.1 Theorem. Let L^^Kβ be countable fragments, T a consistent set of sen-tences of KB such that for each sentence ψ of KB, Tu{ι/^} is not complete for LA.

There are 2K° distinct LA theories of the form ThL^(50l) for models 9JΪ of T.

Proof. Let K' = KuC = Ku{cπ: n<ω} and let K^ be the set of free substitutioninstances of φ's in Kβ. Note that there is no sentence ι//(c l 5...,cJeK^ such thatT + φ(cl5...,cπ) is complete for LA, for if it were then Γ + 3ι;l5...,3ι;π i/^,...,^)would be complete for L^. Consequently there is no finite set s of K^ such thatTus is complete for LA (for otherwise we would form ψ = /\s). Define

SQ = {TUS\S a finite set of sentences of Kβ such that Tus is consistent}.

The set S0 obviously satisfies (Cl) — (C7) so let S be the smallest consistencyproperty containing S0, by 2.3. To simplify notation let us suppress T altogetherand write s for Tus in what follows. We wish to construct a "tree" of membersof S such that

(1) any "branch" through the tree gives us a theory 7^ of L^ consistent with T,(2) distinct branches lead to incompatible theories, and(3) there are 2K° distinct branches.

Since this is our first tree argument, and since this is an important kind ofargument, we give the proof in more detail than is usual. Our tree consists ofall finite sequences d of O's and 1's arranged as illustrated below.

V V000 001

V V101 110

8. Another Look at Consistency Properties 111

A branch through this tree is just an infinite sequence b of O's and 1's. (Thebranch <01000...> has its nodes circled in the tree drawn above.) We wish toplace at each node d of length n an element sdeS which is one of the s*'sreferred to in the introduction to this section. At the empty node place T andlet s0 = T in the notation from the proof of the Model Existence Theorem.Before defining s1? pick some φe\-A not decided by s0, i.e.

and sg*

are consistent and hence in 8. Let sd = s$ for d = <0>, sd = s$* for d = <l>.Given d of length n we go on to find sn + ί^sd just as in the proof of the

Model Existence Theorem. Then, given sw + 1, choose a φeLA such that

are in S, and let

j*'=s*+ί9 if d'=dO

— <?** if A' — A\~ 5 n + l > Π M — αl,

where dO is the sequence d followed by 0. Now let b be any branch through thetree. The set sb = (J{sd\d a node on b} is one of our sω's so it has a model 9Wfc.Since s0 = T, $)lb\=T. If b l9 b2

are distinct branches then there is a φeLA suchthat we have put φesbl, ~Ίφesb2 at the point where bί and b2 split. Thus

and 9Jlb2^~Ίφ, where φeLA. There are 2K° branches so the sets

form 2N° distinct complete theories of LA. D

We will apply the following corollary of Theorem 8.1 in Chapter IV. Let LA

be an admissible fragment of L^. A structure 9JI is decidαble for LA if ThLA(^)is Δ! on A. The structure 9P1 could be a structure for some language K properlycontaining L.

8.2 Corollary. Let L A <=K A be countable admissible fragments, let T be a con-sistent theory of KA which is Σί on the admissible set A such that T has no modelwhich is decidable for LA. There are 2K° distinct theories of the form ThLA(9JΪ)with Wl\=T.

Proof. If there are fewer than 2N° such sets then there is a ^eKA such thatT + ψ is complete for LA. But then any model $R of T + ψ is decidable for LA since

Wl\=φ iff T\=ψ-+φ,

iff

which makes ThLA(9M) a ΔA set by the Extended Completeness Theorem. D


For Lωω and Kωω the theorem and its corollary are old indeed. Here theproof of the theorem is even easier since one no longer has to go back to theModel Existence Theorem but can use the Compactness Theorem for Kωω.

8.3—8.4 Exercises

8.3. Show that if KB=Kω ω, LA=l_ω ω then the hypothesis of Theorem 8.1 canbe weakened to:

_A (T + ψ not complete for LA).

Prove this directly from the Compactness Theorem for Kωω.

8.4. Let L have constant symbols 0,1,..., n,... and a unary predicate P. Finda consistent theory T = {φ} of a countable fragment LA such that ψ has onlyNO non-isomorphic models, but for each ψe Lωω, T+ψ is not complete for Lωω.This shows that the strengthening of 8.1 carried out in 8.3 is not possible in general.

8.5 Notes. The results of this section are new here. They are suggested by, andimply, the theorem of recursion theory that any Σ} set of subsets of ω with lessthan 2K° members is actually a subset of HYP. See § IV.4 for proofs of this andrelated results.

Chapter IV

Elementary Results on

We have seen, in Chapter III, how admissible sets provide a tool for the studyof infinitary logic by giving rise to those countable fragments which are especiallywell-behaved. In this chapter we begin the study of HYP^ by means of thelogical tools developed in Chapter III.

1. On Set Existence

Given $R we form the universe of sets Wm on 9JΪ and speak glibly about arbitrarysets aeVm. In practice, however, one seldom considers the impalpable sets ofextremely high rank. There is even a feeling that these sets have a weaker claimto existence than the sets one normally encounters. Without becoming toophilosophical, we want to touch here on the question: If we assume $R as given,to the existence of what sets are we more or less firmly committed?

IHYP^oj is the intersection of all models 21 of KPU+ and is an admissibleset above 90Ϊ. There appears to be a certain ad hoc feature to JHYP^, however,since it might depend on the exact axioms of KPU+ in a sensitive way. Youwould expect that if you took a stronger theory than KPU+ (say throw in Power,or Infinity or Full Separation) that more sets from V^ would occur in all modelsof this stronger theory. That, for 9DΪ countable, this cannot happen, lends con-siderable weight to the contension that IHYP^ is here to stay.

Of the two results which follow, the second implies the first. We presentthem in the opposite order for expository and historical reasons.

A set S^afl is internal for <Hm = (yjl; A,E,...) if there is an aεA such that

1.1 Theorem. Let $R = <M, #!,..., #,> be a countable structure for L. Let T bea consistent theory ffinitary or infinitary) which is Σί on HYP^ and which has amodel of the form ^^ = (501; A,£,...). Let S^M be such that S is internal forevery such model of T . Then

114 IV. Elementary Results on

Proof. The proof is a routine application of Completeness and Omitting Types.Given the above assumptions we see that there can be no model 9IOT of

where T is T plus

(i) vv[u(v)^yp€Mv=ji]Diagram (9W)

and Φ is the set of formulas

{pφv\peS}v{pev\pφS}9

for then S would not be internal for 91 . The formulas in T and in Φ(t ) aremembers of the admissible fragment LJ of L£ω where A = ΉYPaR = ($R; 4,e),and where we have introduced p by some convention like p = <0,p>. By theOmitting Types Theorem there is a formula σ(v) of L| such that T' + 3ι;σ(ι?)is consistent but such that :

r t= Vt; [σ(ι;)->pet;], for all

T> Vt; [σ(ι?Hp <£*;], for all

But then

S = {pe2R|ri=Vι;(σ(ι;)->peι;)}

so S is Σ! on HYP^ by the Extended Completeness Theorem for LJ. Similarly-ιS is Σ! on HYP^ so S is Δx on HYP^. Thus SelHYP^ by Δt Separation. D

Before stating our next result we need a more sophisticated notion of whatit means for a set aeW^ to be internal for 91 = (Sft A, £,...).

1.2 Definition. A set aeW^ is internal for 91 , = (2R A, E, . . .) if α e 7(9K X, £),where we again identify i^/(^R',A,E) with its transitive collapse.

Note that for a^M this is equivalent to the existence of an xeA witha = x. Also notice that if a is internal and bea then b is internal.

1.3 Theorem. Let ^01 be countable and let αeV^ be a set which is internal forevery model

of some consistent theory T, βnitary or not, formulated in L* = L(e, . . .), KPU+ c: T.If T is Σί on IHYP^, then αeHYP^.

Proof. We prove the theorem by e-induction. By the comment above, if a isinternal for every model 91 of T, so is every be a. By e-induction, each of

1. On Set Existence 115

these b is in HYP^. That is, a^MYPm. A routine modification of the proofof 1.1 shows that a is on JHYP^. If we can prove that αcL(5DΪ,β) for somej5<o(HYPaϊί) then, by Δ: separation, aeMΎPm. Assume, on the contrary, that

O(9Jl) -the least ordinal β such that a^

In any model (Άm of T there would be a unique ordinal x such that

aO TN"x = least ordinal β such that αc

By Σ Reflection in 31 and, by the absoluteness of L( , ), this x must beHence T+ the following theory pins down 0(5)1), contrary to Corollary HI. 7.4.

Diagram (9Jt),

" < is the order type of the e-precedessors of c",

(2) "c is the first ordinal such that L(9Jl,c) is admissible" (if α>ω)

or

(3) "c is the first limit ordinal9 (if α = ω).

This theory is formulated in L(e,..., <,c,p)peΛf. (The reason for the two casesis that we do not yet know how to write "x is admissible" by a finite formula.)We can write (2) as

Thus we see that no matter how we strengthen KPU+ to an axiomitizabletheory T, we cannot assure that any set in Y^ — HYP^ should be internal toevery model ^^ of T.

One could consider HYP^ as a new structure 91 and form ΉYP^ but it ismore natural, and essentially equivalent, to proceed differently.

1.4 Definition. Let AOT = (SBl;X,e) be transitive in VOT. Then HYPίA^) is thestructure (2R; B,ε) where

B = (~}{B'\(MvA)eB',((mιBf,e) admissible}.

We consider IHYP^ as a special case of HYP(Aαn).


1.5. Show that MΎP(Am) is admissible.

1.6. Show that every element aEMΎP(Ayn) has a good Σί definition withparameters from Mu,4u{M,,4}.


1.7. Show that the obvious generalizations of 1.1 and 1.3 are true.

1.8. Let yΓ = <ω, +, ,0> and let X<^ω. Show that there is aKPU+^T such that X is in every model 91 of T. This shows that the con-dition that T be Σ1 on HYP^ is necessary in 1.1 and 1.3.

1.9. Show that the hypothesis KPU+cΓ can be dropped from Theorem 1.3.[Hint: add a new e-symbol and a function symbol used to denote an e-iso-morphism.]

1.10 Notes. Theorem 1.1 is a modern version of the Gandy-Kreisel-Tait Theo-rem: For any consistent Πj T set of axioms for second order number theory, ifa^ω is internal to every model of T, then a is hyperarithmetic.

Theorem 1.3 was announced by Barwise in Barwise-Gandy-Moschovakis[1971]. The part of it contained in Theorem 1.1 is due independently to Grilliot[1972]. The improvement in 1.9 is due to Ville [1974].

2. Defining Π} and Σ} Predicates

Let $R = <M,R1 ?...,KZ> be a fixed infinite structure for a language L. An rc-aryrelation S on 9W is Π{ on $R if it can be defined by a second order formula ofthe form

S(Pl,...,Pn) iff VT1,...,VTk(p(p1,...,Pn,T1,...,Tt),

where φ is a first order formula of L(Ίl9...,Ίk), possibly containing parametersfrom $)ϊ. More formally we should write this as: for all p1?...,pn6M, S(pl5...,pn)holds iff for all relations T^...,Tk on 9K,

The negation of a Π} relation is called Σ\ on 501. Thus S is Σj iff it can be de-fined by

S(p) iff 3T1,...,3Tk^(p,T1,...,Tk)

for some first order ψ. If S is both Π{ and Σ} on 5R then S is said to be Δ} on 9ER.This section is primarily concerned with techniques that can be used to

show that predicates are Πj or Σj on 9JI. The reason for discussing this materialcan be seen by glancing at the next section.

2.1 Examples, (i) // ^Γ = <ω,0, +,•>, then a set is Δ} over Jf iff it is hyper-arithmetic. (This is the classical Souslin-Kleene theorem. See, e. g., Shoenfield[1967].)

2. Defining Π} and Σj Predicates 117

(ii) // 9Ϊ = <ΛΓ,0, +, •> is a nonstandard model of arithmetic then the standardintegers form a Π} set but not, in general, a Δ} set:

x is standard iff VS [S(0)Λ Vy(S(y)^S(y + ί))^S(xf].

(iii) // 9Jl = <G,0, -h> is an abelian group then the torsion part T of G, theset of elements of G of finite order, is Π} on G:

xεT iff V

(iv) // 9JΪ = <G,0, +> is an abelian group then the largest divisible subgroup Dof G is Σ}, but this time it is not so obvious.

xεD iff 3fί [H a subgroup AH divisible Λ/f(x)]

but the clause "H is divisible", meaning

for all integers n, Vye// 3ze#, nz=y

cannot be expressed by a single first order sentence. It is still possible, though,to write D out as a Σ} predicate. The student should try this before going onin order to appreciate the machinery developed below. D

The last example is just the tip of an iceberg. In writing out Π} predicateswe frequently discover that we would like to use an extended first order formulaas defined in §11.2. (In writing out the Σ} predicate in 2.1 (iv) we need theco-extended predicate "H is divisible".) It turns out we can allow ourselves thisfreedom without changing the class of Π} predicates.

2.2 Definition, (i) An extended Πj predicate over S[R is a predicate S ( p ί 9 . . . 9 p i 9

S1,...,Sm,Λ1,...,fl/,P1,...,/i) defined by

(aRΛ^.^ HF^^^

for some extended first order formula φ which may have parameters in it from,,. (We use S, Γfor relations over M; P, Q for relations over MuίHF^.)

(ii) S is co-extended Σ} if it is in the dual class; that is, if it can be defined by

3Q φ(p,a,S,Ύ9 P,Q)

where φ is co-extended.

Thus extended Π} predicates over ΪR are not really predicates over 9JΪ; theyare predicates of points in 2R, relations on $R, sets in HF^ and relations onHR0J. They are important as a tool for showing predicates over 9K are Π}. Forexample, in 2.1 (iv), it is clear that D is co-extended Σ}, so that D is Σ} over Gby 2.8 below.


2.3 Lemma. //S1? S2 are extended Yl\ (respectively co-extended Σ}) so are (8l v S2)and (Sl Λ S2).

Proof. For example,

VT ιA(— , T) Λ VT' VQ θ(_, Γ, Q)

is equivalent to

VT VT' VQ [>(— ,Ό Λ Θ(_,T Q)]

as long as we first make sure T and T are distinct symbols. The part inside thebrackets is still extended first order. D

2.4 Lemma. // S is extended Π} (respectively, co-extended Σ}) then — iS isco-extended Σ\ (respectively, extended Π}). D

2.5 Lemma. // S = S(p1,...,pί, _ ) is extended Πj (co-extended Σ}) then so are

S^P!,. ..,&._!,_) ijff Vpf 8(0!,. ..,?;_!, pί5_),

S2(p1,...,p ί_1,__) zjff apiSKp!,...,^.!,^,—).

Proof. It is hard to see the extended Πj case directly, but we can prove theco-extended Σ} case and then apply 2.4. If

S(p,_) iff 3QιMp,__,Q)then

S^P!,...^-!,—) iff 3Q3PίιA(p,_,Q)and

S2(p1,...,pί_1,_) iff

iff 3Q'Vpi^(p1,...,pί,_,Q'(...,pi))

where the notation indicates that we have replaced the π-ary relation Q(ί ι,...,ίw)by the new n + l-ary Q /(ί1,...,fw,p f) throughout ψ. D

2.6 Lemma. // S = S(α1,...,0/ , _ ) is extended H{ then

is extended Π}. // S is co-extended Σ} ί/zen

S2(α1?. ..,«;_!,__) zjίΓ V^.Sίfl!,...,^-!,^,—)

is co-extended Σj.

Proof. Again we do the extended Σ} case and then apply 2.4. The proof is justlike the "hard" half of 2.5. Note that the easy half does not go through ! D

2. Defining Π} and Σ} Predicates 119

2.7 Lemma. // S = S(p>,S1,...,Sm,α,P1,...,P,J) is extended Π} then so are

V5mS(_S__) and VPΠ S(_, PΛ) .

// S is co-extended Σ} f/zen so are

3SmS(__,Sm,_) am* 3PΛ S(_, Pπ) . D

2.8 Proposition. // S = S(/?1,...,pί) is extended Πj (co-extended Σ}) ami is rea//ya predicate over 9K; i. e. S^M\ then S is Π{ over 9JI (Σ{ ouer 9W).

Proof. It suffices to prove one of these and take negations, so we prove the Σ}case. Typically S has a definition of the form

S(p) iff 3T3Qφ(p9q9a,f,Q)

where a are some parameters from HF^, qeW, and φ is co-extended. Thequantifiers 37] can alway be treated as quantifiers over relations on HF^, sincewe can always say in φ that 7) is a relation of urelements, so we restrict our-selves to

S(p) iff lQφ(p,q,a,Q)

where φ is co-extended. First we need to get rid of the parameter a. But everyaεMFm can be defined over HF^ by some extended formula ψ(x,qί9...,qr) so

S(p) iff Vx[ψ(x,ql,...,qr)^lQφ(p,q,x,Q)']

and the right hand side, by the above rules, is extended Σ{. We are thereforedown to the case

S(p) iff 3Qφ(p,q,Q)

where Q is, say, 3-ary and φ is co-extended. Now the following are equivalent,where ψ is the conjunction of the axioms of extensionality, pair and union andthe empty set axiom:

S(P),

TO,,,

(ΉFm,Q)\=φ(p,q9Q)9 for some β ,

(2ϊ«>β)l=φ(P,9>α), for some (9ϊw,β) with H

(^aR5β)^^(P^,Q)? for some (9ίTO,β) with 2I

The structure 2ITO can be have the same cardinality as 9W in the last two linessince 9JΪ is infinite. The equivalence of the third and fourth lines follows fromthe fact that φ is co-extended so it drops down from 81TO to HF^ by II.2.8. The


equivalence of the fourth and last lines in a consequence of the fact that 21must be isomorphic to an end extension of HF^ if 21 is a model of the axiomsmentioned. The last line can be rewritten as a Σj relation on $R without muciitrouble. Let's assume that $R = <M,K> with R binary, to simplify things. Weintroduce a lot of new relation symbols and define S^(M',R',A,E,F,Q) by

^M, 4nM'=0,

FίΞMxM',

"F establishes an isomorphism between <M,R> and <M',jR'>",

Thus Sx insures that «M',R'>; A,E,Q) is isomorphic to an (51 ,0. LetS2(M',y4,£) assert that this structure satisfies Extensionality, Pair, Union andEmpty set; e. g. Pair can be expressed by

Vx Vy [A(x) v M'(x)) Λ (A(y) v M'(y))-»3z(A(z)Λ Vw [wEz<->w = x v w =

Both Sl5 S2 can be defined by first order sentences over 9Jί in the additionalsymbols. Finally, we let φ'(x,y) result from φ(x,y) by rewritting it in terms ofthe structure «M',R'>,^,£,Q). For example e is replaced by E throughout.Then we have

S(p) iff there are M',K',v4,£,F,Q such that

S2(M'9A,E) and

3p' 3ς[' (F(p, p') Λ F((^, ) Λ φ'(p'9 q'))

which makes S Σ} on 9W. D

2.9 Examples, (i) It is worthwhile going back to look at some of the examplesin 2.1. In 2.1(ii) and 2.1 (Hi) the Πj predicates are actually extended first order.For example, in 2.1(iii),

x is torsion iff

where nx is defined by recursion in HF^ just as usual:

Ox = 0,

2. Defining Π} and Σ} Predicates 121

where the 0 and + on the right hand side are the group 0 and group addition.In 2.1 (iv), D is not co-extended but it is co-extended Σ}, hence Σ} by 2.8.

(ii) Another example that will come up later is where ΪR = <M, ~> with ~ anequivalence relation. Define

x<y iff caτd(x/~)<card(y/~).

This relation is Π}. (This is so simple that the above machinery is of little use.)If each equivalence class is finite then < is also Σ{:

~Ί(x<y) iff M¥m\=^aΆ(a = x/^ Λb=y/~ Λ card (b)*ζ card (α)),

which is extended first order so ~\(x<y) is Π} so x<y is Σ}. D

Let S(p, S) be a predicate of z-tuples p from 9K and m-tuples S of relationsover 5R S is Π} on 9W if there is a φ(p,S, f) such that

S(p,S) iff (9M9S)NVT1,...,VTM^p,S,T).

Some authors refer to such predicates as second order Π} predicates. The proofof 2.8 may be modified in an obvious way to yield a little more.

2.10 Proposition. // S(p,S) is extended Π} then S is Π\ on 9K.

Proof. The extra relations S ride along for free. D

Probably the most familiar example of a Δ} non-elementary set over Jf isthe set of (Gδdel numbers of) true sentences of arithmetic. This kind of exampleis very important. It is contained in the following proposition. Here K is somefinite language which is coded up in HF. To keep the notation (barely) manageable,we restrict the statement of the propositions to the case where K has one binarysymbol r.

2.11 Proposition. Define a predicate S(N,R,φ,s) by the conjunction:(i) ΛΓcM; R^NxNi <p,seWlFm;

(ii) φ is a formula of Kωω, 5 is a function with dom(s)^free variables (φ);(iii) Vxerng(s)ΛΓ(x);(iv) <N,Λ>l=φ[s].

Then S is both extended Π} and co-extended Σ}.

Proof. There is no trouble with (i) — (iii) since (i), (ii) are Δx on HF^ and (iii)is both extended and co-extended first order. The work comes in with (iv). Note,however, that if this particular S is co-extended Σ} then it is also extended Π} since

S(N9R,φ,s) iff (i) Λ (ii) Λ (iii) Λ 3x [x = <^,φ> Λ -ιS(ΛΓ,Λ,x,s)]


and the right hand side is extended Π} by the various lemmas above. We provethat S is co-extended Σ} by introducing another binary relation symbol Sat andfinding a co-extended first order S*(N,R,Sat) such that for N,R,φ,s satis-fying (i)— (iii),

To write out S* we use 5( /1;;) for

sf(dom(s)-{t;ί})u{<ι;ί,p>},

this being a Δt operation of s 9 p and t;f. Now define S*(N9R9Sat) by

Vφ Vs [(i) Λ (ii) Λ (iii) ->

if φ is atomic, say r(vi9Vj), then R(s(Vi),s(Vj))<-+Sat(<p,s)9

i f φ i s <Λ,{^,0}> then Sat(φ,s)*-+Sat(ψ,s)ΛSat(θ9s),

if φ is <~Ί,I/^) then Sat(φ,s)<^->—\Sat(ψ,s),

i f φ i s (B9vi9\l/y then S0ί(φ,s)<->3p[ΛΓ(p)Λ Sat(ιl/,s(p/Vi))]

with similar clauses for equality, \/, V. Note that the only unbounded existentialquantifier comes from the last clause and that quantifier is over urelements soS* is co-extended first order. It clearly has the properties needed to finish ourproof. D

3. Π{ and Δ{ on Countable Structures

We continue to consider a fixed infinite structure 3R = (M9Rί,...,Rly. Our goalhere is to show that if $R is countable then the Δj relations over 2ft are exactlythose relations in IHYP^. In view of II.5, this shows that the Δ} relations over$R are exactly those which are constructible from Sϋt by the time you come tothe first 9W-admissible ordinal.

We split the result in half to isolate the role of countability.

3.1 Theorem. Let 9JΪ be countable. If S is a H\ relation on 9JΪ then S is Σ! on

Proof. Consider the language Lu{P} as coded in HYP^ with p a distinct con-stant symbol for each peM. Suppose S(p) iff 9JIN VP φ(p,q, P). Then S(p) holdsiff (σ-xp(p,q, P)) is valid, where σ is the conjunction of the diagram of $R and

3. Πj and Δ} on Countable Structures 123

Thus S(p) holds iff the following is true in

by the Completeness Theorem for countable, admissible fragments. Thus S isΣ! on HYP^. D

3.2 Corollary. Let 9JΪ be countable. If S is Δ} on 9K then

Proof. Immediate from 3.1 and Δt separation in HYP^. D

The converse does not need the countability assumption.

3.3 Theorem. Let S be a relation on 9K. // S is Σl on HYPOT then S is Π\ on 3R.

3.4 Corollary. // a relation S on 9JI is in HYP^ then S is Δ} on 9ΪΪ.

Proof. If SelHYPaR then S and ~\S are Σί on HYP^. (Remember that parametersfrom HYPvCT are allowed in Σt definitions.) D

Proof of 3.3. Let S(p) be Σi on HYP^. By Proposition II.8.8 we can find a Σ!formula φ(x,q,M) such that the following are equivalent:

S(p)9

(1) 3lswl=<p[>,g,M] for every model 9ITO of KPU+ (of cardinality card(M)).

The last line is true with or without the parenthetical phrase since card(M)= card(HYR0i). Now code up the language L(e) in HF. Call the resultung setK, KeHF. Let kpu+ be the set of codes of KPU+ and let φ = φ(vί9v29v3) denotethe code of itself. Thus φeHF and kpu+ is a Δ! subset ot ΉF by Theorem .11.2.3.Our plan is to rewrite (1) as a Π} relation over $R with the aid of 2.10 and 2.8.Again we simplify notation by assuming 9K = <M?JR> with R binary. Now (1)is equivalent to:

For allM,R,F and all A,E,

(2) if <M',/θ!<M,,R>and «M',.R'>; A,E) is a structure of the appropriate kind, and

(3) if <M',Λ',yl,E>Nkpu+,

(4) then for some p'9q',m, (M',R',A,Eyt=φ(pf,q',m) where F(p) = p', F(q) = q'and me A is such that x Em<^>M'(x) for all x.

Let S^M'.R'.A^E^F) be just as in the proof of 2.8 so that Sx is first order inthe symbols and S: expresses line (2). Let S2(M',R,A,E) hold if

Vψ[\l/eίpu+-+(M'9R',A,E)\=ιl/'].


S2 expresses (3) and is co-extended Σ} by 2.11, 2.6 and other lemmas. (It is notnecessarily extended Π{, though, due to the Vι^ in front.) Line (4) can be writtenin extended Π} form by 2.11. This makes S(p) of the form

where Si is first order, S2 is co-extended Σ} and 83 extended Πj so 5 is extendedΠ} and hence Π} by 2.8. D

3.5 Corollary. For any structure yJl = (M,R1,...,Rιy, countable or not, the rela-tions S on $R in HYPM are exactly the relations definable over 501 by some formulaφ(vί,...,vn,qί,...,qm) of the admissible fragment LA where A^

Proof. If S is defined by

S , , iff

where φeHYP^ then S is Δx since N is Δr Thus SelHYP^ by Δ: separation.To prove the converse, first assume $01 is countable. Since SeHYP^ we

can write

S(p) iff

iff attN=3Γ^(T',p)

for some first order formulas φ,ψ possibly with constants q1 ?...,qm. We mayassume T, T are distinct symbols. Let σ be the sentence

/\ Diagram (OR) Λ Vx \/peM x = p .

The sentence

Vι> l5 . . . , vn [(σ Λ ψ(T, ι;1? . . . , vn)) -+ φ(T, vί9 . . . , t J]

is logically valid since for any T on 9JI, (^R,T')\=φ(p^...,pn) implies S(p1,...,pπ),which in turn implies (3Λ9T)\=φ(pί9...,p^ for any Ton 9Jί. By the InterpolationTheorem of ΠI.6.1 there is a formula 0(u1,...,t;ll)eIHYPaR5 θ involving only thesymbols of L and any constants q in φ such that both

σ Λ ιA(T, vί9 . . . , vn) -> 6(1?!, ...,υn) and

are valid. But then

S(Pl,...9pJ iff

Thus the result holds if 9JΪ is countable.

3. Π j and Δ} on Countable Structures 125

To prove the result for uncountable $01 we apply the Levy AbsolutenessPrinciple of Π.9. The theorem to be proved can be written out as

so we need to see that the part within brackets can be written as a Π predicatein ZFC. Recalling that HYP^^Lΐα)^ for the first α to make L(a)m admissible,we can rewrite it as

The part within brackets here is clearly Δ: since N is Δx. Thus the theorem isa Π sentence and so it suffices to prove it for countable structures 9JΪ. D

There are useful second order generalizations of the above theorems. For ex-ample, generalizing 3.1 we get the following result.

3.6 Theorem. Let S(p,S) be a Πj predicate on a countable structure 9K. For everyadmissible set A with Me A, SnA is Σ! on A. The Σ^ definition is independent of A.

Proof. If S(p,S) holds iff (SR,S)N=VT φ(p,S,T), then S(p,S) holds iff(σ(S)-xp(p,S,Γ)) is valid, where σ(S) is

Λdiagram (9W, S) Λ Vx \/peM (x = p) .

This is a countable sentence of Looω so the proof given in 3.1 carries over. D

The second order generalization of 3.3 is not quite the converse of 3.6.

3.7 Theorem. Let S = S(/?,S) be a second order predicate on 9W which is a Σί

subset of HYP^. Then S is Π{ on 9K.

Proof. A simple modification of the proof of 3.3 suffices. Line (1) becomes

(!') (^OT,S)Nφ|>,g,S,M], for every model 9ΪOT of KPU+ and every S

which results in a modification of (4) to

(4') then for some p',q',m9s, (M',R'9A,Ey\=φ(pf,q',s9m9) where

F(p) = p'9 F(q) = q'9 A(m) Λ Vx [xJEw<->M '(x)] Λ A(s)9

Vx [S(x)<r+3y(F(x) = y* yEs)] . D

3.8 Corollary. The set S defined by

is Π} on $R (as a second order predicate).


Proof. S is Δ0 on HYP^ since

x e S iff HYRm 1= "x is a subset of M"

so S is Π} on ΪR by 3.7. Note, however, that 3.7 will not allow us to concludethat — iS is Π} on 2R since — iS is not a subset of ΉYP^; far from it. D

3.9 Example. Let us return to consider nonstandard models of arithmetic. Weshowed in § 3 that the set of standard integers in a nonstandard model$R = <M,0, + , •> is Π} on ΪR. Sometimes it is Σ{ hence Δ}, sometimes not.Recall that Jf = <ω,0, +,x>.

i) For an 9JΪ where the set of standard integers is Σ} let $R be a minimalelementary extension of Jf\ i.e., Jf<W but Λ^X9K9W implies ^ = $1 or9l = $R. Such $R exist by results of Gaifman [1970]. In such an 2R we can de-fine, for xe9W,

x is standard iff 3M0 [M0 is the universe of a proper elementary submodelofSFΪandM0(x)].

This is extended Σ} by 3.10, hence Σ} by 3.8.ii) For an $R>,/Γ where the set of standard integers is not Δ} hence not Σ},

choose a countable ΪR with 0(331) = ω (by II.8.7). The subsets of 9JI in HYP^are exactly the first order definable sets (by II.6.7) so the set of standard integersare not in HYP^ and hence, by the results of this section, they are not Δ} on SCR.In fact, we see that for countable M , the set of standard integers is Δ} on $R iffO($R)>ω. We will return to this example later. D


3.10. Let ΪR be countable and let S1(p,P),S2(p,P) be predicates of peM, PC M2,each Σ} on $R. Assume that no pair (p,P) satisfies both Si and S2. Show thatthere is a Δ} predicate S(p,P) containing SL but disjoint from S2. [Copy theproof of 3.5 to find a θ(p,P) in LA such that S(p,P) iff (9K,P)t=0(p,P) and thenshow that S is Δ}.]

3.11. Recall Example 2.1 (iv). Let α>ω be any countable admissible ordinal.Let p be any prime. Show that there is a countable p-group G with length (G) = αsuch that G has a proper divisible subgroup but none in HYPG. For such a Gthe largest divisible subgroup of G is thus Σj but not Π}. [Use the FT-Com-pactness Theorem.]

3.12. Generalize the results of this section to show, for A^ transitive,i) If S is a relation on A^ and S is Σl on HYP(Aαϊl) then S is Π} on A^.

ii) If ASH is countable then the converse of i) holds.

3.13 Notes. Kripke and Platek proved that a subset X of HF is Π} over HFiff X is Σ! over HYP(HF) and hence that X is Δj over HF iff XeΉYP(HF).This was generalized in Barwise-Gandy-Moschovakis [1971] by replacing HFby any countable transitive set A closed under pairs. It is clear from the proof

4. Perfect Set Results 127

given there that Theorem 3.1 holds. It came as somewhat of a surprise that itsconverse, Theorem 3.3, holds without any coding assumptions about the struc-ture 9JI, since the inductive definability approach (discussed in Chapter VI) doesnot work in this complete generality.

4. Perfect Set Results

In this section we give a more sophisticated example of the interplay of modeltheory and recursion theory showing how each subject can shed light on the otherand how logic on admissible sets sheds light on both. The results themselves willnot be used in the remainder of the book.

The following, a classical result on hyperarithmetic sets, is the effective version(due to Harrison) of an even older result in descriptive set theory.

4.1 Theorem. //S^Power(ω) is Σ} on yΓ = <ω,0, +,•> and card(S)<2K° thenS is a set of hyperarithmetic sets.

Compare this with two results from model theory. The first is due to Kueker[1968].

4.2 Theorem. Let 9Jl=(M,Rί,...,Rly be a countable structure for a languageL and let P be an n-ary relation on M. If the set

has card(S)<2K° then

for some formula φ of Lωιω and some qi9 ..., <?meSOt.

(A formula φ is in Lωιω if it is in LA for some countable fragment LA of Looω.)The next result is a theorem of Chang [1964], Makkai [1964], and Reyes [1968].

Chang and Makkai had a stronger hypothesis.

4.3 Theorem. Let φ(P) be a finitary sentence of Lu{P}. Assume that for eachcountable model 9JI there are fewer than 2*° relations P such that

Then there are finitary formulas ^(x,;^, ..., yk\ ..., \l/m(x9yl9 ..., ykj of Lωω

such that for every model ($0Ϊ,P) of φ(?\ there is an i, l^i^m, and q^ ..., qk.e3Jlsuch that

128 IV. Elementary Results on IHYP^

The conclusion of 4.3 can be restated as: the sentence

is logically valid.These three results, while incomparable, are obviously quite similar. They

all begin with the assumption that a certain definable or Σ} class S has fewerthan 2*° elements and conclude that each element of S is definable in some way.We want to show these results are more than merely analogous, that they are infact shadows of a single definability result about logic on admissible sets. First,though, we prove a generalization of 4.1, because the proof is relevant to ourgeneral result.

4.4 Theorem. Let 9Jl = <M,R l9 ...9Rty be a countable structure and let S be asecond order Σ\ predicate on 9JI. // card(S)<2*° then S^HYP^ (and hence S iscountable).

Proof. After a trick the result falls right out of III.8.2. Assume S£ HYP^. Thenby 3.8 (and this is the trick), S0 = S — HYP^ is Σ} and non-empty. We provethat S0 (and hence S) has cardinality 2*°. Let us handle the case where S0 is apredicate of one relation :

S0(S)iff(ΪR,S)l=3T<p(S,T).

Let L' = Lu{p:peM}u{S}, K=Lu{T} and let LA, KA be the countableadmissible fragments given by HYP^. If σ is

Diagram(aR) Λ Vx \JpeM(x = p)

then σ Λ φ(S,T) is in KA. We claim that σ can have no model which is decidablefor LA. Such a model would be isomorphic to some structure of the form (9Jί, 5, Γ),where S is Δ x on HYP^ and hence SeHYP^, whereas (2R,S,Γ)t=(p(S,T),implies 5eS0. Thus the result follows from III.8.2. D

We now turn to consider the relationship between 4.2 and 4.4. If we apply4.4 to the situation described in Theorem 4.2 we learn that if there are <2K°Q'swith (aR,P)^(9W,β), then each of these is Δj on (9M,P) which (while interestingand not obvious from 4.2) says nothing about the original P. There are examples(9ΪΪ,P) satisfying 4.2 but where P^HYP^, i.e., is not Δί on ΪR, which rules outone possible strengthing of 4.4 that would yield 4.2. To find the correct generali-zation of 4.2, 4.3 and 4.4 we need a new definition.

4.5 Definition. A Σ\ sentence of an admissible fragment LA is a second orderinfinitary sentence of the form

where £L. is a set of symbols of L, =2e A, and φe LA.


If Ά is finite, the requirement JeA is automatically true, and we could write

or

In the infinite case, however, we should not think of SL as being a well orderedsequence of symbols. Note that even though we have written J, the definitionactually permits function and constant symbols to occur in Ά as well as relationssymbols.

The following result has 4.2 — 4.4 as consequences. For ordinary (as opposed toΣ{) sentences of LA it is due to Makkai [1973]. For 4.4, though, it is the Σ} versionwhich matters. The proof is a minor variation on Makkai' s theme, the Inter-polation Theorem taking the part formerly played by Beth' s theorem.

4.6 Theorem. Let 3 Jφ(P, j2) be a Σ} sentence of the countable admissible fragmentLA. If for each countable structure 90Ϊ there are less than 2K° relations P such that

then there is a sentence σ of LA of the form

which is a logical consequence of <p(P, J), where each φi contains only symbols of

The converse is obvious. In fact, the conclusion implies that every such Pis in any admissible set containing 9Jί and φ so there are ^ X0 such P.

Note that Theorem 4.3 is the special case of Theorem 4.6 where LA is Lωω

and where the Q's do not occur in φ(P, J).Before attempting to prove 4.6 it is good to get some idea of what it says by

applying it to prove 4.2 and strengthen 4.4.

4,7 Corollary. Under the assumption of Theorem 4.4 there is an S'eHYP^ suchthat S^S'.

Proof. Suppose PeSiff(9JΪ,P)l=3Q(/>0(P,Q). Let φ be the conjunction of <p0(P,Q),diagram (9JΪ) and V x YpeM(x = p). The hypothesis of 4.6 is satisfied so let σ be asin the conclusion of 4.6, σ of the form

where each fa is in the language Lu {p| eM}. For each ie/ and qί9...,qm.eMlet


Each Pi-eMYPm by Δ x Separation and, as an operation of ί and q, Pi $ is aoperation in HYP^ so we may form the set

by Σ Replacement and S^S'. D

4.8 Theorem. Let $)l = (M,Rl,...,Rιy be a countable recursively saturatedstructure (i. e. o(]HΎPm) = ω). Let She a second order Σ\ predicate with card(S) < 2*°,say S^ Power (Mn\ There is a finite set of finitary formulas

Ψι(x,yι,...,ymι),...,ιl/k(x,yl9...9ymj)

of Lωω such that for each SeS there is an i, l^i^k, and elements qι,.. ,qmi

of $R so that S is defined by

s(χ) iff

Proof. Using 4.7 choose S' so that S' c Power (MM) and

Since o(HYPaR) = ω we have, by II.7.3,

[I/MS a formula of Lωω,g is an m-tuple of elements of M (where the freevariables of ψ are among ϋ l 9...,ι;π + m) so that for all x1,...,xπeM:

<xl9...,xnyeS iff

Since L is finite we can assume Lωω is coded up on HF. By Σ Collection inthere is a finite set Φ of formulas such that each ψ can be chosen in Φ. D

4.9 Example. Let .yΓ = <ω,0, +,•> ana let $R be a countable recursively saturatedelementary extension of Jf. Then there are 2*° distinct 9J10 such that

(i) 9Jί0-<9JΪ, and(ii) 50Ϊ0 is an initial segment of 501.

Proof. Let

S = {M0 c MI MO is the universe of an 9J10 with (i) and (ii)}.

The techniques of § 2 show that S is Σ} on 501. Suppose, toward a contradiction,that card(S)<2K°. Then since ωeS, there is a formula ψ(x,qι, .9qj with


parameters from 9Jί such that

which is a contradiction. D

Before turning to the proof of Theorem 4.6, we show how 4.8 can be used tostrengthen the Chang-Makkai-Reyes Theorem (4.3). The result is interestingbecause of the light it sheds on the usual proofs of this theorem by means ofsaturated (or special) models.

4.10 Corollary. Let φ(P, Q) be a βnίtary sentence such that for each recursivelysaturated countable model 9JΪ, there are less than 2K° different P with

(9W,P)N=3Q(p(P,a).

Then there is a finite list of finitary formulas ψ^x^y), ..., ψm(x,y) such that

Proof. Suppose that the hypothesis holds but that the conclusion falls. Let T bethe theory

x,;y)], for all

By the ordinary compactness theorem, this theory is consistent. By Theorem II. 8. 8,it has a countable recursively saturated model ($R,P). But this structure 9JI has<2K°F such that (9M,P')t=3Q<p(P,Q) so, by 4.8, each of these P (in particularthe original P) is definable, contradicting the fact that (9W, P) is a model of T. D

4.11. Proof of 4.2 from 4.6. We must cheat a bit by quoting a result, Scott'sTheorem, from Chapter VII. Let 9W,P,S be given as in 4.2 and suppose thatcard(S)<2K°. Let φ(P) be the Scott sentence of (2R,P) so that for all countablestructures (3JΓ, P'),

(The sentence φ(P) involves only constants from Lu{P}.) Thus there are, foreach model 9JΓ, fewer than 2K°P' such that


From 4.6 we get a ^(xl9 ...,xπ, y l 5 . . . ,y m ) such that for some ^1 ?...,

which yields the conclusion of 4.2. D

Having no excuse for further procrastination, we begin the proof of 4.6.

4.12. Proof of 4.6. Since 4.6 implies 4.4 we expect to use considerations similarto those used in proving 4.4, that is, the method of § III.8. The chief differenceis that instead of constructing 2*° distinct models $R we need one model with 2*°distinct P such that

This accounts for the complications in the proof below. We prove the contra-positive, so suppose φ(?,£) does not have any sentence of the desired form as alogical consequence. Let us simplify matters by assuming that 2L has only onerelation symbol Q and, further, that P is unary. The proof will make it clear thatthese assumptions do not really matter. Let

|_°=L-{P,Q}, C = {cjw<ω}, K°=L°uC, K = L u C .

Call a set s of sentences of KA special if the following conditions are fulfilled,conditions (Dl)— (D7) coming from (Cl)— (C7) of 111.2.2 respectively.

(Dl) If φes then ~Ίφφs.(D2) If -ηφes then ~φes.(D3) If /\Φes then φes for all φeΦ.(D4) If Vvφ(v)es then φ(c)es for all ceC.(D5) If \/Φes then φes for some φeΦ.(D6) If 3vφ(v)es then for some ceC, φ(c)es.(D7) If t is a basic term of LA and c,deC then: if (cΞΞd)es then (d = c)es; if

φ(ί), (cΞΞί)es then φ(c)es; for some eeC, (e^i)es.(D8) If φeK A then φes or

In the proof of the Model Existence Theorem we first constructed a set sω satis-fying (Dl) — (D7) and then showed that any set s satisfying (Dl) — (D7) gaverise to a unique canonical model $R by the conditions

9WNR( C l , . . . ,c n ) i f fR( C l , . . . , c π )e5 .

Furthermore, this model was a model of each φes. We shall use these facts here.Note that if a consistency property 5 has the property

(C8) if seS and φeK A then su{φ}eS or su{-Ίφ}eS

then the resulting sω will satisfy (D 8) and hence will be a special set of sentences.


Now recall the notation from § III.8:

d is a typical node on the tree; dO extends d by putting a 0 on right end; dί a 1and 6 is a typical branch.

The level of a node is just its length as a sequence. The plan for the proof is toattach a finite set sd of sentences of KA to each node d of the tree in a way thatinsures the following conditions:

(1) {<p(P,Q}} is placed at the bottom of the tree; i. e.9 s<> = (φ(P,Q)}.

(2) // b is any branch and sb = \J{sd\d a node on b} then sb is a special set ofsentences of KA.

(3) Any two sets sd and sd, on the tree are consistent with respect to the sentencesof Kj; that is, if φeK£ and φεsd then (—\φ)φsd,.

(4) Distinct branches through the tree are inconsistent with respect to thesymbol P; that is, if ί?l5 b2 split at d then there is a constant symbol c so thatP(c) is in sd0, but ~πP(c) is in sdl.

Suppose we contrive to fulfill (1)—(4). The canonical model determined by abranch b through the tree will have the form (Wlb,Pb

9Qb) with φ(P,Q) true by (1),

(2) and the above remarks on special sets. Furthermore, SQl 1 = W*2 for allbranches bί,b2. For if ReL° and R(c1,c2) holds in Wlbί then R(c1?c2)esd forsome d on b1 but then —ιR(c1,c2) is never put into any sd> on b2, by (3), soR(c l 5c2) is in some sd, on b2 by (D8) so R(c1,c2) holds in $R&2. Finally, if bί9 b2

are distinct branches then Pbl φ Pb2 by (4). In other words we have one model 5DΪwith 2K° distinct P each satisfying

and so we will have proved our theorem. Satisfying (1)—(4), though, is not sotrivial.

In order ultimately to satisfy condition (4), we would like to have a symbol Pb

for each branch b thru the tree, but this would make our language uncountable.Instead we introduce new relation symbols Pd, Q.d for each node d on the tree.


We think of Pd as our original P with a ghostly superscript d just barely visible.Our original P, Q are Pd, Qd where d is the empty sequence, d = < >. We de-note this expanded language by 1C7 and the admissible fragment by K^. As usualwe consider only formulas with finite many c's and, this time, only finitely manydifferent P^s ans Q^s. A finite set s of sentences of K^ is g-consistent if all thenodes occuring as ghostly superscripts in s lie on some branch (e. g., P010 andQOioio could both occur in s but poio and Qoii could not) If s is 0_consistent

then s is the result of increasing all superscripts in s to the longest one appearingin s. E. g., if 010 and 01010 are the only superscripts in s then s has all P010 andQ010 replaced by P01010 and Q01010. We define a giant consistency machine Sby (s1,...,sπ}eS iff sί9...,sn are each finite, g-consistent, and ^u us,, doesnot imply any sentence of K| of the form

(*) Vi^" V^, [βy Vx Pdi(x)~ψ(x,y)-]

where each φel!^ and ά{ is the longest node in s f. (Note that if {s^. .jsJeSthen s^u us,, is consistent which will give us (3) above.) Our hypothesis insuresus that

(5) {{φ(P,Q)}}eS.

While S is not really a consistency property, it generates many of them.(6) // (s1,...,sπ,sn + 1}ES then

S = {s|{s l5...,sπ,s}eS}

is a consistency property satisfying (C8) above.

Most of the clauses are routine. Let us check (C 5) and (C 8).

(C5) Suppose \/<9eseS, but that for each ΘeΘ, sv{Θ}φS so that

{Sί,...,sn,sv{θ}}φS.

Since s is g-consistent so is su{0} so the problem comes from (*). We musthave, for each 0e<9, some σθ of the form (*) such that

Now, just as in the proof of the interpolation theorem, we can assume the σθ

is given as a function of θ, a function in our admissible set (σθ will be the dis-junction of the σ's given by strong Σ replacement). But then σ = \/θeθ σθ is againof the form (*), once you rearrange it a bit, and

a contradiction.


(C8) Suppose φ(c1,...,c/ J)eK2, that sεS but neither sv{φ(ci9...,cn)} norsu{— ιφ(c1,...,cJ}eS. Then there are sentences σ1? σ2 of the form (*) such that

but then

vσ 2

and σί v σ2 is equivalent to a sentence of the form (*).We now come to the crucial step which will yield (4) above.

(7) // {s l5...,sπ}eS, if d is the longest node in sn, if dO, di do not occur inSiU us,,, and if c is a constant symbol not in s^ us,, then

{s1,...,sπ_1,sπu{P'i0(c)},snu{^P'ίl(c)}}

is in S.We use the Interpolation Theorem for KA to prove (7). We invite the student

to try the case n = ί for himself before going on. We do the case n = 2 becauseit exhibits the problems that arise in general. Now, if (7) fails, the trouble cannotarise from ^-consistency since

are ^-consistent so it must be that there are sentences σ t, σ2, σ3 where σt is ofthe form

(where Pί is the symbol Pd in S l 5 P2 is Pd°, P3 is Pdl), such that

(8) s1U52u{Pd°(c)}^52u{-ΊP ί ί l(c)}Hσ1 v σ 2 v σ 3 .

We show that (8) implies {s l5s2}£S by finding a sentence σ of the form (*)such that

Rewrite (8) as follows :

Λ s

where s2(Pίί0,Qίί0) indicates the result of replacing ?d by Pdo in s2. Notice thatthe only symbols on both sides of the implication sign are in K°. By the Inter-


polation Theorem there is a ^(c,c1?...,cm) which is an interpolant. We maywrite this as :

51(P1,QjΛ52(P^Qd 0)ΛPd 0(c)-.σ1(P1)vσ2(Pd 0)v (A(c,c1,...,cJ, and

Now replace Pd°, Qd° by Pd, Qd in the top line, Pdl, Qdl by Pd, Qd in the secondline. We obtain

51u52^σ1(P1)vσ2(Pd)vσ3(Pd)v[Pd(c)^ιA(c,c1,...,cJ].

Since c does not occur in s tus2 we get

and hence {sl5s2}£S.Now we are ready to decorate our tree. List the sentences of K| as a sequence

in such a way that any node d appearing in φn is of level ^n. List the termsoccuring in L A :

We work our way up the tree as follows. Place (φ(P,Q)} at < >. Assumewe have placed sets sd at every node d of level n so that d is the longest nodein sd and the set

{sd\d a node at level n}

is in our consistency machine S.

Given sdl we first take care of tn and φn (if φn happens to be ^-consistent with sdl)as in the proof of the Model Existence Theorem, using (6), giving us some

5. Recursively Saturated Structures 137

{s',sd29sd3,sd4}eS.

We then apply (7) to get

{5'u{Pd0(c)},5'u{-lPdl(c)},52,53,54}6S

and we let sdl0 = s'u{Pd°(c)}, sdl l=s/u{"iPdl(c)}. In this way we work our wayalong level n +1 and on up the tree. We see that any finite set of nodes on thetree is in S. This takes care of (3) since, otherwise, they would certainly implya formula of the form (*). Now that there is a set at each node, let the super-scripts vanish and you will discover we have satisfied (1), (2), (3) and (4), provingour theorem. D


4.13. Show that Example 4.9 is not true without the assumption o(ΉYPyΛ) = ω.[Let 901 be a minimal elementary extension of ~/Γ = <ω,0, +, •>].

4.14. Let 50Ϊ = <M,0,+, •> be a countable nonstandard model of Peano arith-metic with o(HYP5p,) = ω. Show that there are 2*° initial segments of 9Jί whichare models of Peano arithmetic.

4.15. Improve 4.14 to get 2*° initial submodels of 9ϊl which are isomorphic to 9JI.[Hint: Use a theorem of Friedman [1973] to the effect that every countablenonstandard model of Peano arithmetic is isomorphic to some initial segmentof itself.]

4.16. Use 4.4 to show that if a countable abelian group G has <2K° divisiblesubgroups then they are all in HYPG and hence there are at most K0 of them.Give a direct group theoretic proof of this fact.

4.17. Extend Theorem 4.6 from simple sentences to Σί theories. Similarly extendthe applications of 4.6 given above.

4.18 Notes. The results of this section are called perfect set results because onealways ends up constructing, by a tree argument, a perfect set of objects, perfectin the topological sense.

5. Recursively Saturated Structures

Having discovered several interesting facts about structures $R with 0(501) = ω,we take time in this section to relate this condition on HYP^ to more traditionalnotions.


Recall that a structure 9M = <M,R1,...,R/> for L is N0-saturated if for everyk<ω and every set Φ ( x 9 v ί 9 . . . 9 v k ) of formulas of Lωω with free variables amongx9vί9...9υk the following infinitary sentence is true in 9JI:

where Sω(Φ) is the set of all finite subsets of Φ.

5.1 Definition. The structure $R = <M,^1,...,^ί> for L is recursively saturatedif the above holds for all k<ω and all recursive sets Φ(x,Vι,...,vk) of Lωω.

Just as in the case of K0-saturated we have the following lemma.

5.2 Lemma. Let 9J? be recursively saturated and let Φ(xl,...,xn,vl,...,vk) be arecursive set of formulas with free variables as indicated. The following infinitarysentence is true in $R:

Proof. The proof is by induction on n9 the case n = 1 being the hypothesis. Itclearly suffices to prove the result for Φ satisfying the condition

Φ0eSω(Φ) implies /\ΦQeΦ,

since we could close Φ under finite conjunctions. Let Ψ(xl9...9xn9vί9...,vk) bethe set of all formulas

for φeΦ. Suppose that qί9...9qke3Jl are such that

for all Φ0eΦ. By the induction hypothesis, there are p l 9 . . . 9 p n G ( S Ά such that

and hence

for all Φ0eSω(Φ), since every such 3xπ + 1/\Φ 0 is in Ψ. But then since 9JΪ isrecursively saturated there is a pn + 1eϊR such that

!,..., !, ,..., ). D


The principal link between recursively saturated structures and admissiblesets is the following theorem of John Schlipf.

5.3 Theorem. Let Wl = (M,Rl7...,Rιy be a structure for L 9JΪ is recursivelysaturated iff

Proof. We prove the easy half first. Suppose that o(ΉΎP,ΰl) = ω. Let Φ(ι;,w1,...,wk)be a recursive set of formulas of Lωco. We may consider Φ as a Δ! subset of IHFby II.2.3. Since ΉF is \ on every admissible set, Φ is also Δj on HYP^. Letg = g1,...,gke

sDίί be such that

We need to find a finite subset Φ0 of Φ such that

Now, since

VpeM 3

we have, by strong Σ Collection, a set b such that

(1) VpeM

and

(2)

From (2) we see that frcφ so let Φ0=b. Φ0 is finite since it is in HYP^, is aset of pure sets, and has finite rank. From (1) we see that Φ0(v,q) is not satis-fiable on 2R.

To prove the other half of the theorem, let 9Jί be recursively saturated. Weneed to prove that L(9^,ω) is admissible; i.e., that it satisfies Δ0 Collection.Call a set αeL(93ϊ,ω) simple if there is a single term &:(vί,...,vk+ί) built upfrom J^,...,J^y, Q) such that each xeα is of the form

for some p^...,pkeM. Assume, for the moment, that we have established (3)and (4):

(3) Every αeL(9Jl,ω) is the union of a finite number of simple sets;

(4) // zeL($0ΐ,ω) and if a simple, then L($ft,co) satisfies

Vxef l 3y φ(x,y,z)-+ 36 Vxeα 3yeb φ(x,y,z)

for all Δ0 formulas φ(x,y,z).


Assuming this, let φ(x,y,z) be a Δ0 formula such that L(50l,ω) satisfies

Vxea 3y φ(x,y,z).

Write a — al\j'"^jam where each at is simple. Since

holds in L($R,ω) there are sets bl,...,bm in L(9ϊl,ω) such that

But then let b = blu- vbm. Then beL($ft,co) and

Vxeα 3yεb φ(x,y,z)

so L(ΪR,ω) satisfies Δ0 Collection.To prove (3) note that in the proof of II.7.7 we showed that for each n there

are a finite number of terms Jrl,...,JΓ/M such that each xeL(9Ji,rc) is of the form

for some ί^m and some pεM. If aeL(9K,n) then α^L(9W,n). Define, byΔ0 Separation, sets aί9...,am by

Then a = al^j "^jam.Finally we prove (4). Let φ be given. By Π.7.7 and Π.7.6 we may assume that

the only parameters in φ are M and some qeM. Given the simple set a letJΓ°(f1,...,ι;n+1) be as given in the definition of simple. Let α = «^"1(r1,...,rk,M)for some r1,...,rfceM. Rather than prove (4) we prove its contrapositive. Let ψbe —ιφ, so that we want to verify that L(9Jl,ω) is a model of

Vέ>

Assume the hypothesis. In particular we have, for each m<ω,

(5)m 3xe

which becomes

(6)m 3Pl,...,

This is a Δ0 formula of p,q,r so, by the effective version of Π.7.8, we can find aformula \l/m(p,q,r) of Lωω equivalent to it. Note that by (5) we have


whenever m^m'. By (6)m we see that

Φ = {ψm(vί9...,vn,q,r)\m<ω}

is finitely satsfiable. Since it is clearly a recursive set (by the exercises at the endof II.7) and 50Ϊ is recursively saturated there are /71?...,pπe9JΪ so that

for all m<ω. Thus for this p, we have, setting x = 3?0(p,M), xεa, and for

all m<ω,

and hence

as desired. D

Schlipf discovered 5.3 by generalizing the results 5.4 and 5.7 below.

5.4 Corollary. // 50i = <M,#1?. ..,#,> is K0-saturated then 0(50l) = ω. D

5.5 Corollary. // 50ί = <M,R1,...,JR/> is recursively saturated and Φ(x,v1,...,vk)is any set of formulas of Lωω which is Σ^ on IH ΎPm then 50Ϊ satisfies :

Proof. The proof that o(HYPaπ)=ω implies 50Ϊ is recursively saturated actuallyproves this stronger result. D

5.6 Corollary. For every infinite $R = <M,Λ1,...,K/) there is a proper elementaryextension 5ft of 50Ϊ of the same cardinality such that 91 is recursively saturated.

Proof. Immediate from 5.3 and II. 8. 6. D

The above corollary shows a contrast between the notions of recursivelysaturated and K0-saturated structures since there is no countable N0-saturatedelementary extension of Jf =<ω,0, +, •>. Of course one could also prove 5.6 bya more standard model theoretic argument using elementary chains.

The following result shows that 5.3 can be improved for countable structures.It shows that if 501 is countable and o(HYPαR)=ω then 501 is saturated for certainsets of Σ} formulas.

5.7 Theorem. Let y$ί = (M,Rγ,...,Rΐ) be a countable structure for L with0(50ί) = ω. Let K=Lu{S l 5...,Sm} and let Φ(x1,...,x l l,t; lJ...,ι;fc,S1,...,SJ be a

142 iv. Elementary Results on

set of formulas of Kωω which is Σl on IHYP^. The following infinitary secondorder sentence holds in TO:

) ,.Proof. We use Theorem III.5.8. Let ql9...,qkeM be given so that

for all Φ0eSω(Φ). We can assume that Kv{cl9...9cn9dί9...9dk} is coded up onHF. Let T be the theory

Φ(c l 5...,cπ,d1,...,dk,S l s...5SJ.

Introduce symbols p for peM as usual and let T' = {ψ} be the conjunction of

ΛDiagram(TO)

For every finite subset 7^ of Γ, TQ\jT' has a model, so TuT' has a model. Thismodel is isomorphic to some

(Wl,Sί9...9Sm9pί9...9pn9ql9...9qk)

with


5.8. Show that every recursively saturated structure is ω-homogeneous.

5.9. Suppose 9JΪ is uncountable. Show that o(HYPan)=ω iff for all relationsT i 9 . . . 9 T k on Wl there is a countable recursively saturated structure 91 with

5.10. Show that the predicate

"TO is recursively saturated"

is absolute (ΔJ for models of KPU + Infinity but that the predicate

"TO is K0-saturated"

cannot be expressed by a Σ formula.


5.11 (J. Schlipf, J.-P. Ressayre). Let α be an admissible ordinal and let A = L(α).Let L be a language with a finite number of symbols. A structure 9JΪ for L isα-recursively saturated iff for every set Φ(x9vί9...,vk) of sentences of LA whichis Δ! on A the following sentence holds in 9JZ:

where SΔ(Φ) = {Φ0

(i) Prove that if Lΐα)^ is admissible then 9JΪ is α-recursively saturated.(ii) Prove that O(9Jl) = the least α such that α is recursively saturated. (This

result, due to J. Schlipf, strenghtens a special case of a theorem of J.-P. Ressayre.Schlipf s proof uses notions from Chapters V and VI.) Makkai has translatedRessayre' s result into our setting to show that for α countable, admissible andgreater than ω, Lfa)^ is admissible iff $R is α-recursively saturated and satisfiesthe following condition: Suppose φβ^(vί,...,vn) is an α-recursive function of β, y.Suppose further that for some /?!,..., pπe$R and some /?0<α:

Then there is a y0

< α sucn tnat

5.12. Show that Wi is K0-saturated iff(i) 0(IHYPOT) = ω.

(ii) for every X^ω, (HYP^,^) is admissible.

5.13. In this exercise we sketch some interesting connections between recursivelysaturated models of Peano arithmetic and models of nonstandard analysis. Tosimplify matters, we identify analysis with second order arithmetic (a standardperversion among logicians). Thus we add to the first order language of numbertheory new second order variables Xί9X2,... and a membership symbol e whichcan hold between first order objects and second order objects ((x^Xj) is aformula but (J^-ex,) isn't). The axiom of induction asserts:

(Warning: when working in systems weaker than the one described here it isoften necessary to replace this single axiom by an axiom scheme.) The axiomof comprehension asserts the following, for every formula φ(x,yί9...,yk):

Vy 3x [Vx(xeX*-*φ(x,y1,...9yk))'].

By analysis we mean the usual axioms of Peano arithmetic plus the axiom ofinduction and the axiom of comprehension. (Of course there is no need to includethe first order form of induction since it follows from our second order axioms.)A model of analysis consists of a pair (9l^f), where Jf is a collection of subsets


of the first order structure 91, which makes all the axioms of analysis true. Anysuch model of analysis gives rise to a model 91 of Peano arithmetic, but notevery model of arithmetic can be expanded to a model of analysis. A model ofnonstandard analysis is a model (91, ffl) of analysis with 91 not isomorphic to thestandard model of arithmetic.

i) Prove that if (91, Jf) is a model of nonstandard analysis, then 91 is recursivelysaturated.

ii) Let 9Ϊ be a nonstandard countable model of arithmetic. Let

tf = ΠI tf IW ) 1= analysis} .

Prove that either ffl is empty or that Jf consist of exactly the definable subsetsof 91. [This is easy from (i) and Theorem 1.1.]

5.14. Prove that there are two nonisomorphic countable recursively saturatedelementary extensions of yΓ = <ω, +,x>.

5.15 Notes. It is not known whether or not there is a complete theory T in anfinite language such that all models of T are recursively saturated but T is notK0-categorical.

6. Countable 9JI-Admissible Ordinals

Since this chapter concerns the interplay of model theory and recursion theory,it seems appropriate to discuss one of the first applications of infinitary logicto the theory of admissible ordinals.

Let yΓ = <ω,0, +, •>. Most countable admissible ordinals α (other than ω)that arise in recursion theory are of the form

for some relation R on ω. The question arose: Is every countable admissibleα, α>ω, of the above form? Sacks eventually answered this in the affirmativeby means of "perfect set" forcing. His proof remains unpublished since Friedman-Jensen [1968] presented a simple proof of the result by means of the BarwiseCompactness Theorem. We extend this theorem as follows.

6.1 Theorem. Let 9Jl = <(M,JR1,...,JR ί) be a countable infinite structure and let αbe a countable ordinal The following are equivalent:

(i) α is ^-admissible;

(ii) for some relation S on SOΐ,

6. Countable ΪR-Admissible Ordinals 145

(iii) for some linear ordering •< of 9K,

and the order type of the largest well-ordered initial segment of < is α.

Proof. The implications (iii)=>(ii) and (ii)=>(i) are obvious. To prove (i)=>(iii)we borrow a fact from Section V.3:

(1) // râxa is a linear ordering, r an element of an admissible set A, andif β is the length of a well-ordered initial segment of r then

This could be proved now, but it is easier to wait for the Second RecursionTheorem. Let α be 501-admissible. Then there is a countable admissible setA = A<m above 501 with

α =

by II.3.3. Let K be the language L* plus new constant symbols c, r, and x foreach xeA^. Let KA be the admissible fragment of Kr jθω given by A^. Let Tbe the theory which asserts :

KPU +

Diagram (A^)

"M is the set of all urelements"

Vv \_vea -+\/xeav=~x] (for all αeA^),

"c is an ordinaΓ

c>β (for all j8<α),

"r is a linear ordering of M of order type en(c x c)".

T has a model of the form

(aR HKê.α,!-)

for any well-ordering r of M of order type α. By IΠ.7.5 T has a model

with QL = oiT/(3R'9B,E). Let A'm = iT/(Wl'9B,E) which is an admissible set bythe Truncation Lemma. Since r^MxM, reA^ so A^ is actually admissibleabove ($R,r). Hence α^0(HYP(aR >r)). But r has an initial segment of order type α(by T) so, by (1) applied to HYP(αR5r), αô(HYP(TOfr)). We let X be r. D

6.2—6.5 Exercises

6.2. Let (2R, X) be as in 6.1 (iii). Show that HYP(OTf< } is a model of ^Beta.


6.3. Prove (1) above.

6.4. Let Agtn be countable, admissible above 9JΪ with o(Aari)>ω. Find a largeradmissible set B^ above 50Ϊ with the same ordinal such that B^ is locallycountable; i. e.,

ΪBm\=Va ("a is countable").

[Hint: Use the YΫ Compactness Theorem and Theorem Π.7.5.]

6.5 (Schlipf). Prove that for every countable admissible ordinal β there is anelementary extension SR of yΓ=<ω,0, + ,x> such that p = o(WL\Pm). [Hint: i)Show that if 501 is not recursively saturated and the set {n<ω\Wl\="n divides fe"}codes a well-ordering of ω, and if α is the length of the well-ordering, theno(HYP9[ϊί)>α. ii) Show that if 501 is a model of Peano arithmetic generated by asingle element /c, usually written 50ϊ = yΓ[/c], then 50} is not recursively saturated.]

6.6 Notes. Theorem 6.1 and Exercise 6.4 are just two of many results that canbe proved by either forcing arguments or by compactness arguments. See theappendix for a few references. Kunen has recently removed the hypothesis ofcountability from 6.5.

7. Representabίlίty in Wl-Logic

One of our principle results in this chapter, Theorems 3.1 and 3.3, identifies therelations on 501 which are Σj on HYP^ as the Π} relations on 501, as long as Mis countable. In Chapter VI we will search for the absolute version of this result.The results of this section will be of central importance in this search.

The reader should recall the notions of representability used to characterizethe r. e. and recursive sets. The following are the infinitary analogues.

7.1 Definition. Let 50Ϊ be an L-structure, Ta set of finitary sentences of L+ whichare consistent in 50ϊ-logic, φ(vί9...,υj a finitary formula of L+ and S an π-aryrelation on -501.

i) We say that φ(vl9...,υn) strongly represents S in T by the 9W-rule if, forall ,...,

S(ql9...,qn) implies Tt-mφ(qί9...,qJ, and

-iS(gl5...,4n) implies Γh-aR-κp(q1,...,qn);

whereas it weakly represents S in T using the $R-rule if for all

S(qi9...,qn) iff

7. Representability in 90?-Logic 147

ii) We say that φ(υi9...9υj invariantly defines S in T in 9Jί-logic if for all

S(ql9...,qn) implies

-\S(qί9...,qn) implies

where as it semi-ίnvarίantly defines S in T in 9Pΐ-logic if for all

%!,..., αj iff ΓNaπφ(q1,...,qJ.

The following is an immediate consequence of the 9Jl-Completeness Theorem.

7.2 Proposition.

Strongly representable => invariantly definable

weakly representable => semi-invariantly definable

and, if $R and L+ are countable, the converses hold. D

These are excellent examples of notions which agree in ordinary recursiontheory but which diverge, yield two interesting distinct notions, in generalizedrecursion theory.

7.3 Theorem. Let Wl = (M,R1,...,Rιy and let S be a relation on 9K.i) IfS is Σl on HYP^ then S is weakly representable in KPU+ using the Wl-rule.

ii) If SeΉYPw then S is strongly representable in KPU+ using the ^ft-rule.

Proof. Our language L+ for SR-logic consists of Lu{p|peM} as in III.3.2(ii).We prove the results for countable 9K. In Chapter VI we will show that theresults are absolute. We prove (i) first. Choose φ(xl,.,.,xn,pί,...,pk,M) as inII.8.8. We can rewrite this using the relation symbol M in place of the singleset M. Thus we have, for

%1;...,αJ iff

which, by 7.2, gives the desired result.Now we prove (ii). Let us assume S is unary to simplify notation. Using II. 5. 15

let <?(*,/?!,. ..,/?„, M) be a good Σΐ definition of S so that

and

for all models M^ of KPU+, and hence

148 IV. Elementary Results on HYPTO

by the $R-completeness theorem. We claim that S is strongly represented by theformula ψ(v) given by

3x[φ(x,p1,...,pπ,M)Λi;Ex].

If S(q) holds then 8ίOTN^(q) for all models Wm of KPU+ so KPU+If -ι%) then, for any 9laϊ ίNKPU+, since <Άm\=φ(S)Λ 3!x<p(x),SIOTN-ι^(q) andhence, KPU+ h-OT-ι^(q). D

We now prove a strong converse to Theorem 7.3. The first time through thisresult the student should think of T as KPU+ or some strong extension of itin L* given by an r. e. set of axioms.

7.4 Theorem. Let T be a set of βnίtary sentences of L+ which is Σί onand is consistent in tyJl-logic. Let S be a relation on 9JΪ.

(i) // S is strongly representable in T using the Ώl-rule then SeΉYP(ii) // S is weakly representable in T using the Wl-rule then S is Σx on

Proof. First note that (ii)=>(i) since S strongly representable implies S and -ηSare weakly representable so S and ~iS are Σί on HYP^, so S is Δr and henceSelHYP^ by Δ Separation. We prove (ii) for the case where $R and L+ arecountable leaving the absoluteness of 7.4 to Chapter VI. Let φ(vί,...,vn) weaklyrepresent S in T. Then we see that the following are equivalent:

S(ql9...,qn),

I.e., the infinitary sentence ^(q1?...,qπ) is a logical consequence of T, where(A(q1 ?...,qw) is

/\ Diagram (9K) Λ Vv [M (v)^\/peM v = p] -> φ(q1? . . . ,qπ) .

The sentence ^(q1,...,qπ)eHYPS[ϊί and the map (^1,...,^n)ι-^^(q1,...,qπ) is Σt

definable so, by the Extended Completeness Theorem, S is Σ: on HYP^. D

It should be obvious from the proof of 7.4 that there was no real reason todemand that T be a set of finitary sentences. It is just that we only bothered todefine 1- for finite sentences. T could have been a set of sentences each iaHYR0J as long as T is Σ1 on HYP^ and the proof would go through unchanged.

One might well ask about what happens to invariant and semi-invariantdefinability in the uncountable case where they no longer coincide with therepresentability notion. They turn out to be significant classes of predicates,ones we study in Chapter VIII.

7. Representability in 501- Logic 149

7.5 Exercise. Let 9Jί = <M,^1,...,.R/> be a structure for L Let L+ be as in 7.3.i) Assume that we have added a Σ function symbol F to L* for the operation

F(x,y) = xv{y} and a constant symbol 0 for the empty set. Show that eachxeHFgpj is denoted by a closed term tx of L+.

ii) Show that S^ HF^ is Σ1 on HYP^ iff S is weakly representable in KPU +

using the 5Dl-rule.

7.6 Notes. The representability approach to the hyperarithmetic sets goes backto Grzegorczyk, Mostowski and Ryll-Nardzewski [1961].

PartB

The Absolute Theory

"... the central notions of modeltheory are absolute, and absoluteness,unlike cardinality, is a logicalconcept."

G. Sacks, fromSaturated Model Theory

Chapter V

The Recursion Theory of Σx Predicateson Admissible Sets

There are many equivalent definitions of the class of recursive functions on thenatural numbers. Different definitions have different uses while the equivalenceof all the notions provides evidence for Church's thesis, the thesis that the con-cept of recursive function is the most reasonable explication of our intuitivenotion of effectively calculable function.

As the various definitions are lifted to domains other than the integers (e. g.,admissible sets) some of the equivalences break down. This break-down providesus with a laboratory for the study of recursion theory. By studying the notionsin the general setting one sees with a clearer eye the truths behind the resultson the integers.

The most dramatic breakdown results in two competing notions of r.e. onadmissible sets, notions which happen to coincide on countable admissible sets.We refer to these as the syntactic and semantic notions of r.e. and study the formerin this chapter. The semantic notion is discussed in Chapter VIII.

1. Satisfaction and Parametrization

In view of Theorem II.2.3 (which shows that r.e. on ω is just Σ! on IHF ) it isnatural to ask oneself what properties of r.e. and recursive lift up to Σ1 and Δ^on an arbitrary admissible set. Luckily, the more important results, results likeKleene's Enumeration and Second Recursion Theorem, lift to completely arbi-trary admissible sets.

1.1 Definition. Let A be admissible and let R be a relation on A.(i) R is A-r.e. if R is Σ^ on A.

(ii) R is ^-recursive if R is Δ^ on A.(Hi) jR is A-finite if RE A.(iv) A function / with domain and range subsets of A is ^-recursive if its

graph is A-r.e.If A = L(α) then we refer to these notions as α-r.e., en-recursive and α-finite,respectively.

154 V. The Recursion Theory of Σl Predicates on Admissible Sets

As in ordinary ω-recursion theory, a total A-recursive function will have anA-recursive graph.

The first result of ω-recursion theory we want to generalize is Kleene'sEnumeration Theorem.

1.2 Definition. Let S be a collection of n-ary relations on some set X. Let Y X.An n + l-ary relation T on X parametrizes S (with indices from Y) if S consistsof all relations of the form

as e ranges over Y.

1.3 Theorem. Let A = (9JΪ; A,e,...) be an admissible set. There is an A-r.e. rela-tion Tn which parametrizes the class of n-ary A-r.e. relations, with indices from A.

To prove this theorem we make use of our earlier formalization in KPU ofsyntax and semantics. The proof is more important than the theorem itself.

There is a systematic ambiguity which has served us well until now. We havebeen using φ,ψ,... to range over formulas of our metalanguage L* as well asover formulas of formalized languages. We must avoid this confusion in thissection.

Let L* = L(e,...) be fixed and finite. For simplicity we assume L* has onlyrelation symbols. The extension to the general case is sketched in the exercises.We consider L* here as a single sorted language with variables x l 5x2,... andunary symbols U (for "urelement") and S (for "set"). Let /* be some effectivecoding of L* in HF. For basic symbols like R we let ΓFΓ be the set in HF whichnames R. For definiteness we take ι;II =

Γχπ"1 = <0,n>. To each formula φ of L*

there corresponds its formalized version Γφ"1, an element of /*ω^HF, definedby recursion equations

and so forth.Define, in KPU, an operation 9lfl on sets a by: 9lfl is a structure for /* with

universe TC(α) which interprets the symbols of /* as follows:

Symbol InterpretationΓIΓ {plpeTC(α)}ΓSΊ {b\beΎC(a)}Γe^ {<x,);>|x,yeTC(α),xej;}

Clearly 9ϊα is a Σί operation of a. Recall the notation φ(a} from § 1.4.

1. Satisfaction and Parametrization 155

1.4 Lemma. For each formula φ(x^...,xn) of L* the following is a theorem ofKPU: for all sets a and all x l9...,xneTC(α), if s = {<ι; ί,x ί>|z = l,...,n} then

φ^ ,. ..,*„) iff 5lβNV[s].

Proof. For φ atomic this follows from the definition of 9ΐα. The result followsby induction on formulas. D

1.5 Definition. Let Σ-Satπ be the following Σί formula of L* with variables

3>,xι,...,xπ:

"y is a Σ formula of /* with free variables among vl,...,vn and there is atransitive set a with xί9...9xnea such that

where s = «ι; ί,x f>|z = !,...,«}".

That this can be expressed by a £x formula follows from the results in § III.l.

1.6 Proposition. Let φ(xl,...,xn) be aΣ formula of L*. The following is a theoremof KPU: for all xί9...9xn9

φ(xί9...9xj iff Σ-Satn(Γφ(xi,...,xn)~],x1,...,xn).

Proof. Assume the axioms of KPU. The following are equivalent:

3α3s[Tran(«)Λx1,...,xneαΛ^l f l^Γφ"1[5], where s = {<t;ί,xί>|ί = l,...,w}]

Σ-SatnOπ,xl9...,xJ.

The first two lines are equivalent by Σ Reflection, the middle two by Lemma 1.4and the last two by the definition of Σ-Sat,,. D

Define Tn(e,xi,...,xn) to be the Σ^ formula:

66 e is an ordered pair <ι/^,z> and Σ-Satn+1(^,x1,...,xn,z)".

Proof of Theorem 1.3. Since Tn is Σί any predicate defined by

R(xί9...9xJ iff Tn(e9xί9...,xJ

is A-r.e. To prove the converse, let R be an n-ary A-r.e. predicate. By usingordered pairs it has a Σ^ definition on A with exactly one parameter z, say

...,x iff

156 V. The Recursion Theory of Σx Predicates on Admissible Sets

Then let e = O(x1,...,xπ + 1Γ,z> and apply 1.6. D

1.7 Corollary. Let A be admissible. There is an A-r.e. set which is not A-recursive.

Proof. Just as for ω-recursion theory define

If A — K were A-r.e. there would be an e0 such that for all eeA, eφK iff7^(e0,e), and hence e0φK iff e0eK. D

Let 9Jΐ = <M> be an infinite set with no additional relations. Note that ifX^M is HYP^-r.e. then X is HYP^-finite since by II.9.3, X or M-X isfinite. Thus Corollary 1.7 cannot in general be improved to get a A-r.e. subsetof 9K which is not A-recursive.


1.8. Suppose L* = L(e,f,...) has a function symbol f. Show that under the stand-ard treatment of function symbols as relation symbols, Δ0 formulas transforminto both Σ and Π formulas (but not necessarily into Δ0 formulas). Hence Σjformulas transform into Σ formulas.

1.9. Let L* be a finite language with function symbols. Define Σ-Satπ for L* insuch a way that 1.6 and hence 1.3 become provable.

1.10. Find an admissible set A^ such that the class of A^-r.e. subsets of 9JΪcannot be parametrized by an A^-r.e. binary relation with indices from M.

2. The Second Recursion Theorem for KPU

The Second Recursion Theorem in ω-recursion theory is a mysterious devicefor implicitly defining recursive partial functions, or equivalently, r.e. predicates.The theorem is equally mysterious and equally useful in our setting.

Let L* = L(e,...) be a finite language (as in §1) and let R be a new π-aryrelation symbol, n^ί.

2.1 Definition. The collection of R-positίve formulas of L*(R) is the smallest classof formulas containing all formulas of L*, all atomic formulas of L*(R), andclosed under

Λ , v, Vwei;, 3weι;, V w , 3u

2. The Second Recursion Theorem for KPU 157

for all variables u, υ. We use the notation

to indicate that φ is an R-positive formula.Given a formula φ(R) of L*(R) and a formula ψ(xί9...9xj of L* we use the

notations

φ ( λ x ί 9 . . . 9 x n ι / / ( x ί 9 . . . 9 x n ) )

more or less interchangeably to denote the formula resulting by replacingeach occurrence of an atomic formula of the form R(ί1?...,ίπ) in φ(R) byΨ ( t ι / x ί 9 . . . 9 t j x n ) (unless some ίf is not free for xt in ψ in which case we mustfirst rename bound variables in ψ9 but then we agreed in Chapter I not to men-tion such details). Thus x1?...,xn do not occur free in φ(φ/R) (unless they arefree in φ(R)\ and R does not occur in φ(ψ/R).

2.2 Lemma. // φ(R+) is a Σ formula of L*(R) and if φ(xl,...,xn) is a Σ formulaof L* then φ(φ/R) is a Σ formula of L*.

Proof. By induction on the class of R-positive formulas φ(R+). D

2.3 The Second Recursion Theorem. Let φ(x9y9R+) be an R-positive Σ formulawhere R is n-ary, x = xί9...9xn and 3^ = 3^1,...,^. There is a Σ formula ψ(x9y) ofL* so that the following is a theorem of KPU: for all parameters y and all xί9...9xn

ψ(xί9...9xn9y) iff φ(xl9...9xn9y9λxί9...9xn\l/(xl9...9xn9y)).

Proof. To simplify notation we assume n = k = i. Let Θ(x9y9z) be the Σ formula

φ(x9y9λxΣ-Sat3(z9x9y9z)).

Let e = Γθ(x,y,zγeMF and let \l/(x9y) be θ(x,y,e)9 or rather, the Σ formulaequivalent to it obtained by replacing the constant e by a good Σx definition of e.Then we have, in KPU, that the following are equivalent:

θ(x,y,e)

φ(x9y9λxΣ-Sat3(e9x9y9e))

φ(x,y,λxθ(x,y,e))

φ(x,y9λxψ(x9y)). D

158 V. The Recursion Theory of Σί Predicates on Admissible Sets

Since any Σ formula is equivalent, in KPU, to a Σί formula, we could havedemanded that the φ of 2.3 be Σx.

We give a simple application of the Second Recursion Theorem. In anyadmissible set A^, HF is an AOT-recursive subset since

ae HF iff sp(α) = 0 Λ (rk(α) is a natural number) .

HF^, however, is not always Δ: definable. (The student can find an example ofthis in Exercise 2.6.)

2.4 Proposition. There is a Σ1 formula φ(x) such that in any admissible set Am,

Proof. Let R be unary and let <p(x, R + ) be the Σ formula

(x is a finite set) Λ V yex (if y is a set then

Now apply the Second Recursion Theorem to get a formula φ such that

KPU\-φ(x) <-»(x is a finite set Λ V y e x (y is a set-+φ(y)) .

Now let AW be admissible. A trivial proof by induction on e shows that

αeΉFan iff ATOl=^[α]

for all aεAm. D


2.5. Show that a formula φ(R) is logically equivalent to an R-positive formulaiff the result of pushing negations inside φ as far as possible (using de Morgan'slaws) results in a formula in which — i R does not occur.

2.6. Let $R be a recursively saturated model of Peano arithmetic, KP or ZF. Showthat IHF^ is not HYP^-recursive.

3. Recursion Along Well-founded Relations

In this section we use the Second Recursion Theorem to give a new principle ofdefinition by recursion along well-founded relations. This serves as a usefulwarm-up exercise in the use of the Second Recursion Theorem.

3. Recursion Along Well-founded Relations 159

3.1 Theorem. Let A=AS[rί be admissible, let p be an A-recursive function anddefine a binary relation -< by

*<y Iff χep(y)

for all yedom(p).

(i) The well-founded part of •<, i^/«), is A-r.e.(ii) // G is a total k + 2-ary A-recursive function then there is an A-recursive

F with

such that

for all ze(MvA)k and all x

Proof. Recall that Ί^/K) is the largest subset B of Field «) such that:

, yeB implies xeB, and

< Γ B2 is well founded .

There is such a largest set by II.8.2. Note that pred(x)^ yγ«) implies xPart (i) of the theorem follows from part (ii) but we need (i) in the proof of (ii).Besides, (i) is an easy example of the use of the Second Recursion Theorem.

Define a Σί formula ^(x,α) such that

(1) \l/(x,<x) iff 3z(z=p(x)ΛVyεzlβ<aψ(y,β))

is a theorem of KPU and hence true in A. Since this is only our second use ofthe Second Recursion Theorem, perhaps we should be a bit more explicit. Letη(x,z) define the graph of p; η may have some other parameters which remainfixed throughout (the /s of the Second Recursion Theorem). Let R be a newbinary relation symbol and let φ(x,α, R + ) be the Σ formula

of L*(R). Note that R does indeed occur positively in this formula. Now applythe Second Recursion Theorem to get ψ satisfying (1). We will never again bethis explicit; rather we'll just write an equation like (1) and leave it to the readerto see that the right-hand side is of the appropriate form. Now given , one proves,for αeA,

(2) ANι^(x,α) implies X


by a simple induction on α, using (1). A little less trivial is

(3) xeiT/K) implies

Assume x6^/«). Since Wf(O<^ Field «), p(x) is defined. We may assumeby induction (-< \ i^/«) is well founded so induction over it is legitimate) thatfor each yep(x)

and hence

so by Σ Reflection there is an αeA with

Combining (2), (3) we have

= {χeA| AN3αι/φc,α)}

which makes i^/«) an A-r.e. set.To prove (ii) we use the Second Recursion Theorem again. We want to define

the graph of F by a Σί formula ι/^(z,x,w). Let us suppress the parameters sincethey are held fixed throughout. We want

ιA(*,w) iff

iff x

The Second Recursion Theorem gives us a Σ{ ψ so that

^(x,w) iff xe^7(-<)Λ 3/[/ /s α function Λ

is true in A for all x, w. Using Σ Replacement one shows by induction on -< [that

xe^y«) implies Al=3!w^(x,w)

so we may use ^(x, w) as a definition of an A-recursive F. One then checks that Fsatisfies the desired equation, again by induction on •< \ Wf«\ D


3.2 Definition. Let -< be a binary relation with nonempty wellfounded part.Define the ~<-rank function p<,(or xeW/(-<), by

Define the rank of -<, /o«), by

3.3 Corollary. Let A be an admissible set with -< an element of A and α=o(A).(i) p«)<α.

(ii) // ^(-OeA (/or example, if -< is we// founded) then(iii) // ^/K)eA then

Proof. To apply Theorem 3.1 define an A -recursive function p by

(This is the reason we assumed -<eA.) Then the definition

falls under 3.1 (ii) so always gives values in A. This proves (i).

If ^/(-<)6A then we may use Σ Replacement to form

in A. This gives (ii). To prove (iii), suppose p(^<) = βeA, and let us prove. For y<β let

be defined by Σ Recursion for y < β. But then

is in A by Σ Replacement. D

While the most useful results of ω-recursion theory lift to an arbitrary ad-missible set, many of the more pleasing facts of recursion theoretic life on ω carryover only to special admissible sets. In particular, there are many results of re-cursion theory which use the effective well-ordering of the domain in an essentialway.

3.4 Definition. Let A=(9M;,4,e, ...) be an admissible set with α = o(A). A isrecursively listed if there is an A-recursive bijection of α onto M u A.

162 V. The Recursion Theory of Σt Predicates on Admissible Sets

Lemma II.2.4 shows that HF is recursively listed. We will study the recursiontheory of recursively listed admissible sets in the next section. They are relatedto this section by means of the following result.

3.5 Proposition. A is recursively listed iff there is a total A recursίve functionp such that

iff χ

defines a well-ordering •< of M u A.

Proof. Suppose e:α-»A is an A-recursive enumeration of M u A. Note thate~l is A-recursive. Define p(y)={e(β)\β<e~l(y)} and note that xep(y) iff

Now suppose p is given as above. Note that, by 3.1, p< is an A-recursivefunction. Since -< is a linear ordering, p< is one-one so we can let e be the inverseof p<. By Σ Replacement, p< has range α so e has domain α. D

Recall the definition of L(α) given (in KPU) in II. 5.

L(0) = 0

) = y α < λ L(α) for λ a limit ordinal

where

We have shown that if α is admissible then L(α) is the smallest admissible set Awith o(A) = α. There is a natural well-ordering of L(α) given by putting everythingin L(β) before everything in L(δ) for β<δ and ordering the elements a of L(/? + l)— L(jβ) according to which ^t(x,y) = a. To make this precise define, in KPU,a Σ! formula ψ(x,y\ which we write as x<Ly, as follows. First let

F(x) = the least α (x e L(oc + 1 ))

O if xe^(L(F(x))),

G(χ) =< the least i, i^i^N , such that x = JΓ

ί(z1,z2) for some

z 1? z2 e 5^(L(F(x))) otherwise .

2 > < α < W l > W 2 > ΐf

^LWi or

x e L(α) Λ w t = L(α) or

1 = w1 and,

Z 2 <L W 2 OΓ

z2eL(α)Λ w2 =


Given that <L already wellorders L(α), <α wellorders the pairs zl9z2 fromy (L(α)) = L(α) u (L(α)} "lexicographically after putting L(α) itself at the end of thealphabet".

Now define < L by

x< L y if xeL and yeL and F(x)<F(y) or

F(x) = F(y)ΛG(x)<G(y) or

F(x) = F(y) Λ G(x) = G(y) and there is a pair

zl9z2e^(L(F(x))) such that x = #Γ

G(jc)(z1,z2)

but for all w1,w2e^(L(F(x))), if .y = G(x)(wι>w2)

then <z1,z2><F ( x )<w1,w2>.

We could define <L explicitly, if we really had to, but for our purposes here wecan be content to use any such formula given by the Second Recursion Theorem.To see that the Second Recursion Theorem applies we need only observe that,once <α is replaced by its definition, the right-hand side is a Σ formula and that< L occurs positively.

3.6 Lemma (of KPU). For each α, <LfL(α) x L(α) well orders L(α) in such a waythat for β < γ < α, if xe L(β\ y e L(y) - L(β) then x < Ly.

Proof. By induction on α. D

3.7 Theorem. If α is an admissible ordinal then L(α) is a recursively listed admissibleset.

Proof. Since, for x,yeL, ~ι(x<Ly) iff x = yvy<Lx, we see that < L i s Δ 1 w h e nrestricted to L. Also we can define

= {yεL\y<Lx}

for all xeL(α) so p is α-recursive, and

x<Ly iff xep(y)

so we may apply Proposition 3.5. D


3.8. An admissible set Am is resolvable if there is an A^-recursive function /with άom(f) = o(Am) such that Am = (Jrng(f).

i) Show that if A^ is resolvable then there is a function / with the aboveproperties which also satisfies : /(/?) is always transitive and β < y implies f(β)Such an / is a resolution of A^.


ii) A well-founded relation -< is a pre-wellordering if for all x, ye Field (~<\

p<(x)<ρ<(y) implies

Show that an admissible set A^ is resolvable iff there is a total A^-recursivefunction p with Aan = yrng(p) such that

iff xep(y)

defines a pre-wellordering of A^.

3.9. Show that every admissible set of the form L(α,α) is resolvable. In particular,is resolvable.

3.10. Let L(α,α) be admissible and assume that there is a well-ordering -< of a,-< an element of L(α,α). Modify the definition of <L to show that L(α,α) is recur-sively listed. In particular, if Λ^= <ω, +, x > and if L(α)^ is admissible then itis recursively listed. Hence HYP^ is recursively listed.

3.11. Let A be admissible, <eA, -< not well-founded but

<is well founded" .

(In other words, every subset X of Field(-<) which happens to be an element of Ahas a -<- minimal element.) Show that

4. Recursively Listed Admissible Sets

In this section we show how the elementary parts of the theory of r. e. sets gen-eralize from ω-recursion theory to any recursively listed admissible set.

4.1 Theorem. Let A=Aα n be a recursively listed admissible set, with α = o(A),and let B be a nonempty subset o/A. The following are equivalent:

(i) B is A-r.e.(ii) B is the range of a total Ik-recursive function.

(iii) B is the range of an A-recursive function with domain α.

Proof. We have (iii)=>(ii) since there is an A-recursive bijection e of α ontoM u A. Clearly (ii) => (i) so we prove (i) => (iii). Let

iff

4. Recursively Listed Admissible Sets 16 J

where φ is Δ0. Fix x0eB. Define an A-recursive / by

f(β) = ί*e(β) if φ(lsie(βl2nde(β))

= x0 otherwise.

Then B = mg(f) and α = dom(/). D

4.2 Reduction Theorem. Let A=Am be a recursively listed admissible set. Forany pair B,C of A-r.e. sets there is a pair B0,C0 of disjoint A-r.e. sets with B0^B,C0<ΞC and #0

Proof. We may assume B and C are nonempty. Use 4.1 to choose A-recursivefunctions F, G with domain o(A) such that

B = rng(F), C = rng(G).

Define B0 and C0 by:

Q iff

0 iff

Then clearly B0 and C0 are disjoint A-r.e. sets with B0^B, C0^C. If xeB — Cthen xeB0. If xεC-B then xeC0. If xeBnC then let j8 be the least ordinalwith F(β) = x, y the least with G(y) = x. If β^y then xe50 but if β>y then

so βuC^β 0uC 0. D

4.3 Corollary (Separation Theorem). Let A=Am be a recursively listed ad-missible set. For any pair B,C of disjoint H^ sets on A there is an A-recursive setcontaining B but disjoint from C.

Proof. Apply 4.2 to A-£, A-C to get disjoint sets B0,C0 with B^B0, C^C0,BQ vC0 = A. Then B0 is A-recursive. D

4.4 Uniformization Theorem. Let A=Ayjl be a recursively listed admissibleset and let R be an A-r.e. binary relation. There is an A-recursive function F with

(i) άom(F) = {x\lyR(x,y)}(ii) for xedom(F),

R(x,F(x)).

Proof. Let e be an A-recursive bijection of o(A) onto A. Let R be given by

R(x,y) iff 3zS(x,y,z)

166 V. The Recursion Theory of Σj Predicates on Admissible Sets

where S is A-recursive. Define F by

F(x) = y iff 3jS[S(x,lst^),2nde(j?))ΛV7<j8nS(x,

y = l«e(fi]. D

The passage, in 4.4, from the Σί definition of R to the Σx definition of F wasexplicitly given, so we can get the following more complicated but stronger result.For zeA we let

W2

z={(x9y)\T2(z9x,y)}

where T2 is the A-r.e. relation which parametrizes the A-r.e. binary relations,as it was defined in § 1.

4.5 Theorem. Let A be a recursively listed admissible set. There is a total A-re-cursive function G such that for all zeA:

(i) WQ(Z) is the graph of an A-recursive function,(ii) W2

G(z}^Wl and(iii)

Proof. See 4.4 and remarks following it. D

Using this we get the following analogue of Kleene's Γ-predicate for recursivepartial functions.

4.6 Theorem. Let A be a recursively listed admissible set. There is an A-r.e.predicate T* of three arguments which parametrizes the collection of all partialA-recursive functions, with indices from the ordinals of A.

Proof. Let e: o(A)-»A be a recursive listing and let G be as given in 4.5. Define

Tϊ(β,x,y) iff T2(G(e(β)\x,y).

Then for each β,

fβ={<χ,y>\τmχ,y)}

is a partial function with Σl graph (by 4.4i). If /= W2

Z then pick β so that e(β) = z.Then since

by4.4,/,=/. D

4.7 Corollary. Let A be a recursively listed admissible set. There are disjointA-r. e. sets which cannot be separated by an A-recursive set.

4. Recursively Listed Admissible Sets 167

Proof. Let B,C be the disjoint A-r.e. sets defined by

B = {β\TΪ(β,β,Q)}

C = {β\TΪ(β,β,ί)}

where Γf is given in Theorem 4.6. Suppose D were an A-recursive set with B^D,= Q. Let

g(x) = ί if

= 0 if xφD

so that g is A-recursive. Pick β so that

g(x) = y iff

If jβeD then g(β) = ί so Tf(ftft l) which implies βeC, but CnD = 0. Ifthen 008) = 0 so Tf (β, ft 0) which implies jSeβ, but B^D. But βeD or βφDso we have a contradiction in either case. Thus there can be no such D. D

It is an open problem to determine whether the conclusion of 4.7 holds forarbitrary admissible sets.


4.8. Let SD ίί=<M,Λ1,...,JR/> be countable and suppose there is a well-orderingof M which is Δ} on 9K. Prove the following:

(i) Let B be a Π} subset of 9M. There is a function F with domainsuch that

and for each β<o(ΉYPm)

is Δ} on M. [Pick an F which is HYP^ recursive.](ii) (Reduction) If B,C are Π} subsets of SDΐ then there are disjoint Πj subsets

B0^B, CQ^C with £ 0uC 0=£uC.(iii) (Separation) If J5, C are disjoint Σ} subsets of 301 then there is a Δ} set D with

(iv) (Uniformization) If R^MxM is Π} on 9W there is a Π} subrelationQ^R such that


If άom(R) = M then RQ is a Δj relation.

4.9. Show that for any admissible Am and any B c A^, B is A^-r. e. iff B = dom(f)for some A^-recursive function /.

4.10. Show that if A^ is resolvable then the Reduction and Separation Theorems,4.2 and 4.3, still hold. In particular, show that 4.8(i), (ii), (iii) hold without thehypothesis that 9Jί has a Δ} well-ordering.

5. Notation Systemsand Projections of Recursion Theory

An important stimulus in the earlier development of admissible ordinals was thedesire to understand the analogy between Πj and r.e. sets of natural numbers.The metarecursion theory of Kreisel-Sacks [1965] explained this by developinga recursion theory on coj, the first nonrecursive ordinal, with the property thata set of natural numbers in Π} on ω iff it is ω^-r.e. The theory was developed byusing a notation system for the recursive ordinals to define the notions of ω\ -re-cursive, ωc

rr.e. and ω\ -finite.The development by means of admissible sets proceeds the other way around.

Instead of using known facts about Π} sets to develop a recursion theory on ω\by means of a notation system, we have a recursion theory given on ω\ (it is thefirst admissible ordinal >ω; see 5.11) and then transfer the results to Π} subsetsof ω via a notation system.

5.1 Definition. Let A=Am be admissible.

(i) A notation system for A is a total A-recursive function π such that ifx^y then π(x) and π(y) are disjoint non-empty sets. (We think of π(x)as a set of notations for x.)

(ii) The domain of a notation system π, Dπ, is defined by (!)

Oπ = \JxeA π(x). (Thus Dπ is the set of all notations.)

(iii) Associated with a notation system π is a function | |π with domain Dπ

and range A u M defined by

\y\π = x iff yεπ(x).

(Thus, for any notation y, y is a notation for \y\π.)(iv) A is projectible into C if C is A-r.e. and there is a notation system π with

5. Notation Systems and Projections of Recursion Theory 169

It is best to think of the notation system as the triple Dπ, | |π, π even though thefirst two can be defined in terms of the third. We require C to be A-r.e. in (iv) onlybecause that is the only kind of C that interests us in this context.

5.2 Lemma. Let π be a notation system for the admissible set A.

(i) π is a one-one function.(ii) Dπ is A-r. e. but not A-finite.

(iii) The graph of | |π is an A-recursive relation. In particular, \ \π is an Ik-re-cursive function.

Proof. The only part which is not absolutely immediate is the fact that Dπ is notA-finite. But if DπeA then, by Σ Replacement, the range of | |π would be an ele-ment of A whereas this range is all of Mu A. D

Our plan for this section is to first exhibit some useful notation systems and thenuse them to transfer results.

5.3 Theorem.

(i) For any structure 90Ϊ, HYP^ is projectible into HF^.(ii) For any admissible set A, HYP(A) is projectible into A.

The theorem is a simple consequence of the following lemma, an effectiveversion of Theorem II.5.14.

5.4 Lemma. Let L be a finite language, let 901 be a structure for L, let L* = L(e)and let aeWm be a transitive set with M^a. Let L'= L*u {x | xeαu {a}} be theusual language with constant symbol x for x. Let α be the least ordinal such that

is admissible and assume L is coded up on A^ in a way that makes the syntacticoperations of Lωω all A^-recursίve. There is a total Am-recursive function π suchthat for each xeA^, π(x) is a set of good Σl definitions of x with parameters fromau {a}.

Proof. We already know, from Theorem Π.5.14, that each xeAm has a good Σl

definition with parameters from a\j{a}. The object here is to use the SecondRecursion Theorem to show how we can go A^-recursively from x to a set π(x)of good Σ! definitions of x, by reexamining the proof of II.5.14. If we look backat that proof we see that this is really pretty obvious. We write out clauses in thedefinition of π. In each case it is assumed that none of the earlier cases hold.We also arrange things so that v is the only free variable in any formula considered.

Caselne. If xeau{a} then π(x) is the set whose only member is the L'ωω

Δφ formula

v=x.


Case 2 wo. If x = β + 1 then π(x) is the set of formulas

3w [υ = <f(w) Λ φ(w/ι;)]

where φ(v)Eπ(β) and w is the first variable not in φ(v).

Case 3hree. If π(β) is defined then π(L(a,β)) is the set of formulas of the form

3w [υ — L(a", w) Λ φ(w/v)~\

where φ(v)eπ(β). We may use "t; = L(1Γ, w)" since L( , ) is a Σx operation symbol.

Case4our. If xeL(α,β + l) — £f(L(a,β)) then π(x) is defined as follows. Findthe least ί, 1^/^Λf, such that for some y,zeL(α,/J)u (L(α,β)},

Then π(x) is the set of all formulas of the form

3wt 3w2[ι; = «^'ί(w1, w2) Λ

where, for some y,ze^(L(a,β)\ x = #r

i(y,z) and φ(v)eπ(y) and ^(ι )eπ(z) andw l 5 w 2 are the first two distinct variables not appearing anywhere in φ or ψ. Theset of all such formulas exists by Σ Replacement. This clause in the definition ofπ(x) = y is Σ, as can be seen by writing it out.

Case5ive. If β<a is a limit ordinal then π(β) is defined A^-effectively asfollows. Find the first Δ0 formula φ(x,y,zί, ...,zn) of L* (first in some effectivewell-ordering of HF, say that given by Π.2.4 or 3.7 of this chapter) such that forsome rf,z1,...,zΠeL(α,j8)

(1)

but

(2)

Now given φ let Θ(β)( = θ(β,d,zί9...9zj) be formed from φ just as in the proofof II.5.14. Let π(β) be the set of all formulas of the form

3w, w1? . . . , wn[θ(ι;, w, w1? . . . , wn) Λ ^(w/ϋ) Λ /\J= x σ/w/i;)]

such that for some rf,zl5...,zneL(0,/?), (1) and (2) hold and ψeπ(d) and, forl^j^H, σ^i JeπίZj ). Again, this clause in the definition of π(x) = y can be seento be Σ and so, by the Second Recursion Theorem, π is an A^-recursive func-tion. D

5. Notation Systems and Projections of Recursion Theory 171

Proof of 5.3. For (i) simply note that Lωω can be coded up on HF^ in this case.For (ii) we can code L'ωω on Aw itself. The admissibility of A^ comes in onlyin that this coding can be done on Am and is far stronger than we need. D

We will see in §VI.4 that if 501 has a "built in pairing function" thenis projectible into SR.

5.5 Corollary. Let yΓ = <ω, +,•> be the structure of the natural numbers.is projectible into Ji* .

Proof. The simplest proof is just to observe that in this case the coding used inthe proof of 5.3 (i) can be done on yΓ itself. An alternate explicit proof will appearin §VL4. D

We now give some examples of the use of notation systems. Combined with5.5 and the results of §IV.3, the next two results show that, over Jf, the Π} rela-tions are parameterized by a Π} relation, that there are Πj sets which are notΔj, and that there are Δ{ sets which are not first order definable over ΛΛ

5.6 Theorem. Let A be an admissible set which is projectible into C.(i) For n^i there is an (n + l)-ary A-r.e. relation S on C which parametrizes

the class of all n-ary relations on C which are A-r.e.(ii) There is subset of C which is A-r. e. but not Ik-recursive.

Proof, (ii) follows from (i) just as in the proof of 1.7. To prove (i) let π be a notationsystem for A with Dπ^C. Let Tn be the (rc + l)-ary relation on A which para-metrizes the n-ary A-r. e. relations. Define

S(y,xί9...,xn) iff xί9...9xneC9 yεDκ and

S is A-r.e. since C and Dπ are A-r.e. and | |π is A-recursive. Now let R^Cn beA-r. e. Pick a z such that

«(*!,. ..,*„) iff Tα(z,xl9...,xJ.

Then for any yeπ(z)9

R(xl9...,xJ iff %,*!,...,*„). D

5.7 Theorem. Let A be an admissible set with o(A)>ω. Let 91 = <7V,...> be astructure (for a language K) which is an element of A and suppose that A is projec-tible into N.

(i) There is an A-recursive (n + i)-ary relation S on N which parametrizesthe n-ary relations on 91 which are first order definable over 91 (using parameters).

(ii) There is a subset of 91 which is A-recursive but not first order definableover 91.


Proof. As usual (ii) follows from (i) by diagonalization. To prove (i) define

SoCy>Xι» - >xn) iff y = (φ>sy where φ(vί9 ..., vn,wί9 ..., wm) is a formula ofKωω and 5 is an assignment with values in 91, 5(1;.) = . all iXn, and

S0 is clearly Δ t on A. Since o(A)>ω the set Jf of all relevant pairs <φ,s> is anelement of A. Let π be the notation system for A with Dπ^N. Define

S(z,x1?. ..,*„) iff

Since XeA, the quantifier on y is bounded so S is indeed A-recursive. It clearlyparametrizes the relations definable over 9ί. D

We now turn to a result, Theorem 5.9, which will allow us to identifyA notation system π is unίvalent if each π(x) is a singleton, that is, if it assigns aunique notation to each xeA.

5.8 Proposition, (i) Let A be a recursively listed admissible set projectible into C.There is a univalent notation system which projects A into C.

(ii) HYP^ has a univalent notation system which projects into Jf .

Proof, (i) If π projects A into C then define π l 5 the univalent notation system, by

πιM = {y} where y is the first member of π(x) .

Part (ii) follows from (i) and 3.10. D

5.9 Theorem. Let A be an admissible set which is projectible into C.

I -< is a well-founded relation, -< c C2, -< e A}ί5 β pre-wellordering, -< c C2, -< eA} .

// ίfeere is α univalent notation system projecting A wίo C f/iew

a well-ordering, <^C2,

Proof. Every well-founded relation -<eA has p(-<)<o(A) by 3.3 (ii) so we needonly show that each jSeA is of the form p(<) for some pre-wellordering•< C2. Let π be a notation system projecting A into C. Let b = \Jrng(π\Now b^Dπ^C and bis the set of all notations for ordinals y<β. Define <<^bxbby

iff W π <!y | π .

Then •< is a pre-wellordering of fc of length β and it is a well-ordering if π happensto be univalent. D

6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals 173

5.10 Corollary. O(Jf} = {p«}\< is a Δ} well-ordering, <^N

Proof. The first equality is immediate 5.9, 5.10 and §IV.3. The second followsfrom the first and the result from ordinary recursion theory that every Δ j well-ordering of Jf has order type some α<ωc

t. D

The reader unfamiliar with the result used in the above proof can take

ωι — \P«}\< is a A} well-ordering, •< <^Jf x JV*}

as the definition of ω\.

5.11 Corollary. ω\ is the first admissible ordinal greater than ω.

Proof. ω\ is admissible by 5.10. Let α be the least admissible >ω so that L(α)is admissible and ω\ ^α. But if ω\ >α then α is the order type of some Δ} well-ordering < of yΓ and hence of some Δ} well-ordering -< of ω. But then -<eL(α)by §IV.3 which contradicts 3.3 (ii). D


5.12. For any 9W = <M,Λ 1,...,R /> show that

0(2)1) = {ρ«)\< is a pre-wellordering, <eHYPOT, -<^IHF^}.

5.13. Let A be a recursively listed admissible set. Show that there is a single-valued notation system with domain o(A). Hence the recursion theory of A canbe transfered to o(A).

5.14 Notes. Notation systems are standard tools in ordinal recursion theorybut don't seem to have been treated systematically before over arbitrary admis-sible sets. The definitions used above are stronger than those of Moschovakis[1974]. In the case where A is projectible into some CeA (the only case ofinterest to Moschovakis) they are equivalent.

Corollary 5.11 is due to Kripke and Platek, but with more complicated proofs.

6. Ordinal Recursion Theory:Projectible and Recursively Inaccessible Ordinals

In the final sections of this chapter we return to the origins of the theory of ad-missible sets, recursion theory on admissible ordinals. We are thus in the domainof admissible sets without urelements.


Let τβ be the βih admissible ordinal; that is, let

τ0=co,

τβ = least α [α is admissible Λ α > τy for all y < /?] .

In this section we begin looking at the sequence of admissible ordinals and therelationships between various members of it.

6.1 Definition. An admissible ordinal α is projectible into β (where β^α) if thereis a total α-recursive function mapping α one-one into β. The least β such that αis projectible into β is called the projectum of α and is denoted by α*. If α*<αthen α is said to be projectible; otherwise α is nonproj edible.

If α is admissible then L(α) is recursively listed so we see that α is projectibleinto β in the sense of 6.1 iff L(α) is projectible into β in the sense of 5.1 (iv). Similarly,if β is also admissible then α is projectible into β (in the sense of 6.1) iff L(α) isprojectible into L(β) (in the sense of 5.1(iv)).

6.2 Proposition

(i) If κ^ω is a cardinal then K is nonproj edible.(ii) For any β,τβ + ± is projectible into τβ.

(iii) // τβ is projectible into τy nnd τy is projectible into δ then τβ is projectibleinto δ.

Proof, (i) is obvious by cardinality considerations since otherwise K would havethe same cardinality as some β<κ. For (ii), note that L(τβ+i) = ΉYP(L(τβ))so L(τ0+ J is projectible into L(τ^) by 5.3. Part (iii) is obvious. We simply composeprojections. D

From this proposition we see that there are many projectible ordinals. Wealso see that τ*=ω for all rc = 0,l,2, ... .

As we mentioned at the beginning of this chapter, one use of generalizedrecursion theory is as a laboratory for understanding ordinary recursion theory.One important aspect of ordinary recursion theory is the number of differentversions of the notion of finite that arise. For examples, a set B^ω is finite iffany one of the following hold: B is recursive and bounded, B is R.E. and bounded,or B is bounded. By defining a set £^L(α) to be α-finite if £eL(α) we havechosen to use the first. This means that when we meet some use of a differentversion of "finite" in ordinary recursion theory we may have trouble lifting thisto α-recursion theory. The following theorem shows us that if α is projectiblethen there are going to be α-r.e. subsets of ordinals β<α which are not α-finite.Thus, for projectible ordinals we may expect some aspects of ordinary recursiontheory to become more subtle. This is particularly true in the study of α-degrees,a subject not treated in this book.

6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals 175

6.3 Theorem. Let α be admissible. The following are equivalent:(i) α is nonprojectίble.

(ii) L(oc)\=Σ1 Separation.(iii) // β <α and B is an α-r. e. subset of β then B is a- finite.

Proof. We first prove (i)=>(ii). Suppose B^aεLfa) and B is Σj definable on L(α).We wish to prove BeL(α). Pick β0<oί such that αeL(j80); hence B<^L(β0).The recursive listing / of L(α) given by < L puts everything in L(/?0) before every-thing in L(α) — L(/?0). If we show that the set

C = {γ\f(y)εB}

is an element of L(α), then £eL(α) by Σ Replacement. But C^β^ for someβl<a. Use 4.1 to pick an α-recursive function G mapping α onto C and defineH by Σ Recursion as follows :

H(β) = G(least y [G(y) φ (H(δ) \ δ < /?}]) .

Now H is α-recursive, one-one, and is defined on some initial segment of α. Itcannot be defined for all β<α, however, for this would give a projection of αinto βι<(x and α is nonprojectible. Let β2 be the least ordinal for which H is notdefined. The only reason H(β2) can be undefined is that

C = {H(β)\β<β2}

so that CeL(α) by Σ Replacement.The implication (ii)=>(iii) is trivial. We prove (iii)=>(i) by contraposition.

Thus, let p: α-»β be an α-recursive one-one mapping of α into β, β«x, and letB = rng(p). Then B is α-r.e., B^β but B cannot be α-finite, since

and p"1 is α-recursive. D

6.4 Corollary. // α is projectible into β then there is an α-r. e. subset of β which isnot α- finite. D

6.5 Corollary. // α is nonprojectible then L(u)\=Beta.

Proof. L(α)l=Σ1 Separation, and Σ1 Separation implies Beta. D

6.6 Corollary. Let K be an uncountable cardinal. For every β<κ there is a non-projectible α between β and K.

Proof. L(κ)^=Σ1 Separation, so apply Theorem II.3.3 with Am

The resulting admissible set satisfies the axiom V = L (i. e. Vx L(x)) and so is L(α)for some α<τc. Since L(α)ΞL(κ;), L(oί)\=Σ1 Separation and hence α is non-projectible. D


Now that we know there are lots of nonprojectible ordinals we can ask howbig the first one is. So far, all we know is that it is bigger than τn for each n<ω.Is it τω? To shed some light on the size of the first nonprojectible we introduce therecursively inaccessible ordinals.

6.7 Definition. An admissible ordinal α is recursively inaccessible if α is the leastupper bound of all admissibles less than α.

6.8 Theorem. // α is nonprojectible and greater than ω then α is recursively in-accessible.

Proof. Assume that α is admissible, α>ω but that the ordinal

β = sup (y < α I y is admissible}

is less than α. We will prove that α is projectible into β. Let e: α->L(α) be the re-cursive listing of L(oc) given by <L. Since β is a sup of admissible ordinals, e\ βis the canonical listing of L(β) by ordinals < β. Thus, if L(α) were projectible intoL(β), then it would be projectible into β and so α would be projectible with α* β.But L(α) is the smallest admissible set with L(β) as an element, i. e. L(α) = HYP(L(β))so L(α) is projectible into L(β) by Lemma 5.4. D

If we combine Theorem 6.8 with the next result we see that the first non-projectible is fairly large, much larger than τω.

6.9 Theorem. // τα is recursively inaccessible then τα = α, and conversely.

We isolate part of the proof of 6.9 which will be used again.

6.10 Lemma. Define G(β) = τβ for β<a. Then G is a τ^-recursive function.

Proof. The result is literally trivial if α = 0. For α>0 we can define G by

G(0)=ω,

for j8<α. Since KP is an ω-recursive set of axioms, it is in L(τα) so this is a Σ Re-cursive definition of G. D

Proof of 6.9. Note first that τα^α for all α, by induction. Suppose τα = α. Thenfor each β < τα, β τβ < τα so τα is the sup of all smaller admissibles. Now supposeτα is recursively inaccessible, but that τα > α. Note that α is a limit ordinal, sinceτβ+i can never be recursively inaccessible. Let G be as in Lemma 6.10 and observethat

7. Ordinal Recursion Theory: Stability 177

But this is a contradiction, for G is τα-recursive and hence the right-hand side ofthis equality is in L(τα) by Σ Replacement. D

We see, by 6.9, that none of the following are recursively inaccesible and,hence, all are projectible:

What are their projectums? We will show in the next section that all are projectibleinto ω by showing that projectums are always admissible.

The interest in projectums stems largely from the following property whichis quite useful in priority arguments involving α-degrees.

6.11 Theorem. Let α be admissible and let α* be its projectum. If B is a-r.e., B^βfor some β<α*, then B is a- finite.

Proof. The proof is like the proof of (i) => (ii) in Theorem 6.3. Define an α-recursivefunction F by

.e.

F(y) = yίh member of B

F(γ) = G (least δ(G(δ) φ (F(ξ) \ ξ < γ } ) )

where G maps α onto B. Now, since β < α*, F cannot be a one-one mapping of αinto β. Thus F(y) is undefined for some y<α. If y0 is the least such thenB = {F(γ):γ<γ0} so £eL(α) by Σ Replacement. D

To prove stronger facts about nonprojectible ordinals we need to use thenotion of stable ordinal introduced in the next section.

6.12 Exercise. Let α be a limit of admissibles. Prove that L(α)t=Beta even ifα is not admissible. This is an improvement of 6.5.

6.13 Notes. The concepts and results of this section are all due to Kripke andPlatek. The student interested in the uses of the projectum in the study of α-degreesshould consult Simpson's excellent survey article, Simpson [1974].

7. Ordinal Recursion Theory: Stability

Given structures land every x !,..., x

^ , we write

iff

if fore very Σ t formula < p ( v i 9 . . . , v n )


7.1 Definition. An ordinal α is stable if L^-^L. The sequence of stable ordinalsis defined by

σ0 = the least stable ordinal,

σy =the least stable ordinal greater than each σβ for β<y .

The first theorem shows that there are lots of stable ordinals and that they arebetter behaved under sups than the admissible ordinals.

7.2 Theorem, (i) If λ>0 is a limit ordinal then σλ = sup{σβ\β<λ}.(ii) Every uncountable cardinal is stable.

(iii) // ω β < K, where K is a cardinal, then there is a stable ordinal α, β < α < K.(iv) // K is a cardinal then κ = σκ.

Proof. To prove (i) let λ>0 be a limit. Since σλ^sup{σβ\β<λ} by definition,it suffices to prove that the ordinal

y = sup{σβ\β<λ}

is stable. Let φ be a Σl formula, let x l 5 ..., xneL(y) and suppose

Pick β<λ such that xl9...,xneL(σβ). Then l^(σβ)\=φ[_xl,...9xn\ by stability,and then L(y)l=<p[x1,...,xπ] by persistence of Σ: formulas. (What we are reallyproving here is that the union 9Ϊ = UJJ<A9Ϊ^ of a chain of -<: -extensions 31^ is a-< ! -extension of each 21 .)

Now let κ>ω be a cardinal and suppose that xί9 ..., xneL(κ) and that

Ll= 3) [>!,..., xj

where ψ is Δ0. We need to see that L(κ ) satisfies the same formula. But, for largeenough cardinal A,

so, by Π.3.5,

and so there is an α</c such that L(α)t=3^^[xl5 ..., xn~] and henceI4κ)t=3yιl/\_xί9 ...,xw], as desired.

To prove (iii) we apply Theorem II.3.3. Note that we need only prove the resultfor K regular since every singular K is a limit of regular cardinals. Let a0 =Given απ apply II.3.3 to get an admissible set B such that

card(απ) = card(B) < K ,

L(/c)Nφ[3c] iff


for every formula φ and every x1?...,xπ6L(απ). Now since lB = L(κ\ B = L(γ)for some admissible y<κ and we let απ + 1 be this y. Let α = supπ<ω απ<τc. Weclaim that α is stable; i.e., that L(α)-<1L. It suffices to prove that L(α)-<1L(κ;)since K is stable. Let φ be Σ1 and L(κ)N=φ[Λ:1,...,xπ], where x1,...,xπeL(α).Pick k<ω so that x1 ?..., xπeL(αk). Then Lία^+^Nφ^j,...,^] by choiceof α f c + 1 and then L(a)t=(p[x l5..., xj by persistence of Σx formulas.

Part (iv) follows from (i) and (iii). In fact, if / is any continuous increasingfunction on the ordinals such that for all cardinals κ>ω, /(α)<κ; implies/(α + l)<κ;, one always has for all κ>ω, f(κ) = κ. First assume K is regularand consider the set B of β such that f(β) <κ. Bis an initial segment of the ordinalsand has no largest element so B is a limit ordinal λ. But then, by continuity,

but f(λ)<£κ since λ φB so f(λ) = κ. Since /c is regular, λ = κ. Now for singularK the result follows by continuity since every singular K is the sup of regularcardinals. For if κ = supβ<yλβ, where the λβ are regular, then f(κ) = supβ<γf(λβ)

λ = κ. D

There is a useful relative notion of stability.

7.3 Definition. An ordinal α is β-stable if α^β and

Since we have allowed a = β there is always at least one jS-stable ordinal.

7.4 Proposition, (i) // α^jδ^y flnd α is γ-stable then α is β-stable.(ii) // α is β-stable and β is γ-stable then α is y-stable.

(iii) // β is stable and a<β then α is stable iff α is β-stable.(iv) // B is a nonempty set of β-stable ordinals and α = supJ5 then α is β-stable.

Proof. These are all simple consequences of the definition and the persistenceof Σ! formulas. D

7.5 Theorem. // a<β and α is β-stable then α is admissible. In particular, everystable ordinal is admissible.

Proof. Suppose α</J and l.(y)^^(β). Note that since the operations ^ ...9&N

all have Δ0 graphs, and for x,yeL(α),

we have

180 V. The Recursion Theory of Σ t Predicates on Admissible Sets

so L(α) is closed under the operations J^, ..., <FN. Thus L(α), in addition to beingtransitive, is closed under pair and union and satisfies Δ0 separation. It remains tocheck Δ0 Collection. Suppose

L(α) N Vx E a 3y φ(x, y, z)

where φ is Δ0 and α,zeL(α). Then, letting 5 = L(α)eL(β), we have

U(β)\=Vxea3yebφ(x9y,z)

soEb φ(x, y, z)

and so, by L^

y,z). D

7.6 Corollary, (i) // β is admissible, α<β and α is β-stable then α is recursivelyinaccessible.

(ii) Every stable ordinal is recursively inaccessible.

Proof, (i) Let β = τy and u = τδ where δ^a and δ<γ. We need to see thatδ = x. Suppose (5<α. Then L(/?)N=3x[x = τJ, so, by Lemma 6. 10, and L^^L^S),L(α)t= 3x[x = τj (one needs to observe that no parameters occur in the definitionof G in 6.10) from which we have τδeL(α), which is ridiculous since a = τδ andα<£L(α). Part (ii) follows from (i). D

The definition of α is β-stable appears to be model theoretic until one reformu-lates it as follows: If / is a ^-recursive function then whenever x l 5 ...,xπeL(α),if /(x1? ...,xj is defined then /(x1? ...,xπ)eL(α). This reformulation suggestsa way of generating the β-stable and the stable ordinals. First, however, a lemma.

Notice that we did not assume 21 end S^ in the definition of $lan<ι3?9i.We are going to apply the notion to a case where we do not know, ahead of time,that this holds.

7.7 Lemma (Tarski Criterion for ^J. If 9ITOc$Bw then &&<&* iff the fol-lowing condition holds for every Δ0 formula φ(v1,...,vn) and every xί9 . . . , x Λ _ 1

then there is an xnetyim such that

Proof. ^ίαu-<ιSί« clearly implies the condition. To prove the converse, one firstuses the criterion to prove

«»l=^[x1,...,xB] iff 93<^<AI>ι?. •-,*„]


for all Δ0 formulas ψ and all *!,..., xΛ62IOT, by induction on ψ. The atomiccases hold by Wm c 33 , the propositional connectives take care of themsevlesand the criterion gets us past bounded quantifiers. Now suppose 3vnφ(vί9 ...,vn)is a Σ! formula, xί9 ...,xn_^G^im. If <St9Ji^=3vnφ[xl9...9xn,ί] then there is anxnεMm such that Mmt=<p[xi9...,xn-i9xn~] so &9iϊ=φ[x1,...,xn-ί,xn]9 since φis Δ0, and hence

The proof of the converse first uses the criterion to pick xne*Άm and then appliesthe result for Δ0 formulas. D

We now come to the main theorem on the generation of stable ordinals. Theproof is rather amusing since we use the Collapsing Lemma to collapse a set thatis already transitive.

7.8 Theorem. Let β be an admissible ordinal and let Q^γ<β. Let A be the set ofthose aeL(β) for which there is a Σ1 definition of a in L(β) using parameters <y(Σj definable as elements in the sense of II. 5. 13). Let α be the least ordinal not in A.Then

(i) A = L(ct\ and(ii) α is the least β-stable ordinal ^ y.

Proof. It is not transparent that A is even transitive, let alone admissible. The firststep in the proof is to show

(1) (Λen^XXLlAe).

We use the Tarski Criterion. Suppose L(β)\=^yφ[a^ . ..,#„], where aί9...,aneA.We need to find a beA such that L(β)\=φ\_a^...,an,b~].

Since each ateA is Σ: definable by a formula with parameters <y, we mayreplace each ai by its definition and assume all the parameters are ordinals < y,except that φ now becomes Σί instead of Δ0. Write φ as 3z\l/(vί9 ..., vm9y,z), sothat L(β) \= 3y 3z ψ( λ ί9...9 λm,y,z\ where λί,...,λm<γ. Let b = l*\c) where cis the least pair <y,z> in L(β) (least under the ordering <L) such thatL(β)\=\//(λl9...9λm,y,z). Then b is Σ{ definable in L(β) with parameters λί9...,λm

so beA and L(β)\=φ(a1,...,an,b). Which proves (1).Let B = clpse(,4) so that B is transitive, and

Let τ be the least ordinal not in B and note that τ^β since there is an embeddingof τ into β. We claim that

(2) βci


The predicate (of x and δ)

xeL(δ)

is Δ! in KP so we can find a Σl formula equivalent to it in KP:

xeL(<5) iff lyθ(x,δ,y)

where θ is Δ0. Now pick any xeB. We will show that xeL(δ) for some δ^τ.Write x as cA(a) for some at A. Since A^L(β) there is an ordinal λ<β suchthat αeL(Λ). By (1), A is a .model of

Iλ9yθ(a9λ9y).

Hence B is a model of

but then x really is in L(δ) for some (5e£, proving (2).Next we prove that

(3) A=B.

To prove this it suffices to prove that cA(ά) = a for all aeA. Since yÂ, cA(λ) = λfor all λ<γ. Let αeA be Σ1 definable in L(β) by the Σ! formula φ(x,λ^...,λn)where λl9...,λn<γ,

If we can prove that l4β)^=φ[cA(a)9λl9 ...,/iJ then we will have a = cA(a). Butfrom (1) it follows that (A9enA2yt=φ[a,λl9...,λn~]9 so BNφ[cx(fl),cx(λ !),..., c^ΛJ],As we mentioned, cî) = Λ>; so B\=φ[cA(a)9λί9...9λn~]. By (2), B^L(β) soL(β) 1= φ [cx(α), Λ l 5 . . . , Aπ] by persistence of φ. This proves A = B.

Since £ is transitive and B^ L(jβ), it follows that B is admissible and thatB = L(τ). But of course τ = α so y4=β = L(α). Thus α is j8-stable. Since 7^^4,y^α. If γâ'^β and α' is also β-stable then every element of A must be in L(α')so α α'. Hence α is the least jS-stable ordinal ^ y. D

7.9 Corollary. The stable ordinals are generated as follows.(i) σ0 = {α|α is Σί definable in L without parameters},

L(σ0) = {xeL|x is Σ^ definable in L without parameters}*,(ii) σy+ 1 = {α|α is Σ1 definable in L wiί/i parameters =ζσy},

L(σ);+1) = {xeL|x is Σj definable in L wiί/i parameters ^σy};(iii) // A is a limit ordinal then


Proof. For (i) apply 7.8 with β = ω^ y =0. For (ii) apply 7.8 with β equal to somecardinal > σγ and the y of 7.8 equal to σy + 1. Part (iii) is just a restatement of partof 7.2 included for completeness. D

We will study σ0 in some depth in the next section and give a classical de-scription of it. Part (i) of the next theorem will play a crucial role.

7.10 Theorem, (i) σ0 is projectible into ω.(ii) σy+1 is projectible into σr

(iii) // λ is a limit ordinal then σλ is nonprojectible.

Proof. Let's dispose of (iii) first since it's fairly trivial. We prove that

L(σλ)\=Σ1 Separation

and then apply Theorem 6.3 to see that σλ is nonprojectible. Let 0eL(σΛ), letφ be Σx and form the set

Pick γ<λ large enough that a and the parameters in φ are members of L(σy).Then, by IXσ^-^IXσJ, we have

so beL(σλ) by Δ Separation.Now for (i). The idea is that we want to assign to each α<σ0 some Σί de-

finition of α, thus projecting σ0 into HF. The trouble is that

is not a σ0-r. e. predicate of the formula φ. To get around this we use the Uni-formization Theorem, Theorem 4.4.

Recall the Σ1 formula Σ-Ssit1(z,y) from § 1. Let F be given by 4.4 so that F isσ0-recursive,

) = {ψ(x)\ψ(x) is a Σ! formula Λ L(σ0) ϊ= 3y \l/(y}}

^"z is Σ! Λ

and for each ψ(x)edom(F), L(σ0)ϊ=ψ(F(ψ)). Now whenever a is Σx definablethere is a ψ such that

so F(ψ) = a. We may project L(σ0) into HF by

π(a) = least Σx formula ψ such that F(ψ) = a,


where by least we mean in some well-ordering of ΉF as given in II.2.4. The prooffor (ii) is similar, using Σ-Sat3 instead of Σ-Sat1? once we observe that everyαeL(σy + 1) is definable by a formula

with foeL(σ) and no other parameters, by just using b to code a finite sequenceof ordinals. Now apply Uniformization to get a σ y + 1 -recursive F such that

is Σ and

and, if F(ι/φc,y,z),b) is defined then

Then define

π(α) = least pair <^,fc> such that F(ψ,b) = a .

This π projects L(σy+1) into L(σy). Since L(σy) is recursively listed, this amountsto projecting L(σy+i) into σr D

The use of Uniformization in 7.10 is very typical of more advanced work in L.We also use it to prove the next result.

7.11 Theorem. Let β be an admissible ordinal whose projectum β* is not ω. Thenβ* is the limit of smaller β-stable ordinals. Hence β* is β-stable and admissible.

Before proving 7.11 we state some of its consequences.

7.12 Corollary. // α>ω is nonprojectible then α is the limit of smaller a-stableordinals. D

Next we present the result promised at the end of the last section.

7.13 Corollary. For any admissible ordinal α, α* is admissible and nonprojectible.

Proof. By 7.11, α* is admissible if α*>ω. But if α*=ω it is also admissible.Nonprojectibility is obvious. D

Thus, if α is an admissible ordinal less than the first nonprojectible thenα*— ω. We saw that the first nonprojectible ordinal was recursively inaccessible.We can iterate this result using 7.11. We give only a sample result which showsthat the first nonprojectible is much larger than the first recursively inaccessible.

7.14 Corollary. Let pβ be the βth recursively inaccessible ordinal. If α is nonpro-jectible and α>ω then a = p0ί.


Proof. Assume that α*=α but that a = pγ for some y<α. Apply 7.12 to findan α-stable ordinal λ,y<λ<a. The predicate

y < α and y is recursively inaccessible

is α-recursive since it holds iff

y is admissible Λ Vx < y 3τ <λ (x < τ Λ τ is admissible) .

Define H(x) = px for x<y. Then H is α-recursive and

so

L(λ)\=Vx<yly(H(x) = y)

since λ is α-stable. But this is ridiculous for λ itself is recursively inaccessible by7.6, so λ = H(x) for some x<y. D

Some authors refer to ordinals α such that α = pα as being recursively hyper-inaccessible.

We now return to prove Theorem 7.11.

7.15 Lemma (Π2-reίlection). Let α>ω be admissible and let Vx3yφ(x,y) be asentence which holds in L(α), where φ is Δ0. Then for every y<α there is a λ,

u such that

Proof. Let y^Λ 0 <α where all parameters in φ are members of L(Λ0). Let λn+ί

be the least ordinal such that for all xeL(/LM) there is a yeL(λn+l) such thatφ(x,y). There is such a λn+1 by Σ Reflection. The sequence (λn:n«x>y is α-recursive so

is less than α. D

Proo/ o/ Theorem 7.11. For several years all that was known about the projectumβ* of an admissible ordinal was that

(4) β* is admissible or the limit of admissibles .

For suppose β*<β but that there is an admissible ordinal τy such that

186 V. The Recursion Theory of Σ! Predicates on Admissible Sets

But by Theorem 6.8 (and its proof) τy is projectible into supδ<yτδ and hence so is β,contradicting the definition of /ϊ*.

For the purposes of this proof we call an ordinal γ nice if <JL(y) x L(y) hasorder type y so that the function enumerating L, definable in KP, maps y ontoL(y). We know that every admissible ordinal is nice. The only point of proving (4)was to prove that

(5) if ω^ξ<β* then there is a nice limit ordinal y, ζ^y<β* .

If β* is admissible, this follows by Π2 Reflection. If β* is the limit of admissibleswe pick y to be an admissible.

We are now ready to prove that if ω y < β* then there is a j?-stable α, y α < β*.By (5) it suffices to prove this for nice y. Let α be the least /J-stable ordinal ^y.Now L(α) is the set of αeL(jS) definable by Σ t formulas with parameters <y.Since y is nice, though, we can code all these parameters into one so that

L(α) = {αeL(β) | for some Σl formula φ(vί9υ2) and some ξ<y, L(β)

As in Theorem 7.10, let F be a β-recursive function uniformizing Σ-Sat2. Notethat Σ-Sat2 and hence the graph of F are j8-r.e. definable by Σl formulas withoutparameters. Thus

L(α) = rng(Fr(ΉFxy)).

Since y is nice we can identify HF x y with y and apply Theorem 6.11 to see thatdom(Ff(lHFxy))eL(β) since it is, essentially, a β-r.e. subset of y<jδ*. But if thedomain of a β-recursive function is in L(β), so is its range, so L(α)eL(/J). That is,α<β. We need to see that α<β*. Suppose β*tζtt. The inverse of Ff(HFxy)maps L(α) into y so we could then project L(β) into y, contradicting the definitionof β*. Thus α < β* so we have proven that β* is the sup of smaller β-stable ordinals.Thus β* is itself ^-stable and admissible. D


7.16. Prove that every stable ordinal is the limit of smaller nonprojectible ordinals.In particular, the first nonprojectible ordinal is less than the first stable ordinal,even though the first stable ordinal is projectible into ω.

7.17. Compute σ*. [Hint: <τ* + 1=ω.]

7.18 (Jensen). Show that α is admissible iff α is a limit ordinal and L(α) satisfiesΔ! Separation.

7.19. Prove the converse of Lemma 7.15. That is show that a limit ordinal α isadmissible and >ω iff every Π2 sentence Vx1yφ(x,y,z) true in L(α) is true inL(jβ) for arbitrarily large β<α.


7.20. Let α be admissible, α>ω. An ordinal β<α is an en-cardinal if there is no/e L(α) mapping β one-one into an ordinal y < β.

(i) Show that if α* (the projectum of α) is <α then α* is an α-cardinal.(ii) Prove that every α-cardinal κ>ω (if there are any) is α-stable. [Show

that the proof of 7.2 (ii) can be effectivized so as to hold inside L(α).](iii) Prove that if K is an α-cardinal > ω and y<κ then there is an α-stable

ordinal β,γ<β<κ. [Modify the proof of 7.1 1.]

7.21. Suppose α is admissible and ω<α*<α. Show that any α-r.e. subset ofsome L(β), for /?<α*, is α*-finite, not just α-finite as stated in 6.11. [Use 7.11.]

7.22. Assume that there is an α such that L(α) is a model of ZF. Show that the leastsuch α is less than σ0.

7.23. Let A^lBvn be countable, admissible sets and suppose that A^ is Bw-stable; i.e., that

Let T be a theory of LB which is definable over by a Σj formula with parametersfrom Am. Show that if every T0 T with Γ0e Aw has a model then T has a model.[Use the Extended Completeness Theorem.]

7.24. Let Am be countable, admissible. Show that the following are equivalent:(i) AaR<1HYP(AαR);

(ii) Am is Π} reflecting; i.e. if Φ(v) is a Π} formula and A^NΦfx], thenthere is an admissible set A^eA^ such that A^NΦ[x]. In particular, if a<ω1

then τα is τα+1-stable iff L^J is Π} reflecting. [Use the Completeness Theorem.]

7.25. An admissible ordinal α is recursively Mahlo if every α-recursive closedunbounded subset of α contains an admissible ordinal. (This is the "effectiveversion" of the definition of Mahlo cardinal. See Chapter VIII.)

(i) Prove that if α is recursively Mahlo then it is recursively inaccessible,recursively hyperinaccessible, etc.

(ii) Prove that if α is the limit of smaller α-stable ordinals then α is recursivelyMahlo.

(iii) Prove that if α is nonprojectible then it is recursively Mahlo.

7.26 Notes. The stability of uncountable cardinals is due to Takeuti [I960].The concepts and other results in 7.1 — 7.10 are due to Kripke and Platek, indepen-dently. Theorem 7.11 (and hence 7.12, 7.13, 7.20, 7.21) are due to Kripke. Thestudent interested in further similar results should study Jensen's theory of thefine structure of L as presented, for example, in Devlin [1973]. Exercise 7.23appears in Barwise [1969]. Exercise 7.24 is due to Aczel-Richter [1973] and,in an absolute form, to Moschovakis [1974]. Exercise 7.25 goes back to Kripkeand Platek.


Putting admissible ordinals α > ω in their place.

Cardinal >ω

istable (i.e.,

Ilimit of smallernonprojectibles

nonprojectible (iff Lαl=Σ1 Separation)

recursively MahloI

Irecursively hyper-hyper-inaccessible

Irecursively hyper-inaccessible (i. e. α = pα)

Irecursively inaccessible (i. e. α = τα)

1admissible

Notes:

1. No arrows are missing.2. No arrows reverse.3. The first stable ordinal σ0 is projectible into ω; the β + 1st stable ordinal σβ+ί

is projectible into σβ.4. For /I a limit, σΛ is nonprojectible.5. If α is projectible then its projectum α* is admissible and nonprojectible.

8. Shoenfield's Absoluteness Lemma and the First Stable Ordinal 189

8. Shoenfield's Absoluteness Lemmaand the First Stable Ordinal

In § 5 we saw that the first admissible ordinal τ t >ω is the least ordinal not theorder type of a Δ} well-ordering of HF and that a set X^ HF is τ^r.e. iff X isΠ{ on HF. In this section we prove an analogous result for the first stable ordinal

σ0.A relation R on HF is Σ2 if it can be defined by a second order formula of the

form 3St VS2φ, where φ is first order:

R(x) iff <HF,6>I=3S1 VS2φ(x, S1,S2).

If the complement HFπ-β of R is Σ2, then R is said to be Π2. If R is both Σ2

and Π2 then R is Δ^.At first glance the step from Δ} to Δ2 seems a small one. We will show, however,

that it is an enormous jump, taking us from τ± past the first recursively inaccessible,past the first nonprojectible all the way to σ0, the first stable ordinal. The precisestatement is contained in Corollary 8.3 below. The main step in the proof is thefollowing theorem, known as the Shoenfield-Levy Absoluteness Lemma.

8.1 Theorem. Any Σ! sentence without parameters true in V is true in L.

Warning: this does not say that L< jV because parameters are not permitted.Some extensions with parameters are discussed in the exercises.

We defer the proof of 8.1 to the end of the section (Corollary 8.11) since itleads away from our chief concern.

8.2 Theorem. Let σ0 be the first stable ordinal and let R be a relation on HF.(i) R is Σ\ on <HF,e> iff R is σ0-r.e.

(ii) R is Δ* on <HF,e> iff #eL(σ0).

Proof. As usual, (ii) follows from (i). We first prove the (<=) half of (i). Let R be Σx

on L(σ0). We know that L^-^L by the definition of σ0 and that every xeL(σ0)is Σ! definable (as an element) in L by a formula without parameters (by 7.9).It follows that every xeL(σ0) is Σ1 definable in L(σ0) by a Σl formula withoutparameters. Thus any parameters in a Σ1 definition of the relation .R can beeliminated so we may assume that

R(x) iff L(σ0)\=3yφ(x,y)

where φ is Δ0 and contains no parameters. But then we claim that

(1) R(x) iff 3α[α admissible Λ L

The proof of (=>) in (1) is trivial since we can let α = σ0. The other half (<=) of (1)follows from L(σ0)<1L for L(α)t= 3yφ(x,y) implies Lϊ=3yφ(x,y) and hence


L(σ0)\='Byφ(x9y) by stability. Using (1), it is not too difficult to rewrite R to beΣ£ over <HF,e>. Namely, R(x) holds iff <ΉF,e> satisfies

(2) 3E,F[<IHF,£>NKP + V = L Λ £ is well founded ΛF is an isomorphismof <HF,£> onto an initial submodel of <HF,£> Λ <HF,E>*=3;yφ(F(x),;y)]

since any such <HF,£> is isomorphic to L(α) for some admissible α. By thetechniques of IV.2, everything inside the brackets is Δ} in e,F,F except the con-dition

E is well founded.

But this is Π} by the very definition:

Hence the whole of (2) has the form

3E,F[— ]

where [---] is Π} so (2) is Σj. (If you insist, you can always collapse the two existen-tial second order quantifiers to one.)

To prove the other half of (i), let R c HF be Σ^ over <HF,e>, say

R(x) iff <HF,e>N3S1VS2φ(x,S1,S2).

For each relation S1 on <HF,e>, let σ(Si) be the infinitary sentence

Thus St occurs in σ(Sί).We claim that R(x) holds iff L(σ0) is a model of

(3) aS^Pp^HFΛP isaproofof(σ(Si)-+φ(x9Sl9S2))']9

which will show that R is Σί over L(σ0). To show this, first suppose R(x) holds.Then there is an Sx such that

(HF,e,S1)l=VS2φ(x,S1,S2)and hence

is logically valid. It is a countable infinitary sentence, so it is provable. Hence (3)holds in V. The only parameter in (3) is x and it is Σί definable, being in HF.Thus (3) holds in L by 8.1 and hence in L(σ0). Thus R(x) implies L(σ0)l=(3). Toprove the converse, suppose (3) holds in L(σ0). Then there is an S1eL(σ0) suchthat


is provable, and hence, is logically valid. Thus,

so R(x) holds. D

8.3 Corollary. The first stable ordinal is the least ordinal not the order type ofsome well-ordering which is Δ2 on <HF,e>.

Proof. Every Δ2 well-ordering R is in L(σ0) so its order type is less than σ0, by 3.3.To prove the converse, recall that σ0 is projectible into ω by Corollary 7.10.

Let p be some one-one σ0-recursive map of σ0 into ω. For β<σ0 let

RP={<p(χ),p(y)>\χ<y<β}

which is in L(σ0) by Σ Replacement. But then Rp is a well-ordering of order type βand Rβ is Δ^ by Theorem 8.2(ii). D

We can now project the recursion theory from σ0-r. e. sets of ordinals to Σ2

sets of integers using Section 5. We state some of the simplest results.

8.4 Corollary, (i) For any Σ2 subsets B,C of HF there are disjoint Σ2 sets B0,C0

with BO^B,CQ^C and BvC = B0vC0.(ii) Any two disjoint Π2 subsets of HF can be separated by a Δ2 set.

(iii) There are disjoint Σ2 subsets of HF which cannot be separated by a Δ2 set.

Proof. These are translations and projections of results we know about σ0. D

8.5 Corollary. Every Σ2 subset of HF is constructible.

Proof. IΪR is Σ2 on HF then it is Σ1 on L(σ0) and hence an element of L(σ0 + ω). D

It follows, of course, that every Π2 subset of HF is constructible, but this isas far as one can go. It is consistent with ZFC to assume there is a nonconstructibleΔ\ subset of HF, where Δ^ means expressible in both the forms

3S1VS23S3(— ),

VS13S2VS3(— ).

We now turn to the proof of Theorem 8.1. We need several preliminary lemmas.A finitary formula φ(χl9 ..., xn) is an V - formula if it has the form

where ^ is quantifier-free.

192 V. The Recursion Theory of Σx Predicates on Admissible Sets

8.6 Lemma (Skolem V3 normal form). Let K be a language and let Ψ be a finiteset of formulas of Kωω. There is an expansion

L=Ku{S1,...,Sπ}

by a finite number of new relation symbols and an VB-sentence φ of Lωω with thefollowing properties:

(i) Every ^-structure $01 has a unique expansion W = ($01, SΊ, . . . , Sn) with Wΐ t= φ.(ii) For each formula φ(y^ ..., yn) in Ψ there is a quantifier free formula ι/f0 of

Lωω such that

Proof. We may assume Ψ is closed under subformulas. Introduce, for eachΨ(yι> •••> yn)

EΨ a new relation symbol S^(yif ...tyn). Let φ be the conjunction of theuniversal closures of the following:

S^ι,...,yJ^ιA(yι,...,yn) if ψeΨ is atomic,if -ΊψεΨ,Sίl,(yί9...,yJ ifS^1,...,yπ) if

ί9...,yj if (BySv^yi- -^ ^J^V^S^y !,..., ym) if

Here we use yι,...,ym,...,yn to denote 3>ι> >)>m-ι) 'm+ι> ---^m if ^^^ todenote yl,...,yniϊm>n. Now φ clearly has the desired properties. D

8.7 Corollary. Let ψ be any sentence of Kωω. There is an expansion L of K by afinite number of new relation symbols and an \f3-sentence φ' of Lωω such that

(i) Every model $R of φ has a unique expansion to a model 9W' of ψf.(ii) // W\=\l/' and 501 is the reduct of 501' to a ^-structure then

Proof. Let Ψ = {ψ} and apply 8.6. Let φ be as given there and let 0 be quantifierfree such that

The desired ψ' is (φ Λ ι/r0), or rather, the V3-sentence equivalent to it after onemoves the quantifiers in φ out front. D

The next lemma gives us an easy way to construct models of V3-sentencesand accounts for our sudden preoccupation with them.

8.8 Lemma. Let φ be an W3-sentence of Lωω, say

where ψ is quantifer-free. Let


be a chain of L-structures. Suppose that for each l<ω and each x l 5 . . . ,x π 6SER / 5

IfVΛ = \Jl<ωml9 then

Proof. Trivial, since yRι + ιt=3yψ(x,y) implies yjl\=3y\//(x,y). 0

The next lemma contains the secret to proving a number of important results,including Theorem 8.1.

8.9 Lemma. Let (X, -<> be a non-wellfounded partially ordered structure whichis constructίble (i.e. is an element of L). There is a sequence <xπ>n<ω

m L such that

for all n<ω.

Proof. The hypothesis is that <JΓ, X>eL and that

V N < Jf , -<> is not well founded .

We claim that

(4) L |= < Jf, -<> is not well founded .

For otherwise, since Lϊ=Beta, there would be a function /eL such thatf(x)={f(y)\y<x} for all xeX. But then <X,<> really would be well founded(see Exercise 1.9.9). Now since (4) holds, there is a nonempty X0eL, X0^Xsuch that

But then, using the axiom of (dependent) choice in L, there is a sequence of thedesired kind. D

8.10 Theorem (of ZF). Let φ be a finitary sentence in a language L containing eand some other relation symbols R 1 , . . . ,R / . // φ is true in some structureA = <y4,e,jR !,..., Rty where A is transitive, then there is a transitive structureB — <£, e, R'ί9 . . . , R[y which is constructible and a model of φ.

Proof. We may assume that extensionality is a consequence of φ since it holdsin A. By 8.7 we may also assume that φ is V3, say

where φ(x) is

194 V. The Recursion Theory of Σ! Predicates on Admissible Sets

and θ is quantifier free. Let β = rk(A). We define a non-wellfounded structure<JΓ,OeL. The set X consits of all pairs <95,/> such that S = <5,£,K1,...,R/>is a finite structure with B^ω, f:B-+β and xEy implies f ( x ) < f ( y ) . We define

<»ιJιX<»o,/o>

to mean that S0^S1? /0^/ι and for every xe£0,

Now the definitions of Z and -< are absolute so <X, -<>eL. We claim that

(5) <X, -<> is not well founded.

Assuming (5) for a moment, let us finish the proof of the theorem. By Lemma 8.9there is a sequence

\^«5 Jn/n<ω

in L such that

for each w. Let 93 = 1J93Π. By Lemma 8.8, 95 \=φ. Let /=UΠ/M. Then, if95 = <B,£,R'1,...,Rί>, then f'.B^β and x£y implies f ( x ) < f ( y ) so £ is wellfounded. Now since LNBeta, there is a transitive structure BeL isomorphicto 95. This B satisfies the conclusion of the theorem.

Now let's go back and prove (5). Let X0 be the set of those (ϊ&9f)eX such thatthere is an embedding i of 23 into the original A such that

/(x) = rk(ΐ(x))

for all xεB. The set X0 is nonempty since <2I0,{<0,0>}>eX0 where 910 is thesubstructure of A with universe {0}. It remains to show that X0 has no •< minimalmember. Let <930,/0>eX0 with i0:930->A the associated embedding. Let3I0— A be isomorphic to 330

γia *o Since

there is a finite structure 9Il5 Slo^gij c A, such that for all

Now choose 951 = <B1,£1, ...> extending 230 with β^ω so that for somei^ extending ί0,


Let /1(x) = rk(i1(x)) for xeBv Then <»ι,/!>^«»oJo> and <»ι >/ι>e*o 0

Theorem 8.1 is the informal version of the next result.

8.11 Corollary. For any Σ sentence φ of set theory

is a theorem of ZF.

Proof. We work in ZF. Assume φ. Then there is a transitive <X,e>N=φ. But thenby Theorem 8.10, there is a transitive <£,e>eL such that <B,e>N<p. And<β,e>c:end<L,e> so <L,e>N</>; i.e., φ(L\ D

Some extensions of these results are sketched in the exercises.


8.12. Show that if

<HF,e>t=3S13S2VRφ(S1,S2,R,x)

where φ is first order, then there is an Sί eL(σ0) such that

<HF,e,S1>l=3S2VRφ(S1,S2,R,x).

This is the original version of Shoenfield's Absoluteness Lemma. A proof of itcan be discovered inside the proof of Theorem 8.2.

8.13. Show that there is a Σ2 well-ordering of a subset of ω of order type σ0.

8.14. Improve 8.11 by replacing ZF by KP + Beta.

8.15. Improve Theorem 8.10 as follows. Let φ, A be as in 8.10, let β = τk(A).Let α be the least admissible τy>β if L(τy)t=Beta, otherwise let α = τy + 1. Showthat there is a transitive model IB of φ which is an element of L(α).

8.16. Let α be a limit of admissibles. Show that any Σl sentence (without para-meters) true in V(α) is true in L(α). [Use 8.15.]

8.17. Let T be a countable set of finitary sentences true in some transitive struc-ture A = <A,6,JR1,Λ2, ...>. Show that there is a transitive model IB of T whichis an element of L(T) and is countable in L(T). [Hint: Modify the definition of<Λ", •<> in the proof of 8.10 so that bigger structures take care of more of thesentences in T.]


8.18. Prove that the following is a theorem of ZF (by using 8.17): for eachformula φ(v)

8.19. Let α be the constructible N l 9 i.e. the ordinal which, in L, is the first un-countable cardinal. Prove that

It is consistent with ZFC to assume α is countable. Prove that if α is countable andif β>κ then

8.20. Let Φ be a Σ^ sentence true in some countable structure 50Ϊ. Prove that thereis an 9JleL(σ0) which satisfies Φ. If Φ is Πj you can improve this bound. How?

8.21 Notes. The original Shoenfield Absoluteness Lemma (Exercise 8.12) wasproved in Shoenfield [1961]. Theorem 8.1 appears as Theorem 43 in Levy [1965].The proof given in this section and some of the generalizations found in the Exer-cises appeared in Barwise-Fisher [1970], Exercise 8.16 is due to Jensen-Karp[1972].

Theorem 8.2 and its Corollary 8.3 are due to Kripke [1963] and Platek [1965]in the form stated here. Their content, however, goes back to Takeuti-Kino [1962].

Chapter VI

Inductive Definitions

"Let X be the smallest set containing ... and closed under ---." A definitionexpressed in this form is called an inductive definition. We have used this methodof definition repeatedly in the previous chapters; for example, in defining thenotions of Δ0 formula, Σ formula, infinitary formula, provable using the 9Jl-rule,etc. In this chapter we turn method into object by studying inductive definitionsin their own right. We will see that their frequent appearance is more than anaccident.

1. Inductive Definitions as Monotonίc Operators

Let A be an arbitrary set. An π-ary inductive definition on A is simply a mappingΓ from n-ary relations on A to n-ary relations on A which is monotone increasingi. e. for all π-ary relations R, S on A

R^S implies Γ(R) c Γ(S) .

If Γ(R) = R then R is a fixed point of Γ.

1.1 Theorem. Every inductive definition on A has a smallest fixed point. Indeed,there is a relation R such that:

(i) Γ(R) = R,(ii) for any relation S on A, if Γ(S)<^S then R^S.

Proof. Let C = {S^An\Γ(S)^S}. Since AneC, C is non-empty. Let # =Since (ii) now holds by definition it remains to prove (i), that is, that Γ(R) = R.Let S be an arbitrary member of C. Since R c S and Γ is monotone we haveΓ(R)^Γ(S\ but Γ(S)^S, so Γ(R)^S. Since S was an arbitrary member of C,and Λ = f)C, we have Γ(R)^R. To show that R^Γ(R) it suffices to provethat Γ(R)eC. But since Γ(R)^R we have, by monotonicity, Γ(Γ(R))^Γ(R)so Γ(R}εC. D

The proof of 1.1, while correct, tells us next to nothing about the smallestfixed point of Γ and is certainly not the way we mentally justify a typical inductivedefinition. Let us look at an example.

198 VI. Inductive Definitions

1.2 Example. Our very first use of an inductive definition was the definitionof the class of Δ0 formulas. We defined it as the smallest class containing theatomic formulas and closed under Λ , v, —ι,Vweι;, 3weι;. How do we convinceourselves that there is such a smallest set? We simply say: start with the atomicformulas and close under (i.e., iterate) the operations Λ, v, ~ι,Vweι;, 3weι;. Wecan turn this process into a much more instructive proof of Theorem 1.1. (By theway, to make the class of Δ0 formulas fall under 1.1 we let A be the class of formulasof L* and define the 1-ary Γ by

Γ(U) = {φeA\φ is atomic or φ = (ψ/\θ) for some ψβεΌ or ...or φ =for some

Motivated by the above example we make the following definitions.

1.3 Definition. Let Γ be any n-ary inductive definition on a set A.(i) The ^-iterate of Γ, denoted by ΓΓ, is the π-ary relation defined by

(ii) IΓ = \JΛΓΓ, where the union is taken over all ordinals.

We will show that IΓ is the smallest fixed point of Γ referred to in Theorem 1.1.We use the notation

to simplify some equations.

1.4 Lemma. Let Γ be any n-ary inductive definition on a set A.(i) /? = Γ(0),

(ϋ) I « = Γ ( I Ϊ Λ ) for all a,(in) α^β implies Ia

Γ^Iβ

Γ, and(iv) I«Γ

+i=Γ(ΓΓ) for all a.

Proof. Parts (i) and (ii) are immediate from the definitions. Part (iii) follows frommonotonicity since

implies

Part (iv) follows from (ii) since

1. Inductive Definitions as Monotonic Operators 199

1.5 Theorem. Let Γ be an n-ary inductive definition on a set A.(i) There is an ordinal y (of cardinality ^ card (,4")) such that

jy — τ<y1Γ — 1Γ

and hence

(ii) IΓ is the smallest fixed point of Γ.

An

Fig. 1 A. Building up the smallest fixed point IΓ

Proof. First notice that the relations form an increasing sequence of subsets of A",

and hence the sequence must stop strictly increasing for some y of cardinality^ card (An\ i.e.,

But then 1} = !^ for all α^y so IΓ = Ify- To prove (ii), note that

by using (i) repeatedly. Hence IΓ is a fixed point and it remains to show that IΓ

is the smallest such. Let Γ(S)^S. We prove ΓΓ^S for all α, by induction. Theinduction hypothesis asserts that Iβ

Γ^S for all β<oc so IpΛ^S. By monotonicitywe have

S. D


1.6 Definition. Given an inductive definition Γ, the least ordinal y such thatIyr = Ify is called the closure ordinal of Γ and is denoted by ||Γ||.

Most of the inductive definitions we have used in the previous chapters havehad closure ordinal ω so that

One of the most important however, the set of sentences provable using the9Jl-rule, will in general have closure ordinal greater than ω. (In fact, this inductivedefinition has closure ordinal 0($R). See Exercise 3.19.)

Our interest in this chapter is in inductive definitions which are definable overan L-structure 9W or over an admissible set A^. In order to insure the mono-tonicity condition on Γ we need the notion of an R-monotone formula.

1.7 Definition. Let 91 be a structure for some language K (usually L or L* inapplications). A formula φ(x1? ..., XΛ, R) of Ku{R} (possibly having parametersfrom 91) is R-monotone on 91 if for all x1,...,xne9t and all relations R^^R2

on 91,

(9l,K1)

implies

Recall the notion R-positive and corresponding notation φ(R+) from V.2.1.

1.8 Lemma. // φ(jc1? ...,xπ, R + ) is an R-positive formula of K then it is R-mono-tone for all ^-structures 91.

Proof. Fix 91 and prove the result by induction following the inductive definitionof R-positive. D

Most inductive definitions are actually given by R-positive formulas becausemost inductive definitions do not really depend on the particular structure 91and any formula which is R-monotone for all structures 91 is equivalent to anR-positive formula (see Exercise 1.14).

1.9 Notation and restatement of results. Let 91 be a structure for a language K.Let R be a new π-ary relation symbol and let φ(x1? ..., xπ, R) be R-monotone on 91.

(i) The n-ary inductive definition given by φ, denoted by Γφ, is defined by

R) iff (

(ii) We let Iφ denote IΓφ and similarly for Pφ and /^α. Thus Iφ is an n-aryrelation on 91 satisfying


Furthermore, if .R is an n-ary relation on 91 satisfying

then Iφ^R. Iφ is called the smallest fixed point of the inductive definition Γφ

and Iφ is called the αth stage of Γφ. It satisfies

1.10 Proposition. Let 91 foe any ^-structure and let φ(xiy ..., xw R) foe R-monotoneon 9ϊ, vv/zere R is a new n-ary symbol. The fixed point Iφ is a Π\ relation on 91.

Proof. By 1.9 we see that (xί9...,xjelφ iff VR[Γφ(R)c R^R(χ1 ? ...,xj] whichbecomes

when written out in full. D

Let ^Γ = <ω,0, +,•>. Spector [1959] observed that Kleene's analysis of Π}relations on Jf showed that every Π} relation or could be obtained by means of aninductive definition. This result will follow from more general results in § 3. Wepresent the classical proof, nevertheless, since it is attractive and illustrates severalimportant points.

1.11 Theorem. Let ^Γ = <ω,0,+, > ana let S be an n-ary Π\ relation on Jf .There is a formula φ(xί9 ...,xn,y,R+) with R n + l-ary such that

for all x l J . . . ,

Proof. We prove the result for n = 1 and use the following normal form of Kleenefor Π} sets S:

S(x) iff VflnP(xJ(n))

where the following are assumed:

P is recursive,

f(n) is a number s coding up the sequence </(0),...,/(n — l)>,

sl<s2 means that sί is a sequence (code) properly extending s2,

P(x,s2) and sί<s2 implies P(x9sί)9

1 codes the empty sequence,

if s codes <x l 9...,xΛ> then sy codes <x1,...,xπ,j;>.


The desired inductive definition φ is given by

s is a sequence code and, P(x,s) or Vy R(x,sr];).

We first prove that

(1) Iφ(x,s) implies V/ [if f extends s then 3nP(x,/(n))].

Let R be the set of pairs (x,s) satisfying the right side of (1). Note thatP(x,s)->#(x,s). It suffices to prove that Γφ(R)^R. If (x,s)eΓφ(R) then eitherP(x,s) or else VyR(x,fy). But then R(x,s) since every function extending sextends 53; for some y.

Next we prove the converse of (1), or rather, as much of it as we need:

(2) V/3nP(xJ(n)) implies /φ(χ,l)

If P(x,l) then (x,l)e/° so we may assume — ιP(x,l). Assuming the left sideof (2) consider the set S of all s such that — ιP(x,s). This set is well founded(under -<) since any infinite descending sequence would produce an / with— ιP(x,/(tt)) holding for arbitrarily large π, and hence for all n. Let us write, inthis proof, p(s) for p<ΓS(s); ρ(s) is defined for all seS since 5 is well founded.We prove by induction on ξ that if p(s) = ξ then (x,s)e/|+1. (Since leS wethen have (x,l)e/|+1 where ξ=ρ(i).) Observe that

p(s) = sup{p(ίy) + l\-ΊP(x,sy),yEω}.

Now for each y, if P(x,fy) then (x,s]r)elφ, and if -iP(x9fy) then (x,sy)el* + l

for some β<ξ by the induction hypothesis. In either case

But then by the definition of φ,

as desired. Combining (1) and (2) yields the theorem. D

One of our goals in this chapter is to prove some generalizations of thisresult to arbitrary structures. Looking at the above theorem and its proof, weare struck by three facts.

The most prominent fact is that the proof uses a normal form for Π{ pre-dicates on Jf which has no generalization to Π} over arbitrary structures. Ifwe can ignore this unsettling fact, however, we can go on to make two usefulobservations.

First, and very typical of the whole subject of inductive definitions, is thatthe Π} relation S was not defined as a fixed point but rather as a "section" ofa fixed point :

S(x) <=>/,(*,!).


The proof makes it clear that the last coordinate of Iφ is where all the work isgoing on. It is only at the very last minute that we can set 5 = 1. (To clinchmatters, Feferman [1963] proves that not every Π} set over Jf is a fixed point.)This motivates the next definition.

1.12 Definition. Let K be a language, 91 be any structure for K and let Φ be aset of formulas such that each φeΦ is of the form φ(x1?...,x , R) for some n andsome n-ary relation symbol R not in K) and is R-monotone on 91.

(i) If S = Iφ for some φεΦ then S is called a Φ- fixed point.(ii) A relation S of m arguments is Φ-ίnductive if there is a Φ fixed point S'

of m + n arguments (n^O) and yί9...9yne9l such that

S(xl9...,xJ iff S'(xl9...9xm9yί9...9y.J

for all xί9...9xme9l. S is called a section of S'.

Combining 1.10 and 1.11 (and the triviality that a section of a Π} relationis Π}) we see that a relation S on Jf is Πj iff it is first order inductive.

A final point on the proof of Theorem 1.11. We made heavy use of codingin the proof, coding of pieces of functions by sequences and sequences by num-bers, not to mention the coding which goes into the proof of the normal formtheorem. In an admissible set, coding presents no trouble. In an arbitrary struc-ture Sffϊ, however, we may be out of luck. In this case we have two options. Oneis to restrict ourselves to 9W which have built in coding machinery (this amountsto Moschovakis [1974]'s use of "acceptable" structures). The second option,more natural in our context is to replace induction on 501 by inductions on HF^.We study both approaches in the latter parts of this chapter.


1.13. Let K be a language with only relation symbols. One form of the LyndonInterpolation Theorem asserts that if φ,^eKωω, if φ or ψ is R-positive, and if

then there is a Θ which is R-positive and has symbols common to φ and ψ such that

H<P->0) and

Prove a generalization of this to arbitrary countable, admissible fragments KA.

1.14. Prove that if ^(x^^^x^R) is R-monotone for all models 91 of sometheory T of Kωω (T not involving R, of course) then there is an R-positive^(x l5...,xπ,R+) of Kωω such that

[Use the Kωω version of 1.13.]


1.15. Let Γ be an inductive definition, i. e. a monotonic increasing operation onn-ary relations on some set A. Show that Γ has a largest fixed point.

1.16. Let Γ be an rc-ary inductive definition on A and define

and for α>0,

Let

Jr = (V?

Show that JΓ is the largest fixed point of Γ refered to in 1.15.

1.17. Let Γ be an π-ary inductive definition on A and let Γ be defined by

Π(R) = An-Γ(An-R).

Prove that Γ' is an inductive definition. Prove that, for each α,

xeΓΓ iff xφJ«Γ

and hence that

IΓ. = A"-JΓ.

1.18. Let Φ1? Φ2 be classes of formulas R-monotone on a structure 9t, closedunder logical equivalence and such that

1 iff

A relation S on 91 is Φ^-coinductίve iff for some φeΦ1 and some parametersyί9...,yneK

S(x!,....,xm) iff (xί9...,xm,yl9...,yn)eJφ

for all x l9...,xn69l. Show that S is Φ^coinductive iff ~ι S is Φ2-inductive.(Hence every coinductive relation on 91 is Σ}. You can also prove this directly.)

1.19. Let G be an abelian p-group. Define Γ(H), for #^G, by

= {px\xeH}.

Show that JΓ is the largest divisible subgroup of G. In this case the least ordinal αsuch that JΓ = Γ\β<oίJr is usually called the length of the group G. It plays akey role in the study of p-groups.

2. Σ Inductive Definitions on Admissible Sets 205

1.20 Notes. We have built monotonicity of Γ into our definition of "inductivedefinition". There are also things called "non-monotonic inductive definitions"which have interesting relationships with admissible sets. For references on theseoperators, we refer the reader to Richter-Aczel [1974] and Moschovakis [1975].

All the results of § 1 are standard.

2. Σ Inductive Definitions on Admissible Sets

Let Σ(R+) be the collection of R-positive Σ formulas of L*(R) and let Σ+ be theunion of the Σ(R+) as R ranges over all relation symbols not in L*. ApplyingDefinition 1.12 (with K=L*, 91 = 21 and Φ = Σ+) we have the companionnotions of Σ+ fixed point and Σ+ inductive relation. These notions are the primaryobject of study of this section. The proofs, however, give information about awider class of relations.

Let JΓ be a class of L*-structures and let Σ(Rt<>O be the collection ofΣ formulas <p(x l5...,xπ, R) of L*(R) which are monotone increasing on eachstructure in jf. We let Σ(tJf) be the union of the Σ(RίJf) as R varies. (Read"Σ increasing on JΓ" for Σ(JJΓ).) Given a structure ^I^eJΓ we have corre-sponding notions of Σ(f JΓ) fixed point on 91 and Σ(|JΓ) inductive relationon Slg,,. If jf = {91TO} then we write Σ(t«lsw) for Σ(pf)

Note that by Lemma 1.8, Σ+cΣ(|jf) for all jf. If jf is the class of allstructures for L* which are models of some theory T then Exercise 1.14 tells usthat Σ+ = Σ(pf), up to logical equivalence. In the results below, however,Jf* is usually a single admissible set or a class of admissible sets.)

We have already studied the most important Σ+ inductive definition at somelength back in Chapter III. Let KA be an admissible fragment and let Thm& bethe set of theorems of KΔ. By definition, T/ιraA is the smallest set of formulasof KA containing the axioms (Al)—(A 7) and closed under (Rl)—(R3). This is,of course, a typical example of an inductive definition. Let Γ0 be this inductivedefinition.

2.1 Proposition. Using the notation just above we have(i) Γ0 is a Σ+ inductive definition, and hence

(ii) Γ/ιmA is a Σ+ fixed point.

Proof. We simply write out the definition of Γ0 to see that it is in fact Σ+. Let Rbe a new unary symbol and recall that

xeΓ0OR) iff xeK A Λ[(A)v(Rl)v(R2)v(R3)]

where we have used

(A) "x is an instance of (Al)—(A7)".

(Rl) 3y[yeR/\(y->x)eR].

(R2) "x is of the form (ψ-*Vvθ(υ)) where v is not free in φ and (\l/-*θ(υ))eR".

(R3) "x is of the form (ψ->/\φ) and, for each φeΦ, (ψ


We can rewrite this schematically in the form

xeΓ0(R) iff Δ 1 Λ[Δ 1 vΣ 1 (R + )vΔ 1 (R + )vΔ(R + )] ,

so Γ0 is indeed a Σ(R+) inductive definition. D

Now one of the primary aims of § III.5 was to prove that T/ιmA was in factΣ! definable on A. In this section we use this fact to prove that every Σ+ in-ductive relation on an admissible set is Σί on that admissible set. For A countable,even more is true.

2.2 Theorem. Let A be a countable admissible set. Every Σ(|A) inductive rela-tion on A is Σ! on A.

Proof. It clearly suffices to prove that every Σ(| A) fixed point on A is Σ! sincethe Σx relations are closed under sections. Let φ(x l9...,xn, R)eΣ(|A). The proofgoes back to the Extended Completeness Theorem for countable admissiblefragments and, hence, to our analysis of Γ0 carried out in § 111.5. Let K be theformalized version of L*(R)u{x|xeA} and let KA be the fragment given byA (=A2n). Let Tbe the KA theory:

Diagram (A),

Vi; [vE~a <-> \/*eα v = *] f°Γ a^ 0G A ,

Vϋ t, . . . , vn [φ(υί9 . . . , vn9 R) -> R(i?1? . . . , t?J] .

We claim that

(1) (xl9...9xjelφ iff TNROq,...,*,,)

from which the conclusion follows by the Extended Completeness Theorem.The (<ί=) half of (1) follows from the observation that

when R is interpreted by Iφ. To prove (==>) suppose that (93 ,1 ) is an arbitrarymodel of T. We need to prove that whenever (x1,...,xπ)e/<p, we have

If we let R0 = R\Am then we note that (up to isomorphism)

so what we need to prove is that Iφ^R0. This will follow (from 1.5(ii)) if we

prove that Γφ(K0)^^o); i e ?

that

(2) (Aw,


So suppose that y1,...,yneAm and

Since φ is a Σ formula and (95^,^) is an end extension of (A^,/^) we have

, R)\=φ(yl9...,yn9R)9

and so, by the last axiom of T, R(yί9...,yn) holds, and hence #()()>!,...,}>,,). Thisestablishes (2) and hence the theorem. D

Let φ(xί,...,xn,υί9...,vk,R) be a fixed Σ formula of L*(R). The followingremarks are intended to lift much of Theorem 2.2 to arbitrary admissible setsby means of the Absoluteness Principle.

2.3 Remark. The Σ1 formula defining Iφ in Theorem 2.2 is independent of Aexcept for the parameters occuring in φ. More fully, let

denote the smallest fixed point defined on A by Γφ when v1=yί,...,yk = yvided <?(*!,..., xπ, }Ί,..., yk,R)eΣ(Rt A)). There is a Σ1 formula ^(Xi,...^^^,...,^)of L* such that for all countable, admissible A and all yly...,ykeA,

(3) if φ(x1,...,xn,y1,...,yk, R) is R-monotone on A then for all x1,...,xnelk,

(x1,...,x/l)e/φ(A,);1,...,};fc) iff ANi/φq,...,*,,,;^,...,^).

Proof. Let ψ be the formula which expresses

3p [p is a proof of σ-» R(xί9 . . . ,"xj where σ is a conjunction of members of T],

where T is as in the proof of 2.2, and examine the proof of Theorem 2.2. D

2.4 Remark. The operation /£(A, yl9..., yk), is a Σ operation of A, yι,...,yfc, sinceit is defined by Σ Recursion on α. In ZF we proved the existence of an α (de-pending on A9yί9...9yk) such that

(This step takes us outside KPU since it requires some form of Σx Separation.)Thus, in ZF, the predicate

is a Δ! predicate of A, xl9...,xn,yί9...,yk. It is expressed by the Σx formula


and the Γ^ formula

V α C / ίA,^,...,;)^^^

(The characterization of Iφ(Jk,yl9...,yώ as smallest fixed point of Γφ gives an-other Πi definition.)

2.5 Remark. The conclusion of line (3) above is a predicate of A, yl9...,yk. Thehypothesis, however, is a Πt predicate of A, y1?...,yk which makes (3) of theform Πi -> Δ! and hence a Σ! predicate of A, y1? . . . , yk. To apply the AbsolutenessPrinciple we would need the result to be Π^

We are now ready to lift Theorem 2.2 to the uncountable. We give two proofsbecause each contains information not available in the other (see the two corol-laries 2.7 and 2.8).

2.6 Gandy's Theorem. Let A be any admissible set. Every Σ+ inductive relationon A is Σ! on A.

First Proof of Theorem 2.6. Fix ςφc l5...,xπ, ι?l5...,ι;k, R)eΣ(R+). Since φ isR-positive it is R-monotone for all structures for L* and hence for all admissiblesets. The troublesome hypothesis of line (3) is thus superfluous and we see thatwe have proved for all countable A:

if A is admissible then for all yί9...,ykeA and all x1,...,xII£A

The displayed part is Δx so by the Levy Absoluteness Principle, the result holdsfor all A. D

2.7 Corollary. Let tf be a class of admissible sets which is Σt definable in ZFC.Then for any AeJf , every Σ(|Jf ) inductive relation on A is Σί on A.

Proof. The hypothesis asserts that there is a Σt formula θ(x) without parameterssuch that

iff 0(A),

ZFC h- Θ(A) -» A is admissible.

Replace "A is admissible" by "Θ(A)" in the above proof. D

For example, the Jf in 2.6 might be the class of all admissible sets or theclass of L(α) where α is recursively inaccessible or nonprojectible.

Second Proof of Theorem 2.6. This proof is more traditional in that it uses theSecond Recursion Theorem. For simplicity we let n = i and we suppress


parameters y1}...,^k entirely since they are held constant in this proof. Tosimplify notation, whenever S is a relation on A and φ(x,R)eΣ(R+), we write

A\=φ(x,S)

instead of the more accurate

Now let φ(x,R)EΣ(R+). Use the Second Recursion Theorem to define a Σί

Formula ψ of L* such that

(More precisely,

To fit thus into Second Recursion Theorem, first let S be a new binary symboland let φ'(x,β,S) be φ(x, 3y</?S( ,y)) and then apply the Second RecursionTheorem.) We claim that

(4) for β<o(A)

xεlβ

φ iff ANι/φc,β).

The proof proceeds by induction on β. The induction hypothesis gives us,for y<β,

so, taking unions,

Then for any xeA we have

iff

iff

iff Aϊ=φ(x,3γ<βψ( ,y))

iff A\=ψ(x,β).

Let α = o(A). From (4) we obtain

(5) i;« = {


Now we claim that

(6) rφ(i<«) =/;«.

It suffices to prove Γφ(/^ *)£/<«, so suppose xeΓφ(/φ

<α), i.e., that

By (5) this becomes

By the Σ Reflection Theorem and Lemma V.2.2 there is a δ<α such that

which, by (4), is equivalent to

Thus xeΓφ(/**) = /£. But /£c/<« so xe/φ

<α as desired. But (6) immediatelyimplies that Iφ = Iφ*, so ||ΓJ|<α and

which proves that Iφ is Σ! on A. D

2.8 Corollary (Second half of Gandy's Theorem). Let A be admissible and letφ(x l5...,xπ, R+) be a Σ formula with parameters from A. Let α = o(A).

(i) IIΓJKα.(ii) For all β, Iβ

φ is Σ, on A.

Proof. Part (i) was explicitly mentioned in the second proof of 2.6. For (ii) wehave the result for jβ^α by 2.6 and for β<α by line (4) above. D

The results mentioned in 2.8 also hold for arbitrary R-monotone φ(x,R) ifthe admissible set A is countable. The proof of this, however, must await astronger reflection principle, the s — Tl\ Reflection Principle.

For sets of the form L(α) the conclusions of 2.6 and 2.8 are actually equivalentto the hypothesis of admissibility. This will follow from Theorem 3.17 in thenext section.


2.9. Let 3ΪOT be a nonstandard model of KPU. Show that i^/(Mm) is a Σ+ fixedpoint which is not first order definable over 91 . What is the length of the in-ductive definition?

3. First Order Positive Inductive Definitions and HYP^ 211

2.10 (Stavi). Show that there are pure transitive sets which are not admissiblebut such that every Σ+ inductive relation is Σ!. [Hint: Let A = L(τ1)nV(y) forsuitably nice α,ω<α<τ1.]

2.11. Let φ(x1,...,xπ,R+) be a Π formula and let Jφ be the largest fixed pointof Γφ on an admissible set A. Show that Jφ is Π?.

2.12 Notes. The fact that, over an admissible set A, a Σ+ inductive definition Γφ

has a Σ! fixed point and closure ordinal ||ΓJ|<0(A) is usually called Gandy'sTheorem. He proved this theorem in lectures at the UCLA Logic year in 1968by adapting the proof-theoretic approach used to prove the Barwise Com-pleteness Theorem. A similar approach is taken in Gandy [1974]. We have giventwo new proofs for this theorem, one which shows that the result can be derivedfrom the Barwise Completeness Theorem, the other a much more standardrecursion theoretic approach using the Second Recursion Theorem.

The recursion theoretic approach to Gandy's Theorem suggests an alternateapproach to the material in this book. One could prove Gandy's Theorem (bymeans of the Second Recursion Theorem) and then quote it to prove that theset T/zwA of theorems of an admissible fragment KA is Σί on A. This wouldsuffice for many applications of the Completeness Theorem, but not all. Someapplications actually need the notion of KA-proof used in § III.5, since there isimportant information coded inside the proof.

The approach taken here also has the advantage of stressing the interplay ofall branches of mathematical logic, which is one of the attractive features ofadmissible set theory.

3. First Order Positive Inductive Definitionsand

We have seen various ways in which HYP^ is a mini-universe of set theoryabove 901. For countable 9JΪ, we have seen that the relations on 9W which areelements of IHYP^ are exactly the Δ} relations. This characterization breaksdown for uncountable 9M (see Exercise VII. 1.1 6) so we are left with two problemsin the general case:

To characterize the relations on 9JZ which are elements of HYP^, andTo characterize the Δ} relations on $R in terms of HYP^.

The first of these two problems is solved by Theorem 3.6 below. The secondproblem is solved at the end of § VIII. 2.


3.1 Definition (Moschovakis). Let K be a language and let Φ be the set of allfinitary formulas of the form φ(R+), for any new relation symbol R. Let 91 bea structure for K and let 5 be a relation on 31.

(i) 5 is a (first order positive) fixed point on 91 if S is a Φ-fixed point (inthe sense of 1.12) on 91.

(ii) S is inductive on 91 of S is Φ-inductive on 91.(iii) S is coίnductive on 91 if —iS is Φ-inductive on 9ί.(iv) S is hyperelementary on 91 if S is inductive and coinductive on 91.

(For more intuition into the notion of coinductive, the student should doExercises 1.15—1.18.)

The theorems of this section are suggested by the following classical result.

3.2 Theorem. Let Jf = <ω,0, +, •) ana let S be a relation on Jf.(i) S is Πj on Jf iff S is inductive on ΛΛ

(ii) S is Δ} on Jf iff S is hyperelementary on J f .

Proof. We proved (i) in 1.10 and 1.11; (ii) is immediate from (i). D

Thus we see that for relations on Jf,

Σt on HYP^ = inductive on Jf,

element of HYP^ = hyperelementary on Jf.

We would like to generalize these equations from Jf to an arbitrary struc-ture 901. We would like to, but we can t because the generalization works onlyfor 9JI which have some built in coding machinery. We discuss just how muchcoding is needed in the next section. For now we simply state one special casewhere all goes smoothly, and then take a different tack.

3.3 Theorem. Let A be an admissible set and let S be a relation on A.(i) S is Σ! on HYP(A) iff S is inductive on A.

(ii) S is an element of HYP(A) iff S is hyperelementary on A.

Proof. We merely sketch a proof since this result is a special case of Theorem 3.8and the results of the next section. The proof sketched here is more direct. Asusual, (ii) follows trivially from (i). We first show that if S is inductive on A thenS is Σ! on HYP(A). It clearly suffices to prove the result for the case where Sis a fixed point Iφ of some first order positive inductive definition Γφ. Since φis first order over A it is Δ0 in HYP(A) so Γφ is, in particular, a Σ+ inductivedefinition over HYP(A), hence by Gandy s Theorem, Iφ is Σi on HYP(A). (Amore direct proof which works here but not in 3.8 is to observe that /£ is aHYP(A)-recursive function of β, for jS<o(HYP(A)), and use Σ Reflection toprove that ||ΓJ^0(HYP(A)). This would give the following ΣL definition of S:

S(x) iff

3. First Order Positive Inductive Definitions and IHYP^ 213

To prove the other half, suppose 5^ A is Σx on HYP(A). By Theorem IV.7.3(or, more precisely, Corollary 3.14 below) S is weakly representable in KPU'using the A-rule, where KPU' is the theory

KPU,

diagram (A),

xeΆ (all xeA),

3α Vt;

But the set CA(KPU') of consequences of KPU' using the A-rule is clearly aninductive subset of A. Thus we have

S(x) iff /(x)eCA(KPU')

for some A-recursive function /. An easy exercise (Exercise 3.20) establishesthat S is inductive on A. D

We have been deliberately sketchy in the above proof to give the student afeel for the main idea. This must be gone into in more detail to prove Theorem 3.8below, the main result of this section. First, though, let's draw some easy corol-laries of Theorem 3.3.

3.4 Corollary. If A is a countable admissible set then

Πj on A = inductive on A,

Δ} on A = hyperelementary on A.

Proof. This is an immediate consequence of Theorem 3.3 and the results of§ IV.3. D

3.5 Lemma. Let A be admissible.(i) There is an (n + ί)-ary inductive relation on A which parametrizes the class

of n-ary inductive relations on A.(ii) There is an inductive subset of A which is not hyperelementary.

Proof. By V.5.3, HYP(A) is projectible into A. Thus the lemma is just a restate-ment using 3.3. D

Using these results we can show just exactly how one gets from one ad-missible ordinal τα to the next admissible ordinal τα + 1. Namely

τα+ι = SUP {\\Γφ\\ '• Γφ ^s a first order positive inductive definition over L(τα)} ,

and this sup is actually obtained. This is a special case of the following result.


3.6 Corollary. Let A be admissible and let α = o(IHYP(A)). Then α is equal ttothe sup of all \\Γφ\\ where Γφ is a first order positive inductive definition over A,and this sup is actually attained.

Proof. We know that any first order positive inductive definition Γφ over A isΣ+ over HYP(A) (in fact "Δ0+") so ||ΓJ|<α by the second half of GandysTheorem. To show that α is such an ordinal \\Γφ\\, use 3.5(ii) to choose an inductivesubset S^A which is not hyperelementary. Then S is a section of some fixedpoint Iφ. Clearly Iφ is not hyperelementary either. We claim that ||ΓJ|=α. Asmentioned in the proof of Theorem 3.3, Iβ

φ is a HYP(A) recursive functionof β, for β<a. Hence /£eHYP(A) for all β<a. But then, if \\Γφ\\=β«*9

/φ = /£eHYP(A) which makes Iφ hyperelementary, a contradiction. D

As we'll see in the next section, the hypothesis that A is admissible is fartoo strong for the above results. All we really need is a reasonable amount ofcoding apparatus.

What we are really after, though, is a characterization of the relations on 50Ϊin IHYPg^ which works for all structures 501, not just those with built in codingmachinery. The best way around this is to slightly strengthen the notion of in-ductive definition, so that one can do the coding needed in the inductive de-finition itself.

3.7 Definition. Let Φ be the set of extended first order formulas φ(R+) of L*(R)as defined in II.2.7, p. 50. Let 501 be a structure for L and let S be a relationon 501 (or even HF^).

(i) S is extended inductive (written inductive*) on 501 iff S is Φ inductive on

(ii) S is extended hyperelementary (written hyperelementary*) on 50Ϊ iff S and—\S are inductive* on 50Ϊ.

Our second, and principal, generalization of Theorem 3.2 is the followingresult.

3.8 Theorem. Let 3K = <M,R1,...,R/> be a structure for L and let S be a rela-tion on 50Ϊ (or even on HF^).

(i) S is Σ! on HYP^ iff S is inductive* on 501.(ii) S is Δ! on HYP^ iff S is hyperelementary* on 501.

Its corollaries are analogous to those of 3.3.

3.9 Corollary. Let 9M = <M,R1,...,Rί) be a countable structure for L.(i) Π} on 501 = inductive* on 501.

(ii) Δ} on 501 = hyperelementary* on 501. D

3.10 Lemma. Let 501^<M,R1,...,KZ> be a structure for L(i) There is an (n + i)-ary inductive* relation on HFOT which parameterizes the

class of n-ary inductive* relations on HF^.(ii) There is an inductive* relation on HF^ which is not hyperelementary*.

3. First Order Positive Inductive Definitions and IHYRp, 215

Proof. HYP^ is projectible into HF^, again by V.5.3, so the results followsfrom V.5.6 and 3.8. D

We use these corollaries to get the most intelligible description yet of(and hence of HYP^ since HYP^-Lία)^ where α-

3.11 Theorem. // SBl = <M,R1,...,Λ/> is a structure for L then

O(9K) = sup{||ΓJ| I Γφ is an extended inductive definition over $R}

and this sup is actually attained.

Proof. The proof of 3.11 is exactly like the proof of 3.6 when 0(9W)>ω forthen HF^elHYP^. Suppose HYP^ has ordinal ω. Let Γφ be an extended firstorder inductive definition on 9JΪ. As we will see in the proof of Theorem 3.8,Γφ is Σ+ on HYR0J, so ||ΓJ|<ω by the second half of Gandy's Theorem. It issimple to give an example of extended first order inductive definitions oflength ω, e. g.,

xeΓ(R) iff "x is a natural number Λ Vy<x R(y)"

defines ω in HF with

so ||Γ||=ω. D

It is worthwhile digressing to compare 3.8 with the following consequenceof 3.3, just to make sure the student is not confusing two distinct things.

3.12 Corollary. Let 9Jl = (M9Rί9....9Rly be a structure for L which is not re-cursively saturated. Let S be a relation on $R (or even

(i) S is Σt on HYP^ iff S is inductive on(ii) SeHYPan iff S is hyper elementary on

Proof. Since 90Ϊ is not recursively saturated, o(HYPaϊϊ)>ω soBut then HYP(MFsw) = IHYPsW since HYP(HFM) is the smallest admissible setwith HF^ as an element. Thus 3.10 is a special case of 3.3. D

The student must be clear about the difference between inductive* definitionson 501 and inductive definitions on HF^. The latter are, in general, much morepowerful since they allow unbounded universal quantification over sets in HF^iri addition to the unbounded existential allowed by inductive* definitions.

We have already done most of the work for proving Theorem 3.8 back in§ III.3, the section on 9Jl-logic and the 90ΐ-rule.

In the discussion below we let 5tR = <M,JR1,...,.R/> be a jϊxed L-structureand we let L+ be an expansion of L with a new unary symbol M and symbols p


for each pe50l, just as in our discussion of 501-logic in § III.3. We assume thatL+ is coded up in an effective way on ΉF^.

3.13 Proposition. Let T be a set of sentences of L+ω which is Σx on HF^. LetCyχ(T) be the set of formulas of L*ω which are provable from T using the SOΪ-rw/e.Then CM(T) is inductive*.

Proof. We simply write out the original definition Γ of C^T) given in III.3.4and observe that it has the correct form. Let R be a new unary symbol anddefine Γ by

xeΓ(R) if x6liωΛ[(l)v v(5)]

where (1)...(5) are given below.

(1) (Logical Axioms) "x is an axiom of first order logic";

(2) (Nonlogical Axioms) xeT;

(3) (Modus Ponens) ly[yeR*(y^>x)eR']9

(4) (Generalization) "x is of the form ( -> Vi? θ(v)) where υ is not free in if/and (^-»θ(t;))e!Γ;

(5) (501-rule) "x is of the form Vt;0 [M(ι;0)->0(ι;0)] and for all peM, θ(p/ι;0)elΓ.

Clearly Γ defines COT(T), i. e., QR(Γ) = /Γ so that G^T) is actually a fixed point.Γ is definable over HF^ by an R-positive formula; the only unbounded universalquantifier is in (5) and it is a quantifier over M. D

The reader may remember that we left a couple of proofs unfinished in § IV.7,the section on representability using the 9W-rule. We proved IV.7.3 and IV.7.4in the countable case but left the absoluteness of those results until later. Prop-osition 3.13 allows us to finish these proofs.

3.14 Corollary. Assume the notation of Proposition 3.13.(i) xeCOT(T) is a Δt predicate of x, T and 9JI, Δx in the theory ZF.

(ii) Consequently, the proofs given in § IV.7 of IV.7.3 and IV. 1 A for the countablecase, together with Levys Absoluteness Principle, yield the general results.

Proof. Part (i) is a consequence of Remark 2.4. For (ii), the proofs ofIV.7.3 and IV.7.4 are quite similar. Since IV.7.3 is the more important for ushere (we apply it in the next proof) let us treat it in some detail. Again 7.3 (i)and 7.3 (ii) are similar so we prove (i). Suppose, as in the proof of (i), thatφ(x1,...,xπ,p1,...,pk,M) i saΣi formula with the property that for all qί9...,qnεM

WίPn*=φ(ql9...9qn9p9M) iff KPU+

Now, if 9W is countable we use the 9Jl-completeness theorem to write

ΉYPn\F=φ(ql9...9qn9p9M) iff KPU+ \-


I.e., we have for all countable SDΪ and all ql,...,qnεM:

t=φ(qι,...,qn9p,M) iff φ(q 1?..., .qB,p,M)eQDI(KPU+).

We claim that this is a Δ! predicate of ΪR, Δt in ZF. The right hand side of theiff is Δx by (i), and the left hand side is Δx since satisfaction is Δx and since HYP^is a Σ! operation of 9JI by the argument given in IV.3.5. By Levy Absoluteness,the result holds for all 9W. D

Theorem 3.8 will follow from Proposition 3.13 given the next lemma. It is aspecial case of the Combination Lemma of Moschovakis [1974].

3.15 Lemma. Let l/cHFTO be inductive*, let /: HF^ -> HF^ be Σ1 onand let P be defined by

P(x1?. ..,*„) iff /(x1,...,xπ)eL7.

Then P is inductive* on 9JΪ.

Proof. Suppose U is a section of the fixed point Iφ where φ(vί9v29R+) is ex-tended first order positive on 951, say

U(y)»((y,z0)elφ).

We define an rc + 3-ary inductive* definition Γφ so that a section of /| (with i = 0)imitates /J and the section with ί = l takes care of /. Define ^(ι,x1,...,xπ,ί;1,ι;2,S+),where S is π + 3-ary, by the following, where ί1,...,ίπ,z1,z2 are arbitrary butfixed elements of

i = 0 Λ x = ΐ Λ ^(t;!, ι?2, Λ,W! w2 S(0, ίl5 . . . , ίπ, w1? w2)/R) , or

A simple proof by induction shows that

(6) φl9υ2) iff IΪ(0,tl9...,tΛ9Όl9v2)

so that

U(y) iff ^(0,ί1,...,ίπ,};,zo).

Another proof by induction, using (6), shows that

(/(x1,...,xπ),z0)e/J iff (l,x1,...,xll,z1,

Thus

P(xl9...,xπ) iff (l,x1,...,xn,z1,


so P is a section of Iψ. The only universal quantifiers in ^ are those in φ so iffis extended first order positive. D

We now return to prove the main theorem of this section, Theorem 3.8.

3.16 Proof of Theorem 3.8. (i) Let Γφ be an extended first order inductive de-finition over 9JI. Since HF^ is a Σ^ subset of HYP^, relativizing the unbounded(existential) set quantifiers in Γφ to HF^ and relativizing the unbounded quan-tifiers over 30Ϊ to the set M turns Γφ into a Σ+ inductive definition over HYP^and hence Γφ has a Σ1 fixed point Iφ, by Gandy's Theorem.

To prove the other half, let us consider a relation S on 9W which is Σl onHYP^. By Theorem IV.7.3, S is weakly representable in KPU+ using the 9R-rule.Thus there is a formula φ(vi,...,υn) of L* such that for all x1,...,x f l6M,

S(xl9...,xJ iff φ(x1,...,x l l)eQw(KPU+).

Now, by 3. 1 3, Cαϊί(KPU+) is inductive* over SR. Let f(x^ , . . . , xj = φ(xjvί , . . . , xn/vn).Then

S(x1?...,xJ iff /(x1,...,

so S is inductive* by Lemma 3.15. The same proof works if S^ΉF^ except thatExercise IV.7.5 replaces Theorem IV.7.3. Part (ii) follows from (i) as usual. D

The final results of this section show that for nonadmissible sets of the formL(α)sw (for example), Σ+ inductive definitions are just as strong as arbitrary firstorder inductive definitions, and that they are just as long. The results thus yieldpartial converses to the results of § 2 by showing how necessary the assumptionof admissibility was for those results.

3.17 Theorem. Let M^a where a is transitive in WM and let β be any limitordinal such that

is not admissible.(i) A relation S on A^ is Σt on HYP(Aarι) iff S is Σ+ inductive on A^.

(ii) The ordinal o(HYP(Aαri)) is equal to

sup{||ΓJ| I Γφ is a Σ+ inductive definition on Am}

and the sup is actually attained.

3.18 Corollary. Let M<Ξα where a is transitive in Ψ^ and let β be any limitordinal. Let


The following are equivalent, where & =(i) Am is admissible.

(ii) Every Σ+ inductive set on Am is Σ: on A^.(iii) For every Σ+ inductive definition Γφ on A^, ||ΓJ|^α.

Proof. By the results of the previous section, (i)=>(ii) and (i)=>(iii). To prove(ii) => (i), suppose A^ is not admissible. Let S be a subset of A^ which is Σ: onHYPίAan) but not HYP^^-finite; such an S exists since HYP^) is pro-jectible into A^. But then 5 is Σ+ inductive on A^ by 3.17. S cannot be Σx onASH for then it would be Δ0 on ΉYP(Aml hence in HYPίA^). Thus -ι(i) => -ι(ii).For the same reason, the length ||ΓJ| of an inductive definition of S could notbe ^ α so ~ι(i) => ~i(iii). D

The proof of Theorem 3.17 uses ideas similar to those used in the proofs ofTheorem 3.3 and 3.8. We leave a few of the details to the student.

Proof of Theorem 3.17. We prove (i) assuming A^ is countable, leaving the exten-sion (via Levy's Absoluteness Principle) to the student. The (<=) half of (i) is ob-vious, so let S be a relation on A^ which is Σ1 on HYPΐA^). Every xeHYP(Aan)has a good Σ: definition with parameters from L(α,jS)u{L(α,β)} by Π.5.14. SinceAm is not admissible, β and hence L(a,β) also have Σx definitions on HYP(Aaϊl)with parameters from L(a,β) by the last step in the proof of II.5.14. Thus everyxeWYP(A^ has a Σ^ definition with parameters from L(α,β). But then S hasa Σ! definition (as a subset, now, not an element) with parameters from L(α,β)since the other parameters can be defined away. Thus suppose that for all

S(x) iff MYP(Am)\=φ(x,y)

where yeL(a,β) and φ is Σx. By the Truncation Lemma S(x) is equivalent to

(7) for all 23^ =>endAm, if ^t-KPU then 95W N φ(x,y).

Since β is a limit, L(a,β) is closed under pairs, union and Δ0 Separation so wemay code up K=L*u{x |xeA a R } on A^. Let KA be the (nonadmissible) frag-ment of K^,^ given by A^. Let T^ KA the the theory

KPU

Diagram (A^)

Vϋ[ι;ea<->\/Jceaι; = x], for all

Every model of T is isomorphic to some ®αn^end^scn so (7) is equivalent to


By Theorem ΠI.4.5 (really III.4.6) this is equivalent to saying that φ(x,y) is inthe smallest set of sentences of KA containing T and (Al)—(A 7) which is closedunder (Rl)—(R3). This clearly amounts to a Σ+ inductive definition Γφ such that

R(x) iff φ(x,y)e/Γ.

Therefore R is Σ+ inductive by Exercise 3.21.To prove (ii) we need only find a Σ+ inductive definition on A^ with length

o(]RΎP(Am)). Let RÂm be HYPίA^-r.e. but not an element of HYPCA^).There is such an R since ΉYP(Ay^) is projectible into Am by V.5.4. Then R isa section of Iφ, where Γφ is some Σ+ inductive definition. But now the argumentused earlier, in the proof of 3.6 for example, shows that ||ΓJ| = o(HYP(Aw)). D


3.19. Let Cm(KP\J+) = Iφ, where Γφ is extended inductive, by 3.13. Show thatO(9K) = ||ΓJ|. Thus, for example, 0($R) is just the least ordinal not assigned toa proof using the ΪR-rule, under the usual assignment of ordinals to proofs.

3.20. Let 9X be a structure, let U be inductive on 91 and let /: An-+A be firstorder definable. Modify the proof of 3.15 to show that

P(x) iff l/(/(x))

defines an inductive relation on 51.

3.21. Let 5ί be a structure, let l/â be Σ+ inductive on 91, let /: AnÂ havea Σ! graph and define P by

P(xl9...,xJ iff /(x1?...,xπ)6l/.

Show that P is Σ+ inductive on 91. [Mimic the proof of 3.15.]

3.22. Give the absoluteness argument for lifting Theorem 3.17 from the countableto the uncountable.

3.23 Notes. The main results of this section are from Barwise-Gandy-Moschovakis [1971], at least in the case of pure admissible sets. Theorem 3.17and its corollaries are new here.

4. Coding HFOT on

A pairing function on a set M is simply a one-one function mapping M x Minto M. An n-ary function / on a structure 9JI is inductive (or hyperelementary)if its graph is an (rc + l)-ary inductive (or hyperelementary, respectively) relationon 501. In this section we show how to code HF^ on 50ί using an inductive pairingfunction on 501. Our goal is to prove the following theorem.

4. Coding MF^ on <0l 221

4.1 Theorem. Let 9Jί = <M,Λ1,...,R/> be a structure with an inductive pairingfunction. The inductive and inductive* relations on SOΐ coincide.

We give the applications of this theorem (and a couple of related resultsobtained along the way) in the next section by showing how a great many resultson inductive relations on 9K can be obtained in a simple fashion by projecting therecursion theory of HYP^. In so doing, we tie up the theory of admissible setswith the theory of inductive relations as developed in Moschovakis [1974]. Sinceour aim in these sections is to relate our theory to Moschovakis theory, we feelonly mildly apologetic for using without proof two results (4.2 and 4.3 below)from Chapter 1 of Moschovakis [1974]. The proofs are sketched in Exercises4.17 and 4.18.

A relation P on 501 is defined from Q by hyperelementary substitution if thereare hyperelementary functions fly...,fk so that

P(x1? . . . , xj iff Q ( f ί ( x ί 9 . . . , xj, . . . , /k(xls . . . , xj)

for all x l 5...,xne9W.

4.2 Theorem. The inductive relations on SJi contain all first order relations andare closed under Λ , v , 3, V and hyperelementary substitution. Hence, the hyper-elementary relations on 501 contain all first order relations on SDΪ and are closedunder ~Ί, Λ , v , 3, V and hyperelementary substitution.

Proof. This result follows easily from 4.3. See Theorem 1D.1 of Moschovakis[1974] or Exercise 4.18. D

The inductive relations on 901 are closed under induction in a sense madeprecise by 4.3.

4.3 Theorem. Let Sl9...9Sk be relations on 50Ϊ and consider an inductive defi-nition Γφ over the expanded structure (^ΰt9Sl9...9Sk)9 where φ is of the formφ(x1,...,xπ,R+,S1,...,Sk) in Lv{R,Sl9...,Sk}.

(i) // Sl5...,Sk are hyperelementary on 9JΪ then the fixed point Iφ defined on(3)l9Sί9...9Sk) is inductive on the original structure SOΪ.

(ii) If Sί9...9Sk are inductive on 50ί then the conclusion of (i) still holds providedφ is Si-positive for i = !,...,&.

(iii) In either case (i) or (ii), Iφ is a section of a fixed point 1^ for some^(x1,...,xm,R+)eLu{R} with \\Γψ\\>\\Γφ\\.

Proof. See Theorem 1C.3 of Moschovakis [1974] or Exercise 4.17. D

There is one simple consequence of 4.2 that deserves mention. If / is an in-ductive function on ΪR and if its domain D is hyperelementary (e. g., if / is total)then / is hyperelementary, since

iff (x1,...,x^/)v3z/(x1,...,xn) = z A


Thus, if 9W has an inductive pairing function p, p is actually hyperelementarysince p is total.

The plan for the proof of Theorem 4.1 is simple. Fix an inductive pairingfunction p on 9JΪ. We are going to use p to assign notations to the elements ofHFM. The set T of notations will be inductive on 9W but not, in general, hyper-elementary. An extended first order formula of the form

will translate into

3x(xeTΛ )

which will keep us within the class of inductive relations since the inductiveset T occurs positively. On the other hand, a quantifier of the form

VαeHFw(...)

would translate into

Vx(xφTv~)

which is not permitted since T occurs negatively. The only complications in theproof are caused by the following two facts. Since {p,q} = [q,p] we are not goingto be able to have unique notations for the elements of HF^. Secondly, we mustfind some way to handle bounded universal quantifiers in a positive way. (Thisaccounts for the relation S used below and most of the other complications.)

The notation system used is based upon the fact that HF^ is the closure ofMu{0} under the operation

Define a hierarchy HF^} as follows:

This hierarchy grows more slowly than the HPm(n) hierarchy used in § II.2 butit eventually gets the job done.

4.4 Lemma. HF^U^HFff.

Proof. Suppose there were some set αeHF^ which did not appear at any stageof our new hierarchy. Among such sets a choose one of least rank and, amongthose of least rank, choose one of smallest cardinality. Since OeHF^, a is non-empty so we may write

4. Coding ΉFTO on $ΐ 223

Let α0 = {x l5...,xk}. Since rk(α0)^rk(α) and card (α0) < card (a), α0 is formedin our new hierarchy, by choice of a. Since rk(xk+1)<rk(α), xk+l is also formed.Pick n so that both a0 and xk+1 are in HF$}. Then a = S(a0,xk + 1) is in HF^ + 1). D

Let M be an infinite set with pairing function p: M x M ->M. Let x0, χ l s x2

be distinct elements of M. We use the following notational conventions.

0 for P(XO,XO),

x for p(x1?x),

xόy for /?(x2,p(x,y)).

4.5 Lemma. The functions /15 /2 defined below are one-one, they have disjointranges and 0 is in the range of neither. They are HF(arι ^-recursive and hyper-elementary on (9Jl,p):

fι(χ) = χ f2(χ,y)

Proof. This is immediate since p is one-one and x0, x l 5 x2 are distinct. D

We use these functions to define two sets of closed terms: the ur-terms denoteelements of M; the set-terms denote hereditarily finite sets over M.

4.6 Definition, (i) For each xeM, x is an ur-term and x denotes x, written

The set of ur-terms is called Tu.(ii) The set Ts of set-terms and the function | | mapping 7^ onto HFM are

defined inductively:a) 0 is in 7^ and 0 is a notation for 0, i. e.,

101 = 0.

b) If x is in 7^ and y is in TuvTs and if \y\φ\x\ then xόy is in 7^ and

\xόy\ = \x\u{\y\}.

(iii) The set T of all notations is TuuTs.

We require |y|^|x| to keep the set of notations of each αeHFM finite.

The definition of 7^ is an inductive definition, not over (9K,p) but rather overΊHF(arifp). One of our tasks is to show that 7^ is actually inductive over (2ft, p)after all.


Note that by Lemma 4.4, every αeHFM is |x| for some xeTs. Define thefollowing relations on M:

iff x,yeT and |x|e|y|;

x<ί y iff yeT and if xeT then

x&y iff x, yeT and |χ| = |j;|;

x£y iff yeT and if xeT then

4.7 Main Lemma. The sets Ts, T and the relations $, δ, «, and & are all in-ductive on (M,p). The set Tu is definable on (M,p).

Proof. It is clear that Tu is definable on (M,p) since

yeTu iff

We will give an informal simultaneous inductive definition of the six other rela-tions as well as two auxiliary relations R and R. First, however, let N be thesmallest subset of M containing 0 and closed under

if xeΛΓ then (xdx)eJV.

Thus N is inductive on (M,p) and N contains a unique notation for eachnatural number. We will confuse a natural number with its notation in thisproof. Define

R(n,x) iff neN and xeT s and |x|eHFjS?;

R(n,x) iff neN and if xeTs then \x\φΉF$ .

The following clauses constitute a simultaneous inductive definition of all theabove relations. It should be pretty obvious to the reader how one could turnthis into one giant inductive definition over (M,p) and then extract the givenrelations as sections. (If he needs help, the student can consult the SimultaneousInduction Lemma on p. 12 of Moschovakis [1974].)

(1) xe'ζ iff x = 0 or there is a yeTs and a zeTuvTs such that z$y andx is yόz.

(2) xeT iff xeT^ or xeT^.

(3) xS'y iff ye Ts and y is of the form uόv and x$u or x«t;.

(4) xS'y iff yeT and y is 0 or ye Tu or y is of the form uόv and x<f uand x£ι;.

(5) x&y iff x,yeT and x=y or x,yeTs and for every z (z$xvz$y) and(z^yvzS'x).

(6) R(Q,x) iff x = 0;

4. Coding HF^ on SR 225

l,x) iff x e 7^ and R(n,x) or else x is of the form yόz whereand (zeTu vR(n,z)).

(7) £(0,x) iff

R(n + l,x) iff £(rc,x) and either x is not of the form u 6v (for all u, υ) orelse x is of the form uόv but one of the following holds:

υδu, R(n,u), R(n,v).

(8) x&y iff there is an neN such that R(n,x) but R(n,y) or there is an nεNsuch that R(n,x) and R(n,y) (in which case x is in 'ζ) and there is a z such that

((ZS'X Λ Z<ί }>) V (z<f y Λ Z<?X)) .

It takes a bit of checking to see that in each case the induction is pushed back,but this checking is best done on scratch paper. D

The relations R, R used above are needed only to prove the Main Lemma.They should not be confused with other relations R used later on.

We are now ready to fill in the outline of the proof of Theorem 4.1. Forsimplicity of notation let us suppose our language L has only one binary sym-bol Q. Let R be a new relation symbol for use in inductive definitions. We con-sider L*(R)=L(e,R) as a single sorted language with unary symbols U (forurelements) and S (for sets) with bounded quantification as a primitive. We letK be a new language with atomic symbols

Q, U, S, R,f,£, *, &.

We define a mapping Λ from L*(R) into K as follows: given φeL*(R), firstpush the negations inside as far as possible so that the only negative sub-formulas in φ are negated atomic. Replace each positive occurrence of xey byx$y, each occurrence of —\(xey) by xSy, each positive occurrence of x = yby x&y, each occurrence of ~~\(x = y) by x&y, each bounded quantifer

Vxey(...) by Vx(x^y v...),

) by

Thus, in φ, all occurrences of <f , /, «, £ are positive. If φ is extended first orderthen S also occurs positively in φ since it only appears in the contexts

3x(S(x)Λ...)and

3x((U(x)vS(x))Λ...).

Let M be the infinite set with pairing function p used above. Let Q be anybinary relation on M. Define Q on Tu by

Q(p,q) iff Q(p,q)


for all p,qεM so that map t-»|t| gives an isomorphism of (TU,Q) onto9W = (M,Q). We let 90Ϊ be the structure for K with universe M and with inter-pretations given by

symbol: U S Q. δ δ « «

interpretation: Tu Ts Q £ β w *

Thus U , Q are interpreted by (hyper)elementary relations; the other symbols(which will occur positively in φ whenever φ is extended first order) are inter-preted by inductive relations so things are set up to apply Theorem 4.3 (i), (ii).

Given an n-ary relation R on HF^ we define R on T by

R(tl9...9tn) iff tfdίj,...,^!), for tl9...9tneT.

4.8 Lemma. For any formula φ^,...,^, R)el_*(R), any relation R on HF^, andany ί1,...,ίkeT w

Proof. By induction on formulas φeL*(R). For atomic and negated atomicformulas, it follows by the definitions. The induction step is immediate sinceevery xeHF^ is denoted by some term ί. D

4.9 Lemma. Let φ(xί9...,xn9 R+)eL*(R). For each a and each ί1 ?...,ίπeT we have

(lίj,...,!^!^ iff (ίl9...Λ)e/!,

where the induction on the left is over HF^, that on the right over Ώΐ.

Proof. By induction, of course. The induction hypothesis asserts that

(|f1 |,...,|fj)e/φ

< β iff (tl9...,tjel£*9

i.e., that (Ί^) = I^. But then

iff (n^J<^φ(\tll...,\tnlR+)

iff (m9I?«)\=φ(tl9...9tn9R + ) (by 4.8)

iff (ί l 5...,ge/. D

We are now ready to prove Theorem 4.1. The following result comes out ofthe proof.

4.10 Corollary. Let Wϊ be a structure for L with an inductive pairing function.If Γφ is an extended first order inductive definition over 501 then there is a firstorder inductive definition Γψ over SOΐ with ||/^||^||/^||.

4. Coding M¥m on 9CK 227

Proof of Theorem 4.1 and Corollary 4.10. Let 50ΐ = <M,Q> be an L-structure andlet p be an inductive, hence hyperelementary, pairing function on 50Ϊ. By 4.2 (i),50Ϊ and the expanded structure (501, p) have exactly the same inductive and hyper-elementary relations. Thus TU,Q are hyperelementary on 50i, and 'ζ, <?,<?, «,and £ are inductive on 501. Let S<^M" be inductive*. Choose an extended firstorder inductive definition Γφ and parameters y l5...,)>keMuIHFOT such that

S(xί9...,xn) iff (xί9...,xn,y)elφ.

Now consider the inductive definition Γ^ over 501. By the above lemma \\Γφ\\ =\\Γ$\\and, for any f1,...,t I I + keT,

(ί l5...,fπ+/c)e^ iff (\t,\,...,\tn

By Theorem 4.3 (ii) and the remarks above about the relations Ts, $, $, « and «all occuring positively in φ, I-φ is inductive over the original 50Ϊ. Choose ί l 5...,f k

with |ίιl=3Ί,...,|ί f c |=^k. Then, for all x1,...,xπeM,

S(x1?...,xn) iff (x1,...,xn,ί1,...,ίk)6/^

so 5 is obtained from the inductive set /^ by hyperelementary substitution and,hence, is inductive. By 4.3 (iii) there is an inductive definition Γφ over 501 withl l^l l^l l^l l = l l^ l l» so this also proves the corollary. D

The notation system we have been using can be seen to be a notation systemin the precise sense of § V.5. This follows from the next lemma. We assume thenotation from above.

4.11 Lemma. Define a function π on HF(M p) by

Then π is a total HF(5[R ^-recursive function.

Proof. Given a set a of cardinality ^1, we call a pair (α0,x) a splitting of a ifa — α0u{x} but xφa0. Let

Spl(a) = {(a0,x)\(aQ,x) is a splitting of a}

for all αeHFM. It is a simple matter to check that Spl is IHF^-recursive. Wefirst define π more explicitly and then discuss the method used to see that thedefinition is HF(an p)-recursive. The definition of π parallels the proof of 4.4.

π(p) = {p} for

π(0) = {0}.


For nonempty sets α, π(a) is defined by a double induction, first on rk(α) and,among sets of the same rank, on card(α). So suppose π(x) is defined for all xεaand all xâ with card (x)< card (a). If α = {x1,...,xll} with nî then we lookat any splitting (α0,x) of a. Now π(α0), π(x) are defined and, for ί0eπ(<20), |ί0l

= floand for ê φc), |ίιl = x so \t06t1\=a0v{x}=a. Thus we may define

π(α) = {ί0dί1: for some (α0,x)eSpl(α), ί0eπ(α0) and tίeπ(x)}.

With this definition π is clearly HF(an>p)-recursive by the Second RecursionTheorem. D

4.12 Theorem. Let 50ί = <M,R1?. ..,#,> be a structure for L(i) // 501 has an HF^-re cursive pairing function then HF^ is projectίble

into 501.

(ii) // 50ΐ has a ΉΎP^-re cursive pairing function then IHYP^ is projectibleinto 50Ϊ.

Proof, (i) The sets in HF^ depend only on M, not on the whole structure 501,so if we add a pairing function p to 501, HF(9R>P) has the same sets as HFW. ByLemma 4.11, HF(SW>p) is projectible into 501; i.e., there is an HF(αn p) recursivenotation system π with Dπ^M. But then, if p is HF^-recursive, π is also HFM-recursive. The proof of (ii) is similar. Let p be a HYP^-recursive pairing functionso that HYP^ and HYP(aKfp) have the same universe of sets. By V.5.3 we havea notation system π0 for HYP^ with Dπ ^HFW. By 4.11, there is a HYP(M p)-recursive map πγ on HF^ with πί(x)^M, π1(x)nπ1(y) = 0 for x^y. Let π bedefined by

Then π is a notation system for HYP^ with Dπ<^M. D

The following special case of 4.12 (ii) will be of great use to us in the nextsection.

4.13 Corollary. Let SDl = <M,R1,...,R ί> be a structure for L with an inductivepairing function. Then ΉYP^ is projectible into 50ί.

Proof. If p is an inductive pairing function on 501 then it is hyperelementary andhence an element of HYP^. Thus 4.12 (ii) applies. D


4.14. Let 5DΪ = <M, ~> where ~ is an equivalence relation on M which exactlyone equivalence class of each finite cardinality. Define

x<y iff card (x/~)< card ( y / ~ ) .

4. Coding HFOT on Sϊl 229

(i) Prove that < is Σx on HF^ and hence is extended inductive on 9JΪ.(ii) (Kunen). Prove that < is not inductive on 9K.

(iii) Prove that o(HYPaϊl)>ω.

4.15. This exercise introduces the Moschovakis[1974] notions of acceptable andalmost acceptable structures. A coding scheme %> for a structure SCR consists of:

(a) a subset N* of M and a linear ordering <* of N^ such that

<AT*, <*>^<ω, <>, and

(b) an injection < >* of the set of all finite sequences from M into M .Given a fixed coding scheme ^ we use 0, 1,2,... to indicate the appropriate

members of N^ as ordered by <*. Associated with a coding scheme # there aresome natural relations and functions.

Seq*(x) iff x = < >* or x = <x1,...,xll>* for some π and some x1?...,xπ.

/Λ*(x) = 0 if= ή if Seq^(x) and x =

q^(x,m) = xm if for some x l 5...,xw, x = <x1,...,xπ>*' and= 0 otherwise.

A structure 90Ϊ is almost acceptable (or acceptable) if M has a coding scheme #with all of N^, <*, Seq^, Ik*, q* hyperelementary (or first order, resp.).

(i) Show that every almost acceptable structure has an inductive pairingfunction.

(ii) Let 9M be a structure with an inductive pairing function. Show that M isalmost acceptable iff M is not recursively saturated. [It is easy to see that if SDΪis almost acceptable then o(HYPaw)>ω. To prove the converse use Corol-lary 4.10.]

4.16. Show that all models of Peano arithmetic, KPU and ZF have definablepairing functions, even the recursively saturated ones.

4.17. Let 50l = <M,R1,...,Λ ί> be an infinite structure and let Γψ be an inductivedefinition over 90Ϊ, say ^ = ,...,114,8+). Now let 90ΐ' = (aR,S) where S isdefined by:

S(xl5x2) iff (x1,x2,al,a2)elφ.

Let φ(ι;1,...,u3,S+,T+)eLu{S,T}, where S is binary (to denote S) and T is 3-ary(to be used in an induction) and let Γφ be the natural inductive definition over($R,S) given by φ. We are going to outline the proof from Moschovakis [1974]that Iφ is inductive over the original structure 9W, thus proving Theorem 4.3.


Let 0,l,w1,...,I74,"y1,...,F3 be constants from M with 0^1. Let Q be a new8-ary (8 = 1+4 + 3) relation symbol and define θ(ί,M1,...,M4, ι? l 9...,ι?3,Q+) by

[l=θΛ ^(M 1,...,M 4, Q(0, , , , •,U1,^2,^3)/R] V

r^ι,β2Λ^2^3)/S,Q(l,w1,ΐ72,i73,ΐI4, , , )/T].

Consider the induction definition Γθ over $R.(i) Prove that for each α,

(M 1 9 . . .,M 4 )6/; iff (0 ,M 1 , . . . ,M 4 , t? 1 , . . . ,

and hence

(M l 5...,M4)e/^ iff (0,M1 9...,M4, ?!,...,

(ii) Prove that if (l,!^,...,^, ^,...,1)3)6/0 then (u l5...,ι;3)e/J.(iii) Prove that if (ι;l5...,ι;3)e/5 then for some jβ, (1,M1 5...,M4, t; l5...,t;3e/£, by

induction on α, using (i).(iv) Use (ii), (iii) to conclude that Iφ is a section of Iθ and hence is inductive

on an.(v) Show that IIΓJ^HΓJ.

(vi) Prove Theorem 4.3.

4.18 Use Theorem 4.3 to prove Theorem 4.2 [For example, show that if Sl5 S2

are inductive on Wl then 8^82 is inductive on (9Jl,Sl5S2) with an inductive de-finition in which S l5S2 occur positively.]

4.19 Notes. The fact that an inductive pairing function suffices for codingon 9JΪ goes back, indirectly, to Aczel [1970]. The proof of Theorem 4.1 givenabove owes much to ideas of Aczel and Nyberg.

5. Inductive Relations on Structures with Pairing

Inductive and coinductive definitions appear in most branches of mathematics.Spector [1961] was the first to focus attention on them as objects worthy of studyin their own right, but then only over the structure Jf of the natural numbers.The development over an absolutely arbitrary structure 9JΪ was not carried outuntil Moschovakis [1974] produced his attractive and coherent picture. Ourobject in this section is to view portions of Moschovakis picture as projectionsof ΉYP^.

Let us summarize the results at our disposal.

5.1 Theorem. Let 9K = <M,R1,...,R ί> be a structure with an inductively definablepairing function. Let S be a relation on $01.

5. Inductive Relations on Structures with Pairing 231

(i) S is inductive on 9M iff S is Σ1 on(ii) S is hyper elementary on 9W iff

(iii) 0($0ΐ) is equal to

sup{||ΓJ| I Γφ is first order positive inductive on 501}

and this sup is attained.(iv) HYPjCT is projectible into 50Ϊ.

Proof. Part (i) follows from Theorems 3.8 and 4.1; (ii) follows from (i). Part (iii)follows from Theorem 3.11 and Corollary 4.10. Part (iv) is Theorem 4.12 (ii). D

We want to use this theorem to obtain some of the results in Moschovakis[1974]. In order to facilitate comparison we use the same names for theoremsas in Moschovakis, even when our theorem is a little more or a little less general.

5.2 Corollary (The Abstract Kleene Theorem). // 9M = <M, Rί9 . . . , Rz> is a counta-ble structure with an inductively definable pairing function then the Π\ relationscoincide with the inductive relations on 9M.

Proof. Both classes of relations coincide with the class of relations on 9JΪ whichare Σ, on HYP^ by 5.1 and § IV.3. D

Notice that this result makes no reference to admissible sets; it is only inthe proof that they appear. The same remark applies to many of the resultsbelow. In order to make this more obvious we use Moschovakis notation

K** = sup {||ΓJ| I Γφ is a first order positive inductive definition over M} .

Thus κm = O(50ΐ) if 50Ϊ has an inductive pairing function. In this section 50Ϊ alwaysdenotes a structure (M,^,...,!^) for the language L.

5.3 Proposition (The Closure Theorem). Let 9JΪ have an inductive pairing functionand let φ(xί9...,xn9 R+) define Γφ over 9JZ.

(i) For each α</caϊϊ, Pφ is hyperelementary on 9JΪ.(ii) Iφ is hyperelementary iff \\Γφ\\<κm.

Proof. I* is a HYP^-recursive function of α, for αeHYP^. Hence each/^elHYP^ for αeHYP^ and is thus hyperelementary by 5.1 (ii). This proves (i)and the (<=) half of (ii). Consider the map pφ defined on Iφ by

pφ(x) = least β(xefy.

This is clearly ΉYP^-recursive. If /^elHYP^ then, by Σ Replacement

exists in HYP and is thus less than κm. D


One of the awkward points in the theory of inductive definitions (when notdone in the context of admissible sets) is that one needs to deal with ordinals butthe ordinals are not in your structure. To get around this difficulty, Moschovakisintroduces the concept of an inductive norm. A norm on a set S is simply amapping p of S onto some ordinal λ. We use

to indicate that p is a norm mapping S onto λ. Given p: S-^ A, define

x^py iff xeSΛ(yφSvp(x)^p(y)),

x<py iff xεS/\(yφSv p(x)<p(y)).

A norm p:S^>λ is inductive on 9ΪI if the relations ^p and <p are inductiveon 9JI. Notice that if p:S-++λ is inductive then S is inductive since S(x) iff x <px.

The most natural inductive norms are those on fixed point Iφ defined by

(To see that this norm p = pφ is inductive observe that

x^py iff

x<py iff

and the relations on the right are clearly Σ: on HYP^, hence inductive on 9JI.)One of the most useful lemmas on inductive definitions is the Prewellordering

Theorem which asserts that every inductive set has an inductive norm. In termsof admissible sets, this is a consequence of the fact that HYP^ is resolvable, in fact

where α = O(9Jl). Most of the consequences of the Prewellordering Theorem inMoschovakis [1974] are actually obtained more easily from this equation. Seefor example, Exercise 5.19 for the Reduction and Separation Theorems.

5.4 The Prewellordering Theorem. Let $R have an inductively definable pairingfunction. Every inductive relation S on Wl has an inductive norm.

Proof. Let S be Σl on IHYP^ , say

S(x) iff

where φ is Δ0 and α = o(HYR0ί) = κ:απ. Let R be the HYP^-recursive predicategiven by

R(β,x) iff lzeL(β)wψ(x,z)so

S(x) iff 3βR(β,x).


Now the map f on S defined by

/(*) = least βR(β,x)

is not onto an ordinal so it is not a norm. Define p on S by

p(x) = {yeM\3γ<f(x)R(γ9y)}.

Now

y<x iff yep(x)

is a well-founded relation so its associated rank function p = p< is a norm. Weclaim it is inductive on SOΪ. To see this observe that

y<px iff yeS and Vβ*ζf(y)-ιR(β,x),

y^px iff yeS and xφp(y)

so both relations are Σl on HYP^, hence inductive on 9Jί. D

The Closure Theorem shows that every fixed point Iφ is the uniform limit ofhyperelementary sets, the Iβ

φ. The Prewellordering Theorem allows us to extendthis from fixed points to arbitrary inductive sets. If p : S ->-> λ then p endows Swith stages Sβ

β in a natural way:

The Boundedness Theorem, Corollary 5.6, is the natural generalization of theClosure Theorem.

5.5 Theorem. Let 9Jί be a structure with an inductive pairing function. Letp\S^+λ be an inductive norm on a relation S.

(i) ^oOHYP^) and p is WLYP^recursive.(ii) For each α<o(HYPαR), ^eHYP^ and, as a function of α, S* is a MYPm-

recursive function.

Proof. Define a function p with domain S by

p(x) = {yeM\p(y)<p(x)}.

For xeS,

so p(x)eWYPyjl by Δt Separation. Further, p is HYP^-recursive since its graphis Σ! definable:

p(x) = z iff


Now we may apply V.3.1 to p. Define

-<x iff

and note that -< is well founded since y-<x implies p(y)<p(x). But then p isIHYR^-recursive by V.3.1 since

This proves (i). To prove (ii) first define Q(β,x) by

Q(j8,x) iff β<λ and

We claim Q is HYP^-recursive. The clause β<λ causes no trouble since eitherλ = o(\RΎPm) in which case the clause is redundant or else A<o(HYPαR) inwhich case "β<λ" is Δ0. But for β<λ

Q(β,x) iff

-ιQ08,x) iff

so Q is Δ! on HYP^. But

so SjeHYPjtf by Δ! Separation. The graph z = Sβ

p is Σl since it is equivalent to

so (ii) holds. D

5.6 Corollary (The Boundedness Theorem). Let Wl be a structure with an induc-tive pairing function. Let p : S -H> λ be any inductive norm.

(i) Kfc*.(ii) For each α<κ:ari, S£ is hyper elementary.

(iii) S is hyper elementary iff λ<κm.

Proof. The only part left to prove, after Theorem 5.5, is that if S is hyper-elementary then every inductive norm p: S->-»λ has λ<κm. This follows byΣ Replacement since

λ = sup{p(x)\xeS}

and p is HYP^-recursive. D

The next result, the Covering Theorem, is one of the most useful consequencesof the Boundedness Theorem. We state only the special case that we need inthe Exercises.


5.7 Corollary (The Covering Theorem). Let $R be a structure with an inductivepairing function. Let S be an inductive subset of 9J? and let TC S be coinductίveon 9JΪ. Let p\S^+λ be any inductive norm on S. Then T is a subset of one ofthe hyperelementary resolvents Sβ

p for β < κm.

Proof. Suppose that the conclusion failed. Then we could write

which makes M — S a Σί subset of HYP^ and hence SeHYP^ since S is alsoΣ! on IHYPj,,. But then S = Sλ

p and λ<κ™ by 5.6, so T is, after all, a subset ofthe hyperelementary resolvent S*. D

We now return to more familiar matters.

5.8 Theorem. Let 9JΪ be a structure with an inductive pairing function. For eachn^ί there is an inductive relation of n + ί arguments that parametrizes the classof n-ary inductive relations.

Proof. In view of 5.1(iv), this is just a restatement of V.5.6. D

As always, we have the following corollary, to be compared with 5.13 below.

5.9 Corollary. // 9JI is a structure with an inductive pairing function, then notevery inductive relation is hyperelementary. D

Some further uses of HYP^ in the study of inductive relations are sketchedin the exercises, see especially 5.19, 5.23 and 5.24.

We can get an excellent feeling for the inductive, coinductive and hyper-elementary relations on a structure by returning to infinitary logic.

Let α be an admissible ordinal, let A = L(α) and let LA be the admissiblefragment of L^ given by A. We refer to the elements of LA as the α- -finite formulas.

Let 9JΪ be a structure for L. A relation S on 90Ϊ is defined by an α- finite formulaif there is an α-finite φ(x l5...,xn, >Ί,...,^fc) and there are # l5...,g fce$)t such that

(1) S(xl9...,xJ iff aR^*!,...,*,,,^...,^]

for all x1?...,xneSR. S is defined by an (^-recursive n-type if there is an α-recursiveset Φ(xι,...,xn,yι,.. ,yk) of α-finite formulas and there are qί9...,qkeW, such that

(2) S(XI,...,XΛ) iff Wl\=/\φeφφ[xi,...,xn,q1,...,qk]

for all x1,...,x l l6SDΪ. Replace the infinite conjunction in (2) by an infinite dis-junction

(3) S(xί,...,xJ, iff


and we say that S is defined by an ^-recursive n-cotype. Notice that S is definedby an α-recursive type iff —\S is defined by an α-recursive cotype. The studentshould compare 5.10 with Theorem II.7.3. (Another version holds without thepairing function assumption; see Exercise 5.29.)

5.10 Theorem. Let 9JI be a structure for L with an inductive pairing function andlet α = O(SR).

(i) A relation S on 9JI is hyper elementary on SCR iff S is defined by an on-finiteformula.

(ii) A relation S on 9Jί is inductive on 50Ϊ iff S is defined by an oί-recursive cotype;S is coinductίve on 95Ϊ iff S is defined by an tt-recursive type.

Proof. We first prove the (<=) parts of (i) and (ii). SinceIXoO^HYRpj, so every α-finite formula is in HYP^. Thus any relation definedby an α-finite formula is in HYP^ by Δx Separation and, hence, is hyper-elementary. It suffices to prove either half of (ii) so suppose that Φ is an α-recursive(or even α-r.e.) set of α-finite formulas and S is defined by (3) above. ThenS(xi9...,xJ iff the following is true in

This makes S a Σt set on JHYP^ so S is inductive on 501 by 5.1.We now prove the (=>) parts of (ii) and (i). Suppose S is inductive on Sffl, say

S(x) iff (x,q0)elφ

where φ(vί9v29q9R+) has R binary and has an extra parameter q. Since u = κm,

*φ = \Jβ<a. *φ

We define formulas ψβ by recursion on β as follows, where 0(f/R) denotes theresult of replacing R(ί1? t2) by ίt Φ tl Λ t2 Φ t2 :

ovί9υ29v3 s

Ψ i f a ί 9 υ 2 9 υ 3 ) is φ(vl9v29υ39 \/γ<β ψ ( , ,

A simple proof by induction shows that

(x,3>)e/£ iff

and, hence,

(x,y)elφ iff

Then we have

S(x) iff


Thus it remains to check that the set

is an α-recursive set. The function f(β) = ψβ is clearly definable by Σ Recursionin L(α) so Φ is at least α-r.e. Define a measure of complexity of formulas, sayc(Θ), by recursion as follows:

c(θ) = l if θ is atomic,

c(θ) = c(\l/} + 1 if θ is -ι\l/, 1v ψ or Vv ψ,

c(θ) = sup{c(ψ) + l\ψeΘ} if θ is /\<9 or V ® -

Then c(ψβ)^β so

iff 3jS

which shows that Φ is α-recursive. This finishes the proof of (ii), but what happensif S is actually hyper elementary? Then SeHYP^ and we can define a function

with dom(0) = S by

g(x) = least β(Wl\= ψβ(x9 qθ9 q)) .

Let y = sup(rng(#)). Then y<α by Σ Replacement in HYP^. Then

S(x) iff Ώt\=\/p^^x9qθ9q)

so S is defined by an α-finite formula. D

The converses of Theorem 5.10 (i), (ii) also hold. We prove the converse of (i)and leave the other as Exercise 5.22. First a lemma.

5.11 Lemma. Let 50Ϊ be an L-structure with an inductive pairing function, let LA

be an admissible fragment which is an element of IHYP^, and let

S"= {S^Mn\for some <peLA, and some qί9...9qkeM,

Wl)r=φ[xί,...9xn,qί,...9qk~] iff S(xi9...,xJ

for all x l 5...,xπeM}.

(i) The collection S" can be parametrized by an n + i-are hyper elementaryrelation, with indices from M.

(ii) There is a hyperelementary set which is not in S1.

Proof, (ii) follows from (i) by the usual diagonalization argument. The proofof (i) is a routine modification of Theorem V.5.7 since HYP^ is projectibleinto 9JI. D


5.12 Theorem. Let Wl be a structure for L with an inductive pairing function andlet α be an admissible ordinal If the hyperelementary relations on 9JΪ consist ofexactly the relations definable by a-finite formulas, then a = κm.

Proof. Lemma 5.11 shows us that if every hyperelementary relation is definableby an α-finite formula then K™ ^α. We now show that if every relation definableby an α-finite formula is hyperelementary, then α^K^. Suppose, to prove thecontrapositive, that a>κm and let 5 be any inductive relation which is nothyperelementary. By 5.10, S is definable by a /c^-recursive cotype. But then,5 is definable by an α-finite formula since α>κ9W, so not every relation definableby an α-finite formula is hyperelementary. D

It is interesting to compare the following corollary of 5.10 and 5.12 with aresult in Moschovakis [1974].

5.13 Corollary. Let $R be a structure with an inductive pairing function. The fol-lowing conditions on ΪR are equivalent:

(i) Wl is recursively saturated.(ii) Every hyperelementary relation is first-order definable.

(iii) κm = ω.

Proof. Since κ;ro = o(HYPaR), we proved (i)<=> (ii) back in § IV.5. We have theimplication (iii)=>(ii) by 5.10 or by II.7.3. By 5.12 we have (ii)=>(iii). D

Moschovakis assumes that his structures are acceptable (see Exercise 4.15),a stronger condition than having an inductive pairing function. Corollary 5B.3of Moschovakis [1974] asserts that if ΪR is acceptable then there is a hyper-elementary relation that is not first order definable. Since an acceptable struc-ture $R always has κm>ω (by 4.1), this follows from 5.13. But 5.13 also showsus that the restriction to acceptable structures rules out many of the most inter-esting structures, model theoretically interesting at any rate.

The general version of 5.13 reads as follows.

5.14 Corollary. Let 30Ϊ have an inductive pairing function and let α be an admissibleordinal. The following are equivalent:

(i) 9JI is a-recursively saturated and not β-recursively saturated for any ad-missible β<0ί.

(ii) The hyperelementary relations are just those definable by en-finite formulas.(iii) κm = oί.

Proof. We have (ii) <=> (iii) by the theorems above and (i) <=> (iii) by ExerciseIV.5.11 and the equality κm = o(WLΎP<m). D

Let 9JI have an inductive pairing function and let o/ί = κm. By 5.14 we seethat the hyperelementary relations on 9JΪ are just the relations explicitly definableby α-finite formulas. One could imagine stronger notions of inductive and hyper-elementary where one allowed an α-finite or even a HYP^-finite formula


φ(x1,...,x , R + ) to define an inductive operation Γφ. Refer to these notions, forthe time being, as oc-inductive, a-hyper 'elementary, MΎP^-inductive and HYP^-hy φer 'elementary . The next result shows that the notion of inductive on 9CR is"stable" in that it coincides with α-inductive and HYP^-inductive.

5.15 Theorem. Let 9JΪ have an inductive pairing function and let a = κ'm.(i) The inductive, oc-inductive and MΎPm-inductίve relations on SCR all coincide.

(ii) Hence, the hyper elementary, a-hyperelementary and ΉYP^-hyperelementaryrelation on 5CR all coincide with the relations explicitly definable by ^-finite formulas.

Proof. It suffices to prove that if φ(xί,...,xn,R+) is a formula of LA, whereA = HYRER, then Iφ is inductive on SJΪ. The proof uses the ideas from the twohalves of 5.10 (ii). First note that /£ is a HYP^-recursive function of β, for β<u,since it is defined by Σ Recursion in HYP^. As before, the Σ Reflection theoremshows that ||ΓJ|^α. Now define the formulas φβ as in the proof of 5.10:

ιA0(x1? . . . , xπ) = φ(xί9 . . . , xn9 f/R) ,

ιj/β(xί,...,xn) = φ(xί,...,xn, \/γ<βφ (...)/R)

so that (*!,.. ..xJe/S iff a»N^[xl9...,xJ. Thus

(Xΐ,...9xjelφ iff aRNV^.^i,-,^].

But the set of HYP^-finite formulas {\l/β\ β«y] is α-r.e. (actually α-recursive) soIφ is Σ! on HYP^ and hence inductive on 9JI by 5.1(i). D


5.16. Show that each of the following structures has a definable pairing function.(i) = <ω,0,+, >.

(ii) Any model of Peano arithmetic.(iii) Any model of ZF, KP or KPU.(iv) L(a,λ) for any limit ordinal λ.(v) ^ = <ωωuω,ω,0, +, ,App>, where ωω is the set of all functions mapping

ω into ω and

App(/,rc,w) iff f(m) = n.

5.17. Show that no nonstandard model of Peano arithmetic is acceptable. Showthat some nonstandard models of Peano arithmetic are almost acceptable andthat some are not. [Show that if (91, 9C) is a model of nonstandard analysis then91 is not almost acceptable.]

5.18 (Moschovakis [1974]). Let 501 = <α, <> where α is any ordinal ^ω. Showthat 9W has an inductive pairing function. This is not easy. First assume


5.19 (Moschovakis [1974]). Let 501 be a structure with an inductive pairing func-tion. Prove the following results using Theorem 5.1.

(i) κm = sup{p(-<)\< is a hyperelementary pre-wellordering of 501}.(ii) If 50Ϊ has a hyperelementary well-ordering then

κm = sup{p(X)K is a hyperelementary well-ordering of 501}.

(iii) (Reduction). Let B, C be inductive on 501. Show that there are disjointinductive sets B0^B, C0^C such that £0uC0 = £uC. [See V.4.10.]

(iv) (Separation). Let B, C be disjoint coinductive subsets of 501. Show thatthere is a hyperelementary set D containing B which is disjoint from C.[Use (iii).]

(v) (Hyperelementary Selection Theorem). Let S(x,y) be an inductive rela-tion on 501. Show that there are inductive relations S0, St such that

5.20. We give an application of the covering theorem; in fact, the original versionof it due to Spector. We use the notation from Rogers [1967]. Let

W — {e\ φ2 is the characteristic function of a well-ordering <J .

Let p(e) = the order type of <e, for eεW.(i) Show that W is Πj on JT.

(ii) Show that p is an inductive norm,

(iii) Let B be a Σj set of natural numbers, B^W. Show that sup {p(e)\eeB}< ω{ .

5.21. Show that 5.10(ii) remain true if "α-recursive type" is replaced by any ofthe following:

(i) α-r.e. type,(ii) HYP^-recursive type,

(iii) HYPgjj-r.e. type.

5.22. Let 501 be a structure with an inductive pairing function and let α be anadmissible ordinal. Suppose that the inductive relations on 50ί are exactly therelations defined by an α-recursive cotype. Show that oc = κm.

5.23. Let SOΪ have an inductive pairing function. Let S, Ύ be inductive relationswhich are not hyperelementary.

(i) Show that TeHYP(αriS), and hence that HYP(SWfS) and HYP(αϊl>Γ) havethe same universe of sets. [Show that o(HYP(αϊl s^xXΉYP^) and then use5.10 (ii).]


(ii) (Moschovakis [1974]). Show that the two expanded structures ($R,S) and(9JΪ, T) have the same inductive and hyperelementary relations.

5.24 (Moschovakis [1974]). Let 9JI be a structure with an inductive pairing func-tion and let S be an inductive relation on 95Ϊ which is not hyperelementary. Showthat for any relation T on 9Jί,

S is hyperelementary on (2K,Γ) iff fc^ ^fc*.

5.25. Show that Theorem 5.15 is not true without the hypothesis that 9JΪ has aninductive pairing function. [Use the 90Ϊ of Exercise 4.14.]

5.26. Our proof of the Abstract Kleene Theorem, Corollary 5.2, is a bit roundabout. Prove it directly from the 9Jί-completeness theorem and Proposition 3.13.(This proof, by the way, establishes the second order version given in Moschovakis[1974] without change.)

5.27. Let 9JΪ be a structure for L with an inductive pairing function.(i) Show that C^KPU*), in the notation of Proposition 3.13, is inductive

but not hyperelementary.(ii) Show that κm = closure ordinal of the inductive definition of "provable

from KPU+ by the 9JΪ-rule".(iii) Show that Qn(KPU+) can be used to parametrize the inductive relations

on 9W. [Use the closure of the inductive relations under hyperelementary sub-stitution and some hyperelementary coding of formulas.]

5.28. The following definition, due to Nyberg, will be useful in Exercise VIΠ.9.16and in Theorem VIΠ.9.5. A structure 9Jl = <M,JR1,...,Λk> is a uniform Kleenestructure if for every Π} formula Φ(x,S+) in some extra relation symbols S thereis a first order φ(x,y, R + ,S+) and a yeM such that for all x and all 5

if and only if

where the R in φ is used for the induction over the structure (9Jΐ,S). Prove thatevery countable structure with an inductive pairing function is a uniform Kleenestructure. Let α be any ordinal of cofinality ω. Show that <V(α),e> is a uniformKleene structure. (This last is due to Chang-Moschovakis [1970].)

5.29 (Makkai and Schlipf, independently). Improve Theorem 5.10 as follows:Let 9K be a structure for L and let α = O(9JΪ). Let S be a relation on 2tt. Show that:

(i) SeHYPjH iff S is defined by an α-finite formula;(ii) S is Σ! on HYP^ iff S is defined by an α-recursive cotype. [Hint: Use

the fact that every fleHYP^ is of the form &(pι,...9pn,M,L(λί)m,...,L{λk)m)for some limit ordinals λί9...9λk and a substitutable function ^.]


5.30 (Moschovakis [1974]). Let $R noίhave an inductive pairing function. Provethat κm is admissible or the limit of admissibles. It is an open problem to findan 9JI where K®1 is not admissible.

5.31 Notes. Some of the results discussed above hold without the pairing func-tion assumption. For example, all of 5.3 through 5.6 are proved directly inMoschovakis [1974]. On the other hand, some of the results are false withoutthe pairing function (like 5.2, 5.8—5.12) and those that do hold are much harderto prove without the admissible set machinery. For structures without an in-ductive pairing function we are left with two distinct approaches, inductivedefinitions and HYR^ (equivalently, inductive* definitions). Only time will tellwhich is the most fruitful tool for definability theory.

6. Recursive Open Games

An open game formula is an infinitary expression ^(x) of the form

where each φn is a formula of Looω. Note that ^(x) itself is not a formula of L^due to the infinite string of quantifiers out front. If {φn\n<ω} is a recursive setof finitary formulas then ^(x) is called a recursive open game formula.

For our study, the most important result on game formulas goes back toSvenonius [1965] where he proves that, for countable ΪR, the Π} predicates areexactly those defined by recursive open game formulas (Theorem 6.8 below). Thisresult went largely unnoticed until the formulas were rediscovered by Moschovakis[1971]. He established that for acceptable 9JΪ (of any cardinality), it is the in-ductive relations on 9JΪ which are definable by recursive open game formulas(Corollary 6.11 below). Thus, from our point of view, Moschovakis was provingthe "absolute version" of the Svenonius theorem.

Before going into these results in detail, let's step back to examine the conceptof "absolute version" with some detachment.

We have been using ZFC as a convenient informal metatheory and hencemay construe all our results as statements about the universe ¥ of sets. By aclass C on ¥ we mean a definable class,

xeC iff

for some formula φ(v) of set theory. A predicate P on ¥ is, by definition, given by

P(x) iff

for some formula (

6. Recursive Open Games 243

6.1 Definition. Let C be a class defined by a Σ{ formula without parameters andlet P be some predicate. A relation Pabs is an absolute version of P on C if thefollowing conditions hold:

(i) Pabs is absolute on C (that is, there are Σ and Π formulas Ψι(vl9...,vn),\I/2(V i,..., V^ SUCΓ1

Pabs(x) iff

iff

(ii) P and Pabs agree on CnHfa) (that is, for all x1,...,xπeCn/ί(ω1),

P(x) iff Pabs(x)).

While not every predicate has an absolute version, at least there can be atmost one absolute version.

6.2 Metatheorem. Let C be a Σ^ definable class, let P be some predicate and letPl5 P2 be absolute versions of P on C. Then for all xeC,

P,(x) iff P2(x).

Proof. This is just a special case of the Levy Absoluteness Principle, one wehave used several times in special cases. The hypothesis can be written

The part within brackets is equivalent to a Π formula so the conclusion followsfrom the Levy Absoluteness Principle. D

6.3 Example. Let C be the class of pairs (9W,S) where 93Ϊ is a structure. Let P(9W,5)assert that S is Π} on $R. Let Pabs(2R,S) assert that 5 is Σί on HYP^. Then wehave shown that P and Pabs agree on countable structures and that Pabs is ab-solute. For other examples, see Table 5 on page 254.

The distinction between Pabs and P is the distinction between Part B andPart C of this book.

In this section we apply these general considerations as follows. We firstprove that for all countable 9M=<M,Λ1,...,Λ />, a relation S on 9W is Π} iff itis defined by a recursive open game formula. Next we show that the notion"S is definable on 9Jί by a recursive open game formula" is absolute. It will thenfollow that for any 9JΪ,

S is Σl on IHYPgR iff S is definable by a recursive open game formula

and hence, by Theorem 5.1, that if 9JΪ has an inductive pairing function,

S is inductive on 9JΪ iff S is definable by a recursive open game formula.


(For 5R without an inductive pairing function, we must replace inductive byinductive*.)

The first question to settle is the very meaning of an infinite string of quan-tifiers. Given a relation R(yι,zl9...9yn9zn9...) of infinite sequences from 9K, whatis to be meant by

The sensible interpretation is by means of Skolem functions. The above is de-fined to mean

3F1,F2,...[(TO,F1,...,Fπ,...)t=V^^

For ease in presenting informal proofs it is convenient to rephrase this interms of an infinite two person game, one played by players V and 3. Theplayers take turns choosing elements al,bί9a2,b2, . from 9JI. Player 3 wins ifR(al9bl9a29b2,...)'9 otherwise V wins. Then

is equivalent to:

Player 3 has a winning strategy in the above game.

Formally, of course, a strategy for 3 simply consists of a set {F1?F2,...} of Skolemfunctions such that

For games which begin with a play by 3,

we use the convention that a function of 0 arguments is simply an element of 90Ϊ.We have already defined the notion of an open game formula ^(x)

' Vyi ^Z! . . . VH <Pn(x, J>1, Zl9 . . . , J>π, Zπ) .

The important part here is the infinite disjunction, not the fact that it beginswith V (we could always add a superfluous V if it started with 3) nor the factthe quantifiers exactly alternate one for one (again we could introduce super-fluous quantifiers if necessary). The reason this is referred to as an "open" gameformula is that in any given play

aί9bl9a29b29...


of the game, if 3 wins then he wins at some finite stage n and thus it wouldn'tmatter what he played after stage n. (That is, there is a whole neighborhood ofwinning plays for 3 in the suitable product topology.)

The dual of an open game formula is a closed game formula, one of the form

V^i 3 ι Vj;2 3z2 . . . /\n φn(x, yί9 z1? . . . , yn, zn) .

In a closed game, 3 must remain eternally diligent if he is to win.

6.4 Examples, (i) The simplest example of an important recursive open gamesentence is given by

tyl V^2 ••• Vn<ω Όπ + l ^π) -

This sentence holds in <9W,E> iff £ is well founded. This is a rather boring gamefor 3 since he never gets to play. Once V has played a sequence a1?a2,...,3 winsif it is not a descending sequence. Hence, 3 has a winning strategy iff there areno infinite descending sequences.

(ii) The Kleene normal form for Π} relations on ^Γ = <ω,0, +,•>,

S(x) iff V/3πΛ(/(fi),x),

can be considered as a reduction of Π} relations to recursive open game for-mulas, namely S(x) iff

VJΊ Vy2 . . . \/B 35 [5 codes < j71? . . . , yny Λ R(s, x)] .

(iii) On arbitrary countable structures we must use game formulas in whichboth players get to play if we are to characterize Π} relations. Suppose M iscountable and let 50Ϊ = <M,JR,5> where R, S are binary. Then 9K is a model of

Vyι 3z! Vy2 3z2.../\ntm<ωR(yn,yJ<->S(zn,zm)

iff <M,.R>^<M,S>. Here we have expressed a Σ{ sentence by a recursive closedgame sentence.

Given a game formula ^(x) we write

as shorthand for

not (9WN #(*)).

In general one must resist certain impulses generated by experience with finitestrings of quantifiers. There is no reason to suppose that


implies

That is, just because 3 has no winning strategy in the first game is no reasonto suppose he does have a winning strategy in the second game. One can findK's for which this fails. For open and closed games, however, this temptingmaneuver is perfectly acceptable, as Theorem 6.5 shows. We shall use the ideafrom this proof a couple of times later on.

6.5 Gale-Stewart Theorem. For all 9Ji and x,

iff

Proof. Let game I be the game given by

(we are suppressing the x since they play no role) and let game II be given by

It is clear that 3 cannot have a winning strategy in both games, for then V coulduse 3's strategy from game II to defeat him in game I. Thus we have the (<=) halfof the theorem. (This part does not use the openness hypothesis.) Now suppose3 has no strategy in game I. We show that V has a winning strategy in I whichof course amounts to a winning strategy for 3 in II. Now since 3 has no strategyin I there must be a fixed a± such that 3 still has no strategy in the game

9WN3Z! Vy23z2...\/nφn(aί9zί9...9yn9zJ.

Why? Because if each av gave rise to a strategy s(aί) for 3 then he would havehad a winning strategy at the start; namely

answer Vs play of % by using s ( a ί ) .

Thus Vs first play is to play an aγ such that

Now after 3 makes some play zί=bΐ, V again plays an a2 so that 3 still hasno winning strategy; i. e.


The same reasoning as above shows that such an a2 exists. Now V keeps onplaying at the mth play some am so that

and, in particular

Then, at the conclusion of play we have

a win for V in game I. We have thus defined a winning strategy for V in game I. D

6.6 Corollary. For all 9W, x,

iffWlϊ=3yίVzί...\/n-ιφn(x9yί9zί9...9yn9zn).

Proof. The following are equivalent:

W\=3yίVzί...\/n-}φn(x9yl9zί9...9yn9zn)

not [W\=-\3yίVzί...\/n-\φn(x9yί9zi9...9yn9zn)']

not [aRNVy13z1.../\nn-ιφπ(x,y1,z1,...,yπ,zπ)]

zί.../\φn(x9yi9zί9...9yn9zn). D

A simple application of the Gale-Stewart Theorem is to show that recursiveopen game formulas define Π} sets. We'll improve this later by improving theGale-Stewart Theorem.

6.7 Corollary. Let $(x) be a recursive open game formula of L. There is a Yl\formula Θ(x) such that for all infinite L-structures 501 and all x l 9...,x f ce9ίR,

) iff Wl\=Θ(x).

Proof. Let #(x) be

To prove the corollary it suffices, by the Gale-Stewart Theorem, to find a Σ}formula equivalent to

3yίVzί.../\n-ιφn(x9yl9...9zn).


This expression is equivalent to

3F [F is a function with dom(F) = all finite sequences from M Λ for all n

and all yl9...9yneM9 -\φn(x,yi,F((yίy)9...,yn9F((yί,...9yny))'].

This is co-extended Σ} by Proposition IV.2.11 and hence is Σ{ by PropositionIV.2.8. To see that the same Σj formula works in all structures one simply noticesthat the proofs in § IV.2 were uniform. D

We now come to the theorem of Svenonius referred to above, a partial con-verse to 6.7.

6.8 Svenonius Theorem. For every Πj formula Θ(x) of L there is an recursiveopen game formula &(x) of L such that for all countable structures 501 and allx l 5...,x f ce$0l,

9Wt=#(3) iff ΪRN=β(Jc).

Proof. It suffices, by the addition of constant symbols for the variables x l 5...,xn,to prove the theorem for Π} sentences. We actually prove the dual, that everyΣ} sentence is defined by some recursive closed game sentence in all countablestructures. By the Skolem Lemma of V.8.7, any Σj sentence is equivalent toone of the form

3S1,...,SmVy1,...,y/3z1,...,zkφ(j;,z,S)

where φ is quantifier free with no function symbols. We prove the special case

3S Myγy2 3^z2 φ(yί9y29zl9z29S)9

the general case being only notationally more complicated. We need the fol-lowing fact.

(1) For each quantifier free formula θ(v,S) there is another quantifier freeformula θ°(v) such that

is valid. Moreover, one can find θ° effectively from θ.To prove (1), first write θ(v, S) as a disjunction

where each 0£ is a conjunction of atomic and negated atomic formulas. Since 3commutes with \/ it suffices to prove (1) for formulas which are conjunctionsof atomic and negated atomic formulas. So suppose we have to get rid of the3S from 3S 0(S,S) where θ(v, S) is


and each ι//i is atomic or negated atomic. This just amounts to propositionallogic. First remove all equalities like (x = y) and make up for them by replacingx by y and y by x everywhere they occur (see examples below). Next we simplyinspect the new list of formulas to see if it is consistent in propositional logic.If it is, θ° consists of the conjunction of all the formulas in the original list thatdon't mention S; if it isn't consistent, θ° consists of some false formula likeWe give three examples.

Example 1. Suppose θ(v,S) consists of

R(x,z), S(x), (x = y), -ιS()0.

The new list consists of

R(x,z), R(y,z), S(x), S(y)9 -ιS(y), -ιS(x).

This is not consistent so there can be no such S.

Example 2. Suppose θ(v,S) consists of

R(x,z), S(x), (xϊy), (y = z).

The new list consists of

R(x,z), R(x,y), S(x), (xϊy), (x^z).

This is consistent so there will be such an S iff

Example 3. Suppose φ(v,S) consists of

S(x), (x=y), (y = z)9

The new list will contain (y^y) which is not consistent; there is no such S.These examples should convince the student that the procedure decribed

above actually works. It is obviously effective. This proves (1).Now, using (1), let Ψjίyn9yί2,zn9zi29y2ί9y229z2l9z22,...9ynί,yn29znι9zn2) be a

quantifier free formula equivalent to

and let the closed sentence ^ be


First we prove that:

(2) For any model m, if Wl\=lSVyl9y2=lzί9z2φ9 then

For suppose (9Jl,S)^Vy1 Vy21z1 Iz2φ(yί9 y29zί9z29 S). Let 3 play with thestrategy:

if V plays al9a2 at stage n, then choose bί9b2 so that ( W l 9 S ) \ F = ( p ( a l 9 a 2 9 b l 9 b 2 9 S ) .This clearly presents 3 with a win.

To conclude the proof we need only prove

(3) // 9JI is countable and 9JlN^ then there is a relation S on 9ϊΐ so that

Suppose SOtN^ so that player 3 has a winning strategy. Since 9W is countable,so is M2, so enumerate M2, M2 = {<α f l l,απ2> I n<ω}. Let V play yni = ani and let3 play zni = bnieM using his winning strategy. Thus, we end up with

for each n<ω. Then, by the ordinary Compactness Theorem for propositionallogic

Diagram (3Λ)v {φ(amί9am29bmί9bm29S) \ m<ω]

is consistent. Thus there really is an S such that

&Λ9S)^φ(aml9an29bml9bn29S)

for each m, since φ is quantifier free. Thus, since every pair is mί9am2y for some m,

This proves (3).The proof of the theorem is complete except that ^ is not quite in the form

demanded of a recursive closed game formula. But trivial modifications withsuperfluous quantifiers, renaming variables and renaming the subformulasobviously puts it in the desired form. D

We have carried out half our task by showing Π} is the same as "defined by arecursive open game formula" for countable structures. It remains to show thatit is absolute. We prove more than this in the next two results.

The next theorem can be viewed as an effective version of the main theoremof Keisler [1965]. The proof is rather different.

Given a recursive open game formula ^(x), say,


we define its finite approximations δm(x) by :

δm is Vy^... Vymlzm\/n^mφn(x,y1,z1,...,yn,zn).

It is obvious, from a gamesmanship point of view, that

is true in all structures.

6.9 Theorem. Let 30Ϊ be recursively saturated. Then, using the notation of theprevious paragraph,

Proof. We already have the trivial implication (<-). To prove the contrapositiveof the other direction we imitate the proof of the Gale-Stewart theorem. Weassume

and exhibit a winning strategy for V in the game

Vj/i 3zi Vy2 3z2 ... V» <?»(*> >Ί>zι> •••> ^'z«)

We claim that there is an aί such that, for each m<ω,

Why? Suppose that for every α^M there is an m such that

Now this all holds in IHYP^, which has ordinal ω, so, by Σ Reflection there is afe < ω such that m can always be chosen less than k. (Here we are using the factthat φn is a recursive function of n, so is Σί in HYP^.) But then

contrary to assumption. Thus there is such an a^ and we let V play it. Let 3 playzί=bl. We claim that there is an a2 such that, for all m <ω,

The reasoning is just as for a^. If V continues in this way, do what 3 will, a sequencea\bla2b2 ... will be generated which satisfies

for each n. Hence we have described a winning strategy for V. D


Now, if 9JI is a structure with α = o(HYPan) one would hope to show that,on 9JΪ, (x) is equivalent to the disjunction of its α-finite approximations:

This turns out to be true once one has the correct definition of theLet y(x) be a recursive open game formula, say

Define formulas δn

β(x, yί9zi9...9 ynzn)

δn

λ(x, y 1? . . . , zj is \/β<λδnβ if λ is a limit ordinal .

Let δβ(x) be δ^x). Note that δn, for π<ω, has the same meaning as it did inTheorem 6.9. Also note that δβ is an α-recursive function of β<α, whenever αis an admissible ordinal.

6.10 Theorem. Let <y. = o(MYPm). Then, using the notation of the previous para-graph,

Proof. To prove the easy half (<-) one first proves by a straightforward inductionon β that

for all n. For n = Q this gives the desired result. The proof of the other half is sosimilar to the proof of Theorem 6.9 (a special case of 6.10) that we leave it to thestudent. D

6.11 Corollary. For any structure yn = (M,Rι, ...,Rty and any relation S on $R,the following are equivalent:

(i) S is definable by a recursive open game formula on 9K.(ii) S is inductive* on 9Jί.

(hi) S is Σ! on HYP^.// $R has an inductive pairing function, these are also equivalent to

(iv) S is inductive on Wl.

Proof. It follows from 6.10 that

"S is definable on 9M by a recursive open game formula"

is absolute so the theorem follows from Theorem 6.2. We present a slightly moredirect proof which shows a bit more uniformity.


We see immediately that (i)=>(iii), from Theorem 6.9, since

is Σ! on HYP^. It thus suffices to prove (ii)=>(i). Let φ(x, R + ) be any extendedfirst order formula. Write /φ(9CR) for the fixed point defined on 9JΪ by Γφ. We provethat there is a fixed recursive open game formula ^(x) such that

(4) for all 9«, xe/φ(3W) iff 9MN^(x)

Now Iφ(9Jl) is extended Π} on 3W, hence Π} on 50ί by Proposition IV.2.8, and thesame Π} formula Φ(x) defines /^(SR) for all 2K;

(5) for all OR, xe/φ(9W) iff

Now use Theorem 6.8 to choose ^(x) such that

(6) for all countable SR, ΪR^Φ(x) iff

Now, combining lines (5) and (6) we have

for all countable Stt[xe/φ(SW) iff

and the part in brackets is absolute. Hence, by Levy Absoluteness, we have (4). D

6.12 Exercise. The Interpolation Theorem for Lωω can be stated as follows.Let Φ(x1? ..., xj be a finitary Σ} formula of Lωω and let Ψ(xί9...9xn) be a finitaryΠ} formula of Lωω. If every L-structure 501 is a model of

(*) Vxl9...,xH[Φ(Z)-*Ψ(x)']

then there is a first order formula Θ(x) such that every L-structure 9JΪ is a model of

(**)

We can turn this into a local result as follows.(i) Let 9JI be a recursively saturated countable model of (*). Show that there

is a θ(x) such that 501 is a model of (**). [This is easy from Exercise V.4.8. A moredirect proof goes via Sveiionius Theorem and the Approximation Theorem 6.9.Of course one could also cheat and apply the Interpolation Theorem for LA

with A-HYP^.](ii) Prove the interpolation theorem for Lωω directly from (i).

6.13 Notes. The student would profit from a comparison of our treatment withthat in Moschovakis [1971], [1974]. His proof [1971] makes it clear where theapproximations δβ originate. The model theoretic interest of the Moschovakis-Svenonius results was brought out by the important paper Vaught [1973]. The


student is urged to read this and Makkai [1973] in the same volume. This section(VI.6) of the book is included partly to make these papers more accessible.

Table 5. Absolute versions of some nonabsolute notions

Primitive notion P Absolute version Pabs Relevant class C of objects

1. S i s Π } o n 9 J Ϊ S i s Σ j on HYP^ all structures 9W = <M,K1,...,and relations S on $R

2. S is Π{ on <0ί S is inductive* on ΪR same as (1)

3. S is PI} on <0ί S is inductive on 2R SR, S as in (1) when ΪR has aninductive pairing function

4. S is Π } on SJR S is defined by an open recursive game same as (1)

5. \= φ t— φ all sentences of Looω

6. <0ί 91 9N p 91 (cf. § VII.5) all structures 9K, 91

7. ΪR^9I SWΞftίL^J (cf.§VIL5) same as (6)

8. S is strict Π} on A S is Σ j on A (cf. § VIII.3) all admissible sets A andrelations S on A

9. Wl is rigid every element of <0ί is definable by a all L-structures $R = < M , Λ t, . .(cf. § VII.7) formula of Looωn MYP^ without

parameters

PartC

Towards a General Theory

"The sensible practical man realizes that the questions whichhe dismisses may be the key to a theory. Further, since hedoesn't have a good theoretical analysis of familiar matters,sometimes not even the concepts needed to frame one, he willnot be surprised if a novel situation turns out to be genuinelyproblematic."

G. KreiselObservations on Popular Discussions of Foundations

Chapter VII

More about L ooω

In this chapter we resume the discussion of L^ where we left it in Chapter III.This time, however, we do not restrict our attention to countable fragments butdevelop the beginning of a general theory. In this way we can gain insight into thecountable case by seeing what principles are involved in the general case.

The most useful result, both for model-theoretic applications and for appli-cations to generalized recursion theory, is the Weak Model Existence Theoremof § 2. Its model theoretic applications are discussed in §§ 3 and 4. The applicationsto definability theory can be found in Chapter VIII.

§§ 5,6 and 7 are concerned with Scott sentences of L^ and their approximations.These sections are independent of most of the rest of the book but they do illustratethe importance of L^ and some uses of admissible sets in studying them.

1. Some Definitions and Examples

Once the hypothesis of countability is removed, all the major theorems ofChapter III fail dramatically. This section consists largely of "counter" examplesto these statements. It also contains a number of definitions which will be im-portant in our study.

1.1 Definition. An admissible set A is Σx compact if for each admissible fragmentof the form LA and each Σί theory T of LA, if every subset Γ0 of T which is aelement of A has a model, then T has a model.

The Compactness Theorem of § III.5 states that every countable, admissibleset is Σ! compact.

1.2 Definition. An admissible set A is self-definable if for some language L con-taining the language of A there is a Σ! theory T of LA such that

(i) some expansion (A,...) of A to an L-structure is a model of T.(ii) if (33,...) is any model of T then 95 A.

If T can be chosen to be a single sentence of LA then A is called strongly self-definable.

258 VII. More about !_«,„

We obtain a host of counter-examples to Σ1 compactness by means of 1.3and 1.4. The first is a trivial exercise in compactness.

1.3 Proposition. // A is Σ^ compact then A is not self -definable. D

1.4 Proposition. For all α^O, #(Kα+1) is self -definable.

Proof. Let /l = H(Kα+1) and let T be the theory consisting of the followingsentences :

KP,

Vx(xea<->\/bef lx = b) for all aeA,

Vx 3β 3/ [jβ ωα Λ / maps TC(x) one-one onto /?] .

With the obvious interpretation of the constant symbols, A is a model of T.Suppose <#,£> is some other model of T. The infinitary sentences of T insure thatwe can assume

Let xe£ and suppose ye 5 is such that

Pick β^Nα such that

<£, £> 1= 3/ [/ maps y one-one onto β]

by the last axiom of T. Then there is some F^βx β such that <#, £> is a model of<y,F|»^ <β,F> and hence "</?,,F> is well founded" is true in <£,£>. The crucialstep in the proof is to verify that

(1) <jβ,F> really is well founded.

Suppose that <β,F> is not well founded and let X^β have no F-minimalmember. But card(X)<Kα+1, so XeA^B, and hence <£,£> is a model of"X has no F-minimal element", which is a contradiction. Thus (1) is established.But then the transitive set isomorphic to <β,F> is, on the one hand, (y,E\ y> and,on the other, in H(Kα+1). Thus yeH(Kα+1) so xeίf(Kα + 1). In other words04,6> = <B,E>. D

A strengthening of 1.4 is given in Exercise 1.12.If we had wanted only to prove that H(Kα+1) is not Σ1 compact, we could

have come up with much simpler examples. A good example does more thanjust refute (the function of a counterexample), it makes almost explicit some ofthe ideas needed for understanding and generalizing existing results. Most ofthe examples in this section are good examples.

1. Some Definitions and Examples 259

To understand the above example, the student should consider what happensto the proof of 1.4 if we replace <//(Nα+1),e> by some countable, transitive set<^4,e> elementarily equivalent to it. Something must go wrong since A is Σί

compact. If he works through the proof he will see that the only step that failsis the proof of (1). This suggests the following proposition.

1.5 Proposition. Let A be admissible.(i) If A is self-definable then there is a Σ1 theory T(<) of LA which pins down

ordinals greater than those in A.(ii) If A is strongly self-definable then there is a single sentence φ(<) of LA

which pins down ordinals greater than those in A.

Proof. We prove (i); the proof of (ii) is the same. Let Γ0 be a theory which self-defines A and let T=T0 + "<=et ordinals". Then every model 95Ϊ of T has<9CR of order type o(A). D

Thus, self-definable admissible sets show that the theorems of § III. 7 on theordinals pinned down by Σ1 theories of LA cannot go through in general; forexample, they fail when A = /f(K1). To get an example where a single sentencepins down large ordinals, we need some strongly self-definable admissible sets.

A set A is essentially uncountable if every countable subset X^A is an elementof A.

1.6 Proposition. Let A be an essentially uncountable admissible set and letIB = HYP(A). Then B is strongly self -definable.

Proof. Let ψ be the conjunction of the following:

ΛKPU,Vt;3α[xeL(A,α)],

Vα 3r[r c= A Λ r is a pre-wellordering of type α] .

Since MYP(A) is projectible into A, HYP(A) is a model of the last conjunct andhence of ψ. The well founded models of the first four conjuncts are isomorphicto HYP(A) so it remains to see that all models of φ are well founded. Using therank function we see that if <£',£> is a non-wellfounded model of ψ then thereis a descending sequence of ordinals in <£',£> so it suffices to see that the ordinalsof <#',£> are wellfounded. Let aεB' be an "ordinal" of <£',£>. Apply the lastconjunct of ψ to get an re A such that

<£',£> !="r has order type a" .

260 VII. More about Lw ω

We need to see that r really is well ordered. Suppose

...rxπ + 1rxπr...rx 1

is an r-descending sequence. Let b = {xn\n<ω}. Since b is a countable subsetof A, be A. But then beB' and b has no r-minimal element, contradicting

<£',£> N"r is well ordered". D

For example, if cf(κ)>ω then A = ΉΎP(H(κ)) is strongly self-definable.Hence LA is not Σί compact and there is a single sentence of LA which pins down

Our next examples have to do with attempts to generalize the Completenessand Extended Completeness Theorems of § III.5 to arbitrary admissible fragments.

1.7 Definition. Let A be an admissible set.(i) A is validity admissible if the set of valid infinitary sentence of A is Σ^ on A.

(ii) A is Σ! complete if, for every Σί theory T of LA, the set

is Σ! on A.Don't forget, in reading 1.7, that the extra relations which may be part of A

count in the definition of Σt. It is also important to notice that Σ^ completenessimplies validity admissibility.

1.8 Proposition. Let A be admissible.(i) // A is self-definable then A is not Σj complete.

(ii) // A is a strongly self-definable pure admissible set then A is not evenvalidity admissible.

Proof. Recall, from § V.I, that there is a Γ^ subset of A which is not Σ t. Hence,there is certainly a Π} subset of A which is not Σ^ Thus the result follows fromthe following lemma. D

1.9 Lemma. Let A be admissible, let T be the theory which self-defines A in 1.8and let X 9ΞA be Πj on A. There is an A-recursίve function f such that for everyxeA we have xeX iff /(x)eCn(T).

Proof. Suppose

xeX iff A N V R φ ( R , x ) ,

where R is a symbol not in the language of T. In case (i) of 1.8 we may assume that Tcontains the diagram of A. Then xeX iff (p(R,x)eCn(Γ).

In case (ii) we settle the question "xeXT by checking whether the conjunctionof T and the diagram of TC({x}) implies φ(R, x). D

1. Some Definitions and Examples 261

1.10 Corollary. If A is pure and strongly self-definable then there are valid sen-tences of LA which are not provable by the axioms and rules of Chapter III.

Proof. The set of provable sentences is a Σ1 set. D

Thus, #(Kα+1) is never Σ1 complete, even if α = 0, and HYP(/f(Kα+1)) isnever validity admissible.

We conclude this section with a counterexample to the interpolation theorem.It has a rather different flavor and will not be used in the following sections.

1.11 Proposition. Let A be an admissible set with an uncountable element ando(A)>ω. The interpolation theorem fails for LA.

Proof. Let φ(<) characterize <ω, <> up to isomorphism and let ψ be

where aeA is uncountable. (All we reed about φ is that it has only uncountablemodels and has no symbols in common with φ.) Then </>,^eA and N=φ-»> — \ψ.If the interpolation theorem held for LA then there would be a sentence θ in-volving only equality such that l=φ->θ and t=^->— iθ. Thus θ is true in allcountable infinite structures since such structures can always be turned intomodels of φ. Similarly, ~\θ is true in all structures of power ^card(α). But thiscontradicts :

(2) A sentence θe Looω involving only equality is true in all infinite structuresor in none.

The proof of (2) is easy, given some notation and results of § 5, which we assume.Let 9K = <M, =>, 91 = <ΛΓ, => be infinite. Let / be the set of all finite one-onemaps from M0^M onto N0^N. Then

so S R s S l ί L . Thus W\=θ iff


1.12. Suppose 0<α<Kα and card (501) <Kα. Show that H(KΛ)m is self-definable.This includes 1.4 and //(KJ as special cases.

1.13. A sentence φ(<) (or theory Γ(<)) pins down α exactly if φ has models andevery model 50ί of φ has <αίϊ of order type exactly α.

(i) Prove that if A is self-definable (strongly self-definable) then there is a Σ1

theory T of LA (sentence φ of LA) which pins down o(A) exactly.(ii) Let A be a resolvable admissible set and let T be a Σί theory of LA which

pins down o(A) exactly. Show that A is self-definable.

262 VII. More about !_„„

1.14. Let A = HYP(H(Xα+1)). Show that there is a sentence of LA which pinsdownKα + 2.

1.15. Show that the results of § IV.l fail in the uncountable case.

1.16. Show that if A is essentially uncountable then every inductive relation on Ais Δj. Conclude that not every Π{ relation on A is inductive on A, for A essentiallyuncountable.

1.17. Improve 1.8 (ii) by allowing A^ admissible above ΪR.

1.18 Notes. Counterexamples to compactness go back to Hanf [1964] and earlierunpublished work of Tarski. Karp [1967] showed that, for cf(α)>ω, the set#(Kα) is not validity admissible. The results on pinning down large ordinals (1.14for example) are due to Chang [1968]. The counterexample to interpolation is dueto Malitz [1971]. We have tried to unify the various examples by centering themon the notion of self-definable, admissible set. Our notion is suggested by, andequivalent to, that of Kunen [1968].

Kreisel [1968] has observed that the counterexample to interpolation hasthe defect that it might disappear by some reasonable strengthening of the logicLA or L^. The other examples of this section do not have this defect. The situationwith compactness, say, could only get worse if we were to increase the expressionpower of the logic by introducing some new quantifier or connective. Ratherthan strengthen LA we must look for strengthenings of the notion of admissibilitywhich coincides with the old notion in the countable case. This is taken up inChapter VIII.

2. A Weak Completeness Theoremfor Arbitrary Fragments

The model theory of second-order logic is totally unmanageable and seemsdestined to remain so. Infinitary logic is an attempt to dent second-order logicby studying logics which have greater expressive power than Lωω but still have aworkable model theory. The examples of § 1 show that uncountable fragmentsbehave more like second-order logic than do countable fragments. This makes theproblem of developing a theory which handles arbitrary admissible fragmentsvery intriguing.

In spite of, or because of, the "counter'-examples, the model theory of arbitraryadmissible fragments is becoming a rich subject. In this section we present somebasic tools for studying these logics. In particular, we prove an analogue of theExtended Completeness Theorem of § III.5. Recall our line of attack on theproblem of completeness in Chapter III:

(1) We defined the notion: validity property for L A .(2) We proved that if LA is countable then a sentence φeL A is valid iff φ

is in every validity property.

2. A Weak Completeness Theorem for Arbitrary Fragments 263

(3) We showed that if LA is an admissible fragment then the intersection ofall validity properties is a validity property which is A-r.e., that is, Σί on A.

When we drop the assumption that LA is countable step (2) breaks down.In general, a sentence may be true in all models without being in every validityproperty (i.e., without being a theorem of LA) as Corollary 1.10 shows. In thissection we attack the problem of completeness as follows:

(Γ) We define a stronger notion: supervalidity property for LA.(2') We prove that a sentence <peLA is valid iff φ is in every supervalidity

property.(3') In Chapter VIII we will introduce a semantic notion of r. e., called strict Π},

and show that the intersection of all supervalidity properties for LA is a strict Π}set. When A is countable the notion of strict Πj reduces to Σ: on A.

It is convenient in this part of the theory to work with sufficiently rich frag-ments, so-called Skolem fragments with constants.

2.1 Definition. Let LA be a fragment of L^ and let C be a (possibly empty) setof constant symbols of L such that every formula of LA contains at most a finitenumber of constants from C.

(i) LA is a Skolem fragment with constants C if there is a one-one functionwhich assigns to each formula of LA of the form

, 3; !,..., 3>«),

φ contains no constants from C and

y^ ",yn are not bound in φ

an n-ary function symbol

' 3xφ

of L not occuring in φ; it is called the Skolem function symbol for 3xφ(x9yί9 ..., yn).If C = 0 we just say that LA is a Skolem fragment.

(ii) Let LA be a Skolem fragment with constants C. The Skolem theory forLA, denoted by 7Jkolem, consists of all sentences of LA of the form

for all formulas 3xφ(x, _ y l 5 ..., yn) as in (i). An L-structure $01 is a Skolem structurefor LA if

The extra freedom permitted by the set C of constant symbols is crucial formany applications. For now we can barely hint at their use by the followinglemmas.


2.2 Lemma. Let LA be a Skolem fragment with constants C and let & be any

validity property for LA with TSkolem^@. Then for any formula

,y !,..., yπ, C 1 ? . . . ,c f c )

of LA the sentence

is in 2, where F is the Skolem function symbol for

Proof. By the definition of 7^kolem,

is in T^koiem — Using the axioms for V and modus ponens shows that the de-sired sentence is in <2). D

If LA is a fragment and C is a set of new constant symbols we use LA(C) to

denote the fragment which consists of all substitution instances of formulas inLA by means of a finite number of constants from C. If C = {c l9 . . . , cπ} we some-times use LA(c l 5 ..., cπ) for LA(C).

2.3 Lemma. Let LA be a Skolem fragment with constants C0 and let C be a setof new constant symbols. Then LA(C) is a Skolem fragment with constants C0uC.

Proof. Immediate from the definition. D

The next result shows us that we lose nothing (we gain a lot) by restrictingourselves to Skolem fragments and Skolem structures as far as the existence of

models is concerned.

2.4 Proposition. Let LA be a fragment of Looω. There is an expansion L' of L bynew function symbols with the following properties:

(i) Let LA be the set of formulas which result from a formula of LA by sub-stituting a finite number of terms from L'. Then LA is a Skolem fragment. Further-

more, card ( LA )= card (LA) and every Skolem function symbol is in L' — L(ii) Every L-structure 9W has an expansion 501' = (501, . . .) to a Skolem structure

for L;.(iii) // LA is an admissible fragment then we can define L' so that LA is Δ t on

A and such that the symbol P3xφ is an A-recursive function ofIn particular, TSkolem is then an Ik-recursive set of sentences of LA.

Proof. Let L°=L, L A =L A . For each formula

' , y !,..., ym)


of LA in which yί9...9ym are not bound, add a new function symbol

to L" and let LA

+1 be the resulting fragment. Let L' = \JnLn so that L^jJ^L^.

Part (ii) is obvious from thus construction. (See Lecture 13 of Keisler [1971] formore details, if necessary.) Part (iii) is obvious if we just code up F3jc<p by somethinglike <17, 3xφ>. D

We now come to the notion of supervalidity property.

2.5 Definition. Let LA be a Skolem fragment with constants C. A validity propertyS> for LA is a supervalidity property (s.v.p.) for LA (more precisely, for (LA,C)) ifTskoiem £® and the following \J-rule holds.

\J-Rule: If \JΦ is a SENTENCE of LA and \/Φε@ then there is somesuch that

The \/-rule causes supervalidity properties to behave in quite a differentmanner than ordinary validity properties. For example, it prevents the inter-section of all supervalidity properties for LA from being an s. v. p. The next lemmashows just how strong the \/-rule is.

2.6 Lemma. Let LA be a Skolem fragment with constants and let 2 be a validityproperty for LA with ^kolem^^. Then Q) is an s.v.p. iff Q) is complete, that is,iff for each sentence ι//e LA

or (

Proof. Assume Q) is an s. v. p. Since all axioms of LA are in

so the conclusion follows by the \/-rule. Now assume 2 is complete, \/Φ asentence of LA, \/ΦeQ). If for each φeΦ, φφ@, then, for each φeΦ,—\(pE@;so, by the /\-rule K3,

/\ {-\φ I φe Φ} E@ .

But this sentence is just ~ \/Φ. Since Q) is a validity property it cannot have both\JΦ and ~\]Φ as members, so φe@ for some φeΦ. D

Note that if 3) is an s.v.p. for LA and φ(vl9 ..., vn)e LA then

iff \/vί9...9vnφ(vί9...9 vn)

266 VII. More about !_„„,

so that 3f is determined by its sentences. We say that an L-structure 9JI is a model of2 if 9W is a model of all sentences in ® .

2.7 Definition. Let 9Jί be a Skolem structure for the Skolem fragment LA (withconstants). The supervalidity property given by $R, denoted by Q)^ is the setof all φ(ι? !,..., t Je LA such that

In the notation of III.4.2, ®a» = Λw It is clear that @m is an s.v.p. for L^.If a sentence φeLA is in all supervalidity properties then it is in all ®OT; henceit is true in all Skolem structures for LA. This gives the trivial half of the nexttheorem.

2.8 Theorem (Weak Completeness Theorem for Arbitrary Skolem Fragments).Let LA be a Skolem fragment with constants C.

(i) A sentence φ of LA is true in all Skolem structures for LA iff φ is in everysupervalidity property.

(ii) Let T be a theory of LA, φ a sentence of LA. Then φ is true in every Skolemstructure Wl which is a model of T iff φ is in every s.v.p. 2 with Ί^Q).

Proof, (i) is the special case of (ii) where T=0. The proof of (<=) in (ii) is immediateby the remarks following Definition 2.7. Most of the work for proving (=>) wasdone back in the proof of the model existence theorem. We break its proof up intwo lemmas to make this clear and because we need one of the lemmas (2.9) later.

Compare the next lemma with the definition of consistency property on p. 85.

2.9 Lemma (Weak Model Existence Theorem). Let L have at least one constantsymbol and let LA be any fragment of L^. Any set S of sentences of LA whichsatisfies the following rules has a model

Consistency rule: If φ is atomic and φeS then (—\φ)φS.-(-rule: If (-κp)eS then (~φ)eS./\-rule: If /\ΦeS then for all φeΦ, φeS.V-rule: If (Vvφ(υ))eS then for each closed term t of L, φ(t/υ)eS.\J-rule: If \/ΦGS then for some φeΦ, φεS.3-rule: If (Jvφ(v))eS then for some closed term t of L, φ(t(v))eS.Equality rules: For all closed terms ί l5ί2 of L:

if (tl = tz)eS then (t2 = ti)GS, and

if φ(ίι),(fιΞΞf 2)eS then φ(ί2)eS.

Proof. The proof of the Model Existence Theorem was in two stages. We firstshowed how to construct a set sω of sentences having the above properties (plussome others involving constants from C) and then showed how to construct amodel from such a set. The second stage of that proof constitutes the proof ofthis lemma. D


2.10 Lemma (Alternate form of Weak Completeness Theorem). Let LA be aSkolem fragment with constants C. Let Q> be an s.v.p. for (LA,C) and let S be theset of sentences in Q). Then S is true in some Skolem structure for L^; i.e., Q) has amodel.

Proof. Since I^kolem , any model of S will be a Skolem structure for LA. Weneed only prove that S satisfies the rules of Lemma 2.9. Since @) contains theaxioms (Al) — (A 7) and is closed under (Rl) — (R3), these are all routine exceptfor the V and 3 rules. The \/-rule for S follows from the \/-rule for Q). To checkthe 3-ruίe, suppose

By Lemma 2.2,

for the appropriate function symbol F. Thus,

as demanded by the 3-rule. D

Proof of Theorem 2.8 (ii) (=>). Suppose TvTSkolem\=φ. We need to see that if2 is an s.v.p. with T^& then φe^. If not, then —\φe& by Lemma 2.6. Then,appying Lemma 2.10 we would get a Skolem model of T\j{—\φ], a contra-diction. D

We conclude this section with a result which allows us to construct interestingsupervalidity properties and hence, by Weak Completeness, interesting models.It often gives us the effect of the ordinary Compactness Theorem for Lωω. Givena Skolem fragment LA with constants C0 and a Skolem fragment KB with con-stants Cί we write

if L A cK B , CO^CΊ, and if F3xφ is the Skolem function symbol assigned to3x φ(x,yl,...,yn) by LA, then it is also the one assigned to 3x φ(x,y1,...,yj by KB.

2.11 Union of Chain Lemma. Let I be a lineary ordered index set and supposethat, for each ie/, L^ is a Skolem fragment with constants Ct and Q){ is a super-validity property for (Lj^Q. Suppose, further, that for all ijel, with ί<j,

.) and ^<Ξ%

Let KB = U£6/ Lίλ Q =y i6/ Ci9 &„ =(Ji6/ ®{. Then KB is a Skolem fragmentwith constants C , and Q)^ is a supervalidity property for (K^C^).


Proof. Simple checking of the definition shows that KB is a Skolem fragmentwith constants Q. The Skolem theory for (K^CJ is the union of the Skolemtheories for the various (L^C,) so the Skolem theory for (K^C^) is containedin Q)^. Similarly, the axioms (Al) — (A 7) for KB are all in Q)^. It is a trivial matterto check (Rl), (R2) and the \/-rule. This time it is the /\-rule which requires amoment's thought. Suppose /\ΦeK B and that, for each φeΦ, φe^^. Weneed to check that f\Φe&^. Choose i so that /\Φel_A

l ). We claim that, foreach φeΦ, φe^ (so that ΛΦe^ c^J. Otherwise, suppose φ = φ(ι;1,...,ι;ϊl)eΦbut that <ρφ®i. Then

By completeness (Lemma 2.6),

-Ί\fvί9...,vnφ(v1,...9υn)e®i.

But φ(vί,...,vn)e@ao so for some j>i9 φ(vΐ,...,vn)e&j. Hence

But since Q)^Q) p this contradicts the consistency requirement for the validityproperty 3) y D

All known applications of 2.11 follow from the following very special case.It exhibits the role of constants in our notion of Skolem fragment.

2.12 Union of Chain Lemma (Special form). Let LA be a Skolem fragment. LetC = {cπ |0<tt<ω} be a countable set of new constant symbols. Suppose that foreach n, Q)n is an s.v.p. for LA(c l9...,cn) and that Q)n^Q)m for n^m. Let2^=\Jn &„. Then &„ is an s.v.p. for LA(C).

Proof. (LA(c1,...,cJ,{c1,...,cn})c(LA(c1,...,cJ,{c1,...,cm}) for n^m so the re-sult follows at once from 2.11. D

Applications of the results of this section appear in the next two sections aswell as in Chapter VIII.


2.13. Let LA be a fragment if L^ and let 9JΪ, 91 be L-structures. $R is anL -elementary substructure of 91, written

if 9Jίc9l and for every ^^....^^eL^ and every α1,...,απe9Jί

,...,a iff


(i) Prove that if 9JΪ c ϊt then SR X 91 ( LA) iff for every formula 3x φ(χ, , . . . , yn) ε LA

and every α l 5...,απeSOl, if

then there is a beWl such that

(ii) Prove that if

for α</?<y and 9M = (J^<yaR^, then

for all

2.14. Let LA be a Skolem fragment with constants and let 9M, 91 be Skolem struc-tures for LA. Show that if 9M<=9i then 9W<Ή(LA). [Use 2.13 (i).]

2.15 (Downward Lowenheim-Skolem-Tarski Theorem). Let LA be a fragmentof L^ and let κ^card(LA). Let 50Ϊ be an L-structure, X^Wl, κ<card(95ϊ),

κ:. Prove that there is an 91 with

A), card(Ή) = /c, and X^Vl.

[By 2.4 you may assume LA is a Skolem fragment and that 90Ϊ is a Skolem struc-ture for LA.]

2.16. If $R is an L-structure and X^M then Hull^X) is the smallest sub-structure of 9JZ containing X.

(i) Prove

card (HullTO(X)) = max (K0? card(L), card(X)} .

(ii) Prove that if 50Ϊ is a Skolem structure for a Skolem fragment LA andX^Wl then

2.17 Notes. The essential content of the Weak Completeness Theorem is as oldas the Henkin [1949] proof of the completeness theorem for Lωω. As we havetried to suggest in 2.9, it is implicit in the Model Existence Theorem. Only recently,however, has it become clear that the result is useful enough to deserve to becalled a Weak Completeness Theorem. (The perjorative "weak" is there for thesame reason as in § III.4; there is no nice notion of provability to go along with it.)

270

The first explicit statement of the Weak Completeness Theorem appears asLemma 1.5 in Barwise-Kunen [1971], where it was used to attack the modeltheory of uncountable fragment.

Our treatment of Skolem fragments is a modification of that contained inLecture 13 of Keisler [1971]. In particular, the exercises are proven there (in thecountable case).

3. Pinning Down Ordinals: the General Case

Several of the examples in § 1 hinge on our ability to pin down ordinals largerthen o(A) by a Σ^ theory of LA, for certain uncountable admissible sets A. Wewill see, in fact, that a good deal of the model theory of uncountable, admissiblefragments revolves about this question of pinning down ordinals. For this reasonwe choose it as the first application of the Weak Completeness Theorem.

The proof of the next theorem proves more than we state. In fact, it willallow us to compute exactly the ordinals pinned down by theories, once wedevelop some recursion theoretic machinery in the next chapter. For now wecontent ourselves with a crude statement of the result.

3.1 Theorem. Let T = T( <,...) be a set of sentences of Lr^ω. If T pins downordinals then there is a ξ such that all ordinals pinned down by T are less than ξ.

Proof. We may assume that T has models since otherwise ί = 0 will do. Wemay also assume that if T pins down α and β<a the T pins down β, by aremark in § III.7. By 2.4 we may assume that T c LA where LA is a Skolemfragment and that ^koiem^^- We want to set things up to apply the specialform of 2.12, the Union of Chain Lemma, so let C = {cn\0<n<ω} be a set ofnew constant symbols. Let Sn be the set of all supervalidity properties 2 forLA(c l 5...,cn) (this is just LA if n = 0) such that

Ί^Q) and (c2<

(For rc = 0,l, none of the sentences involving the ct occur.) Since T has a model$R, the s.v.p. &m given by 9JΪ is in S0, so S0^0. Let

and put an ordering < on S by

if 2^2' and the (unique) n such that ^'e®π is greater than the unique msuch that ^eSm. (Note that for ^eS, we can tell which n has @εQn byjust seeing what the largest n is such that (cn = cn)e@.) We claim that

(1) <S,-<> is well founded.

3. Pinning Down Ordinals: the General Case 271

For suppose

Let 2ao = (jn^n. By the union of chain lemma, 3f^ is an s.v.p. and hence, bythe Weak Completeness Theorem, there is a model

of Q)^ where an is the interpretation of CΛ. But then 9JΪNT, and an + ί<an forall n<ω which contradicts the hypothesis that T pins down ordinals. Thisproves (1).

Using (1) it is easy to get an upper bound for the ordinals pinned down by T.By (1), each êS has an ordinal rank p(β\

and <S,<> has a rank

We will prove that

(2) if êSn and (Ôl9aί,...9an)^=Q then the <9CW predecessors of an haveorder type ^p(@) when rc>0; if n = 0 then <m has order type

Since every SOΪI=T is a model of ίêSo, and p(^5m)<ξ, (2) gives us:

(3) every model 9)1 of T has <aι of order type less than ξ,

which proves the theorem. We prove (2) by induction on p(2\ Supposeα = p(ί^), (yjl,aί9...,an)\=@ but that the predecessors of an have order type >α.(The case n=Q is essentially the same.) Let an + ί be the αth member of the fieldof <9Cίί as ordered by <m and let Q)' be the s.v.p. given by

Then &fε<5n + ί, and Q)^Q)' so Q)' <Q) and hence p(&')«x. But TO' is a modelof Q)' with the precedessors of an + ί of order type α>p(^'), contradicting theinductive hypothesis. D

Without Theorem 3.1 we could not be sure that the next definition made sense.

3.2 Definition. Let A be an admissible set.(i) /z(A) is the least ordinal not pinned down by some sentence φ( <,...) in

some admissible fragment LA.(ii) /ιΣ(A) is the least ordinal not pinned down by some Σl theory of some

admissible fragment LA.

272 VII. More about L^

In the next chapter we will determine exact recursion-theoretic descriptionsof ftΣ(A) and, in most cases, of fo(A).

Let us collect together remarks made at various places.

3.3 Proposition. Let A be admissible.(i) /ιΣ(A) is the sup of the ordinals pinned down by Σl theories of LA; similarly,

h(A) is the sup of the ordinals pinned down by single sentences of LA.(ϋ) MA)>MA)>o(A).

(iii) // A is countable then

(iv) // A is Σ! compact then

Proof. Only (iv) needs proving. Suppose A is Σ: compact but that /ιΣ(A)>/ι(A).Let T(<) be a Σ! theory which pins down some β^h(A). Add new constantsymbols c l5...,cn,... and let T" be T plus the axioms

Since β^/ι(A), every A-finite subset of T has a model which is not well foundedso every A-finite subset of T has a model. Thus, by Σl compactness, T' has amodel, a contradiction. D

^) is an example of a set A with /zΣ(A)>/z(A) = o(A). HYP^α^)) isan example with /ίΣ(A) = /ι(A)>o(A).

The next theorem is extremely useful in computations which involve hΣ(A)and h(A).

3.4 Theorem. Let A be admissible and let F: Ordπ-> Ord be an n-ary functionon ordinals which is ΣL definable in KPU.

(i) α1? . . . , απ < /ιΣ(A) implies F(αl9 . . . , απ) < hΣ(A).(ii) α1,...,αn</z(A) implies F(α1,...,αn)</z(A).

Proof. We first prove (i) in case n = 2. The case for n^2 is similar. Let

F OrdxOrd^Ord

be Σ! definable in KPU, hence in the stronger KP, say by the Σ! formulaσ(xί9x2,y):

(4)

(5) for all α1,α2¥Nσ(α1,α2,F(α1,α2)).


Suppose α1,α2<hΣ(A) and let /? = F(αl5α2). We need to prove that β<hΣ(A).Let Ti(<1, R x), T2(<2, R2) be Σl theories which pin down α1? α2 respectively, thecase with more relation symbols being similar. We will define a Σx theory T(<)which pins down β, but first let us exhibit its intended model 50Ϊ, the one with<m of type β. Let K be a regular cardinal, α 1 ?α 2<κ, so that β<κ. Let

3R1 = <M1, < 1 ,Λ 1 >NΓ 1 , <! of order type α l 9

50Ϊ2 = <M2, <2,^2>^72, <2 of order type α2.

By the downward Lδwenheim-Skolem Theorem (Exercise 2.15) (and the fact thatisomorphic models satisfy the same sentences) we may assume

α^MjC/c and <. = e|kα ί.

Now let

, <,M1? <1,R1,M2, <2,Λ2,α1,α2,)8>

where < =ef j8 and α l 9 α2 and /? are treated as elements, not as subsets. Then $Ris clearly a model of the following set of sentences, where U f is interpreted as Mp

cf is interpreted as αf and d as j8.

φ ( U l ) for all

φ(U2) for all φeT 2,

KP,

c1? c2,d are ordinals,

"< =efd",

σ(c l9c2,d).

If we call the above set of sentences T(<, ...), then 501 is a model of T with <9cri

of order type β. We need to prove that every model 501 of Γhas <αrl well ordered.Thus, let

be any model of T. Identify the well-founded part of <M ,£> with an admissibleset <£,e> by the Truncation Lemma. Since, for ί = l,2

t is well ordered, so a± and a2 are real ordinals and aί9a2eB. By (4),


By (5), and the persistence of Σ formulas,

and, since

we have by persistence,

, E> 1= σ(aί9 α2, F(aί9 a2))

so that b = F(aί,a2). Since b = F(aί9a2)eB, and <=e\b, < is a real well-ordering.This proves (i).

The proof of (ii) is exactly the same when o(A)>ω, since then we may form/\KP and the rest as a single sentence of LA. If o(A) = ω we must replace KPby a single sentence θ of ZF- Power (and hence true in H(κ) since K is regular)strong enough to insure that the standard part of any model of θ is an admissibleset. We leave this to the student. D

All we will actually need of Theorem 3.4 is the following special case.

3.5 Corollary. Let A be admissible. Then /ιΣ(A) and /ι(A) are closed under ordinalsuccessor, ordinal addition, multiplication, and exponentiation.

Proof. We have shown that all these functions are Σt definable in KP. D

The final result of this section seems almost obvious, but it needs proof.

3.6 Theorem. Let A be admissible.(i) // T is a Σ! theory of LA which pins down ordinals then there is a ξ</ιΣ(A)

such that every ordinal pinned down by T is less than ξ.(ii) // φ is a sentence of LA which pins down ordinals then there is a ξ< h(A)

which is greater than all ordinals pinned down by φ.

Proof. This is a typical example of a proof in soft model theory since the proofworks for any logic. We prove (ii). We may assume that the sentence φ(<) pinsdown an initial segment {β\β<ξ}=ξ of ordinals. We show that some othersentence ψ( -<,...) pins down ξ. As before, before writing down ψ, we describeits intended model 9JI, the one with -<*" of type ξ. To simplify matters we assumeφ = φ(<,R), where R is binary, contains no other symbols. For each β<ξ, let

$R0 = <M0, <β,Rβy, yjlϊ=φ and <β have order type β.

Since isomorphic structures satisfy the same sentences, we can rearrange SJΪ^ abit and assume β^Wlβ and <β=e\β.


Define 9M = <M,C7,<JV,S1,S2> where

N(β,x) iff

β<y iff βεγεξ,

SM9y9z) iff

S2(β,y,z) iff

Thus 9JΪ is a structure where -<m has order type ξ. Let ιA(X, • ••) be the sentencedescribed as follows. Let 0(x,N,S l5S2) result from φ(<,R) by replacing

y<z by S^x^z),

R(y,z) by S2(x,y,z),

) by Vy(N(x,y)-> ), and

by

taking care to avoid clashes of variables. Let φ be the conjunction of:

(6) Vx[U(x)^θ(x,N,S1,S2)];

(7) "U is linearly ordered by -<"

(8) V

It is clear that 501 is a model of ψ since (6) just asserts that each Wβ is a modelof θ. We need to show that any other model

of ψ has •< well ordered. To do this it suffices to prove that for any xe L7, the -<predecessors of x are well ordered. Let

aw, =<M,, <„/?,>

where Mx = {y\N(x,y)}, y<xz iff S^x^z) and Rx(y,z) iff S2(x9y9z). By (6),$Jlx\=φ, so <x is a well-ordering and <x agrees with < on the predecessors of x.Thus ^ does pin down ordinals, ξ among them. D

3.7—3.8 Exercises

3.7. Let A be admissible, o(A) = ω, where A is Σx compact. Show that

276

3.8. Let A be Σx compact and suppose that α = o(A)>ω is such that for somexeA,

α = leastβ (L(x,β) is admissible).

Prove that

fcΣ(A) = o(A).

3.9 Notes. Theorem 3.1 is due to Lopez-Escobar [1966]. His proof, however,was by way of Hanf numbers and gave no clue as to the exact description of /z(A)or /ιΣ(A), even for A = H(Kα+1). The proof given here is taken from Barwise-Kunen [1971]. Theorem 3.4 is also taken from there.

There are, by the way, admissible sets which are Σ! compact but such that/ιΣ(A)>o(A). This follows from Theorem VIII.8.3. It is known that /ι(A) neednot be admissible. It is not known whether /ιΣ(A) is always admissible, though itseems unlikely.

4. Indίscernίblesand Upward Lowenheίm-Skolem Theorems

In this section we show how to use the Weak Completeness Theorem and theordinal hΣ(A) to tackle some model theoretic problems for LA. The material inthis section is not used elsewhere in this book.

The simplest result to state is the following theorem, stated in terms of theBeth sequence. Given a cardinal /c, define the cardinal Uα(κ;) by induction on α:

We write 5α for iα(0), but warn the reader that some authors use 5α forWith our definition, 3Λ = c

4.1 Theorem. Let A be an admissible set, let κ = card(A) and oc = hΣ(A). Let Tbe a ΣI theory of LA. //, for each β<α, T has a model of power ^2β(κ), thenfor any λ^κ, T has a model of power λ.

The proof of 4.1 is given in 4.13 below. Actually the proof of this theorem isno more complicated for uncountable LA; it is just that for countable A weknow that hΣ(A) = o(A). Thus 4.1 gives us the following corollary.

4.2 Corollary. Let A be a countable, admissible set and let T be a Σt theory ofLA //, far each β<α = o(A), T has a model of power ^^(K0), then T has amodel of each infinite power. D

4. Indiscernibles and Upward Lόwenheim-Skolem Theorems 277

If Agfl is not HFgjn then it is easy to show that for each βeAm there is asentence of LA which has a model of power ^(K0) but none larger (see Exercise4.18), so 4.2 is best possible for A^HF^. For A^^HF^, L A =L ω ω so weknow a better result.

For applications, there are more useful upward Lδwenheim-Skolem Theoremsin terms of two cardinal models.

Assume our language L has a unary symbol U. A model 9JΪfor L is a model of type (TC, λ) if

caτd(U)=λ.

A set T of sentences of Looω is said to admit (K, λ) if T has a model 50Ϊ of type (K, λ).

4.3 Theorem. Let LA be an admissible fragment, let τc = card(A), a = hΣ(A). LetT be a Σ! theory of LA. // for each β<a there is a λ^κ such that T admits(2β(λ),λ)9 then T admits (δ,κ) for all cardinals δ^κ.

Theorem 4.1 is an easy consequence of 4.3 by adding a new symbol U to Lwithout mentioning it in the theory T of 4.1. On the other hand, a direct proofof 4.1 is a bit simpler than the proof of 4.3, and since the student may be interestedin 4.1, we will also give a direct proof of it.

4.4 Corollary. Let T be a Σ1 theory of a countable admissible fragment LA.Suppose that for each /J<α = o(A), there is a λ^ω such that T admits (3β(λ),λ).Then T admits (λ,ω) for all

Proof. Immediate from 4.3 since /zΣ(A) = o(A). D

4.5 Corollary (Morley s Two Cardinal Theorem). Let T be a countable theory ofLωιω. Suppose that for each a<ω1 there is a λ^ω such that T admits (2j(λ),λ).Then T admits (λ,ώ) for all

Proof. Immediate from 4.4 by putting T in some countable admissible frag-ment. D

The reader of Keisler [1971] will have discovered many applications ofCorollary 4.5. Some of these have routine generalizations using 4.3.

Two-cardinal models are extremely natural when one is working with modelsof set theory of urelements. How many times have we written a typical modelof KPU as a single sorted structure

In fact, we can use such models to prove that Theorem 4.3 is an optimal resultof its type, except for trivial generalizations using downward Lόwenheim-Skolemarguments.


4.6 Example. Let A be an admissible set with κ = card(A), α = /ιΣ(A). For anyjβ<α one can find a Σί theory T=T(U,...) of Luanda ξ,β<ξ<hΣ(A) such that

(i) T has a model of type (^(κ:),κ:).(ii) // SK is a model of T of type (λ,δ) then λ^5ξ(δ). In particular, T has no

model of type (5α(κ:),κ).

Proof. Let T0 = T0(<) be a Σ! theory of LA which pins down β. Let £</zΣ(A)be greater than all ordinals pinned down by T0, by Theorem 3.6. Before de-scribing T we describe its intended model, the one of type (3β(κ),κ). Let M bea set of urelements of power K. Let

be a model of 7^ where < has order type β. By the Downward Lowenheim-Skolem theorem we may assume card ($R0)< max (K, card (β)) so we may aswell assume M0cMuβ. Now let

2R = (MuVTO(jS),M,e,F,M0, <,...)

where, by definition,

F0(a) = rank of a in VM,

F(ά) = the F0(fl)-th member of < .

The theory T is defined as follows. For each xeA let cx be a constant symbol,so there are K of them. T consists of

cx^cy for all

U(cJ for all xεA,

Extensionality (as in KPU),

φ(Uo) for all φeT 0 .

Here U and U0 are new unary symbols. The theory T clearly holds in M. Onthe other hand, if 9K = <Λ,17,E,F,170, <,...> is another model of T then<L/0, <,...> 1=7^), so < is well ordered of order type <ξ. But then F insuresthat E is well founded and of rank <ξ so <>4, 17, £> is isomorphic to a submodelof V^ξ) and hence has card(>4)<^(card(t7)). D

We now turn to the tools for the proofs of these theorems. Anyone familiarwith the model theory of Lωω is aware of the importance of the Ehrenfeucht-Mostowski method of indiscernibles. It plays an even more important role inthe model theory of Looω.


4.7 Definition. Let LA be a fragment of L^, 951 be an L-structure and let <X, <>be a linearly ordered set with X c $01. We say that <X, < > is a sef o/ indiscernibles(for LA in 95Ϊ) if for every n and any two increasing rc-tuples from <X, <>,

we have

(m9Xl9...9xJ = (m9yl9...,yJ (LA),

i.e. the n-tuples <x1,...,xll>, <y l 5...,yn> satisfy the same formulas φ(v1,...,vn)of LA in 951. If 95ί = <M, t/,...> then we say that <X, <> is a seί o/ indiscerniblesover U if, for every finite set w l 5 . . . ,M m e t/ and all increasing π-typles from (X9 < >

we have

(50l,u1,...,um,x1,...,xπ) = (9K,M1,...,Mm,y1,...,y I I) (LA).

The < relation on X need not be definable on 951 in the above definition.The latter notion is really a special case of the first, for let SB = <M, [/,...>

be a structure for LA, let C = {cjMeC/} be a set of new constant symbols, andlet W = ($Jl9u)ueU be the canonical expansion of ΪR to a model for LA(C). (Thelanguage LA(C) is defined in § 2.) Then <X, < > is a set of indiscernibles over Ufor LA in 9Jί iff <X, <> is a set of indiscernibles for LA(C) in 951'.

Indiscernibles help us build large models, and hence prove our theorems bymeans of the following Stretching Theorem.

4.8 Stretching Theorem. Let LA be a Skolem fragment with constants and let 9JΪbe a Skolem structure for LA. Let (X, <) be an infinite set of indiscernibles forLA. For any infinite linearly ordered set <7, <> there is a Skolem structure 91for LA such that:

(i) <y, <} is a set of indiscernibles for LA in 91;(ii) // x^ ^x,, in (X, <> and Oι< <)O in <Y, <>

(iii) In particular, card (51)^ card (Y) and 351 = 91 (LA).

Proof. Part (iii) is just part (ii) with n = 0. Since the distinguished constants ofLA do not play any role in this proof we simply assume LA is a Skolem frag-ment. Let

C = {cy\yeY}

be a set of new constant symbols and form LA(C) as described in § 2. Then LA(C)is a Skolem fragment with constants C. We define a set Q) of formulas of LA(C)

280 VII. More about !_„„,

as follows. Any formula of LA can be written in the form

(1) φ(vί9...9vn, cyι/vn + l9...9cyjvn + m)

where

yι<~'<ym

in < y ><>

Put the formula (1) into Q) just in case

(2) (m9xl9...9xj\=\fvl9...,vnφ(ΰ9cyι9...9cym)

for some increasing sequence

x1< <xm in <Jr,<>,

where x< interprets cy., of course. We claim that

(3) 3) is a supervalidity property for LA(C).

If (1) is a logical axiom, then (2) certainly holds, so (l)e^. We need to see thatif φ(v9G)e@ then (—\φ(ΰ,£))φ@. If not, then we would have

(3Λ9xl9...9xm)t=Vvl9...9vnφ(ΰ9cί9...9cm),

where xί < < xm9 xί < < x'm in <Jf, < >. But this contradicts the indiscernibilityof (X9 <>. The other clauses are equally trivial. We check the \/-rule and leavethe other three to the student. Suppose ι^(c1,...,cm) = \/Φ is a sentence ofLA(C)and ^(clί...,cje®. Then

(9W,x1,...,xm)l=V*

so, for some φeΦ,

so φe^. Thus 3) is a supervalidity property.

(4) If φ(υί,...9vjG\-9 yι< "<yn, y'i<"m<y'n m <^<> then the following

LA(C) sentence is in 2 :

(*) φ(cyι, . . . , cyn) <-> φ(cy{, . . . , cy.n) .

To see what is going on here, suppose φ is φ(vi,v2,v3) and that

and


To see that the sentence (*) in question is in Q) we must first arrange theseelements of < 7, < > in order. Suppose, for example, that

Thus there are only five elements in this case. Let i//(vl9...,v5) be

The definition of Q) says that (*) is in β> iff

whenever xί <x2< <x5. That is, just in case

,x4,x5 iff

whenever xί< ~<x5. This is obvious from the indiscernibility of <Jf, <>, sothis proves (a typical example of) (4). Apply the Weak Completeness Theoremto get a model (5l,fly)yey of Q). Since (cy^cy )e® for yφy\ we can identify ay

with y. Then 91 has properties (i), (ii) of the theorem. D

Using the Stretching Theorem we can reduce our theorems to proving theexistence of models with indiscernibles, as in the next lemma.

4.9 Lemma. Let LA be a Skolem fragment with constants and let T be a theoryo/L A , TSkolemc:T. Let κ = card(LA).

(i) // T has a model with an infinite set of indiscernibles for LA then T hasa model of any power ^ K.

(ii) // Γ = T(U,...) has a model 9M = <M, I/,...) with <X, <> an infinite setof indiscernibles over U for LA then T admits (Λ,,card(£/)) for all λ^

Proof, (i) is immediate from 4.8 (iii) and the Downward Lόwenheim-SkolemTheorem for LA. To prove (ii) let 9K have <X, < ) an infinite set of indiscerniblesover U. Let

C = {cJ«eI7},

be as usual. Thus, <ΛΓ, <> is a set of indiscernibles for LA(C) in 9JZ'. Givenlet < Y, < > be a linearly ordered set of power λ and let

be as given by 4.8, the Stretching Theorem. By Exercise 2.16, we may assume


since this Hull also has properties (i), (ii) of 4.8. Write 91 as 31 = <N, £/',...>. Weclaim that 17=17'. For suppose aeU'. Then

for some term t of LA, some M 1,...,Mmel7 and some )Ί <•"<)'„ in <Y, <>.But, then,

so, by (ii) of 4.8,

whenever x1< <xπ in <J*f, <>. Pick such a sequence of x's. Then there isa ueU such that

and, hence by (ii) of 4.8,

so

In other words, every member of U' is one of the original members of 17. Thus,card (17') -card (17) but

card (Stt) = card ( L A(Q) + card ( Y)

= /c + card(C) + /l

= >L D

To construct a model with an infinite set of indiscernibles, we use the Erdos-Rado theorem of cardinal arithmetic (Lemma 4.10) to construct "coherent setsof /c-variable indiscernibles" and the Weak Completeness Theorem to piece themtogether to get a model with a set of indiscernibles.

We use X n to denote the set

4.10 Lemma (Erdόs-Rado Theorem). Let K be an infinite cardinal and let 0 < n <ω.Let X be a set with card(X)>5π_1(/c) and suppose [ ]n is partitioned into ^Ksubsets, say \_X~\n = \JieICt where card(/)^κ. There is an X0^X and an i0e/such that

card(X0)>/c and [X0]π^Cίo.


Proof. If the reader is not familiar with this result, he can find a proof in mostadvanced books on set theory, in Keisler [1971] or in Chang-Keisler [1973]. D

Let SOΪ be a structure for L and let <ΛT, <> be linearly ordered with X^Wl.Let k<ω be fixed. We say that (X, <> is a set of k-variable indiscernibles forLA in 5DΪ if, for all increasing /c-tuples

x^ ^Xfc, J>ι< <J>fc

in (X, <>, we have

Thus < Jf , < > is a set of indiscernibles iff it is a set of /c-variable indiscerniblesfor each k<ω. Also note that if <X, <> is a set of /c-variable indiscerniblesthen (X, <> is a set of /-variable indiscernibles for all l<k. Any linearly ordered(X, <> with Xc9JJ is a set of 0- variable indiscernibles. The notion of set ofk-variable indiscernibles over U (when SOΪ = <M, 17, . . .» is defined in the same way.

As a first simple use of the Erdόs-Rado Theorem we can prove a result whichis useful when /ιΣ(A) = ω.

4.11 Proposition. Let LA be a fragment of Laoω with card(LA) = κ. Let 0</c<ωbe fixed and let 501 be a structure for L

(i) // card (501) ><Dk(/c) then there is an infinite set (X, <> of k-variable in-discernibles for LA in 50Z.

(ii) // 2B = <M,l/,...> w/ιm> card(ί/)^κ: and card (M)>Dk (card (L/)), ί/zercί/zere is απ infinite set (X, <> o/ k-variable indiscernibles over U for LA in 50Ϊ.

Proof, (i) Let < be a linear ordering of M and, for each /c-tuple x = xt < < xk

from M, let

This partitions [M]k up into ^2K distinct sets, since there are ^2* differentsets of formulas of LA. Since

the Erdόs-Rado Theorem tells us that there is an X^M (of power >2K>K0)such that every element of [X]* is in one fixed member of the partition. Thatis, Tχ = Tf whenever x=Xι < <xk, y = y^ <"'<yk an(3 Xι,...,x f c, yi,...,ykeX.Thus (X9 <\Xy is a set of /c-variable indiscernibles in 501. To prove (ii), letC = {cjMel/} and apply (i) to LA(C) and W = (Wl,u)ueυ with K replaced bycard(LA(C)). D

Theorem 4.1 follows easily from Lemma 4.9 (i) and the following theorem.


4.12 Theorem. Let LA be an admissible fragment, let card (A), α = /zΣ(A). Let Tbe a Σ! theory of LA. // for each β<α, T has model of power ^3β(κ), then Thas a model with an infinite set of indiscernibles.

Proof. We may assume by 2.4 that LA is a Skolem fragment, that ^koiem— Tand that LA is Δt on A. We assume that T has models but no model with a setof indiscernibles for LA and prove that, for some j?<α, T has no models ofpower ^ 3β(κ). Let LA be a Skolem fragment containing LA and two new sym-bols X, <. Let

C = {cπ |0<n<ω}

be a set of new constant symbols. We will be concerned with all the languages

LA(C), L'A(C).

These are all Skolem fragments with constants. For n^O define Θn to be theset of all supervalidity properties 0 for LA(c1?..., cw) with the following properties:

(a) TCΞ^;

(b) "X is linearly ordered by < and has no last element" 60;

(c) "cteXΛCi<ci + 1" e@ for

for each φ(ι;1,...,ί;π)eLA, when

It follows immediately from the Weak Completeness Theorem that

:>„ iff <2 is an s.v.p. for LA(c1,...,cπ) given by some structure

(1)

where 9Jl^Γ, <X, <> is an infinite set of n-variable indiscernibles forLA in $R and a1< <an in (X, <>.

Let S = yπSπ. Note that each ^e6 is in exactly one ®n for Oθ; this nis called the level of 2 and we can determine the level n of 2$ by seeing whether(cπ = cje^ but (cn + 1=c n + 1)<^. Let 1(2) be the level of 2. We define anorder < on S by

iff /(0')>/(0) and

Thus, if 0X0 then 0 and 0' contain exactly the same formulas from thelanguage LA(cί9...9c^9 n = /(0), but not necessarily from L^c^. ^cJ.

The crucial step in the proof is to realize that

(2) <6,-<> is well founded.


Suppose that it were not well founded and let

be an infinite descending chain. If ^eSn and n>m then ^nLA(c1,...,cm)e®m

so we may suppose that the level of $)n is n. Let ®J=®πn LA(c l 5...,cπ)and let ° =(JΠ^°. By the union of chain lemma, ®£ is an s.v.p. for LA(C).Let (50Ϊ, #!,..., απ,...) be a model of ί^°, by the Weak Completeness Theorem.Then 501 NT and X = {al9a2,...} is an infinite set of indiscernibles for LA in 50Ϊwhen ordered by α^α/ if i<j. This proves (2).

Using (2), we can define the usual rank function on S:

= sup

Since 60^0, p(S)>0. We will prove later that p(S)<ΛΣ(A).

(3) Assume p(S) = n<ω. Then no model 501 o/ T has an infinite set of n-variableindiscernibles.

For suppose 50ί N T and (X, < > is an infinite set of n- variable indiscernibles.Let, for

and let $)m be the s.v.p. for LA(c1,...,cJ given by 50ίw. Then ^meSm and

so p(^o)^^1 and hence p(®)>n, contrary to hypothesis.From (3) and Proposition 4.11 (i), we immediately obtain

(4) // p(S) = n<ω then T has no model of power >3n(

If p(S)^ω then we cannot put such an a priori upper bound on the "size" nof a set <A', <> of n- variables indiscernibles, but we can put a bound on card(X).

(Suppose 501 NT, <Jf, <> is a set of n-varίable indiscernibles for LA in 50ί(5)<and that a1< ~<an in <JΓ, <>. Let & be the s.v.p. in 6Π given by

We prove (5) by induction on β using the Erdδs-Rado Theorem as in4.11(i). So suppose we know the result for ordinals γ<β (β>0) and suppose

286 VII. More about !_«,„

(κ). For each increasing n + l tuple x = xl<- '<xn<xn+ί

from <X, <>, let

This partitions [X]π + 1 into ^2K sets. Since 2κ^'Dωβ + 1(κ) and

we can apply the Erdδs-Rado Theorem to find an X0^X with card(AΓ0)> 3ω^(κ;)such that every member of [X0]"

+1 ϋes in one member of the partition. Thatis, for tt + 1-tuples x1< <xn+1 from X09

so that (X0, <> forms a set of (n + l)- variable indiscernibles in 9W. Letal< -<an+i be chosen from X0 and let 0 be the s.v.p. given by

Then Q)Q<$) so p(^0)<jS. But then 9W0 contradicts the inductive hypothesissince card(X0)>lDω/s(κ:)>5ω(y+1)(κ:) where y=p(®0) This contradiction proves(5) for jS>0. The case for j5=^0 is easier and is left to the ideal student.

From (5) we get at once:

(6) Every model 9JI of T has power <5ωβ(κ), where β = p(S).

For let X = M and let < be any linear ordering of X. Recall that <X, <>is a set of 0-ary indiscernibles for 50Ϊ. Then, if ® is the s.v.p. for L'A given by

(W9X9<)

then p(@)<β and card (5K) = card (-SO<3ω(pW + 1)(ιc) which is ^Sωβ(κ).Finally, we claim that

(7) p(S)<ΛΣ(A).

To see that this concludes the proof, we see that if hΣ(Jk)=ω then the resultfollows from (4). If p(S)=)8 and ΛΣ(A)>ω then ωβ<hΣ(A) by Corollary 3.5,so the conclusion follows from (6). (This is the only use of anything remotelyapproaching admissibility in the entire proof.)

It remains only to prove (7). We will see in § VIII.6 that <S, -<> is a Π de-finable well-founded tree of subsets of A and that every such tree has rank less


than /ιΣ(A). That is probably the simplest proof of (7). It's good for the soul,though, and gives added appreciation of the machinery developed in § VIII.6,to give a direct proof. We present a sketch to be filled in by the student.

Our goal then is to write down a Σl theory T'(<) of LA which pins downβ = p(&). As is our custom, we first describe the intended model 9JI of T'(<),the one where <αίϊ has order type β. Let 5R be the following structure:

<Λf ;)»,<; A; Power(A),£;S,<F,G,x>X6A

where

M = /?uAuPower(A),

= level of 2 for

= some constant φω, otherwise,

= p(2) if ^e6

= some constant φβ otherwise,

x Power (A)).

Now suppose that

satisfies all the finitary first order sentences true in M and that

Ac:endSΓ.

We will show that <£, <'> is well ordered. The proof will show that the set offinitary sentences we actually use is Σί on A so that will conclude the proof.

By the axiom of Extensionality for Power (A), we may assume that

PC Power (21'), F = en(2ΓxP), and S'cp.

Now suppose that the linear ordering <£, <'> is not well ordered, so that thereis a subset B0^B with no <'-minimal element. Let

and let

where we must remember that G'(^) might be a nonstandard integer. It is notdifficult, though tedious, to see that ®o^S5 since each £#e®' claims to bean s.v.p. for liA(cl9...,cG(g)) of the appropriate kind, and the relevant quantifiersare all universal. So ®'ό must have a minimal element 2. By chasing 3> back


into B, a contradiction easily results by considering the cases G'(^) standardand G'(^) nonstandard separately. D

4.13 Proof of Theorem 4.1. Again, using 2.4 we may assume LA is a Skolemfragment and that ^kolemcT. Then 4.1 follows from 4.12 and 4.9 (i). D

4.14 Corollary. Let LA be an admissible Skolem fragment with /zΣ(A) = ω. Let Tbe a Σ! theory of LA. // for each /c<ω, T has a Skolem model with an infiniteset of k-variable indiscernibles, then T has a Skolem model with an infinite set ofindiscernible s for LA.

Proof. See line (3) of the proof of Theorem 4.12. D

We next turn to the analogous theorem for two cardinal models.

4.15 Theorem. Let LA be an admissible fragment with τc = card(A), α = /ιΣ(A).Let T=T(U,...) be a Σ^ theory of LA. If for each β<α, there is a λ^κ suchthat T admits (3β(λ),λ), then T has a model 501 = <M, [/,...> with an infinite setof indiscernibles over U for LA.

Proof. We indicate the changes necessary in the proof of Theorem 4.12. We mayagain assume that LA is a Skolem fragment and that ^kolemcT. We may alsoassume (by adding K new constant symbols and some axioms of the formU(cx), cx^cy to T) that every model 9JI of T has card(L/)^κ.

Let LA(c1,...,cn) be as before and let ^eSn iff <2) is an s.v.p. for LA(c1,...,cn)with properties (a), (b), (c), (d) as before plus

(e)for all terms ί(^1?...,ι;n) of LA.

The analogue of (1) is the one way result:

(Γ) ^eSπ if Q) is the s.v.p. for LA(c1?...,cJ given by some (9JΪ,X, <,α1,...,αjwhere <X, <> is a set of n-variable indiscernibles over Uw for LA andaί<" <an in <X, <>.

Luckily, we never really used the other half of (1).

The relation -< on S = (Jn ®n is defined just as before. Again we have, as-suming T does not have a model ΪR with a set of indiscernibles over U931,

(2') <6,-<> is well founded.

This is just a bit trickier than (2). Suppose


Again, we may assume that each Q)n has level n. Let = nnLA(c1?...,cn) andlet @%=\Jn@°. βy the Union of Chain Lemma, ° is an s.v.p. for LA(C). Let(9010,α1,α2,. ..,«„,...) be a model for ° let X = {α1,α2,...}, α^α,. iff i<j. LetaW-Hull^X). By Exercise 2.16, 2Rl=®£. Thus,9ϊϊis a model for Γand <X, <>is a set of indiscernibles for LA in $R. We need to see that (X , < > is a set ofindiscernibles over U9". Thus suppose i/eU9^. We need to see that increasingn-tuples from (X, <> satisfy the same formulas in (95ϊ,w). (The case with morethat one u is similar.) Since ΪJΪ = Hull (X), there is a term ί(ι;l5...,t;TO) such that

Then, by (e)

whenever x1< <xm in <X, <>. Now suppose rc<ω, < <xπ, <" <yn

in <X, <>. We need to see that for all formulas φ(vl9...,vn9vn + ί), if501 \= φ\_x^. . . , xπ, w] then <ίΰl\=φ\_yί,...,yn9u]. Pick an increasing m-tuple w± < - < wm

such that W!>Λ:Π, Wι>yn. Now consider the formula ψ(vί9...9vn,vn + ί,...9vn+m)given by

Then, since M = ί(w l5...,\vm),

and hence,

by the indiscernibility of <X, <> in 9K. Thus

Thus <X, < > is indiscernible over 17, proving (2).Define p(^), p(S) as before.

(3') v4sswme p(S) = π<ω. T/zβπ no model Wl of T has an infinite set ofn-varίable indiscernibles over (Jm.

The proof of (3') is just like the proof of (3).Using (3') and 4.11(ii), we get

(4') // p((S) = n<ω then T has no models of type pn+1(A),/l) for any λ.


Corresponding to (5) we have

(Suppose M\=T, (X, <> is a set of n-varίable ίndίscernibles over U( = \(5')<and that aί<"-<an in <X, <>. Let Q) be the s.v.p. in ®B given by

[(aR,X,<,α1?...,απ) and let β = p(&). Then card(X)<5ω(/?+1)(card(l7)).

The proof is by induction on β and uses the Erdόs-Rado Theorem. The proofit too similar to the proof of (5) to present. From (5') we get

(6') // art^T then card (9W)<3ω/ϊ (card (U9*)) where β = p(<S).

The proof is concluded by showing that

(7) p(S)</ιΣ(A).

The proof of (7') is just like the proof of (7). D

Theorem 4.3 follows from 4.15 just as Theorem 4.1 followed from 4.12.


4.16. Prove that if α is admissible then

3α(K0) = 2α if α>ω,

3β(No) = («o)

= 3ω+ω if α = ω.

4.17. Let AN be admissible above 91, κ0 = card(9l), α^o(A). Prove that

Let κl=carά(A9l). Prove that if hΣ(A) = β>a then

4.18. Let A be an admissible set, α = /ι(A). Prove that the Hanf number forsingle sentences of LA is at least

for some

That is, show that for λ0<λ there is a sentence φ of LA which has models ofpower ^λ0 but none of power ^λ. [Given XeA, β<h(A), formalize Vx(β).~\Prove that the Hanf number is always of the form 5λ for some limit ordinal λ.


4.19. Let A be an admissible set with o(A)>ω.(i) Prove that each φeA can be put in a Skolem fragment LβeA in such

a way that every model of φ (not just those in A) can be expanded to a modelof Tskoiem [Use Infinity to carry out the proof of 2.4 inside A.]

(ii) Prove that the Hanf number for single sentences of LA is

where α = /z(A). That is, prove that if φeLA does not have a model of everypower ^ card (A) then there is an XeA and a β<h(A) such that φ has nomodel of power ^3β(caτd(X)). [The set X will be the LB of (i). Modify theproof of 4.12.]

(iii) Prove that if A is a pure admissible set then the Hanf number for singlesentences of LA is 5h(A), even if o(A) = ω.

4.20. Let A be an admissible set, let α = /zΣ(A) and let

some

Theorem 4.12 states that the Hanf number for Σx theories of LA is ^λl.(i) Prove that this Hanf number is ^λ0.

(ii) Prove that if A is countable and /HF^, or if /zΣ(A)>o(A), then λ0 = λ ί f

It is an open problem to describe this Hanf number in general. Is it λ0 or λί orsomething in between?

4.21 Notes. Morley [1965] shows that the Hanf number for single sentences ofLωιω was 5ωι. (This follows from 4.2.) Morley [1967] showed that the Hanf num-ber for single sentences of ω-logic was 3α where α = ω{. (The hard half of thisfollows from 4.2 with A = L(α).) Barwise [1967] generalized this to obtain theHanf number for any countable, admissible fragment. This was generalized inBarwise-Kunen [1971] to obtain 4.19 (iii). The theorems of this section are areworking of the ideas from Barwise-Kunen [1971] so that they apply to theories,not just single sentences. Theorem 4.3 is a generalization of Morely's Two Car-dinal Theorem of Morley [1965]. The student should consult lectures 16 and17 of Keisler [1971] for a different proof of the countable versions of these results.

The student should be aware of a difference between the results of this sectionand those in Chapter III. The use of admissible sets was absolutely essential inChapter III to obtain our results. Here they provide a convenient setting butweaker assumptions would do. Of course we need to know that the countableset A is admissible to know that hΣ(A) =


5. Partially Isomorphίc Structures

Having seen in the previous sections that the model theory of uncountable frag-ments is not completely beyond our control, even if it is less tractable thanfor countable fragments, we now investigate some uses of uncountable sentences.

One way to appreciate Looω is to see the role it plays in algebra, but this isnot the book to discuss such topics. We can only give a few exercises. The topicswe discuss are of a more logical nature. These final sections are completelyindependent of the first half of the chapter. Admissible sets will not appear inan essential way until § 7.

A partial isomorphism f from 501 into 91 is simply an isomorphism

where 9Jί0, 910 are substructures of 9JI and 91 respectively. A set / of partialisomorphisms from 9JΪ into 91 has the back and forth property if

(1) for every /e/ and every xe$R (or ye9ΐ) there is a gel with f^gand xedom(g) (or yerng(g), resp.).

We write

if / is a nonempty set of partial isomorphisms and / has the back and forthproperty. If there is an / such that /: 9K^p9t then we say that 9JZ, 91 are partiallyisomorphic and write 501 p 91. (Some authors prefer the more picturesque ter-minology potentially isomorphic, to suggest that $R and 91 would become isomor-phic if only they were to become countable, say in some larger universe of settheory.) Note that if /: 501^91, then {/}:9Jϊ^p9l.

5.1 Examples, (i) The canonical example is given by two dense linear orderings9K = <M, <> nd 9t = <Λf, <> without end-points. Let / be the set of all finitepartial isomorphisms from $R into 91. Then

regardless of the cardinalities of $R and 91. This is quite easy to verify. Combinedwith Theorem 5.2, this shows that the theory of dense linear orderings withoutend points is K0-categorical, i. e., that all its countable models are isomorphic.

(ii) If 50Ϊ, 91 are dense linear orderings with first elements x0, y0 respectively,but without last elements, then 9W^p9l but the set / used in (i) no longer hasthe back and forth property. Let

Then

5. Partially Isomorphic Structures 293

(iii) We can generalize (i), (ii) as follows. Let LA be a countable fragmentand let T be an X0-categorical theory of LA. Then for any two infinite modelsw, 9i of r,

an 91.

A proof of this will be given in 5.5 below.(iv) We can get a different generalization of (i) and (ii) by looking at

Ko-saturated structures 9JΪ and 9ί. If 9K^9l (LωJ then Wl^pW. The set / is de-fined as follows: Consider those partial isomorphisms

where 9Ϊ10 is finitely generated by some aί9...,an. We will let fel iff

(^a^...,an)^(^f(a^...J(an)) (Lωω).

A simple use of K0-saturation shows that / has the back and forth property.Traditionally, the back and forth property has been used for constructing

isomorphisms of countable structures.

5.2 Theorem. Let 9Jί, 9ΐ be countable structures for the same language and let/:SDΪ=p9l. For every /Oe/ there is an isomorphism

with /o<Ξ/.

Proof. Enumerate <3R = {xί,x2,...}9 9l = {y1,j;2,...}. Define

/2n+ι=some gel with f2n^g, xwedom(#),

/2n + 2 = some 9^1 with /2« + ι — #> .y«erng(0)

by using the back and forth property (1). Let / = (Jn /„. Then / maps 50ί onto91 and preserves atomic and negated atomic formulas so /:9JΪ^9l.

The examples and Theorem 5.2 should suggest to the student of the previouschapter that ^p could be the absolute version of ^. After all, they agree oncountable structures and ^p does not seem to depend on cardinality. At firstglance, though, it is not obvious that ^p is absolute, but merely that it is Σi:

where the part within brackets is Δ0. This is no better that ^, itself a Σί notion.The Πi equivalent of =p is given by the next result. There is, of course, no Γ^equivalent of ^. This result as well as 5.7 appear in Karp [1965].


5.3 Karps Theorem. // $R, 91 are structures for the language L, then 9Jϊ^p9ii^aBΞΞϊiα^j.Proof. We first prove (=>). Let /:SR^p9l. We prove, by induction on formulasφ(vί9...9vj of L^ that if /G/, x1,...,xπGdom(/) then

OT^IX,...,*,] iff 9tNφ[/(xO,...,/(x,,)].

(The theorem follows by considering those φeLaoω which are sentences.) Ifφ is atomic, the result follows from the fact that each /G/ is a partial iso-morphism and so preserves atomic and negated atomic formulas. The casewhere φ is a propositional combination of simpler formulas is immediate bythe induction hypothesis. The back and forth property (1) comes into play onlyin getting past quantifiers. Suppose φ is 1vn+ί\l/(vl9...9vn+ί). Let f,xi9...9xn begiven. We assume 9JtNφ[x !,...,*„] and prove yi\=φ\_f(xί),...9f(xn)']9 the otherhalf being similar. Thus, there is a ye9Jl so that

W\=ιl/[_xl9...9xΛ9y].

Use (1) to get a gel with /£#, .yedom(g). Then, by the induction hypothesis,

so

and X = (X) so

as desired. Since Vt;w^<-»-n3t;w— 11 , we need not treat V separately.Now assume $R = 9l (L^J. What should our set / be? The proof of the first

half of the theorem tells use. Let /G/ iff

where $R0 is finitely generated by some x l 5...,xπ and

by which we mean that x1?...,xπ satisfies the same formula of L^ in M thatf(xΐ ),..., f ( x n ) satisfy in 91. (Note that we need Πj Separation to define 7 sothat we cannot carry out this proof in KPU.) Since $01 = 91 (L^J, the trivialpartial isomorphism is in /. We claim that 7 has the back and forth property.Let /G/ be as above and let xn+1 be a new element which we need to add tothe domain of/. It suffices to find a y e 91 so that

5. Partially Isomorphic Structures 295

for then we may set g(xn+1) = y and extend to the substructure generated byXi, . . . ,x w +ι in the canonical fashion. So suppose there is no such y. Then, forevery ye9l there is a formula φy(vi9..., vn+l) such that

Let ψ(vl9 ...,*;„) be

Then 9WN^[x 1 ? ...,xn] by letting uπ +ι = xπ + 1 but

This contradicts fel. D

This theorem has a number of important uses. Here we state those having to dowith absoluteness.

5.4 Corollary. ^p is the absolute version of ^.

Proof. 501 = 91(1. ) is a Πx predicate of 9M,9t, by the results of § III.l, so ^p is A t .It agrees with ^ on countable structures by Theorem 5.2. D

5.5 Corollary. Example 5.1 (iii) is true.

Proof. Let T, LA be as in 5.1 (iii). We need to show that

VSR V9l[9Jϊ, 91 infinite Λ9Jt^TΛ9l^T -» 9Jl^p9ί].

By 5.4, the part within brackets is absolute (in the countable parameter T), so weneed only verify the result for 9M, 91 countable. But for such 9JΪ, 91, the resultfollows from the hypothesis that T is K0-categorical. D

This result (5.5) shows us that if a countable theory T is X0-categorical, thenwe should be able to prove this by a back and forth argument.

5.6 Corollary. Let SR,9t be partially isomorphic structures for a finite language L(i) For alia, L(α)OT p L(α)«.

(ii) For all α, α z's -admissible iff α is Wl-admissible.(iii)

Proo/. (i) This is a Π t condition on 301,91 which clearly holds when 9W, 91 arecountable since then they are isomorphic. Part (ii) follows immediately from (i)since α is 9Jt-admissible iff L(oί)^\= KPU + . Part (iii) follows from (ii). D

296 VII. More about Lcoω

One of the advantages of Theorem 5.3 is that it allows us to approximate therelation 9R^p9l by approximating

Define the quantifier rank of a formula φ of Laoω recursively as follows:

) = 0 if φ is atomic,

qr(3ι;φ) = qr(Vι φ) = qr(φ) 4- 1,

qr(/\Φ) = qr( N/Φ) = sup (qr(φ) | φ e Φ} .

Thus qr(<p) is an ordinal number. Since qr is defined by Σ Recursion in KPU,we have qr(φ)<o(A) whenever φ is in the admissible fragment LA.

We write

if for all sentences φ of L^ with

iff

Thus SRΞίlίLaJ iff for alia, SHI = "31.The following is a refinement of Karp's Theorem also due to Karp [1965].

5.7 Theorem. Given structures 9W, 91 for L, 9JΪΞΞα9l i/f ί/ie following conditionholds: There is a sequence

where each Iβ is a nonempty set of partial isomorphisms from 9JΪ into 91 and suchthat whenever β + l^α, felβ+ί and xe9Jl (or yε 91) there is a gεlβ such thatf^g and xedom(g) (resp., yerng(gf)).

Proof. The proof is a routine refinement of the proof of Karp's Theorem. To prove(<=), one shows that if

then

2Rl=< 3o[x 1,...,xJ iff

To prove (=>), let /^ be the set of those finitely generated partial isomorphisms /which preserved satisfaction of formulas φ with qr(φ) β. D

6. Scott Sentences and their Approximations 297


5.8. Prove that if a theory T of Lωω is K0-categorical then every model of T isX0-saturated. [Use 5.1(iii), Theorem 5.3 and the fact that K0-saturation can bedefined by a conjunction of sentences from Lωιω.]

5.9. Let SOΐ, 91 be partially isomorphic structures for a finite language. Show thatfor every α, the pure sets in L(α)aR and IXα)^ are the same.

5.10. Let λ be a limit ordinal. Prove that if Wi = β(3l for all β<λ then $R = A9l.[Each sentence of quantifier rank λ is a propositional combination of sentences ofsmaller quantifier rank.]

5.11. Show that the following notions are definable by a single sentence of Looω.(i) G is an K^free group.

(ii) G is an abelian p-group of length ^ α (for any ordinal α).

5.12. (i) Show that if G is a reduced abelian p-group and G = H (L^J then H is areduced abelian p-group.

(ii) Show that the notion of a reduced abelian p-group is not definable by asingle sentence of L^. [Hint: There are reduced p-groups of every ordinal length.Show that if the notion were definable then there would be a sentence whichpinned down all ordinals, contrary to Theorem 4.1.]

6. Scott Sentences and their Approximations

One of the tasks the mathematician sets for himself is the discovery of invariantswhich classify a structure 9JΪ up to isomorphism (homomorphism, homeomor-phism, etc.) among similar structures. In this section we consider the problemof characterizing arbitrary structures up to ^p. We will associate with eachstructure 9JΪ, in a reasonably effective manner, a canonical object σm such that

9Ή^p9l iff <7OT = <r»-

Hence, if 9JI, 91 are countable we will have 50Ϊ 91 iff σyn = σ^. Our invariantswill not be cardinal or ordinal numbers, though, as is often the case. Rather,they will be sentences of Looω with the additional properties:

and

implies 9W^ p9ϊ.

The sentence σ^ is called the canonical Scott sentence of 9M.The canonical Scott sentence is built up from its approximations defined

below. We use s to range over finite sequences <x l 5 ...,*„> from 9JΪ and S Λ Xto denote the extension <x l 9..., xπ,x> of s by x.


6.1 Definition. Let $ίl be a structure for a language L. For each ordinal α andeach sequence s = <x l 9 . . . , xπ> we define a formula σ%(vί9 . . . , vn\ the ^-characteristicof s in 9K, by recursion on α:

(i) σ*(υί9...9vj is

/\{φ(vί9...9vj\φ is atomic or negated atomic and 9WN=φ[s]}.

(ii) σf + ί(υί,..., vn) is the conjunction of the following three formulas

(1) σζ(υl9...9vj;

(2) VVn+l\/XeW°^X(Vl>~">Vn);

(3) Λ*e^ 3^+l σK(^ ! > • • • > O

(iii) If Λ>0 is a limit ordinal then σ^(v^ ..., i J is

If we need to indicate the dependence on 9JΪ we write σ*mtS) for σ*. If 5 is the emptysequence we write σα or σ^.

6.2 Lemma. Fix 9JΪ,α and s = <x l 5 ...,xπ>.(i)

(ii)(iii)

(iv) // /c is an infinite cardinal and card(!0l)<κ:, card(L)</c and a<κ thencard(sub(σ^))<κ;.

Proof. A simple induction on α proves all these facts. D

The crucial properties of the α-characteristics are given by the next result.In this section we write

(and

to indicate that all <x1? . . . , xπ> satisfies the same formulas φ(vί9 . . . , vn) (of quantifierrank at most α) in 9Jί that <)>ι, ..., }O satisfies in 9ί.

6.3 Theorem. Let 9K,^ fee Lrstructures, s = <x l 5 ...,xπ) α sequence from 9JΪ,ί = <y1? ..., };„) α sequence from 91. Tfe following are equivalent:

(i) (9W,x1,...,xπ)Ξ

α(9d,y1,...,3;π).(ii) 9lf=σ^>s)[f].

(iii) Γ/ie ^-characteristic of s in 5R is identical with the ^.-characteristic of tin 91.


Proof. The proofs of (i)=>(ii) and (iii)=>(ii) and both trivial. The first followsimmediately form 6.2 (i), (ii). The second implication also follows from 6.2 (ii),since ΉNσf^O], so if σ^s) = σ^ί)? then

We are left with task of proving (ϋ)=>(i) and (ii)=>(iii). To prove (ii)=>(i), we useTheorem 5.7. Assume

and define, for β^α, a set Iβ as follows: felβ iff

/: arioso, W10^WI, yi0<^9l, where

9JΪ0 is generated by some z1? ..., zk, and

The map /0 generated by sending xf to y. (i = l, ..., n) is in /α by hypothesis. By6.2 (iii), we have

The final condition on this sequence, the one demanded by 5.7, follows immediatelyfrom the definition of σf^^ >Zk).

Finally, we prove (ii)=>(iii) by induction on α. The cases for α^O and α alimit ordinal are trivial. So suppose

By 6.1 (ii), we need to prove that

(4) σ(0«,s) 1S σ(9M) '

(5) for each xeϊR there is a ye9t such that

σ?SR,s x sx) i s σ?»,fX X3')>and

(6) for each ye 91 there is an xe9Jί such that

σ(m,s~X) is σon,r^y)

Now, by the induction hypothesis, (4) is true, (5) reduces to

(5')


and (6) reduces to

(6') 911= Vl^ Vxe«^f»..-χ)(^+l)M

But (5'), (6') are immediate consequences of

by lines (3), (2) respectively. D

If we apply 6.3 to the empty sequence, we obtain the following result.

6.4 Corollary. For all 501,91, the following are equivalent:(i) SRΞα9ί;

(ii) 91 σ^;(iii) σ^ = σ^. D

6.5 Definition. The Scott rank of a structure ΪR, sr(9Jl), is the least ordinal α suchthat for all finite sequences x l 9 . . . , xn, y 15 . . . , yn from 9ϊί,

implies

(9W,x1,...,xn)^a+1(^,yi,...,yM).

We will see, quite soon, that if α = sr(9W) then

(aMι,...,xJ = W3>ι,...,3U

actually implies

It is more convenient to use 6.5 as the definition, though, since then the nextlemma becomes obvious.

6.6 Lemma. // K is an infinite cardinal and card(50l)<κ: then sr($R)<κ:.

Proof. The proof is easy and we will get a much better bound in the next section,so we leave the proof to the student. D

6.7 Definition. Let 9JI be a structure for L, let μ = sr(ΪR). The canonical Scotttheory of 9JI, Sm consists of the sentences below:


for all finite sequences s=<x 1 ? ..., xπ> from SDΪ. The canonical Scott sentenceof 9JI, σ , is the conjunction of the canonical Scott theory of 9JΪ:

σ<m — Λ ^SEK

Note that qr(σαrί) = sr(9JΪ) + ω. Also, from the definition of sr(9Jΐ) we see that

We now come to the main theorem on Scott sentences.

6.8 Theorem. Given structures 501,91 for a language L, the following are equivalent:(i) m^p9l;

(ii) $i\=σm;

(iii) σ<m = σ<n

Proof. We already know that Wl^p$l iff m= rjoω^l. Since 9KNσOT we see that(i)=>(ii) is immediate. Similarly, since ^l\=σ^ (iii)=>(ii) is immediate. To prove(ii)=>(i) define Iβ, for all β, just as in the proof of 6.3. The hypothesis that 91 Nσ^insures that Iμ+l=Iμ so

/ 9Ή~ 911 μ. JJl — p Jl .

Finally, we prove that (i)=>(in). Assume that 501=^91. Then sr(9M) = sr(9l).Let μ = sr(9Jί). For each xl9 ...,xπe9M there is a sequence yί9...9ynE9l such that

and vice versa. Then, by 6.3, every σf^ s) is some σ(Vo and vice versa. Thus 8^ = 8^and σ^^σ^. D

The remainder of this section is devoted to corollaries of Theorem 6.8. Firstwe have Scott's original result.

6.9 Corollary (Scott's Theorem). Let L be a countable language and let Wl be acountable structure for L. The Scott sentence σm is a sentence of Lωιω with theproperty that

9JΪ^9l iff Vl\=σm

for all countable L-structures 91.

Proof, σ^ is in Lωιω by Lemma 6.2(iv). The result is then an immediate con-

sequence of Theorem 5.2 and 6.8. D

An π-ary relation P on 9K = <M,K1,...,R/> is invariant if for every automor-phism / of 9JΪ and every x l 9 ...,xπ

iff ^/(xj,..., /(*„)).


From now on (in this section) we assume L is countable. Whenever we refer towe assume L is finite.

6.10 Corollary. // ΪR is a countable structure for L and P is an n-ary relation on 501,then P is invariant iff P is definable by some formula φ(vl9 ..., vn) of Lωιω (withoutadditional parameters) :

P(xl9...,xJ iff 9W^(p[x 1,...,xJ.

Proof. If P is defined by φ then P must be invariant since /:$0ΐ^9W and9Jί^φ[x l9...,xπ] implies yR\=φ\_f(x1\ ...,/(*„)]. Now assume P is invariant.Let φ(vl9 ...,!;„) be

where μ = sr(ΪR). It is clear that P(xl9 ...,xπ) implies 9Kt=φ[x l 5 ...,xπ].To prove the converse, suppose that $X\=φ\_yl9 ...,yn]> so that

[yl9...,y^\ for some x1? ...,xπ with F(xl9 ...,xn). Then

xί9 . . . , xn) = ω (9W, yί9 . . . , yn) ,

so that there is an automorphism / of 9JI with f ( x t ) = yt by 5.2. Since P is invariant,P(y !,..., yn) holds. D

6.11 Corollary. Let 9JI be a countable structure for L and let x£ΪR be an elementfixed by every automorphism of 9JΪ. Then x is definable by a formula φ(v) of Lωιω:

Conversely, a definable element of $R is fixed by every automorphism.

Proof. Apply 6.10 with P-{x}. D

A rigid structure is one with only one automorphism, the identity map.

6.12 Corollary. // 9Jί is α countable structure for L then $R is rigid iff every elementx of ail is definable by a formula φ(v) of Lωιω:

These results will be improved in the next section.

7. Scott Sentences and Admissible Sets 303


6.13. Let ΪR be a countable structure with x l 9 . . . , xneWl such that ($01,x^, ...,xjis rigid; i.e., no nontrivial automorphisms of 501 fix, x l 5...,xπ. Show that 501has ^K0 automorphisms.

6.14. Let 501 be a countable L-structure with <2K° automorphisms.(i) Prove that there is a finite sequence x ί,..., χn from 501 such that (501, x t,..., χn)

is rigid. [Hint (P. M. Cohn): Let σn fix x l 9 ...,xπ but move, say, xπ + 1. Letσ = ... σε

n

n... σε

2

2 σ^1 where ε£ =0 or = 1. Show that this gives 2*° automorphisms.](ii) Show that for all 5R, 9ll=σOT implies 501 5R; i.e., that there are no un-

countable 91 with 5DΪΞΞ9l(Lωιω).

6.15. Show that if G is an K t-free abelian group then G = pH iff H is N1-free.Thus the notion of free group is not definable in Looω.

6.15 Notes. Scott's Theorem and Corollary 6.10 were announced in Scott [1965].A proof, in the context of invariant Borel sets, appears in Scott [1964]. The Scottsentences used here are derived from Chang's proof of Scott's Theorem in Chang[1968]. The presentation follows that used in the survey article Barwise [1973].Exercises 6.13, 6.14, 6.15 are due to Kueker. They are proved in Barwise [1973].

7. Scott Sentences and Admissible Sets

The first systematic study of the relationship between α-characteristics, canonicalScott sentences and admissible sets was undertaken by Nadel in his doctoraldissertation. His idea was to use α-characteristics and Scott sentences as approx-imations of models, asking to which admissible sets the formulas σm9σξΛ belongas an alternative to asking to which admissible sets 501 itself belongs. This hasproven to be a fruitful idea. In this section we delve into the more elementaryparts of the theory.

To simplify matters we assume the underlying language L of Lαoω has nofunction symbols. Since function symbols can always be replaced by relationsymbols, this is no essential loss. (The sole point in this restriction is that if L isan element of an admissible set A then the set of atomic and negated atomicformulas of the form

φ(vl9...,vn)

(for fixed n<ω) is a set in A if L has no function symbols, or if o(A)>ω, butnot if L has a function symbol and o(A) = ω).


7.1 Proposition. The formula

is definable in KPU as a Σ^ operation of $R,s,α.

Proof. Consider sequences s as functions with dom(s) some n<ω and rangecan. Let

If we write out the definition of F as given in 6.1 it takes the following form:

F(3W,s,α) = y iff (i)v(ii)v(iii)

where(i) α = 0 Λ Δ0(9Jί, s, y) (a Δ0 predicate of 9Jί,s and y);

(ii) α = /ί + l for some β<a and y = /\{θί,θ2,θ3} where

Θ2 is Vu π + ! \/Φ where

R,sΛx,β) = z,

,sAx,jβ) = z, and

is similar to Θ22

(iii) Lim(α)Λy =This definition clearly falls under the second recursion theorem. D

7.2 Corollary. // LA is an admissible fragment and 9JΪ is an L-structure in the ad-missible set A then, for any L-structure 91,

$R = 9t(LA) implies $0ΪΞΞα9l

where α = o(A).

Proof. By Exercise 5.10 it suffices to prove that

for all β<α. But for β<α, σ^eLA by 7.1 and »l^σ^ so 91 Nσ^. But then9JίΞ^9l by Corollary 6.4. D

If σgjj were definable as a Σ^ operation of 9JI in KPU then we could extend 7.2to read

3[R = 9l(LA) implies 951=^91,


since then σm would be in LA. This, however, is not true. Unlike its approximations,the canonical Scott sentence σm is not definable in KPU as a Σl operation of 9JΪ.The problem is that sr(9Ji) may be just a bit too big; that is, sr(9#) may equalotHYPgfl). (See Exercise 7.13, 7.14.) This is as big as it can get, though, as we seein Corollary 7.4.

7.3 Theorem. Let LA be an admissible fragment of LaQω and let 9W,9Ϊ be L-structureswhich are both elements of the admissible set A. Then

aW = 5R(L A ) implies 501 = 91.

Proof. By 7.2 we see that 9JlΞΞα9ΐ where u = o(A). Let / be the set of finite partialisomorphisms /={<x1, ιy1>,..., <xπ,);II>} (0^rc<ω) such that

(1) (Wxι,...,xJ = *(W,yι,...,yJ.

Since 9JΪΞΞα9l, the trivial map is in / so 7^0. We will prove that

Suppose (1) holds and that a new xn + 1e$R is given. We need to find asuch that

By Exercise 5.10 it suffices to insure that

for each β«x. Suppose that no such yn+ί exists. Then

where S = <X I , . . . ,X B + I >. By Σ Reflection in A, there is a y<α such that

and hence

so

contradicting


This establishes the "forth" half of the back and forth property; the "back" halffollows from the symmetry of 9JΪ and 91 in the theorem. D

Theorem 7.3 is sometimes called NadeΓs Basis Theorem. The reason for callingit a basis theorem is seen by stating the converse of its conclusion: If there is asentence φ of L^ true in 501 and false in 91, then there is such a sentence in LA.

Our first application of 7.3 is to get the best possible bound on sr(50l). Anotherproof of this can be given by means of inductive definitions.

7.4 Corollary. Let 9JI be a structure in an admissible set A. Then

Proof. Let oc = o(A). Let x l 9 ..., xπ, y1? ...,3/πe9Jl be such that

But then

so, by 7.3,

...,y). D

The remainder of this section deals with uses of NadeΓs Basis Theorem toimprove the results of the previous section.

7.5 Theorem. Let 9JΪ be an L-structure and let Pbea relation on $R which is definableby some formula of Looω without parameters. Let A be any admissible set with(9K,P)eA. Then P is definable by a formula of LA without parameters.

Proof. Let us suppose, for convenience, that P is unary. We assume that P is notdefinable by any formula of LA. If we can find an x,y such that

O, and (SW,x)Ξ=(SR,j,)(LA)

then, by 7.3,

so P is not definable by any formula of L^. To find such an x,y we proceed asfollows. Define, for β<α, φβ(v) to be the formula

VM(ι;)|xeP}.

Then ψβ(v)£ LA by 7.1 and

(2)


for β^y. Since 9Mt=^[x] for all xeP, and φβ does not define P (nothingin LA does) there must be some yeM — P such that 9Ml=^[);].

We claim that there is a fixed yεM-P which works for all β<u:

(3) 3yeM

For otherwise we would have

But then by Σ Reflection there is a γ <α such that for all yeM — P

and hence by (2),

a contradiction. Thus (3) is established. Let y be as in (3). For each β there is anxeP such that

by the definition of φβ. By an argument entirely analogous to the proof of (3), wesee that

For any such x we have (3R9x) = *(Wl,y) and hence (5TO,x) = (9K,j;)(LA), as de-sired. D

7.6 Corollary. Lei $K= <M,#l9 ..., /^> be α countable structure for L A relationP on 9Jί is invariant on $01 zjff iί z's definable by a formula in Laoω n HYP(OT P).

Proo/. Combine 6.10 with 7.5. D

7.7 Corollary. Lef LA fee απ admissible fragment of L^. // 9W z's an L-structure,SDΪeA, ί/z^« £i>£ry element of SPΪ definable by some formula of Looω is definableby a formula of LA.

Proof. Apply 7.5 with P = {x}. D

7.8 Corollary. Let 9M = <M,Λ 1 , . . . ,R / > fo? α countable L-structure. Then 9W isrigfid i/f every element of 9W is definable by a formula of L^

Proo/. Combine 6.12 with 7.7. D


7.9 Corollary. // 3Λ = (M,Rί9 ..., Rty is a countable rigid structure thensr(50ϊ)<O(50l).

Proof. By 7.8 we know that

Let β(x) be the least such β. Then σβ

x

(x\υ) is a HYP^-recursive function of x so,by Σ Replacement,

is in HYPgg, and every element of M is definable by some member of it. Lety = sup{jβ(x)|xeM}. We claim that sr(50l)^y. For suppose

Then

so Xi = yt for i = l,...,n, and hence

(aR^i,...^^^^^^!,...,^). D

We can improve 7.9 by replacing the requirement that 50Ϊ is rigid by the re-quirement that 501 have <2X° automorphisms. See Exercise 7.15.

We end this section by returning to our old favorite, recursively saturatedstructures, to see what some of our results say in this case.

7.10 Corollary. Let $R=(M,Ri,...,Rly be a recursively saturated L-structureand let P be a relation on 501 definable by some formula of L^. Then (501, P) isrecursively saturated iff P is definable by a βnitary formula of Lωω.

Proof. The (=>) half follows from 7.5 with A = HYP(SW>P). To prove the (<=) half,note that if P is definable by a formula φelHYP^ then PeHYP^ by Δj Separa-tion so o(HYP(aΛ>P)) = ω. D

Note that if 9W is recursively saturated then so is (501, x) for any xeϊR so 7.10also applies to relations definable by a fixed finite number of parameters. Thesame remark applies to the next result.

7.11 Corollary. Let 501 = <M,K1? ..., Rty be an infinite recursively saturatedL-structure and let

= {yeM\y is definable by some formula φ(v) of Looω without parameters} .


Then we have the following:(i) Every element of @/(Wl) is definable by a finitary formula of Lωω.

(ii) ^/($R) is Σ! on HYP^, hence inductive* on 50Ϊ.(iii) // ^/(9JΪ) is hyperelementary* on SR (i.e., if it is in HYPOT) then @/(Wl) is

finite.(iv) 2R-0/(2R) is infinite.

Proof, (i) follows from 7.7 and (i) => (ii). To prove (iii) suppose thatLet

Then, exactly as in the proof of 7.9, Φ is an element of ΉYP^. But Φ is a pureset and o(HYPΪR) = ω so Φ is finite. Thus ^/(9Ή) must also be finite, since everymember is defined by a formula in Φ. Part (iv) is immediate from (iii), for if

is finite then S^Λ)eΉΎPm. D

7.12. Example. Let tΛr' be a nonstandard model of Peano Arithmetic and let

xeN' be a nonstandard integer. Let yΓ[x] be the submodel of Jf' with universe

The axiom of induction insures that

Corollary 7.11(iv) (applied to (^Γ[x],x)) shows that models of the form Λ^[x]can never be recursively saturated. Hence, the standard integers of jV[x] forma hyperelementary subset of J^[x] by VI.5.1(ii). From this it follows that suchmodels can never be expanded to a model of second order arithmetic, by Exer-cise IV.5.13.


7.13. Let M be countable, α a countable admissible ordinal, α>ω, and let ηbe the order type of the rationals.

(i) Prove that if <ί is a linear ordering of M of order type a,(i+η) than,setting aK1 = <M, <!>,

N"^ is well founded",

[See the proof of IV.6.1.](ii) Let aRo = 7(9ftι) Let LA be tne admissible fragment of L^ given by

, where L = {<}. Prove that


[Use the Tarski Criterion for LA (Exercise 2.13) and the fact that any x in thenon-wellfounded part of <l can be moved by an automorphism of <SJlί.~\

(iii) Prove that

(iv) Prove that sr(9Jϊ1) = α.(v) Conclude that σ^ is not definable in KPU as a Σί operation of SCR.

7.14. Prove that sr (501) and σM are Σ^ definable in KPU + Infinity + Σt Separation,as operations of 9JI.

7.15. Use 6.14 (i) to improve 7.9 to the case where 9JI has <2*° automorphisms.

7.16. Prove that if o(HYPTO)>ω and sr(2R)<o(HYPOT) then σ^elHYP^.

7.17. Prove that the absolute version of

"P is invariant on 9JI"

is

"P is definable by a formula of LooωnHYP(aRfP)".

7.18. Prove that the absolute version of "501 is rigid" is "Every element of 9JΪis definable by a formula of LooωnHYPαϊl".

7.19 Notes. There are a number of interesting and important results which couldbe gone into at this point, but they would take us too far afield. The student isurged to read Makkai [1975] and Nadel [1974].

Theorem 7.3 is from Nadel [1971] (and Nadel [1974]) as are Collaries 7.7and 7.8. Theorem 7.5 is new here but it is a fairly routine generalization ofNadeΓs 7.7. The important example 7.13 is also taken from Nadel [1971]. Thelast sentence of Example 7.12 is a theorem of Ehrenfeucht and Kreisel. [Addedin proof: A recent paper by Nadel and Stari called "The pure part of HYP"(to appear in the Journal of Symbolic Logic) has a number of interesting andhighly relevant results. In particular, they characterize the pure part of HYP^ interms of the sentences σ^ for β

Chapter VIII

Strict Π{ Predicates and Kόnig Principles

1. The Kόnίg Infinity Lemma

In this section we discuss some of the uses of the Infinity Lemma in ordinaryrecursion theory. The applications chosen for discussion are those which be-come important new "axioms" or Kόnig Principles, when stated in the abstract.

Let T = <T, •<> be the full binary tree, as pictured below.

11

The set T is the set of nodes (finite sequences of O's and Γs) ordered by

d'<d

if the sequence d' properly extends the sequence d. If S^T is such that d0εSand do^di implies d^eS, then S = <S, <fS> is called a subtree of T. If S isa subtree then any maximal •< -linearly ordered subset b of S is called a branchthrough S.

1.1 Kδnig Infinity Lemma. Let S = <S,-<tS> be any subtree of the full binarytree. The following are equivalent:

(i) S has no infinite branch,(ii) S is well founded,

(iii) 5 is well founded and has finite rank,(iv) S is finite.

312 VIII. Strict Πj; Predicates and Kόnig Principles

Proof. Each of the implications (iv) => (iii) => (ii) => (i) is completely trivial so weneed only prove (i)=>(iv), or equivalently, —ι(iv) => —ι(i). Suppose S is infinite.Let d0 = ( >eS. Either 0 or 1 has infinitely many predecessors in S so let dv

be the least of these which has infinitely many predecessors in SA Let^ dneS haveinfinitely many predecessors in S and let dn+1 be the first of dβ, <0 which hasinfinitely many predecessors in S. One of them must. Then

b = {dθ9dl9d29...}

is an infinite branch through S. D

One can generalize 1.1 trivially by allowing each node to have any finitenumber of immediate predecessors, instead of exactly two, but once you allowinfinitely many, the theorem becomes false, as the following tree shows.

Indeed, the Infinity Lemma is so tied to the notion of finiteness and the integersthat it is difficult to generalize in a really useful way. So, rather than generalizethe Infinity Lemma itself, we go back and look for useful consequences of theInfinity Lemma. Three of these consequences have turned out to play importantroles when generalized to other admissible sets. In this section we prove thesethree results.

A predicate P ( x , f ) of integers x and number theoretic functions / is r.e. iffthere is a recursive predicate R(x,y) of integers such that

P(x,f)~3nR(xJ(n))

where f(n) is (a code for) the finite sequence </(0),...,/(n —1)> and R(x, f(n))implies R(x,f(m)) for all m^n. (This may be taken as the definition or verifiedeasily from any other reasonable definition. This is the natural extension of r.e.to predicates P ( x , f ) of numbers and functions.)

1. The Kόnig Infinity Lemma 313

1.2 Definition. A predicate S(x) on the integers is strict-Π{ (or s-Π\ for short) ifit can be written in the form

where P is r.e.Here we use 2ω to denote the set of characteristic functions, i. e., those func-

tions mapping ω into 2 = {0, 1 } . The word "strict" refers to the fact that / rangesonly over 2ω, not over all number theoretic functions.

Our first application of the Infinity Lemma is to prove the following result.

1.3 Theorem (s-Π} = r.e., on ω). A predicate P on ω is strict-Ill iff P is r.e.

Proof. To prove (<=) just add a superfluous function quantifier. To prove (=>)write

(1) P(x)~V/e2ω In R(xJ(n))

where R is recursive and satisfies

For /e2ω, each f(n) is a sequence of O's and 1's and so is really just a node onthe full binary tree T. The condition on R above asserts that

R(x,d)*d'<d=>R(x,d')

or, turning it around,

Thus, Sx={d\-\R(x9d)} is a subtree of T. If we restate (1) in terms of trees, itbecomes

(2) P(x) iff Sx has no infinite path,

which becomes, by the Infinity Lemma,

P(x)<r+Sx is finite

^3ΛΓW of length N,dφSx

<-+3ΛΓW of length N,R(x,d)

where #' is recursive. More informally, P(x) holds iff you can find a finite subtreesuch that R(x,d) holds for every end-node d on S. D

314 VIII. Strict Π} Predicates and Kδnig Principles

In proving 1.3 we also proved the next theorem. This (or a relativized formof it) is what Shoenfield [1967] refers to as the Brouwer-Kόnig Infinity Lemma.

1.4 Theorem (s-Π} Reflection for ω). Let

define a strict-Til predicate. Then for any x

Proof. This is contained in the proof of 1.3. D

We will see in § 4 that the equation "s-Π} =r.e." can be viewed as abstractformulation of the completeness theorem for Lωω and that "s-Πj Reflection for ω"corresponds to the compactness theorem for Lωω. The fact that our proof of 1.3also gives 1.4 corresponds to the fact that most proofs of the completeness theo-rem yield compactness, but not vice versa.

Our final application of the Infinity Lemma is to the notion of implicit ordinal.We state the definition in general to save repeating the definition in § 5.

1.5 Definition. Let SR be a structure for some language L, let R, S be two newn-ary relation symbols and let <p(R,S) be a sentence of L(R,S), possibly con-taining parameters from 501. Let α be an ordinal. The sentence φ(R,S) implicitlydefines α over 501 if the relation -<φ defined by

R<φS iff (9Jί,R,S)t=φ(R,S)

is well founded and α is its rank, i.e., a = ρ(^φ).Our final application of the Infinity Lemma shows that if a Π° relation

φ(R,S) on ω implicitly defines an ordinal, then that ordinal is finite. In § 6 wewill learn that any α implicitly defined by even a Σ} sentence on ω is just theorder type of a recursive (explicit) well-ordering of ω. These two facts explainwhy the notion of implicitly defined ordinal does not arise explicitly in ordinaryrecursion theory.

A predicate φ(R,S) on ω is Π? iff ~Ίφ(R,S) is r.e. (To fit this into our de-finition of r.e. replace R, S by their characteristic functions.)

1.6 Theorem. Let φ(R,S) be a 11° predicate of n-ary relations on ω. If the rela-tion <φ defined by

iff <P(R,S)

is well founded, then its rank is finite.

315

Proof. By use of pairing functions we can assume n = l, i.e., that R, S rangeover subsets of ω. Assume -< is well founded so that

(2)

For any /e2ω let (/)π = {χ|/(2x3") = l}. We can restate (2) as

(3)

Since φ is Π?, the predicate ~ιφ((/)π+ι,(/)J is an r.e. predicate of /, n. Bys-Πj reflection there is an N <ω such that

which says that there is no sequence

RN + 1 ^φ RN ^φ ' ' ' φ *M ~^φ 0

Thus p K K A Γ + l. D


1.7. Prove the relativized version of the theorems of this section. (The fact thats-Π} Reflection holds relativized to any relation R on ω is expressed by sayingthat ω is strict-I\\ indescribable)

1.8. Let R<S iff R,S<^ω and the least member of R is less than the leastmember of S. Show that this is an r.e. predicate of R, S and that it implicitlydefines ω.

1.9. Let R-<S iff R.S^ωx ω, R, S are well-orderings and R is a proper initialsegment of 5. Show that this is a Π} relation which implicitly defines the firstuncountable ordinal.

1.10. Let -< be a well-founded relation on subsets of ω defined by a Σ} sen-tence φ. Show that the rank of •< is <ω\. [Hint: Show that ρ«) can be pinneddown by a sentence of Lωc.]

1.11 Notes. The equation "s-Π|=r.e., on ω" was first observed by Kreisel inthe proof of the Kreisel Basis Theorem (cf. p. 187 of Shoenfield [1967]).

2. Strict Π{ Predicates: Preliminaries

Over ω, or HF, the strict-Π} predicates coincide with the r.e. predicates (by 1.3)so it is difficult to see the exact role that the notion of strict-Π } plays in traditionalmodel theory and recursion theory. In general, however, strict-Π} does not

316 VIII. Strict Π; Predicates and Kόnig Principles

coincide with Σl. By studying the s-Πj predicates in the general case, then, wesee more clearly the role they play over ω.

Let L* = L(e,...) be the language for KPU. We assume that there are onlyrelation and constant symbols in L*, no function symbols. (This is not an essentialrestriction — see Exercise V.I. 8.) Let R1 ? R 2, be an infinite list of new relationsymbols, an infinite number of arity n for each n<ω. Let L*(R) be the expandedlanguage.

2.1 Definition, i) The strict-Hi formulas (s-Π} for short) of L*(R) form the smallestclass containing the Δ0 formulas of L*(R) closed under Λ , v, Vwet;, 3weι;, 3wand the clause

if Φ(R;) is strict-Πί so is VR^(R;).

The strict-Π} formulas of L* consist of those s-Tl\ formulas of L*(R) which haveonly quantified occurrences of the new relation symbols R l 9 R 2,... .

ii) The strict-Σ\ formulas form the dual class; that is, they form the smallestclass containing the Δ0 formulas closed under Λ, v, VWEI;, 3uev, Vw, 3R f .

There are two essential restrictions in the definition of strict-Π { formula.First, only existential quantifiers over individuals are permitted. Second, onlyuniversal second order quantifiers are allowed, and then only over relations, notover functions. If we were to allow universal second order quantification overfunctions, then we could build in first order universal quantification (by themanipulations discussed in § IV.2). These observations are summarized by thediagram:

strict-Πί formulas

ϋ ^Σ formulas Π} formulas

^ c,first order formulas

All inclusions are proper.Don't forget that L* may have extra relation symbols (like a symbol for the

power set relation) which are allowed to occur in Δ0, hence in s-Π}, formulas.Satisfaction of s-Π} and s-Σ} formulas is defined in the classical second order

manner. Thus

SK^VRΦ(R)

means that for every relation R on 9K (of the correct number of places)

The study of s-Πj predicates is one of the few places in logic where the dif-ference between relation symbols and function symbols really matters. In § 1we defined s-Πj over ω in terms of quantification over characteristic functions,

2. Strict Π{ Predicates: Preliminaries 317

rather than the relations they describe, just to fit with standard practice in ordinaryrecursion theory. Here the approach with relation symbols is more natural.

The following simple lemma expresses one of the most crucial properties ofstrict-Π} formulas.

2.2 Lemma. Strict-Til formulas persist upwards under end extensions. That is,If ^^ ®$n are ^-structures with %0ιênd®<n> and If Φ(vl,...,vn) is a s-Π} for-mula of L* then

q,...,*,,] implies

for all x^.êSlspt.

Proof. We need to prove a bit more to keep the induction on s-Π} formulasgoing. Let 2Iαnênd®<R be jμven. We prove by induction on s-Π} formulasΦ(Rί,...,Rm,v1,...,vn) of L*(R) that for all relations R^...,Rm on SK and all

implies

The proof is just the proof of persistence of Σ formulas with a new case for VSthrown in. Suppose

(1)

Let S be any relation on 93^ of the correct number of places. By (1)

(2) (^,Rl\^,...9RJ[

so, by the induction hypothesis,

as desired. Notice that if we had allowed quantification over function symbolsstep (2) would fail; just because S is a total function on 93^ is no reason tosuppose that St%0ι is a tota^ function. D

Let ί be a structure for L*. A relation P on 91 is s-Π} if it can be definedby a s-Π} formula of L* with parameters from Vlm. P is s-Σ} if P can be definedby a s-Σ} formula with parameters. P is strict-Δ} if P is both s-Π} and s-Σ}.

A function is s-Π}, s-Σ} on s-Δ} iff its graph is s-Π}, s-Σ} or s-Δ} respectively.

2.3 Lemma. // a total function f on 91 is s-Π} then it is s-Δ}.


Proof. Since / is total,

f(xl9...9xn)ϊy iff 3z[/(x1,...,xJ = zΛ

Replacing f(x) = z by its s-Π} definition gives us a s-Π} definition ofor equivalently, a s-Σ} definition of the graph of/. D

2.4 Examples. Let A be an admissible set. We give three examples of strict-Π}predicates which are not, in general, Σx. Note, however, that if A is countablethen these relations are Σ1 (for rather trivial reasons).

(i) Define P(a,b) iff card (α)< card (b). Then P is s-Π} on A.(ii) Define P(a) iff card (a)< card (A). Then P is s-Π} on A.

(iii) Define P(a,b) iff b = Power (α), the real power set of a. P is s-Π} on A.If A is closed under the power set operation then P is s-Δ} on A.

Proof, (i) We can write card (ά)< card (b) as

VR [R^bxa^Vxeblyea R(x9y)-> 3x,x'eb 3yeα(x^xΆ R(x,y)Λ R(x',y))]

which asserts that no relation on b x a can be a one-one map of b into α.Schematically, we can rewrite this as

VR[Π 1(R)ΛΔ 0(R)^Δ 0(R)].

Replacing -> by v we get

which is s-Π}. The proof of (ii) is much the same. To prove that fc = Power(α)is s-Π}, note that fo = Power(α) iff

c f l)Λ VR

The second sentence of (iii) follows from 2.3. D

A formula is in s-Π} normal form if it is of the form

VRlyί,...,ymφ(υί,...,υn9y1,...,ym9R)

where φ is Δ0. The next lemma states that every s-Π} formula is logically equiv-alent to one in normal form.

2.5 s-Π} Normal Form Lemma. Assume that L* has a constant symbol 0. Forevery s-Π} formula Φ(x,R) of L*(R) there is a s-Π} formula Φ' in normal form,with exactly the same free variables and free relation symbols, such that for allL* structures ^lm:

Vx [Φ(x, R)<->Φ'(X, R)] .

2. Strict Π j Predicates: Preliminaries 319

Proof. We describe five quantifier-pushing manipulations which allow us to putany s-Π} formula in normal form.

(i) VR 1 VR 2 Φ(R 1 ,R 2 )«VSΦ'(S)

where S is rc + w-ary, n being the arity of R1 ? m the arity of R 2, and where Φ'(S)results from Φ by replacing

R!^,...,^) by S(tl9...9tn9Q9...90)9

R2(ί1?...,ίm) by S(0,...,0, *!,...,*„)•

(ii) V R i 3x φ Λ V R 2 3y ψ ++ V R t V R 2 3

as long as the various symbols are distinct. Similarly for v .

(iii) Vx 3R Φ(x, R)<-+3R' Vx Φ'(x, R')

where Φ' results from Φ by replacing R(ί l5...,ίπ) by R'(x9tί9...9t^. Takingnegations on both sides of (iii) we get

(iv) 3xVRΦ(x,R)^VR /3xΦ'(x,R /)

which lets us pull VR out in front of 3x. The bounded existential quantifier stepfollows from (ii) Λ (iv). The only remaining step is the bounded universal quantifier.

(v)<->VU

The part in brackets is Δ0 since it can be written

Vzεa (U(x) Λ U(z)^x = z) -> Vxea (U(x)^Φ(x, R,

It is now clear, by induction on s-Π} formulas, that we can put every s-Π} formulain normal form. D

The Normal Form Lemma is quite useful. We use it in proving the nexttheorem, and repeatedly in this sections to come.

Recall, from § IV.3, that for countable structures 9K

Π} on 9JI - Σ! on HYP^.

We proved an absolute version of this theorem in § VI. 5, by showing that

inductive* on 501 - Σx on HYP^ .

We close this section by proving a different generalization.

320 VIII. Strict Πj Predicates and Konig Principles

2.6 Theorem. Let 9W = <M,Λ1,...,R/> be an infinite structure. A relation S onis Πj on m iff S is strict-Ill on

To see that this is a generalization of the countable result we need to knowthat, for 9W countable,

5-ΠJ on HYRro - Σ! on

This follows from Theorem 3.1 of the next section.

Proof of Theorem 2.6. We first prove the easy half (=>). Suppose

S(x)<->9Ml=VRφ(x,R).

Then

The part within brackets is Σt since all quantifiers in φ(M) are bounded by M,an element of HYP^. To prove (<=) we must reexamine the proof of TheoremIV.3.3, the result that

Σ! on HYRro => Πj on 501

regardless of 9Jΐ's cardinality. Suppose S is s-Π} on HYPOT, S^aR. By theNormal Form Lemma we can write

S(p) iff HYPT ONVP3j;φ(pJ,P,f)

for some z = z1,...,zfcelHYPS[R. As in the proof mentioned above, we can replaceall parameters zi by good Σ1 definitions and so assume all parameters are fromMu{M}. Let's say

S(p) iff HYP^VP3j?φ(p,y,P,4,M).

By the persistence of s-Πj formulas under end extensions, and by the truncationlemma, S(p) is equivalent to

(!') (Wn,P)*=19φ(p39P,q,M) for all P and all models 91 of KPU+

(of cardinality card(5M)).

From here the proof proceeds exactly like the proof of IV.3.3, by coding up (Γ)on M, with the extra VP riding along for free. D

3. Kόnig Principles on Countable Admissible Sets 321

2.7 Exercise. Let Mm be any structure for L* and let Γ be a s-Π} inductive de-finition on 91 ; i. e.

xeΓ(R) iff (9ίs

where Φ is s-Π}. Show that the fixed point IΓ is s-Π} on Mm.

2.8 Notes. The only theorem of this section comes from Barwise-Gandy-Moschovakis [1971].

3. Kόnig Principles on Countable Admissible Sets

Strict-Π} formulas give us a language for expressing important new principles,or axioms, for admissible sets; principles that isolate important aspects of theInfinity Lemma.

In this section we discuss three Kόnig principles and show that they holdon all countable admissible sets. Their role in the general case is discussed inthe remaining sections of this chapter.

K!: An admissible set A satisfies

if every strict-Til relation on A is already a Σl relation on A.

It is important to remember that this equation (s-Π} =Σ t) depends very muchon just what extra relations may be part of our admissible set A=(9W; A,e,...)in those three little dots. Add a new relation to A and you increase both thenumber of s-Π} formulas and the number of Σί formulas. It should also be keptin mind that the Σί formula defining a s-Π} predicate P may have parametersnot appearing in a given s-Π} definition of P.

3.1 Theorem. Every countable admissible set satisfies s-Π} =Σί.

Proof. We will prove more; namely, that every Σί complete admissible set Asatisfies s-Π}=Σ1. Let P be s-Π} on A. By the Normal Form Lemma we canwrite P in the form

P(x) iff A l = V R φ ( x , R )

for some Σί formula φ. Let T be the usual infinitary diagram of A:

diagram (A),

322 VIII. Strict Π} Predicates and Kόnig Principles

By the persistence of s-Πj formulas under end extensions (Lemma 2.2) the fol-lowing are equivalent:

P(χ),

93NVRφ(R,x) for all ®^e n dA,

) for all »3endA and all R^W1,

If A is Σ! complete then the set of x such that T\=φ(R,x) is a Σί set. D

Before stating the second Kδnig Principle, K2, we need to define the notationΦ(α) for second order formulas Φ. To obtain Φ(α) one relativizes all unboundedfirst order quantifiers to a (replace 3w by 3weα, Vw by Vweα) and replaces

...) by 3R[Rc=a"A(.. .)],

...) by VR[Rc f l»->(...)].

Note that if Φ is strict-Πj, or even Π}, then Φ(α) is strict-Π} with free variablesthose of Φ and the new variable a.

3.2 Lemma. For every structure $1 for L* and every s-H{ formula Φ(vl,...,vn)of L*, the following are true in 9IOT:

(i) VaVv1,...,vnεa[Ύτan(a)ΛΦ(a\v)-*Φ(v)~]',

(ii) Va,b,Vvl,...,vnεa[Ίτan(a)Λa<^bΛΦ(a\v)-*Φ(b)(v)'].

Proof. This is just another version of the persistence of s-Π} formulas under endextensions. It can be proved directly or deduced from Lemma 2.2. D

K2: An admissible set A satisfies strict-Hi reflection if for every s-H\ formulaΦ(vl,...,vn) and every x1,...,xneA, A satisfies

Φ(x) -> 3α [Tran(a)Ax l 5 . . . ,x n eaA Φ(α)(x)].

We will see in §§ 4, 6 and 7 that s-Π{ reflection is a strong assumption. Fornow we prove that it holds in all countable admissible sets.

3.3 Theorem. Every countable admissible set satisfies s-Π\ reflection.

Proof. Again we prove more with an eye toward the next section. This time weprove that if A is Σt compact then A satisfies s-Π} reflection. Let Φ(vί9...,vn)be s-Π}. By the Normal Form Lemma there is a formula Ψ(vί9...,vn) in s-Πjnormal form logically equivalent to Φ. It follows that Ψ(a\v^...v^ is logically

3. Kδnig Principles on Countable Admissible Sets 323

equivalent to Φ(a)(vί9...9υn) so it suffices to prove reflection for formulas in s-Π}normal form. So suppose Φ(v) is VR φ(v, R) and that A=Aα ϊ ί and

where φ is a Σ1 formula. Let T be the infinitary diagram of A, as in 3.2. As wesaw in that proof,

T\=φ(xl9...9xn9R).

By Σ! compactness there is a T0^T, T0eA such that

T0\=φ(xl9...9xH9R).

Let a0 = {y\y occurs in 7^}u{x1?...,xπ} and let a = ΊC(a0) so that aeA. Then

so

which is another way of saying that Φ(fl)[x1?...,xJ holds. D

The third Konig principle concerns the notion of implicit ordinal introducedin 1.5 and is suggested by Theorem 1.6.

An ordinal α is a Π implicit ordinal over A if there is a Π sentence φ(R,S),possibly containing parameters from A, which implicitly defines α over A (inthe sense of 1.5). The notion of a s-Σ} implicit ordinal is defined in a parallelfashion. (It will turn out that every s-Σ} implicit ordinal is less than some Π im-plicit ordinal; see 3.11 or 6.3). It is easy to see that every β<o(A) is a Π implicitordinal over A.

K3: The third Konig principle asserts that every Π implicit ordinal over A isan element of A.

One might paraphrase K3 by saying that the Π implicit ordinals over A areexplicitly in A.

3.4. Theorem. Every countable admissible set satisfies the third Kδnig principle.

Proof. Since hΣ(A) = o(A) for countable A, 3.4 follows from 3.5. D

Admissible sets do not, in general, satisfy K3. In general, the Π implicitordinals know new bounds.

3.5 Theorem. Let A be admissible and let a be a s-Σ} implicit ordinal over A.// β = hΣ(A) then a<β.

324 VIII. Strict Π} Predicates and Konig Principles

Proof. Let Jk=Am be admissible and let α be the rank of the well-foundedrelation -<φ where Φ is s-Σj; say

iff (A,

and φ is a Π sentence. We can assume that Q, R, S are all unary (i. e. range oversubsets of A) since the pairing function <x,y> is A-recursise. We need to finda Σί theory T of LA which pins down α. The crucial observation is contained in (1).

lf ATOcendBw and (»^,^,S)^3Q φ(Q,R,S), and if R0 =( ί then R0<ΦS0.

This follows from (A^,#0,So)^end(®<n>^>S) by thes-Σj version of Lemma 2.2.From (1) we get (2).

(2)

Let AaϊicendS5n. The relation -< on subsets of 23^ defined by

Λ<S iff (»w,

is well founded. Hence any subrelation -<" of •< is well founded.

For, by (1), any infinite descending sequence in -<' would give rise to aninfinite descending sequence in -<φ.

It is pretty obvious how to use (2) to pin down the ordinal α by buildingthe hypothesis of (2) into a Σt theory T = T(<,...). The language for T willcontain the symbols of L* = L(e,...); a constant symbol x for each xeA; unarysymbols A (for A\? (for Power(A)), U (for α); binary symbols E (for &r\(A x P)),-< (for <φ), < (for efα); and a function symbol F. The intended model for T,the one with <m of order type α, is:

9W = <4uPower(4)uα;yl,...; Power(A), er\(A x Power(A)),-<φ;α, <,F>

where F(x)=0 for x^field«φ), F(R) = <φ-rank of R for R in field of <φ.Hence rng(F) = α and R<ΦS implies F(R)<F(S). The theory T contains:

Vx[A(x)vP(x)vU(x)],

Infinitary diagram of A,

Extensionality for E,

"Ec=AxP".

(3) Vr,5 [r<5^P(r)Λ P(s) Λ Jy(P(q)Λφ(q9r9s))'].

(4) "< linearly orders U, rng(F) = U, F(x)=0 for x#field(X), andF(s)=<-sup{F(r) + l : rXs} for sefield«)".

3. Kόnig Principles on Countable Admissible Sets 325

In line (3), φ(q,r,s) denotes the result of replacing R(x) by xEr, — \R(x) by~ι(xEr) and similar for Q, S. (We are abusing notation since q does not rangeover urelements here.) To see that T pins down α it remains only to prove thatfor any other model 9W of T, <9W is well ordered. Let $R be any model of T. Wecan obviously assume 90Ϊ has the form

where A^^S^ and P^ Power (33 ). To see that < is well ordered it suf-fices, by (4), to prove that X" is well founded. But this is immediate from (2)and (3). D

3.6 Corollary. // $Jl = (M,Rί,...,Rιy is countable and α is a first order, or evenΣ}, implicit ordinal over 901 then α<O(5[R). In particular, α is countable.

Proof. If α is Σ} over 501 then it is s-Σ} over HYP^ so the result follows from 3.5. D

3.7 Corollary. Every Σ} implicit ordinal over ω is less than ω{.

Proof. Immediate from 3.6 since ω\=O(J/^}. D

As we mentioned in § 1, Theorem 1.6 and Corollary 3.7 together account forthe fact that implicit ordinals seldom appear in ordinary recursion theory on ω.They do appear in parts of mathematics far removed from the theory of ad-missible sets. We present one example suggestive of others.

3.8 Example. Let 5DΪ be a Noetherian module (over a ring with identity), that is,a module with no infinite chain

of submodules. Then

MXM" iff Mr, ΛΓ are submodules, M"cM'

defines a first order implicit well-founded relation. Its rank α = p(-<) is calledthe length of 9W, α = /(ϊR). Thus /(ΪR) is a first order implicit ordinal over M.This ordinal plays an important role in the structure theory of Noetherian modules.


3.9. Prove a uniform version of 3.1. That is, show that for every s-Π} formulaΦ(υl9...,vJ there is a Σ^ formula φ(v^...,vn) such that for every countable ad-missible set A,

326 VIII. Strict Πj Predicates and Kόnig Principles

3.10. Prove directly that every s-Σ} implicit ordinal is ^ some Π implicit ordinalover A.

3.11. Let 9Jl = <M, #!,..., Kj> and let α be a Σ} implicit ordinal over SR. Im-prove 3.5 to show that

3.12. Prove that on the class of admissible sets A and relations P on A, .

"P is Σ! on A"

is the absolute version of

"P is s-Πl on A".

4. Kόnig Principles K^ and K2

on Arbitrary Admissible Sets

To summarize, the three Kόnig Principles introduced in § 3 are :

KK

strict-Πί reflection;Every Π implicit ordinal over A is an element o f ,

These three principles are generalized recursion theoretic statements whichattempt to capture different aspects of the Infinity Lemma. Each of them hasa model-theoretic counterpart for the infmitary logic LA. In this section wediscuss the counterparts of K^ and K2.

The basic tool for the study of all three of these principles is the Weak Com-pleteness Theorem of § VII.2. Our first theorem explains the reason for referringto that result as a completeness theorem.

4.1 Theorem. Let A be admissible and let T be a set of sentence of LA which isstrict-Til definable on A. The set

A: T\=φ}

is also strict-Π{ on A.

Theorem 4.1 will follow from the Weak Completeness Theorem and the nextLemma.

4.2 Lemma. Let A be admissible and let LA be a Skolem fragment which is Δt

on A (in the sense of Lemma VII.2.4). There is a Π sentence φ(D) such that forany ^^A:

Q) is an s.v.p. for LA iff (A,

4. Kόnig Principles KΣ and K2 on Arbitrary Admissible Sets 327

Proof. Since LA is Δx on A, Z^koiem is a Δt subset of A. 2 is an s.v.p. for LA iff(A, ®) satisfies all the following conditions:

Vφ [(φ an axiom (Al)— (A 7) of LJ->(φe

® is closed under (Rl)— (R3),

VΦ [V Φ e 0 Λ ( V Φ a sentence) -> 3φ e Φ(φ e @)~\ .

Each of these conditions is naturally expressed as a Π condition on 2, so thelemma is proved. Note the important role played here by Skolem fragments. Ifwe had to do without 4"ζkoiem^ ®"> we would have to add the clause

= φ(ι?)Λ(3ι;φ(t;))e ^->3ί [φ(t/v)e

which is not Π due to the unbounded 3ί. D

Proof of Theorem 4.1. We may assume, by VII.2.4, that LA is a Skolem frag-ment Δ! on A and that every model of T can be expanded to a Skolem model.By the Weak Completeness Theorem we have T\=φ iff

V® [0 an s.v.p. for L A Λ Γ^ @^φe &\.

By 4.2 this takes the form

where Φ(υ) defines the s-Π} theory T. The hypothesis of the outer implicationis s-Σ} so the whole becomes a s-Π} predicate of φ. D

4.3 Corollary. An admissible set A satisfies s-Tl\=Σ1 iff A is Σx complete.

Proof. The implication (=>) follows from 4.1. The other direction was provedexplicitly in the proof of Theorem 3.1. D

4.4 Corollary. The set of valid sentences of the admissible fragment LA is alwayss-Π} on A.

Proof. Let T = 0 in 4.1. D

At various places in the book we have referred to Σί as a syntactic generali-zation of r.e. on ω and to strict-Π} as a semantic version of r.e. on ω. The nextcorollary of 4.1 makes this precise.


A subset X^A is a complete Σ1 set (or complete sfπcf-Π} set) for the ad-missible set A if X is Σl (resp. s-Π}), and for any other Σί set (resp. s-Π} set)Y on A there is a one-one total A-recursive function F such that

ye Y iff F(y)eX

for all yelk.Recall that T\-φ means that φ is provable from T in the sense of LA. (This

notation occurs in § III.5.)

4.5 Corollary. Let A be admissible. Let L contain L*(R) and a symbol x for eachxeA and let L'A be the admissible fragment given by A. Let T be the infinitarydiagram of A.

(i) The set X0 = {φeL'A | TΊ— φ} is complete Σ^ for A.(ii) The set Xί = {φeL^\T\=φ] is complete s-Π} for A.

Proof, (i) is implicit in much of Chapters V and VI. It can also be obtained simplyas the absolute version of (ii). To prove (ii) note that Xl is s-Π} by 4.1 and thatevery s-Π} set is one-one reducible to X± by the proof of 3.1. D

An analogous proof shows that on "bad" admissible sets, s-Π} can be as farfrom Σ! as is conceivable.

4.6 Corollary. Let A be a self "-definable admissible set. Then H{ = strict-Ill on A.That is, every Π} relation on A can be defined by a strict-Hi formula.

Proof. Let T be a Σl theory of LA which self-defines A. By 4.1, Cn(T) is s-Π}.But by Lemma VII.1.9, every Π} relation on A is one-one reducible to Cn(T)so every Π} relation is s-Π}. D

For example, Π}=strict-Π} on H(Kα+1) for all α^O, by VII.1.4.

We now turn to consider the logical role of strict-Π} reflection.

4.7 Theorem. An admissible set is Σl compact iff it satisfies sίricί-Π} reflection.

Proof. The implication (=>) was proved explicitly in the proof of Theorem 3.3. Toprove the converse, suppose that A is admissible and satisfies s-Π} reflection andthat T is a Σ! theory of LA. Assume further that every T0^ 7; T0ε A has a model.By Lemma VII.2.4 we may assume that LA is a Skolem fragment and that everyT0^T, TOE A has a Skolem model. We will prove that T has a Skolem model.Suppose, aiming at a contradiction, that T has no Skolem model. By the WeakCompleteness Theorem, no s.v.p. Q) for LA can contain T as a subset. Hence(A,T) satisfies the s-Π} sentence Ψ(Ύ) expressing:

V ® [0 an s.v.p. for L A ^ 3 x ( x e T Λ x < £ )].

Let θ(v) be the Σ! formula defining T. The line above becomes

^ an s.v.p. for L A -»3x(θ(x)Λx

4. Kόnig Principles K t and K2 on Arbitrary Admissible Sets 329

which is a s-Π{ sentence Φ(y) with parameters y those of Θ(v) = θ(v,yί9...9yk). By5-Π} reflection there is a transitive set αeA with yea such that Af=Φ(β)[y].Let A0 = (9Knα;α,e,...) and let

so that TJe A by Δ0 Separation and TQ^T since θ is Σx. Since Φ(α)[y] holdswe have

We don't really know what Ψ(Ύ) says on A0, but (A0,7ί))cend(A,Γ0) so, by thepersistence of s-Π} formulas

(A,T0)I=?P(T).

But this says that 7^ is not a subset of any s.v.p. 2 for LA. Hence 7^ has noSkolem model, a contradiction. D

Thus we see that two different aspects of the Kόnig Infinity Lemma, thoseexpressed by K{ and K2, reflect themselves in related but apparently distinctaspects of first order logic. K t is responsible for the Completeness Theorem,K2 for the Compactness Theorem.

One usually thinks of the Completeness Theorem as implying the Com-pactness Theorem. The corollary to the next result shows this to be the casefor resolvable admissible sets. The general case is still open.

4.8 Proposition. The resolvable admissible sets are divided into two disjoint classes:those that are Σ1 compact and those that are self-definable.

Proof. Proposition VII. 1.3 shows that no self-definable A can be Σt compact.Now let A be a resolvable, admissible set which is not Σt compact. We mustshow that it is self-definable. Since A is resolvable there is a total A-recursivefunction J : o(A) -> A such that

α<jS=>J(α)eJ08),

J(α) is transitive, for all α,

Since A is not Σx compact, A does not satisfy s-Π} reflection, by 4.7. Thus thereis a s-Πj formula Φ(t ), and an xeA such that A satisfies:


The second formula is s-Σ[ and so is logically equivalent to a s-Σ{ formula

3Rφ(x,R)

where φ is Π^ Let σ(u,v) be a Σί formula defining J :

J((ή = y iff ANσ(α,y).

The Σ! theory used to self-define A consists of:

The infinitary diagram of A ,

Vw, v [J(w) = ι;<->σ(w, v)~\ ,

Vx3l l [Ord(M)ΛXeJ(M)],

Vιι[Tran(J(u))],

φ(x,R).

Let (33<n,J) be any model of T. We can assume Agt^end^R We need to showthat AαR = 939l. If not, let xeS^ — A^. Then by the axioms on σ,

95wN3α[Ord(fl)ΛxeJ(α)].

Pick such an a. Then a is an ordinal of 93^ but aφA^ for if αeAw thenJ(α)eA which implies xeA. Thus α>β for all jSeA. But then J(β)^J(a)holds in 93*, for each jβeA. Thus Acend<j(α),£> and so Φ(J(fl))(x) holds.This contradicts

since this asserts that Φ(fl) fails for all transitive b, in particular for b = J(a). D

4.9 Corollary. Every Σ^ complete resolvable admissible set is Σt compact. In otherwords, K t implies K2 on resolvable admissible sets.

Proof. If 5-Πj^Σi then s-Π}^Π} and hence A cannot be self-definable,by 4.6. Then by 4.8, A must be Σx compact. D

What is wrong with the following argument? If s-Π^Σj then (since Σreflection holds in all admissible sets) we must have s-Π} reflection. If you tryto fill in the steps in this argument you see that one is missing a certain uniformityin the equation s-Π^Σi. This uniformity is captured by the next definition.

Let A=Am be transitive, let Φ(vlί...,vn) be strict-Π} and let φ(vl9...,vn)be Σ1? where extra parameters from A are permitted in φ. We say that φ isuniformly equivalent to Φ on A if

(1) .

(2) A t= Vα Vt^, . . . , vn € α [Tran (α) Λ ψ(a\v) ->• Φ(a\v)\ .

4. Kόnig Principles K! and K2 on Arbitrary Admissible Sets 331

4.10 Lemma. Let A=Am be transitive, closed under pairs and TC. Let φ(vί9...9vn)be a Σl formula which is uniformly equivalent to the s-Π} formula Φ(vl,...,vn)on A. For all x1 ?...,xπeA, A satisfies:

Φ(x) <-> φ(x)

<r+la [Tran(α) Λ xεa Λ <p(fl)(x)]

<-» 3fl [Tran(fl) Λ xea Λ Φ(fl)(x)] .

Proof. Write φ(vί,...,vn) as Byi/^,...,^,);) where y is Δ0. By (1),

If At=^(x,.y) then let a = ΎC({y,xl9...9xn}). Then αeA and φ(α)(^) holds so

φ(jc) -> 3α [Tran (a) Λ x e φ(fl)(*)] .

By (2), the right hand side of the above line implies

la [Tran(α) Λ xea Λ Φ(fl)(x)] .

By Lemma 3.2 (i), the above implies Φ(x). D

4.11 Definition. An admissible set A satisfies

5-Π}=Σ1 uniformly

if for each s-Π} formula Φ(vί9...,vn) of L* there is a Σl formula φ(vί9...,vn),possibly with additional parameters from A, such that φ is uniformly equiv-alent to Φ on A.

4.12 Theorem. An admissible set A satisfies s-Tl{=Σl uniformly iff A satisfiess-Π\=Σl ands-Π\ Reflection.

Proof. The implication (=>) is immediate from Lemma 4.10. To prove that con-verse, assume that A satisfies K^ and K2 and that Φ(vί,...,vn) is s-Π}. We mustfind a Σί formula φ(vl,...,vn) uniformly equivalent to Φ(v1,...,vn) on A. LetΨ(vl9...9vn9b) be the s-Π} formula

[Tran(fe)Λ ϋ l 9.. .,!?„€& ΛΦ(>!,.. .,!;„)].

Since s-Π}=Σi there is a Σj formula \//(vi9...9vn9b) equivalent to Ψ(vί9...9vn9b)on A. Let <p(vί9...9υn) be

3bψ(vί9...9υn9b).

To prove φ(v) uniformly equivalent to Φ(i ) first suppose that Φ(x) holds in A.By s-Π} Reflection there is a be A such that Ψ(X9b) holds in A. But then ψ(Z9b)


holds so φ(x) holds. To prove (2), suppose that αeA is transitive, that x l 5. . .,xπeαand that φ(a)(x) holds. Then there is a be a such that ψ(x,b)(a} holds in A. Henceψ(x, b) holds in A and so

Tran(ft)Λ *!,..., x n e&ΛΦ ( f c ) (x).

Since bea and a is transitive, b^a so, by 3.2 (ii), Φ(α)(x) holds, as desired. Q

4.13 Corollary. An admissible set A satisfies s-Πj = ΣX uniformly iff A is Σx com-! compact. D

4.14 Corollary. On resolvable admissible sets the condition s-Yl\=Σl is equivalentto the condition s-Tl\=Σ1 uniformly.

Proof. By 4.9 and 4.12. D

The condition 5-11}=^ uniformly clearly captures a great deal of the re-cursion theoretic and logical content of the Infinity Lemma. If you state it inthe "s-Σj =Πί uniformly' version, it even looks like the Infinity Lemma, at leastfrom one point of view. We will use it in § 6 to help us find interesting uncountableΣ! complete and Σl compact admissible sets.


4.15. Let α be admissible but not recursively inaccessible, let A = L(α). Provethat the valid sentences of LA form a complete s-Πj set. Show that for any ad-missible /?^ω l9 β is recursively inaccessible iff the set of valid sentences of Lβ

is ^-recursive.

4.16. A subset X of an admissible set A in bounded if X^a for some αeA.Let A be admissible and satisfy s-Π} Reflection. Let T be a set of sentences of LA

which is s-Πj on A. Prove that if every bounded subset T0^T has a modelthen T has a model. (It is open whether one can improve this by replacing"bounded" by "A-finite".)

4.17. Let A be admissible. Suppose that for every Δ0 theory T of LA, if everyTQ^T, T0eA has a model, then T has a model. Show that A is Σ! compact.[Show that s-Π} Reflection holds.]

4.18. Let A be admissible, α = o(A). A is s-Δ} resolvable if there is a limit ordinaland a s-Δj function J:Λ,-»A such that

β<ξ=>JβeJξ for ]

Jβ is transitive for all β < λ ,

4. Kόnig Principles K! and K2 on Arbitrary Admissible Sets 333

The ordinal λ is said to be the length of the hierarchy J on A.(i) Prove that if α = 3α then #pα) is s-Δ} resolvable. [Hint: If α = Sα then

HpJ = V(α). Let J, = VGB).](ii) Prove that if A is s-Δ} resolvable and if J is as above with A<o(A) then

A is self-definable.(iii) Strengthen Proposition 4.8 to: The class of s-Δ} resolvable admissible

sets are divided into disjoint two classes, the Σ^ compact and the self-definable.(iv) Let κ = 3κ. Show that <#(κ),e,^> satisfies K t iff it satisfies K2.

4.19. Kunen [1968] introduced an invariant definability approach to generalizedrecursion theory on admissible sets by introducing the notions of a.i.d., i.i.d.,and s.i.i.d. (see below) as generalizations of the concepts of finite, recursiveand r.e. In Barwise [1968], [1969 b] we showed that s-Π}=s.i.i.d. (see (ii)). (Thisleads to the formulation of s-Π{ Reflection and the results of this section inBarwise [1968], [1969 b].) Let A be admissible and let P be an rc-ary relationon A. P is

(a) absolutely implicitly definable (a.i.d.) on A ,(b) invariantly implicitly definable (i.i.d.) on A ,(c) semi-invariantly implicitly definable (s.i.i.d.) on A

if there is a finitary first order sentence 0(P,Sl5...,Sfc) of L*(P,...) such that

(A,P)N3S1,...,SJk,θ(P,S1,...,Sfc)

and such that if Ac e n d® and P'c®" satisfies

then

(a) P =

(b) P =

(c) PcΞ

The sentence θ may contain parameters from A.

(i) Prove that P is i.i.d. iff P, ~ιP are s.i.i.d.(ii) Prove that P is s.i.i.d. iff P is s-U\. [One half of this uses 4.1.]

(iii) Prove that if A satisfies s-Πj =Σ: uniformly then

s.i.i.d. — Σ! on A,

i.i.d. = Δ! on A,

a.i.d. — element of A .

(iv) Prove that A is self-definable iff A is a.i.d. on A.(v) Prove that if A satisfies K2 then every a.i.d. subset of A is bounded.


4.20. The notion of uniformity given by Definition 4.11 is really suggested bythe notion of proof. Prove directly, using the Extended Completeness Theoremthat if A is countable and admissible then A satisfies s-Πj^Σ! uniformly.

4.21. A more recursion theoretic version of the uniformity discussed in 4.11 and4.20 was discovered by Nyberg. Prove that the admissible set A satisfies s-Π} =Σ1

uniformly iff for every s-Π} formula Φ(x,T+) in an extra relation symbol T thereis a Σ! formula φ(x,T+) such that for all Σx relations T on A, (A,T) satisfies

Vx[Φ(xJ+)~φ(x,T+)].

[Show that this condition is equivalent to Kt Λ K2. Note that in the proof of 4.7,T occurs positively in

4.22 Notes. See Exercise 4.19 for the way s-Π} predicates found their way intothe subject. Corollary 4.9 was observed by Nyberg. For the record, it is stillopen whether every Σ^ complete admissible set is Σx compact. (Surely not!)It follows from Theorem 8.3 (applied to L(α)) that there are lots of resolvableΣ! compact sets which are not Σj complete.

5. Kόnίg's Lemma and Nerodes Theorem:a Digression

In this section we interupt our study to apply the condition

s-Π} = Σ! uniformly

to notions of relative definability.One of the starkest applications of the Infinity Lemma in ordinary recursion

theory is the proof of Nerode s Theorem:

B is truth table reducible to C iff there is a total general recursive operator gwith %(KC) = KB.

Here B,C<^ω and KB is the characteristic function of B. This says, in effect,that the total general recursive operators rather trivial.

Since Nerode s Theorem uses so little about ω, other than the Infinity Lemma,it becomes a good test case for abstract versions of the Infinity Lemma, thematter which concerns us in this chapter.

Turing reducibility breaks up into many non-equivalent notions over anarbitrary set. We discuss generalizations of Nerode's Theorem for three of these:

<d is "Δ definable from",

^w is "weakly meta-recursive in",

^mr is "meta-recursive in".

5. Kόnig's Lemma and Nerode's Theorem: a Digression 335

5.1 Definition. Let A be admissible and let φ(x,C), ψ(x,C) be Σ formulas,possibly containing parameters from A. Let B, C be subsets of A. We say thatB^dC (via <φ,ιA» if for all xeA:

xεB iff (A,C)N<p(x,C),

xφB iff (A,C)Nι/φc,C).

If, for every C there is a £ such that £^dC via <φ,^> then the pair <φ,ιA> iscalled a general Δ definability operator g over A, and we write 5(Q = B.

By the relativized version of Theorem II.2.3, if A = HF then B^dC iff B isrecursive in C, so that ^d coincides with Turing reducibility.

What is the most obvious way to define a general Δ definability operator?It seems to be captured by the following definition. If A is admissible then Δ0(A)denotes the Δ0 formulas when all total A-recursive functions are denoted byterms of the language.

5.2 Definition. Let A be admissible and let φ(x,C) be a Δ0(A) formula. Thenvia φ if, for all xeA,

5.3 Lemma. Let A be admissible.(i) Every Δ0(A) formula φ(x,C) defines a general Δ definability operator.

(ii) // F is A-recursive and

xeB iff F(x)eC

then(iii) A-(iv) ^d is transitive.

Proof. They are all trivial. For example, to prove (i) you simply replace anyfunction symbol in φ by its definition as in § 1.4. Note that

xφB iff (A,C)N=-ιφ(x,C)

and —ιφ is also a Δ0(A) formula. D

5.4 Theorem. Let A be a resolvable, admissible set satisfying s-Tl\=Σ1 uniformly.Let 3 be any general Δ definability operator over A. There is a Δ0(A) formulaφ(C) such that for all C^A

S(C)<}C via φ.

Proof. Let 0(x,C), ψ(x,C) be Σ formulas such that g is defined by

xeS(C) iff (A,C)N0(x,C),

xφ%(C) iff (A,C)N=^(x,C).


Then, for every x, the following s-Tl{ formula Φ(x) holds on A:

VC[0(x,C)vιA(x,C)].

Let φ(x) be a Σ1 formula uniformly equivalent to Φ(x) on A. Since A is resolvablethere is an A-recursive function J: o(A)-> A such that A = !Jα<0(A) </(α) and J(α)is transitive for all α. Now for each x, Φ(x) holds so there is an αeA such that<p(x)(J(α)). Define

G(x)-J (least α[φ(x)(J(α))]).

Then G is A-recursive and total, G(x) is always transitive and

But then for every x, Φ(G(x))(x), by the uniform equivalence of φ and Φ. Thus,for every C^A, either θ(x, C)(G(x)) or ^(x, C)(G(x)\ We claim that

(1) xeft(C) iff θ(x,C)(G(x)).

For if xeg(C) then (A, C) 1= θ(x, C) so ^(x, C)(G(JC)) cannot hold so φ(x,C)(G(x))

must hold. Similarly, if x^S(C) then 0(x,C)(G(x)) cannot hold. Let σ(υ,C) beθ(υ, C)(F(V». Then σ(ι;, C) is Δ0(A) and g(C) C via σ. D

Since "s — Tl\=Σ1" implies "s — Π^Σ! uniformly" on resolvable admissiblesets, we could have used the weaker condition in the statement of the theorem.This seems to conceal the main point of the theorem, though, since it is the uni-formity which really matters in the above proof. Since the above proof is virtuallyidentical (in outline) to the proof of Nerode's Theorem (in, say Rogers [1967])this gives further support to the feeling that "s — Π^Σ! uniformly" captures agreat deal of the recursion theoretic content of the Infinity Lemma.

The relation ^ d is quite sensible from a definability point of view. It has beenstudied very little, however, because one does not have the tools from ordinaryrecursion theory available. Put another way, the relation B^dC is not sensiblein terms of computation if the expanded structure (A, C) fails to be admissible,for then in checking whether or not xeB one may have to use all of C, not justan A-finite amount of information about C. This never comes up for HF, or forany other H(κ\ κ:-regular, since every expansion of H(κ) is still admissible.

These observations prompt one to define a new notion of reducibility, onewhere an answer to "xeF?" is determined by an A-finite amount of informationabout C. Let Kc be the characteristic function of C :

0 if xeC,

1 if x£C

and let Cht(C) = {feA\f^Kc}. Thus ChA(C) is the set of all A-finite bits ofinformation about membership in C.

5. Konig's Lemma and Nerode's Theorem: a Digression 337

5.5 Definition. Let A be admissible and let φ(x,f\ ψ(x,f) be Σ^ formulas withparameters from A. We say that B is weakly metarecursive in C via <φ,^>, written

C via <<p», if for all xeA

xεB iff 3/eChA(C)[ANφ(x,/)],

*<££ iff 3/eChA(C)[Al=^(x,/)].

If for every C there is a £ such that B^WC via <φ,ι^> then the pair <φ,^> iscalled a general weak metarecursive operator g and we write 5(C) = B.

The notion of tt-reducibility corresponding to ^w is complicated by the fol-lowing observations. On HF one can define a recursive function H by

H(x) = { f \ f is a characteristic function with dom(/) = x} .

Then given any recursive predicate P of finite functions one can "split" it by

) = {feH(x)\P(f)},

Then F,G are recursive and, for each xeHF and each C^HF, ChHF(C) meets(has nonempty intersection with) exactly one of the sets F(x), G(x) (dependingon whether or not Kc\x satisfies P or not). This triviality simplifies a lot of therecursion theory on HF, especially when contrasted with a general admissibleset A where H(x) need not be a subset of A, let alone an element of A.

5.6 Definition. Let A be admissible. An A-recursive splitting is a pair F,G oftotal A-recursive functions such that

(i) for each xeA, F(x), G(x) are sets of A-finite characteristic functions,(ii) for each xeA and each C^A, ChA(C) meets exactly one of F(x), G(x).

5.7 Lemma. Let A be admissible and let F, G be an A-recursive splitting. Define

Then g is a general weak metarecursive operator on A.

Proof. Letφ(x,/)be/eF(x),^(x,/)be/eG(x). Then

iff 3/eChA(C)φ(x,/),

iff

so g(CKwC via<<p», for all C^A. D

The next theorem shows that for some admissible sets, every general weakmetarecursive operator arises as in the above lemma.


5.8 Theorem. Let A be a resolvable admissible set satisfying s — Π { = Σ! uniformly.Let 5 be any general weak metarecursίve operator. There is an A-recursive splittingF, G such that for all C,

Proof. The proof is very much like the proof of Theorem 5.4. Let θ(x,/), ψ(x,f)be Σ! formulas such that g is defined by

iff 3/eChA(Q[Al=θ(x,/)],

iff

Then for each xeA the following 5 — Π} formula Φ(x) holds :

VC3/[/eChA(C)Λ [θ(x,/)

Let φ(x) be uniformly equivalent to Φ(x) on A. Let J: o(A)->A be as in the proofof 5.4 and define

H(x) = J(leastαφ(x)(J(α))).

As in the proof of 5.4 we see that H is a total A-recursive function, that H(x)is always transitive, and Φ(x)(H(x)); i.e.,

(2) VC 3/eH(x) [>ChA(Q Λ 0(x,/) v ^(x,/)](IIW) .

LetF(x) - {/eH(x) |/ is a characteristic function Λ θ(x,

G(x) = {/eH(x)|/ is a characteristic function Λ ι^(x,

We claim that for all x, C

xeg(C) iff F(x)nChA(C)^0,

iff

This will prove that F, G is an A-recursive splitting and the conclusion of thetheorem. First suppose xeg(C). From line (2) we see that F(x)nChA(C)^0.But line (2) also implies that G(x)nChA(CHO for if /eChA(C) Λ (x,/)(H(x))

then ψ(x,f) holds in A, since ψ is Σ1? so x^g(C). The other half is similar. D

The relation ^ w has been studied a fair amount by the Sacks school (on ad-missible sets of the form L(α)). In particular, it has shown that ^ w is not transitive.This is not too surprising given the disparity between the amount of informationused about C (namely /eChA(C)) and the amount of information received

or xφB). Thus Sacks defines

C iff C

6. Implicit Ordinals on Arbitrary Admissible Sets 339

This is equivalent to the existence of a single Σj formula φ(x,f) such that

(3) geChA(B) iff 3/eChA(C)[ANιA(x,/)].

There doesn't seem to be a very elegant notion of tt-reducibility to go alongwith ^mr, but we do get one out of Theorem 5.8.

5.9 Corollary. Let A be resolvable and satisfy s — Yl\=Σί uniformly. Let ψ be suchthat for every C there is a B satisfying line (3) above. There is an Ik-recursive splittingF,G such that for all C,B as in (3)

£) iff F(0)nChA(C)^0. D

5.10 Exercise (R. Shore). Show that if V = L then the conclusions of 5.4 and5.8 fail for A=L(ωί). [For 5.8 define g(C)=A if Cnω is infinite, =0 other-wise. For 5.4 let R^&(ω) by Δx on A but not Δ0 and define 5(C) = A if

), =0 otherwise.]

5.11 Notes. The reader should consult Simpson's forthcoming book in thisseries for more about reducibilities on admissible sets.

6. Implicit Ordinals on Arbitrary Admissible Sets

For the model theory of an admissible fragment LA, the ordinal /zΣ(A) plays amore important role than o(A). For countable A we have /zΣ(A) = o(A). Ingeneral, we will see that this condition again goes back to the Kόnig InfinityLemma.

6.1 Theorem. An admissible set satisfies the third Kδnig principle iff /zΣ(A) = o(A).

Proof. This is an immediate consequence of the next theorem. D

The ordinal hΣ(A) is not an absolute notion. That is, the size (cardinality) ofΛΣ(A) may vary drastically from one model of set theory to another (cf. Theorem4.2 in Barwise-Kunen [1971]). The important point for application, though,is that /zΣ(A) has a precise description in terms of the generalized recursion theory

6.2 Theorem. Let A be admissible:

hΣ(A) = sup{ξ\ ξ is a Π implicit ordinal over A} .

Proof. The inequality ^ follows from Theorem 3.5. To prove the theorem itsuffices to prove that every ordinal β < hΣ(A) is less than some Π implicit ordinal ξ.


We can read this off the proof of Theorem VII.3.1. Since β </zΣ(A), β can be pinneddown by some Σ± theory T of LA. Now consider the proof of VII.3.1 for thisparticular theory T. In particular, consider the well-founded relation <S, •<>constructed there. Since every ordinal pinned down by T is less than the rankξ = /?(-<) of this well-founded relation, it suffices to prove that this ξ is Π implicitover A or, at least less than or equal to some Π implicit ordinal.

Case 1. If o(A) > ω then ξ is a Π implicit ordinal.For if o(A) > ω. then we can write

(1)

out as a Π sentence φ(<2) ',&) using 4.2:

3rc<ω3w<ω[tt>wΛ TC^C^'Λ®' is an s.v.p. for LA(c l 9..., cj,

2 is an s.v.p. for LA(c l5 ..., cm)

All these clauses are Π and the others follows from these. Thus φ(β\Qi) is a Πsentence which implicitly defines ξ = p(-<).

Case 2. // o(A) = ω then ξ^ξ' for some Π implicit ordinal ξ'.Let \l/(@\ί&) be the Π sentence expressing the following:

Q),Q)' are sets of sentences of LA(C),

2 n LA is an s.v.p. for LA ,

Vm[(cm = cm)e®->® n LA(c l9 ..., cj is an s.v.p.

for LA(c l9 ..., cj and

the same sentence for ',

If ^,^'eS and ®X® then ψ(2',3ι\ If

0'<® iff (A,^,^'

defines a well-founded relation then its rank is ^ ξ = p(X) since •< is a subrelation.So we need only prove that -<' is well founded. Suppose not. That is, suppose

Let $)n = <3)nc\ LA(c l9 .„, cπ). Since \l/(&ί9£&0) holds it follows that ^0 is an s.v.p.for LA, that (GI = C1)e^2 and hence that ® t is an s.v.p. forj-^cj. That is, Oe S0,®1e®1 and ^i^^Q. By induction on n we see that ^πeSn and @n+i<@n.

6. Implicit Ordinals on Arbitrary Admissible Sets 341

This contradicts the well-foundedness of •<. Thus •<' is well founded. Since ψis a Π formula, the rank ξ' = p(X') is a Π implicit ordinal and β<ξ^ξ'. 0

Theorem 6.2 would have simplified the proofs of the theorems in §VII.4since it is usually easier to show that a given well-founded relation -< is definableby a Π sentence than to prove p(-<)</zΣ(A).

6.3 Corollary. Every s — Σ\ implicit ordinal over the admissible set A is less thansome Π implicit ordinal over A.

Proof. Immediate from 3.5 and 6.2. D

6.4 Corollary. // A is a resolvable admissible set and hΣ(A) = o(A) then A isΣ! compact; i.e., K3 implies K2 on resolvable admissible sets.

Proof. By 4.8, if A fails to be Σί compact then A is self-definable, and hence/ιΣ(A)>o(A) by Proposition VII.1.5. D

6.5 Corollary. Let a» = <M,Λ1, ...,/?,> and let A^HYP^. Then A is Σ!compact i

Proof. One half follows from 6.4, since HYP^ is resolvable; the other half (=>)from VII.3.8. D

We conclude this section with a theorem that explains why Π implicit ordinalsare so important from a theoretical, not just a practical, point of view.

Let φ(υ, R, S) be a formula with R, S n-ary, v a free variable, which may containparameters from A. For xeA we write -<* for the relation defined by

iff (A,

6.6 Lemma. // φ(v, R,S) is a Π (or even s — Σ}) formula then

P(*} tff <x

φ is well founded

defines a s — Πj predicate P over the admissible set A.

Proof. P(x) holds iff

where Q is n + 1 ary and <p(x,(Q)m+1,(Q)OT) denotes the result of replacing(x1? ...,*„) by Q(x1? ..., xm,m). D

6.7 Theorem. Let A be admissible. There is a Π formula φ(v, R, S) such that

{x: XJ is well founded]

is a complete strict — Πj set for A.

342 VIII. Strict Πj Predicates and Kδnig Principles

Proof. The set in question is always s-Π\ by 6.6. Let X1 be the complete s-Π}set defined in Corollary 4.5: Xί = {φe L^| TN= φ}. We can assume L'A (of 4.5) is aSkolem fragment which is Δ t on A and that every model of T can be expandedto a Skolem model. We will show how to write "Is xeX^T in terms of askingwhether or not a certain tree of theories of L^ is well founded. Let (p(x,T",Γ)express :

T", T" are sets of sentences of L'A,

= Vι;φ(ι;)eT->Vί(ί a closed term of L'->φ(f/u)eT')],

Vy, z [y, z closed terms of L Λ (y = z) e T' -» (z = y) e T"] ,

Vy, z, w [w = φ(ι ) e LA Λ y, z closed terms of L' Λ φ(y/v) E T

If xφXi (i.e. T^x) then Γu {~ιx} has a Skolem model 501. Let T be the set ofsentences true in 9JΪ. Then ψ(x,T', T') holds so <J is not well founded. Now sup-pose <£ is not well founded, so there is a sequence

Let Tω = (JnTn. Then T satisfies all the conditions of Lemma VII.2.9, so Tω hasa model. But Tu{~ιx}cΓω so T^x. Thus

xeX t iff <J is well founded . D


6.8. Let 501 be infinite. Show that if P^Wl is Πj on 501 then there is a first orderformula φ(v, R, S) such that

P(x) iff -< J is well founded .

This is analogous to the normal form for H{ relations on JΛ

6.9. Show that if α is the rank of some well-founded relation on Power(A) thenα<(2card(A))+. Conclude that /zΣ(A)<(2card(A))+.

6.10 (Open). Prove that hΣ(ΉΎPgjl) = sup{ξ: ξ is a first order implicit ordinalover 9JΪ}.

7. Trees and Σl Compact Sets of Cofmality ω 343

7. Trees and Σ1 Compact Sets of Cofίnalίty ω

The results of this chapter would be vacuous if there were no uncountable ad-missible sets sytisfying the Kόnig principles K^ — K3. We exhibit such admissiblesets in this and the next section.

A set A is essentially uncountable if every countable subset of A is an elementof A. All of the Σί compact sets exhibited in this section have cofinalίty ω in thesense that

where each AneA. Hence none of them is essentially uncountable. We give aproof of the existence of essentially uncountable Σx compact sets in the nextsection, though no explicit such sets are known. An explanation for this pheno-menon will be found in § 9.

Let us return to our discussion of trees from § 1, and attempt to give a gen-eralization of the Infinity Lemma soley in terms of trees.

In this section we turn the full binary tree around and think of it as picturedin Fig. 7A.

Fig. 7 A. Another view of the full binary tree

Another tree, one with paths of length ω2, is pictured in Fig. 7 B.

Fig. 7 B. A tree of rank ω2

In general, a tree is a well-founded partial ordering ^~ = <Γ, -<>, with a leastelement (usually denoted by 0), such that for each xeT, the set {yeT:y-<x}

344 VIII. Strict π; Predicates and Kόnig Principles

of predecessors of x is well ordered by -<. A subset C^ T is a chain in y if foreach x,yeC,

x^y or x = y or y^x .

A path thru ?Γ is a maximal chain. Thus every path is well ordered by •<.Let ^ = <T, •<> be a tree. Since •< is well founded we have the usual rank

function p = p^ associated with y :

p(x) = sup{p(y)+l:)Kx}

and 3Γ has a rank ρ(y):

= sup{p(x) + l :xeΓ} .

A branch thru the tree 3~ is a path of length p(^). Not every tree has a branch,as Fig. 7 C demonstrates.

Fig. 7 C. A tree with no branch

This tree has rank ρ(3~) = ω but every path is finite. Thus y has no branch.We call the elements x of a tree 2Γ with p(x) = β the nodes of level β. Thus yhas nodes of every level β < pψ~\ Let lev = lev^- be the function with domain

defined by

Let A be an admissible set. A tree ^~ = <7;-<> isanA-ίreeif T^A, T, Xjlev^are A-recursive and the rank ρ(y) of y is o(A). In particular, for each β < o(A),T has nodes of level β (since p(^~) = o(A)) but the set of all nodes of level β isA-finite (since lev(β)eA).

The Kόnig Infinity Lemma can be restated as follows. Let A = <HF,e,Λ>.Then every A-tree has a branch.

7.1 Theorem. // A is a Σ! compact admissible set then every A-tree has a branch.

7. Trees and Σl Compact Sets of Cofinality ω 345

Proof. It is easy to prove this by means of Σ^ compactness, by constructing anonstandard extension of the tree, picking a node d of nonstandard length andletting the branch be defined by

An even easier proof, though, is by means of s-Π\ Reflection. Let ^r = <T,-<>be an A-tree and suppose y has no branch, i. e. no path of length ρ($~} = o(A).Then A satisfies the s — Π} sentence:

VC[C is a chain -> Jβ Vxelev(β) (xφC)'] .

By s — Π} Reflection there is a yeA such that

(1) VC[C is a chains 3β<y Vxelev(β) (xφC)] .

But then lev(y) must be empty, for if yelev(y) then

C = {xεT\x<y}

would violate (1). But then p(^")< y < o(A), contradicting the definition ofA-tree. D

The hypothesis "every A-tree has a branch" looks like it ought to be calleda Kόnig principle. The next theorem shows that it is in fact too weak to be ofgeneral interest.

7.2 Theorem. Let A be an admissible set whose ordinal α = o(A) has coβnality ω.Then every A-tree has a branch.

Proof. Since this is a direct generalization of the Infinity Lemma it is not sur-prising that the proof is an amplification of the proof of that lemma. Letα = sup{απ: n<ω] where a0<a t < <all< <a. Let ^~ = <T,<> beanA-tree.We claim that we can find x0,x1? ... such that xne\Qv(an) and x0 X x j X . . . .If we can do this, then

B = {yeT: y<xn for some n}

will be a branch thru 2Γ . To find the x's, let x0elev(α0) be such that

(x0<z).

(We must see that there is such an x0.) Given x0, let X^XQ be choosen sothat x^lev^) and

V j8>α13zelev(jS)[x1<z].

346 VIII. Strict Π{ Predicates and Kδnig Principles

Continuing in this way gives the desired sequence of xn's. Let us now provethat x0 exists. (The proof that given xn we can find xn+l as above is almostidentical.) Suppose there were no such x0. Then

Vxelev(α0) 3)S>α0 Vzelev(jS) [x-£z] .

By Σ! Reflection there is a ye A such that

(2)

Let wElev(y). Now w has a predecessor x of level α0 and a predecessor zβ foreach β<y. But then x-<z^, contradicting (2). D

Thus, e.g., A = ffpω+ω) satisfies "Every A-tree has a branch" but it doesnot satisfy s-Tl[ Reflection. Still, we did use the Infinity Lemma to prove s-Yl{Reflection in § 1, so there should be some context in which the tree proof gen-eralizes. If you analyze that proof you see that we also used two other facts:every subset of an αelHF is in HF and, moreover, we can effectively find theset of all subsets of a. It easy to see that a pure admissible set A such that

(sometimes called supertransίtive) must be of the form H(κ) for some TC, so werestrict attention to H(κjs for the time being. In order for H(κ) to be closedunder , the power set, it is necessary and sufficient that K be a strong limitcardinal (λ<κ=>2λ<κ). Note that Hpω + ω) is closed under the power setbut that

is not admissible (for the same reason that L(ω + ω) is not admissible). We write,e,^> rather than the correct

7.3 Theorem. Let K be a strong limit cardinal and suppose that A =is admissible. Then A is Σt compact iff every A-tree has a branch.

Proof. We have (=>) by 7.1. To prove the converse we assume that every A-treehas a branch and prove s-Π} Reflection. Since H(κ) is closed under the powerset and since <H(κ;),e,^> is admissible, the usual definition by recursion ofV(α) shows that α-» V(α) is an A-recursive function of α. The usual "ZF-prooΓthat every set is in some V(α) shows that V(κ) = H(κ). Let

(3) VS3yφ(S,x,y)

be a typical s-Πj formula true in A, where φ is Δ0 and we assume S is unary,for simplicity. Let ξ = τk(x) so that xeV(α) for all a>ξ. We suppose that foreach α, £<α</c,

c V(α) 3yeV(α) φ(S,x,y)

7. Trees and Σt Compact Sets of Cofinality ω 347

and get a contradiction, thus establishing s-Π} Reflection. Thus we are assumingthat for each α, ξ < α < K

(4)

We define a tree " = <T,O by

'> iff α<β and S = S'nV(α).

Each such 5 is an element of /f (κ;), by supertransitivity. The least member of Tis <0,0>. The predecessors of some <α,5'>eΓ are just the pairs of the form<β,SnV(/?)> for β<tt and hence have order type α under -<. Thus the levelof a pair <α,S> is just α. Furthermore, since

lev(α)eA and, by the above comments, lev is A-recursive. Line (4) says thatlev (α) 0 for all α<τc. Thus y is an A-tree. Let B be a branch thru y> thatis, a path of order type K. B is a set of pairs

exactly one pair for each α<τc, linearly ordered by •<. Furthermore, a<βimplies Sα = S^nV(α). Let S = (JΛ<KS0[. Then

for each α. We claim that (A,S) satisfies

-ι3yφ(S,x,y)

contradicting (3). For let ye A be arbitrary. Pick α<κ: such that £<α andyeV(α). Since <α,Sα>eΓ

But (A, S) is an end extension of this structure so it also satsfies the Δ0 formula,x,y), establishing our contradiction to (3). D

A cardinal is said to be (strongly) inaccessible if K: is a regular strong limitcardinal. It follows from Theorem Π.3.2 that if K is inaccessible then


is admissible for all R<^H(κ), so that Theorem 7.3 applies. A simple Lόwenheim-Skolem argument shows that one can find λ<κ such that

is admissible and cf(Λ,) = ω. Alternatively, one can drop all talk of inaccessiblesand prove directly (using the reflection theorem of Levy) that for any definableclass R there are cardinals λ with cf(λ) = ω such that (H(λ),e,0>,R\H(λ)y isadmissible. Thus the hypothesis of the next theorem is not vacuous. It is thistheorem which has been the aim of the first part of this section.

7.4 Theorem. Let K be a strong limit cardinal of cofinality ω and assume that,6,^,K> is admissible. Then A is Σl compact.

Proof. This is immediate from 7.2 and 7.3. D

Exercise 7.10 shows that A is also Σ^ complete. Exercise 7.11 shows that itsatisfies K3.

The urelement versions of 7.3 and 7.4 are not very interesting since 7.3 onlygoes through for <SDt;H(/c)SM,e,^> when card(9Jl)<K, in which case 9JI isalready contained in H(κ), up to isomorphism.

We will return briefly to the notion of tree in § 9. Now we go on to discusstwo rather different examples of Σί compact admissible sets.

The following theorem of Nyberg gives quite concrete examples of Σt com-pact and Σ! complete admissible sets.

7.5 Theorem. Let α be a limit ordinal of cofinality ω, let A be of the form<H(5α),6,R> and let B be admissible with A6 IB, B projectible into A. Then Bsatisfies s-Π} = Σ1 uniformly and hence is Σ1 complete and Σ1 compact. (A is notnecessarily admissible.)

The proof of this theorem is sketched in Exercise 7.16. Note that it appliesto HYP(H(5α)) whenever cf(α) = ω. This is a resolvable admissible set satis-fying s-Π}=Σ1 uniformly (and hence the theorems of the previous sections).On the other hand, if cf(α)>ω then HYP(HpJ) is strongly self-definable,hence not Σ1 complete or Σί compact.

We conclude this section with a different kind of example.A structure 9M = <M,R1,...,JR/> is recursively Σ\ saturated iff for every finite

expansion L' = L(Sl5...,Sk) of L and every recursive (equivalently, r.e.) setΦ(vί,...9vn, S1?...,Sk) of formulas of L'ωω, $01 is a model of:

7. Trees and Σ{ Compact Sets of Cofinality ω 349

It is easy to see that every recursively Σ} saturated structure is recursivelysaturated. Theorem IV.5.7 shows that if 501 is countable and recursively saturatedthen it is recursively Σ} saturated. The following theorem characterizes therecursively Σ} saturated structures among the class of recursively saturatedstructures.

7.6 Theorem. Let Wl = (M,R1,...,Rιy be an infinite recursively saturated struc-ture of any cardinality. Then 501 is recursively Σj saturated iff HYR^ is Σ1 compact.

Proof. The "if part of the theorem is established by the very proof of TheoremIV.5.7. To prove the converse, we assume that 9Jί is recursively Σ} saturated andprove that HYP^ satisfies s-Π} Reflection. Suppose

(5) HYP w t=VR<p(R,jZ)

where φ is a Σ^ formula. Since ΉYPm = L$R9o)), we need to exclude the pos-sibility that for every n < ω

(6)B L(ΪR,tt)^3R^φ(R,x).

Since each xeHYP^ has a good Σl definition in terms of parameters fromMu{M}, we may assume that each xf in the sequence x is either in M or is Mitself. Let us rewrite (5) as

(7) L@Λ9ω)ϊ=VRlyψ(R9p9y9M)

where ψ is a Δ0 formula with no other parameters. We can rewrite (6) as: forevery n<ω

(8)π iχ5W,ω)l=3RVy6L(Af,π)-ι^(R,p,y,M).

Let Φ(p) be the set of formulas in L(U,A,E,F, R,Ri,. . . ,R;) which express thefollowing about 9K, p:

KPU+ relativized to U (for urelements), S (for sets), E (for e),

VyeL(U,Λ)-ι^(R,F(p),3;,U) (for all n<ω).

Every finite subset Φ0(p) of Φ(p) is satisfiable on 9JΪ by choosing relations whichcode up IHYP^ on 501 itself and using (8)π to satisfy the last sentence in Φ0(p).Since 9JΪ is recursively Σ} saturated there are relations on 2R which make thewhole set Φ(p) true:

(9)

;,...,Λ'>;4JE,/ί^^


Let R*-RfIHYR0 l. By (7) there is a yeHYP^ such that

But yeL(M,n) for some n<ω. Since ψ is Δ0 and

we have

contradicting (9), since (9) asserts, among other things, that

9y9M). D

To see that this result gives us lots of uncountable Σ^ compact sets, we mustknow that there are lots of recursively Σj saturated models. We assume thereader is familiar with saturated or special models, referring him to Chang-Keisler [1973] for the relevant definitions and properties.

7.7 Proposition. Every saturated (or even every special) model 9Jl = (M9Rl9...,Rlyis recursively Σ\ saturated.

Proof. If we assume the GCH we can get rid of the requirement that the set offormulas is recursive; the proof not involving the GCH is sketched in Exercise7.17. Let 9ϊΐ be a special model and let Φ(p,S) be a set of sentences such that foreach finite

Then the first order theory T/ϊ(9JΪ,p)uΦ(p,S) is consistent and so has a specialmodel (9K',p',S') of power card(9W), by the GCH.

But then (3R9p) = (3Jl'9pf) (Lωω), and both models are special so

Hence


7.8. Prove that the pure admissible set A is supertransitive iff A = H(κ) forsome cardinal K.

7.9. Prove the following: Let A be pure, admissible, supertransitive and Σt com-pact. There is a cardinal κ = 2κ such that A = H(κ). Let A / = (A,^)). Then A'is admissible and satisfies s-Πj^Σi uniformly. More slowly, prove:

7. Trees and Σl Compact Sets of Cofinality ω 351

(i) A is closed under , using s-Yl{ Reflection.(ii) A' satisfies s-Π} Reflection (0> is s-Δ} on A).

(iii) A' is admissible (using (ii)).(iv) A' satisfies s-Π^Zi.

7.10. Prove that the following are equivalent, where K is a strong limit cardinaland A = (H(κ),E,0>,Ry is admissible:

(i) A is Σ! compact (s-Πj Reflection),(ii) A is Σ! complete (s-Π\=Σl),

(iii) Every A-tree has a branch.

7.11. Let A = <//(κ),e,R> be Σx compact. Prove that hΣ(A) = κ. [Use 7.9 ands-Π} Reflection plus trivial cardinality considerations.]

7.12. Let λ = card (501) and let K: be a limit ordinal. Prove that the followingare equivalent:

(i) (9JI; VOT(κ:),e) is admissible,(ii) κ = ϊκ(λ),

(iii) K: is a cardinal and VyJl(κ) = H(κ),0l.

7.13. Prove in ZFC that there are arbitrarily large cardinals κ = 3κ of cofinalityω such that <//(τc),e,^> is admissible.

7.14. Let K be the Hanf number of second order logic. Show thatsatisfies the hypothesis of 7.4.

7.15. Let α be a limit ordinal, let A be admissible and let V(α)eA. Prove thatΉpJeA. [Consider the set X = {EeV(a)}: E is well-founded}.]

7.16. Theorem 7.5 follows from the following result of Nyberg. Prove that if $Ris a uniform Kleene structure and A^ is admissible above $R and projectibleinto SDΪ then A^ satisfies s-Π^Σi uniformly. [Use the alternate form of"s-Πj^Σi uniformly" given in Exercise 4.21.]

7.17. A structure ΪR^M,^,...,^) is resplendent if for every finitary Σ}sentence 3Sφ(S) with constants from 2R, if 9lt=3Sφ(S) for some 91>9JI,then 9«N3Sφ(S).

(i) Prove that every special model is ω-resplendent (Kueker [1971]).(ii) Prove that every resplendent model is recursively Σ{ saturated. [Use

the techniques of IV.2.](iii) Associate with any finitary Σ} formula Φ(x) a recursive closed game

formula ^φ(x) such that


for all 501 and, for SDΪ countable,

(10)

Such a ^φ is given by (the proof of) Svenonius's Theorem. Prove that if 9JΪ isrecursively saturated then StJΪ is resplendent iff

for all Σj formulas Φ.(iv) Prove that if 9JI is resplendent then Π} on Wl = Σί on(v) (Schlipf). Improve (iv) by showing that if 9JI is resplendent then

satisfies K t.

7.18. (Open). Characterize those 9W such that HYP^ is Σί compact.

7.19. (Open). Characterize those ΪR such that HYPW is Σ1 complete.

7.20 Notes. Theorem 7.4 is due to Barwise [1968] and, independently, and bya completely different proof, to Karp [1968]. Theorem 7.3 is a refinement of aclassical result about weakly compact cardinals, contained in Theorem 9.10.

8. Σ1 Compact Sets of Cofίnalίty Greater than ω

In this section we prove an existence theorem which shows that there are manyΣt compact admissible sets besides those exhibited in the previous section. Inparticular, we prove the existence of essentially uncountable Σt compact ad-missible sets.

Let K be an uncountable regular cardinal. A subset C of K is closed in K iffor each initial segment C0 of C,

(supC0)<κ implies (supC0)eC.

This says that C is closed in the order topology on K. C is unbounded in K if

A set C is c.u.b. in K if C^κ and C is closed and unbounded in K.

8,1 Lemma. Let κ>ω be regular. If C0, Q are c.u.b. in K then so isIn particular, CΌnCΊ is nonempty.

Proof. The intersection C0πC1 is closed since the intersection of closed setsis closed. To see that CΌnCΊ is unbounded, let β<κ be given. Let y±>β be

8. Σj Compact Sets of Cofinality Greater than ω 353

in Q. Let y2

>7ι be in C0. Let y 3>y 2 be in Q and so on for each n<ω. Theny = supM < ωyπ is less than /c since K is regular. Since y = supn < ωy2 n and C0 isclosed^ ye C0. Since γ = supn<ωy2n + ι and Cί is closed, yeQ. Thus /?<y and

). D

Thus, by 8.1, the collection

5 = {C c K; : C0 c C for some C0 c.u.b. in K:}

defines a filter on the subsets of K, called the c.u.b. filter on K. We say that P(α)/zo/ds for almost all α < K if

is a member of the c.u.b. filter on K.

8.2 Lemma. Let λ, K be regular cardinals, ω^λ<κ. If P(α) holds for almostall UL<K then P(α) holds for some α with cf(α) = A.

Proof. Let C be c.u.b. in K: be a subset of

Let y be the λth member of C, enumerated in the natural order. There is sucha λth member since K is regular and C is unbounded in K. It is clear thatcf(y) = d(λ) = λ since λ is regular. D

In reading the next theorem, the student should think of Jα as #(Nα) orL(ωα) or L(α,ωα), since these are the usual applications.

8.3 Theorem. Let K be an uncountable regular cardinal, let R^H(κ) and letJ: κ-+H(κ) have the following properties:

(i) Jα is transitive and closed under pairs and union, for all α</c;(ii) <x<β<κ implies JaeJβ;

(iii) if λ<κ is a limit ordinal then Jλ = \JΛ<λJΛ

(iv) for each α<κ, the structure

satisfies Δ0 Separation. Then, for almost all α< K, Jία is a Σ^ compact admissible set.

Proof. The idea for this proof goes back to the notion of stable ordinal. Forthe purposes of this proof we call an ordinal α β-superstable if a<β<κ andfor every s-Π} formula Φ(vί9...,υJ and every aί,...,anεJΛ,

if Jpϊ=Φ(aί9...9an) then

354 VIII. Strict Π; Predicates and Kόnig Principles

We first prove:

(1) if α is β-superstable then JFα is a Σ! compact admissible set.

So suppose α is β-superstable. Since Δ0 Collection follows from s-Π} Reflection(in fact from Σ Reflection) it suffices to prove that JΓα satisfies s-Π} Reflection.Let Φ(aί,...,an) be a s-Π} formula true in JΛ.

Then

and hence JJ^ is a model of the s-Π} formula Ψ(aί9...9an)

since JaeJβ. But then by superstability, JίΛ^=Ψ(aί,...,an)9 so JJα satisfies s-Π}Reflection, proving (1).

We will prove the theorem by proving that almost every α<κ; is β-super-stable for every β,a<β<κ. To prove this we use normal functions. (A func-tion f:κ-+κ is normal if /is increasing (α<j8<κ=>/(α) </(/?)) and continuous(A a limit <κ=>/(A) = sup{/(α): α<A}). If f:κ->κ is normal then the set offixed points of /,

is always c.u.b. in K, as is easily seen.) We define a normal function / such that/(α) = α implies α is β-superstable for all β between α and K. This will provethe theorem. Let P(α,]8) be the following condition on α, β<κ:

for all β',β^β'<κ, and for all s-Π} sentences Φ(α l5....,απ) with constantsfrom JΛ9 if Jβ,\=Φ(aί9...9aJ then

Note that P(α,j80) implies P(a9βι) for all /?! between jS0 and c. Since card(Jα)<κ:there are <κ s-Π} formulas Φ(α) so a trivial cardinality argument proves thatVα<κ:3j3</cP(α,jβ). Now define / by

/(α) = least β [ β > f ( y ) for all y<α, and P(α,j8)] .

Since K is regular, /(α) is defined for all α<κ. Thus f:κ^κ and / is increasingby definition. Let us prove that / is continuous. Let λ < K be a limit ordinal.Let β = sup{/(α):α<;i}. We need to verify P(λ9β). Thus let β'^β and let Φbe a s-Π} sentence with parameters from Jλ which is true in Jβ,. We must seethat Φ is true in Jβ. But Φ is defined in Jα for some α<A so Φ is true in J/(α)

and hence in Jβ by persistence of s-Π} formulas. Thus / is normal.Now suppose /(α) = α. Then P(α,α) holds so α is /P-stable for all j5'>α,

β'<κ. By (1) this shows that almost every α<κ has Jία Σt compact. D

8. Σ! Compact Sets of Cofinality Greater than ω 355

8.4 Corollary. Let κ>ω be regular. Then for almost all α<κ, L(α) is a Σί

compact admissible set.

Proof. Apply 8.3 with Jα = L(ωα). Then for almost all α<κ, L(ωα) is Σ1 com-pact. But ωα = α for almost all α<τc since /(α) = ωα is a normal function. D

8.5 Corollary. Let κ>ω be regular and let 5Π = <M,JR1,...,R ί> be a structureof power less than K. Then for almost all α<κ;, L(9Jl,α) is Σί compact.

Proof. Similar to 8.4. Since there is isomorphic copy of 50ΐ in H(κ). D

The next result gives us essentially uncountable Σί compact admissible sets,when one applies Lemma 8.2 and the observation that H(κ) is essentially un-countable iff d(κ)>ω. & denotes the power set operation (restricted to H(λ)in 8.6).

8.6 Theorem. Let K be inaccessible, κ>ω. Let R^H(κ). Then for almost allλ<κ, (H(λ\e,0>,RπH(λ)y is Σ1 compact.

Proof. Let Jα = /f(Dα). Then J:κ-*H(κ) since κ = 2κ, and card(H(3α)^iα+1 <κ.Thus, for almost all α<κ, <#pα),e,^,fln//pα)> is Σ! compact. But /(α) = 5α

is a normal function so almost all α<κ: have 5α = α. Thus almost all λ<κ have

Σ1 compact. D

We can reinterpret all of the above by thinking of the class of all ordinalsas an inaccessible cardinal. We can restate Theorem 8.6 in this case as a resultin ZFC.

8.7 Corollary. Let R be any class. The class of λ such that (H(λ),e,0>,RnH(λ))is Σ! compact contains a closed proper class of cardinals. Hence for any regular Kthere are arbitrarily large such Xs of coβnality K.

Proof. The last sentence follows from 8.2. D

A cardinal K: is a Mahlo cardinal if every c.u.b. set C^κ contains an in-accessible cardinal (and hence contains K such inaccessible cardinals λ<κ).

8.8 Corollary. Let K be a Mahlo cardinal and let R^H(κ). There are K inacces-sible cardinals λ<κ such that (H(λ),e,0>,RrιH(λ)y is Σ1 compact.

Proof. Immediate from 8.6. D

8.9 Exercise. Suppose <//(/c),e> is Σί compact. Prove that K is not the firstinaccessible. Prove, in fact, that if K is inaccessible then K is the τcth inaccessible.[Use s-Π{ Reflection.]

8.10 Notes. Theorem 8.3 is contained in Barwise [1969 b].


9. Weakly Compact Cardinals

In this final section we consider weakly compact cardinals and their relationshipto Σ! compact admissible sets.

Let L be a language with ^K symbols coded as a Δ! subset of H(κ). Thelanguage Lκω consists of those φeLaoω with less than K subformulas.

9.1 Definition. A cardinal κ^ω is weakly compact (for Lκω) if for every setT^H(κ) of sentences of Lκω, if every subset T0^T of power <κ has a modelthen T has a model.

This definition is usually expressed in terms of a stronger language Lκκ (de-fined in Exercise 9.14) and it is usually assumed that K is inaccessible in whichcase H(κ) has power K and hence T has power ^ K. We will see that both ofthese apparent strengthenings follow from Definition 9.1. Note that ω is weaklycompact.

9.2 Lemma. Let κ^ω be a cardinal(i) Lκω=LQθωnH(κ).

(ii) // K is regular then Lκω is the least subset of Laoω containing Lωω closedunder ~~ι, V, 3 and

if Φ^Lκ ω and card(Φ)<κ then /\Φ and \/ΦeL κ ω .

(iii) // κ>ω is a limit cardinal then

*-κω = \Jλ<κ *-λω

where the union is over all infinite cardinals λ<κ.(iv) K is weakly compact iff <//(jc),e,.R> is Σl compact for every relation

Proof, (i), (iii) and (iv) are immediate from the definitions. To prove (ii) let L'κω

be the least class described. It is clear that Lκω^Lκω. To prove Lκω=L'κω itsuffices to prove that Lκω is closed under —ι, V, 3 and the clause

if Φ^Lκ ω and card(Φ)<κ then /\Φ, γΦeL κ ω .

The first part is trivial. So suppose Φ^ Lκω and card(Φ)< K. We must verify that

card (sub (/\ Φ)) < K .

But

Since Φ^Lκ ω each sub(φ) has power <κ for φeΦ. But card(Φ)<κ; and K:is regular so card(/\Φ)<κ. Similarly, card(\/Φ)<κ;. D

9. Weakly Compact Cardinals 357

Part (iv) of this lemma shows that the notion of weakly compact cardinalis just the relativization of the concept of Σl compact admissible set to an arbi-trary R^H(κ).

Before we see just how strong the assumption that K is weakly compact anduncountable is, let us stop to examine the plausibility of the existence of suchcardinals. We want to show that the same kind of intuition which prompts oneto admit ω, inaccessible cardinals and Mahlo cardinals as legitimate abstractobjects also prompts one to admit weakly compact cardinals as legitimate ob-jects in the hierarchy of sets.

There was a time when the existence of ω was considered problematic. Onemust accept each natural number, but it took years for the limit, the set of naturalnumbers, to be accepted as a legitimate abstract object, suitable for use inmathematics.

Once one accepts the basic principles of set theory, one sees how to generatemany cardinal numbers, which must be accepted. Only fairly recently have in-accessible cardinals begun to be considered as the natural limit of the accessiblecardinals and hence suitable for use in mathematics.

We saw in Corollary 8.7 that for any class R, almost all cardinals K have theproperty that (H(κ),e,RπH(κ)y is Σί compact. Given any collection 0t ofclasses that can be coded by a single class, we see that almost all K are such that(H(κ),e,RnH(κ)y is Σί compact for all Re&. A natural limiting assumptionis that (H(κ),e,Ry should by Σ^ compact for all R^H(κ). This is the assump-tion that K is weakly compact.

(Another argument that is often given for the existence of weakly compactcardinals, as well as measurable cardinals and strongly compact cardinals,cardinals we can see no argument for at all, is that they should exist "by analogywith ω". This seems like a very weak argument. The results of § 7 suggest thatthe crucial property of κ = ω for compactness is that d(κ) = ω, whereas weaklycompact cardinals are always inaccessible and hence regular. Of course ω is theonly regular cardinal K with cf(κ:) = ω.)

Call K a Σl compact cardinal if <H(κ),e> is Σ1 compact. Call K a Σt(jR)compact cardinal if <//(κ:),e,R) is Σx compact. Thus K is weakly compact iff Kis Σ^R) compact for every R^H(κ). We remind the reader once again that

9.3 Proposition. Let(i) // K is Σί compact then K = 3K.

(ii) // K is weakly compact then K is inaccessible.

Proof. Part (i) is a small part of Exercise 7.9 but we include its proof for com-pleteness sake. Suppose K is Σ: compact. We will first prove that

(1) H(κ) is closed under the power set.

Suppose aεH(κ). Then H(κ) satisfies the s-Π} formula

V17 3b


By s-Π} Reflection, ^(a)^c for some ceH(κ) so 0>(a)<ΞH(κ). For (1) we seethat κ = 30[ for some limit ordinal α. Suppose α<κ. Then H(κ) satisfies thes-Π} formula expressing:

Vjβ<oc 3/ [fun(/) Λ dom(/) = j3 + l

for y<β

for limit

Then 5-Πj Reflection gives a contradiction since one would have an aeH(κ)such that V(α)cα. This proves (i). To prove (ii) we need only see that K is reg-ular. Suppose /:α->/c where α<κ: and K = sup {/(/?): β<α}. We claim that<H(κ),e,/> does not satisfy s-Π} Reflection. In fact it does not even satisfyΣ Reflection and hence is not admissible, since it satisfies the Σ formula

but there can be no bound ξ<κ for the ordinals y. D

There are many characterizations of the class of weakly compact cardinalswhich fall out of our study. An admissible set A is strict-Til indescribable if (A,,R)satisfies s-Πj Reflection for every R^A. K is s-Π} indescribable iff <H(κ;),e>is s-Π} indescribable.

9.4 Theorem. An infinite cardinal K is weakly compact iff it is strict-Til inde-scribable.

Proof. Immediate from Theorem 4.7. D

An admissible set A satisfies Π} Reflection if for every Π} formula Φ(x l9...,xπ),A satisfies

Φ(x)-> 3α [Tran(α)Λx1,...,x I IeαΛ Φ(fl)(x)].

A is Π} indescribable if (A,.R) satisfies Π} Reflection for every R^A. K is Π}indescribable iff <ff(τc),e> is Π} indescribable. HF does not satisfy Π} Reflectionor, for that matter, Π^ Reflection since

HF^Vx3y (xey)

but no finite set can satisfy this sentence. Thus ω is certainly not Πj indescribable.We will see, however, that for K with cf(κ)>ω, s-Π} Reflection implies Π}Reflection and s-Π} indescribability implies Π} indescribability. The secret tounderstanding this and a number of other facts is contained in the followingsurprising result.


Let the language L (of L^J contain a distinguished binary relation symbol E.A well-founded L-structure is an L-structure 9W with E951 well-founded.

9.5 Theorem. Let A be an essentially uncountable Σί compact admissible set.Let T be a Σ1 theory of LA. // every A-fmite T0^T has a well-founded modelthen T has a well-founded model.

Proof. Recall that A is essentially uncountable iff every countable subset of Ais an element of A. We know that A satisfies s-Π} Reflection since A is Σί com-pact. The proof of this theorem is exactly like the proof that s-Π} Reflectionimplies Σt compactness, once we have the following definitions and lemma. D

We may assume that LA is a Skolem fragment which is Δ! on A. Call ans.v.p. Q) for LA well-founded if there is no infinite sequence <ίπ: rc<ω> of closedterms of LA such that (tn+ί Etn)e@ for all n<ω.

9.6 Lemma. Let A be an essentially uncountable admissible set.(i) There is a Π sentence φ(D) such that for all ί^<ΞA,

iff 2 is a well-founded s.v.p. for LA.

(ii) // 9JΪ is a well-founded Skolem structure for LA then the s.v.p. Q)^ given

by 50Ϊ is well-founded.(iii) // 3) is a well-founded s.v.p. for LA then 3) has a well-founded model.

Proof, (i) Since A is essentially uncountable, every sequence <ίπ: n<ω> of termsof LA is actually an element of LA. Thus the condition that 2 be well-foundedis expressed by a universal quantifier over A. The proof of (ii) is trivial. To prove(iii) let 2 be a well-founded s.v.p. By the Weak Completeness Theorem, Q) hasa model 2^. Let SR be the smallest submodel of 2 . Then

3JK3R! (LA).

By Exercise VII.2.14 every element of $R is denoted by a closed term of LA.Thus SPΪ is well-founded and a model of the sentences in 2. D

This lemma can also be used to prove a completeness theorem. See Exer-cise 9.11.

Theorem 9.5 explains why none of the explicitly described Σt compact setsgiven in § 7 were essentially uncountable. The conclusion of Theorem 9.5 is sostrong that it makes such sets very hard to find.

Our first use of Theorem 9.5 is to prove the results referred to above.

9.7 Theorem. Let K be a cardinal with d(κ)>ω.(i) // <H(κ),e,K> satisfies s-Π} Reflection then it satisfies Π} Reflection.

(ii) // K is s-Π} indescribable then K is Π} indescribable.


Proof. Part (ii) follows immediately from (i). To prove (i) let <#(*),£,#> satisfys-Π\ Reflection. By 9.3, κ = 3κ. Since H(κ) is closed under 9, the graph of 9is s-Π} on H(κ) so A = <#(κ;),e,^,K> also satisfies s-Π} Reflection and inparticular, is admissible. Thus the definition of V(α) is A-recursive andH(κ) = V(κ). Suppose

where ψ is first order but that for all α, α0

<V(α),e,ΛnV(α)>l=3S-ι^(S)

where α0 is large enough so that all parameters in ψ are in V(α0). Let T be thefollowing Σ! theory of LA:

KP + Power,

Infinitary diagram of <A,^>,

"c is an ordinal",

(c>j5) for all β<κ = o(A),

Every A-finite subset of T has a well-founded model; one simply interprets cas some large α<κ;. By Theorem 9.5, T has a well-founded model 501. Since itis well founded we can assume it is transitive. But then c501 is a real ordinal β Kand the last axiom of T implies that there is an S^V(κ) such that

D

Theorem 9.7 is really rather remarkable since if K is Σl compact thens-Π{=Σί(0>) and hence s-Π\

9.8 Corollary. // K is weakly compact and greater than ω then K is Mahlo.

Proof. Since K is weakly compact it is inaccessible. Since κ>ω, 9.7 appliesso K is Π} indescribable. Let C^κ be c.u.b. in K. We must prove that there isa λ<κ such that λ is inaccessible and λeC. Let A = <#(κ;),e,^,C> and con-sider the Π} sentence Φ true in A:

(2) VF Vα [F a function Λ dom(F) = α Λ Vβ<α (F(j?) is an ordinal)

(3) Vα

(4) Vα


The VF in (2) is the only second order quantifier; so Φ is Π} (but not s-Π}).By Π{ Reflection, there is transitive BeH(κ) such that Φ(B) holds. Letλ = o(B) = £n Ord. By (2), λ is a regular cardinal. By (3), A is a strong limit cardinal.By (4), λ is the sup of elements of C. Since C is closed, λεC. D

We can connect weakly compact cardinals with trees as follows. A tree2Γ = (Ύ, -<> is a κ-tree if the rank of 2Γ is K and for each α</c, y has lessthan K nodes of level α. A cardinal K has the tree property iff every κ>tree hasa branch, that is, a path of length K.

9.9 Theorem. Lei τc^ω fee inaccessible. Then K is weakly compact iff K has thetree property.

Proof. By Theorem 7.3 we see that, for K inaccessible, K is weakly compact ifffor every A of the form <f/(κ:),e,^,K>, every A-tree has a branch. Clearlyevery such A-tree is a τc-tree. Conversely, if 2Γ is a τc-tree then ?Γ is isomorphicto a tree on H(κ). Thus T is isomorphic to an A-tree for some expansion<H(κ),e,Ry of H(ιc). D

We summarize the characterizations of weakly compact cardinals obtainedin the above by means of the following statement. We say that K satisfiess-Π\(R) = Σί(R) uniformly in R if for every s-Π} formula Φ(vi9...,vΛ9P9R) thereis a Σ! formula φ(vl9...9vn,P9K) such that

<ff(ιc),e,^,Λ>M W [Φβ R)~φ(£, R)]

for all R^H(κ). (This is a different use of the word "uniformly".) We say that Kis weakly compact for Lκω(ι^7) if for every T^H(κ), if every subset of T0 ofpower < K has a well-founded model, then T has a well-founded model.

9.10 Theorem (Summary). Let K be an infinite cardinal. The following areequivalent:

(i) K is weakly compact for Lκω.(ii) K = ω or K is weakly compact for LK(0(iP~/).

(in) K is s-Π I indescribable.(iv) κ = ω or K is Π} indescribable.(v) K is inaccessible and has the tree property.

(vi) K is inaccessible and for every R^H(κ), <H(κr),e,/O has a proper ele-mentary end extension.

(vii) K is inaccessible and κ = ω or else for every R^H(κ), (H(κ),E,Ry hasa proper well-founded elementary end extension.

(viii) K is inaccessible and satisfies s-Π{(R) = Σ1(R), uniformly in R.

Proof. We list below the equivalences which have been already stated or elseare immediate consequences of earlier results.


(i) <=> (ϋ) (=> by 9.5; <= by just adding E to a theory not mentioning it),(i)«=>(iii) (by 9.4),

(iii)<=>(iv) (by 9.7 ϋ),(i) <=> (v) (by 9.3 and 9.9).

The following implications are trivial:(ii) => (vii) (trivial compactness argument),

(vii) => (vi) (trivial for κ>ω, the case κ = ω follows from compactness of

*~ωω)

The remaining implications (vi) => (v), and (iii) <=> (viii) are implicit in earlierresults or proofs, but we will make them explicit. To prove (vi) => (v), let^ = <Γ,<> be a fc-tree. We may assume T^κ. Let A = <#(/c),e,7;<,lev>.We can code up all of T, •<, lev into one R^H(κ) so, by assumption (vi), thereis a proper elementary end extension 33 = <£,E,T', -<,lev'> of A. Let beB bean ordinal, bφA. Let xeT satisfy

Then {yeA y^'x} is a branch through T. To prove (iii) => (viii), letΦ(x,R) = VS φ(x,R,S) be a s-Π{ formula involving an extra relation symbol R.For any R, <H(κ:),e,^,K> satisfies one of the below iff it satisfies all:

Φ(x,R),

VSφ(x,R,S),

3α [Tran(α) Λ x e α Λ V S ^ α φM(χ9 R, S)] (by (iii)),

The last line gives us a Σt formula i/φc,^0, R) equivalent to Φ(x, R) for all JR. Toprove (viii) => (iii), notice that since K is inaccessible, H (K) = V(κ) and thatA = <H(κ),e,^,IO is resolvable, since H(κ) = \Ja<κ V(α). Thus if A satisfiess-Π^Σi then A satisfies s-Πj Reflection by Corollary 4.9. D

Some further equivalences are given in the Exercises.

Looking at this summary, one can hardly fail to be struck by the equivalenceof notions coming to us from model theory, set theory and recursion theory.The summary is slightly misleading, however, in that it hides many important con-siderations which go into its proof, considerations including supervalidity prop-erties, resolvability, essential uncountability, A-trees, and so forth. It is only byunderstanding the earlier results involving these notions that one sees the variousforces at work in Theorem 9.10.


9.11. Let A be an essentially uncountable admissible set and let T be a s-Π} setof sentences of LA. Let

: φ is true in all well-founded models of T}.

Show that Cn^(Γ) is s-Π}.


9.12. Let K be weakly compact, κ>ω. Show that if C^κ is c.u.b. then thereis a Mahlo cardinal λ<κ, λeC.

9.13. Suppose that for every R, <H(κ:),e,^,R> satisfies s-Πj^Σi. Show that,6,^> satisfies s-Π}(R) = Σ1(R), uniformly in R.

9.14. The definition of weakly compact cardinal is often given in terms of Lκκ.We sketch a proof that the two definitions are equivalent. We define L^ to bethe smallest collecting containing L^ closed under — i, /\, \/ and

if φeLaoω and V is a set of variables occurring in φ then 3Vφ andare in L^.

For any K, Lκlc = LQOQOnH(κ:).(i) Prove that Lκκ consists of those φeL^ with <κ subformulas.

(ii) The following are sentences of Lωιωι :

Give a formal definition of 9JlNφ[s] for φe L^ so that these sentences expresswell-foundedness and essential uncountability, respectively.

(iii) Show that every subformula of a sentence of Lκκ has less than K freevariables.

(iv) Let K be inaccessible and let φe\-κκ. Show that if φ has a model thenit has one in H(κ). Let T^H(κ) be a set of sentences of Lκκ. Show that if Γhasa model then it has one of power K. [Modify the usual Lόwenheim-Skolem proof.]

(v) Let K be weakly compact for Lκω. Show that K is weakly compact for Lκκ.That is, let T^LKK be a set of sentences such that every T0^T, card(7^)<κ;,has a model. Show that T has a model. [For κ = ω this is trivial. For κ>ωapply 9.10 (vii) to <#(κ),e,^, Γ>. Use the fact that (iv) holds in H(κ) and hencein any elementary end extension. Also use the fact that H(κ) is closed undersequences of length < K;.]

9.15. Show that K is weakly compact iff κ-+(κ)\\ that is, iff for every partition

of [/c]2 = {{α,/?}: %<β<κ} into two sets, there is a subset C^κ such that[C]2^p. for ί = 0 or i = l. [It is probably easiest to prove that 9.10 (vii)implies κ-+(κ)\ and to prove ω-^(ω)l separately. To prove the other halfshow that κ-^(κ)2 implies 9.10 (v).]

9.16. The parts (vi) and (vii) of Theorem 9.10 do not have significant lightfaceversions; that is, versions without the "for all K" clause, as the following exampleof Kunen shows. Let K be the least inaccessible cardinal such that <H(τc),e> has


an elementary end extension. Show that it has no well-founded elementary endextension.

9.17 Notes. The "weakly" in weakly compact derives from the following. Acardinal K is strongly compact if (yR',H(κ)m,e,Ry is Σt compact for everystructure ΪR = <M,S> and every R^H(K)W, regardless of the size of 501 ascompared to K. We see no convincing argument that strongly compact cardinals>ω are a natural limit of existing cardinals and so we do not study them here.

The equivalence, for /c>ω, of weakly compact with Πj indescribability isdue to Hanf and Scott [1961]. Some authors take Π} indescribability as thedefinition of weakly compact, thus ruling out ω. This seems not only silly (torule out the one concrete example) but positively misleading since, as the proofof 9.7 shows, a number of considerations besides compactness are involved inthe proof of Πj indescribability. The equivalences (in 9.10) (i) <=> (ii) <=> (v)<=> (vi) <=> (vii) are all well known. Similarly for the other equivalences givenin the exercises. Corollary 9.8 and Exercise 9.12, which show that the firstweakly compact κ>ω is much larger than the first inaccessible cardinal, aredue to Hanf [1964]. The last equivalence ((i) <=> (viii)) in 9.10 is a uniform ver-sion of a result in Kunen [1968].

The remarkable argument that strongly compact cardinals exist "by analogywith ω" always reminds me of the goofang, described in The Book of ImaginaryBeings, by Jorge Luis Borges :

The yarns and tall tales of the lumber camps of Wisconsin andMinnesota include some singular creatures, in which, surely, no oneever believed...

There's another fish, the Goofang, that swims backward to keepthe water out of its eyes. It's described as "about the size of a sunfish,only much bigger".

Appendix

Nonstandard Compactness Argumentsand the Admissible Cover

One of the subjects we have not touched on in this book is applications ofinfinitary logic to constructing models of set theory and the relationship betweencompactness and forcing arguments. At one time we planned to include a chapteron these matters, but the book developed along other lines.

In this appendix we present one example of such a result because it leadsvery naturally to the admissible cover of a model 9JI of set theory. We want totreat this admissible set for two reasons. In the first place, it gives an exampleof an admissible set with urelements which has no counterpart in the theorywithout urelements, and it is as different from HYP^ as possible. Secondly, wepromised (in Barwise [1974]) to present the details of the construction of thisadmissible set in this book.

1. Compactness Arguments over Standard Modelsof Set Theory

Let A = <^4,e> be a countable transitive model of ZF. Then A is an admissibleset and, moreover, (A,jR) is admissible for every definable relation R. We cantherefore apply Completeness and Compactness to LA or L(A Λ), for any such R.There are many interesting results to be obtained in this way; we present onehere and refer the reader to Barwise [1971], Barwise [1974], Friedman [1973],Krivine-MacAloon [1973], Suzuki-Wilmers [1973], and Wilmers [1973] for otherexamples. We also refer the reader to Keisler [1973] for connections with forcing.

The axiom V = L asserts that every set is constructible.

1.1 Theorem. Let Ik be a countable transitive model of ZF. There is an endextension 33 = <(#,£> of A which is a model of ZF + V = L.

Proof. Let T be the theory of LA containing:ZF.The Infinitary diagram of A.

We need to see that Tu{V = L} has a model. If not, then

366 Appendix: Nonstandard Compactness Arguments and the Admissible Cover

SO

by the Extended Completeness Theorem of § III.5. Thus A is a model of theΣ! sentence expressing:

(1) 3Φ 3p \_p is a proof of (/\Φ)->(WL) where VxeΦ(xeZF or x isa member of the infinitary diagram)] .

This Σ! sentence contains no parameters. Now let α = o(A) and let A0 = L(α).Then AO is a model of ZF + V = L (it is the constructible sets in the model Aof ZF) and, interpreting Shoenfield's Lemma (Theorem V.8.1) in A, we have:Any Σ! sentence true in A is true in A0.

Thus the sentence (1) is also true in A0. But this means that there is somesubset TQ of the infinitary diagram of A0 such that

which is ridiculous since A0 itself is a model of 7^ + ZF + V = L . D

There are a number of extensions of the above which will strike the reader;most of these are covered by the version contained in Theorem 3.1 of Barwise[1971]. What is not so obvious is how to extend the result from standard modelsof set theory to nonstandard models. For if 31 = <,4,E> is a nonstandard modelof ZF then we have no guarantee that a "proof in the sense of $1 proves any-thing at all. What we need is a new admissible set intimately related to 91 whichwill allow us to carry out the above, and similar, proofs.

What is even less obvious is how to generalize results like Theorem 1.1 tothe uncountable. There are uncountable models of ZFC with no end extensionsatisfying V = L, assuming of course that ZFC is consistent. Is there an un-countable generalization of Theorem 1.1, involving consideration like /ιΣ(A),which explains more satisfactorily why the result holds in the countable case?The same question applies to all the results in Barwise [1971] and Barwise [1974].

2. The Admissible Cover and its Properties

In this section we will be considering models of set theory as basic structures overwhich we build admissible sets. Thus we denote such structures by 95i = <M,E>where E is binary. Recall, for xe9Jl, the definition

xE = {yeM\yEx}.

Let L contain only the relation symbol E; let L* = L(e, F) where F is a unaryfunction symbol. Let (f) be the axiom of L* given by

(t) Vp,x [x Ep^xE F(p)] Λ Vα

2. The Admissible Cover and its Properties 367

An admissible set (for L*), say A^^^; ,4,e,F), is a cover of 501 if A^ is amodel of (|). That is, A^ is a cover of 501 iff

F(x) = xE for

0 for xεA.

The point of the definition is pretty obvious, assuming that we are working inan admissible set A^ with 501 <£ AOT. A quantifier like Vx (x Ey-> . . .) is a boundedquantifier in the sense of L but it is not bounded, in general, in L*. Using theaxiom (|) however, it becomes equivalent to the bounded quantifier Vx e F(y) (...).

In this way every formula φ of L translates into a formula φ of L* with theproperties :

if φ is Δ0 (resp. Σx) is L then φ is Δ0 (resp. ΣJ in L*

We use these remarks below without comment.There are many admissible sets which cover a given structure 50Ϊ. For ex-

ample, if AaR = (SR; A,e) is admissible above 501 (in the sense of L(e)) then wecan define an A^-recursive F by

F(x) = {yeM\yEx}, xeM,

F(x) = 0, xφM,

and then (A^F) will be admissible in the sense of L(e, F) and will cover 501.These admissible sets are not tied closely enough to the intended interpretationof 501 for the applications we have in mind; they are too big with too manysubsets of $R. What we would like would be an admissible set A^ which covers9JΪ and whose only sets of urelements are the sets of the form pE for peSR.

2.1 Definition. Let 9W = <M,E> be an L-structure and let (Cov^ be the inter-section of all admissible sets which cover $R. More precisely,

where:

A = f } { B \ ( Ώ l ' , B , ε , F ) is admissible and covers

0 for aεA.

2.2 Theorem. // 5ΠΪ is a model of KP then Cov^ is admissible. Cov^ is calledthe admissible cover of 501.

Proof. Deferred to § 3. D


If we proved this theorem right now, the proof would look complicated andad hoc. What we shall do instead is to develop further properties of the admissiblecover in this section until, by the end of the section, we will know almost exactlywhat (CovjK looks like. This should make the proofs (in § 3) easier to follow.

The next property of the admissible cover suggests the main step in the proofof Theorem 2.2 and shows us that CovOT really lives in 901. (The corollaries ofTheorem 2.3 are easier to understand than 2.3 at a first reading.)

2.3 Theorem. Let 501 = <M,E> be a model of KP. There is a single valued notationsystem p projecting <CovOT into 501 satisfying the following equations (where we usex for the unique y such that p(x) = {y}, where 0, 1 denote the first two ordinalsin the sense of 50i and where < , ) is the ordered pair operation as defined in $01) :

(i) For xeM,

(ii) for αeCoVgpj, there is a yeM such that

ά = <l,y>

and yE = {x\xεa}.

Proof. Deferred to § 3, 3.1—3.7. D

Call a set a^Wl of urelements 9)1- finite if a — xE for some

2.4 Corollary. Let 501NKP and let α^50ϊ. Then a is Wl-finίte iffHence for any αeCov^, the support of a is $01- finite. In particular,

Proof. Let α^9W, αeCov^. Using the notation from 2.3,

where yE = {x\xea}. But α^9Jl so x = <0,x> for all xeα. Then we can define,inside the model 501, the following set by Σ Replacement, remembering thatΪRt-KP:

and then zE = a. The converse is trivial. D

Corollary 2.4 is very useful in compactness arguments involving Cov^, forit tells us that if Γ0 e (Cov^ is a set of infinitary sentences, then the set

{xeM x is mentioned in T0}

is 50ϊ-finite. Recall that x is the constant symbol used to denote x. D

2. The Admissible Cover and its Properties 369

We can use the projection from 2.3 to identify the pure sets in Cov^ and theordinals

2.5 Corollary. Let 2Rl=KP. Let A0 be the transitive set isomorphic toThe pure sets in Cov^ are exactly the sets in A0. In particular, o(<Covm) =

Proof. Since A0 is admissible (by the Truncation Lemma) it is closed under TCso it suffices to prove that every transitive set 0eA0 is in Cov^ in order to prove

since Cov^ is transitive. Let αeA 0 be transitive and let

where xen^y(9K). Since Cov^ is admissible, by 2.2, we can apply Theorem V.3.1in CoVjoj to see that ae(£ovm. To prove the other inclusion define the followingfunction by recursion in 9DΪ (more precisely, define it by Σ Recursion in KP andinterpret the result in 9JΪ):

(It is only the second clause which is relevant here but we'll use ' again later.)Let η: </^/(90ίl),£> = <A0,e> and consider the following diagram, whereDO = [a I a a pure set in Cov^} :

Pure part (Co v^) -=-> D0cM

We claim that, for every pure set αeCov^, (ά)'e^/(9Jl) and η((ά)') = a, whichwill conclude 2.5. The proof is by induction on e. First, ά = <l,x> wherexE = {b:bea}. But then (ά)' = z where

Thus (ά)E^iT?(W) by part of the induction hypothesis, and hence (a)'Computing η((ά)') we get

= { η ( ( ί ) ' ) \ b e a } .


The other part of the induction hypothesis states that η((b)') = b for bεa so weget

= α. D

Using 2.4 and 2.5 we can give a picture of (Cov^. The dotted line in 9W is the levelat which it becomes nonstandard (if it is nonstandard).

Wl

Fig. 2 A. A model $R of set theory next to its admissible cover

The projection given in 2.3 is ad hoc in that we could have used others. The nextfunction, by contrast, is canonical.

Let Ayn = (yR;A,€,F) be admissible and a cover of $R. A function * is ane-retraction of A^ onto Wl if x* is defined for every xeA^ and satisfies the fol-lowing equations:

(1)\p*=p for

\(a*)E = {b*\bea} for all

We can use the projection given by Theorem 2.3 to prove the following characteri-zation

2.6 Corollary. Let 9JINKP. (Cov^ has an ^-retraction into 9JΪ and it is the onlyadmissible set covering 9JΪ which has such an e-retraction.

Proof. The proof is an elaboration of the proof of Theorem 2.5. It is clear that anyadmissible set A^ covering 9Jί has a function * satisfying (1), simply by the secondrecursion theorem for KPU:

x* = y iff (x is an urelement Λ y = x) v

(x is a set and F(y) = {b* \ b e x}) .

The problem is that x* won t usually be defined for all x. Let us first show thatfor A^^Cov^, x* is defined for all x. Define ' just as in the proof of 2.5. We

2. The Admissible Cover and its Properties

claim that for all

371

(x)' is defined

(p)'=p for peM

((ά)')E = {(x)'\xea} for aeM .

This is proved by induction just as in 2.5 and shows that x* is defined for all xsince x*=(x)'. This proves that Cov^ has an E-retraction onto $R. Let A^ beany other cover

which has a totally defined e-retraction *. Let D be the domain (in the peculiarsense of Definition V.5.1; that is D = rng( )) of the notation system of Theorem 2.3and let

\p\ = the unique x such that x = p

for pεD. Thus | | maps D onto (Cov^. Define an A^-recursive function / fromASK into ®ί using * :

/(P)=<0,p>

See Fig. 2B at this point.

Fig.2B.

A simple proof by induction on e shows that f(x)eD and |/(x)|=x, for allxeAm. Thus A^^Cov^ so (Eovm = A<m since (Cov^ is the smallest admissibleset covering 9JΪ. D

The e-retraction * of Cov^ onto 9W is not one-one, of course, since (α*)* = α*but a*Φa, for any set αeCov^. Otherwise, though, it is far more natural andless ad hoc than the projection of Theorem 2.3. We saw in the proof of 2.6 how toreconstruct the projection from *.

Also note that * is (Cov^-recursive.For applications of (Cov^ we need two more properties of (Cov^. The first

tells us what Σ^ on Cov^ means in term of 9Jί.


2.7 Theorem. Let 9Jlt=KP. A relation S on SK is Σ! on Covw iff S is Σ+ inductiveon 9JΪ; that is, iff S is a section of Iφ where φ = φ(vl,...,υΛ>R+) is some Σ inductivedefinition (in the language L(R)) interpreted over 90Ϊ.

Proof. Deferred to 3.9. D

The last property we need relates the admissible covers of two differentmodels 3W, 31. Let 9K = <M,£>, 91 = <ΛΓ,F> where 9K^9ί. Note that $ϋϊ<Ξend9ίiif (Cov^cCov^. If 9K,9tNKP and 5ϋtcend9l then Cov^ c Cov^, as theconstruction in § 3 makes translucent.

2.8 Theorem. Let W,9ί^KP, SRcend$R.

only if

Proof. The translation φ->φ defined at the beginning of this section makesthe (<=) half of this theorem immediate. The converse follows from the considera-tions of the next section. D

3. An Interpretation of KPU in KP

The proofs of the theorems of §2 all involve interpreting the theory KPU ofL(e, F) in the theory KP of L, in the sense of § II.4, and then applying this inter-pretation to models 501 of K P.

The interpretation is the one suggested by the projection of Cov^ into 5DΪwhich we want to construct to prove Theorem 2.3 :

where

yE = {x\xEa} .

3.1 The Interpretation /. We are dealing with two separate set theories, KPformulated in L with E as a membership symbol and KPU + (|) formulated inL(e, F) with e as the membership symbol, so this must make things a bit confusingno matter what we do. In this subsection we want to work axiomatically withinKP so we use e for membership when we really ougth to use E, just because itseems the lesser of two evils. We use the usual notation for symbols defined in KP,symbols like 0, 1, <*,)>>, OP (for ordered pair).

3. An Interpretation of KPU in KP 373

Define predicates within KP by the following:

xE'y «-> N(x) Λ N(y) Λ (2nd(x)e2nd(y))

Set(x) <-> 3y[x = <!,}>> Λ Vze.y(N(z) v Set(z))]

OP(x) Λ 1 st(x) = 1 Λ ze2nd(x)

The predicates N, E',<ί and F' are defined by Δ0 formulas. The predicate Set isdefined, using the second recursion theorem, by a Σ t formula. We use these todefine our interpretation as follows, where L* = L(e, F) is considered as a one-sorted language with relation symbols U (for urelement), S (for set)

Symbol of L* Interpretation in KP under I

Vx Vx(N(x)vSet(xH...)

U(x) N(x)

S(x) Set(x)

xEy xE'y

xey x$y

F(x) F'(x)

3.2 Lemma. / is an interpretation of KPU + (f) in KP. That is, for each axiom φof KPU + ft), φ1 is a theorem of KP.

Proof. We run quickly through the axioms, beginning with (f). The interpretationof (t) reads

VxVy[N(x)Λ N(y)->(xE';y4-*x<f F'GO)] .

So suppose N(x)ΛlN(y). Let x = <0,x0>, y = <0,y0>. Then the following areequivalent :

xE>,

Extensionality: The interpretation of Extensionality asserts that if Set(x) andSet(y) and

Vz[(N(z) v


then x = y. Assume the three hypotheses. Let x = <l,M>, y = <l,ι;>. Then zS'xiff zew, z$y iff zεv. Since every zewui; satisfies N(z) v Set(z), u = v and hencex = y.

Foundation: Suppose there is an x such that

Set(x)Λ<pJ(x).

Choose such an x of least possible rank. Then since y£>z-+rk(y)<rk(z), we have

Vz[Set(z)Λz<ίx->-V(z)].

Pair: Suppose N(x)vSet(x) and N(y) v Set(y). Let

Then Set(z)Λ(w<ίz<->(w = x v w = >;)).

Union: Suppose Set(x). Let

by Δ0 Separation and let j; = <l,y0>.

Δ0 Separation: Let φ be a Δ0 formula of L(e, F). The formula φ1 is a Δ0 formulaof L* when L* is expanded by the symbols N, E', E, F'. Suppose Set(x), say x = <l,x0>.Let

y0 = {zex0\φI(z)}

by Δ0 Separation and let ^ = <1,^0) Then

zS'y iff z£x/\φ\z).

Δ0 Collection: Suppose φ(x,y) is Δ0, suppose Set(«) and that

Vx^a3y[N(y) v Set(y))Λ φI(x9y)'] .

Let α = <l,Λ0> so that the above becomes

Vxeα 0 3y [(N(y) v Set(y)) Λ φI(x9y)'] .

By Σ Reflection there is a b such that

Vxεa0lyeb[(N(y) v Set(y))Λ φ^y)]^ .

Let


by Δ0 Separation and let b1 = <l,fe0>. Then

y ) . D

3.3 The model 9ft-'. Let 9JΪNKP. Let TV, E', Set, δ, F be the predicates and func-tion defined in Wl by the corresponding symbols of KP. Then, letting 91 = <ΛΓ,F>we have

= 93α, say.

93<n is a model of KPU + (t), by 3.2. The structure 91 is isomorphic to 9W via themap x^<0,x>. If DOT is any admissible set covering 9Jί then

N,E',F' are D^-recursive, as is the isomorphism χπ-><0,x>

Set,

by the remarks at the beginning of § 2.

3.4 The model ^(SDΓ7). Let 9Kt=KP and let 93* be as defined in 3.3.is the largest well-founded substructure of 93 , before being identified with atransitive set this time. Notice that 1 7(93 ) is closed under F' since F'(x) is alwaysa set of urelements. Thus by the Truncation Lemma, ^7(93 ) is a well-foundedmodel of KPU + (t) If ID^ is admissible and covers $R then

Λf,E',Fare D^-recursive, as is the isomorphism x -^<0,x> and

Set n 7(93 ), g\ (Set n (»«)) are Dro-r. e.

The first follows from 3.2. The second line follows from Theorem V.3.1.

3.5 The admissible set isomorphic to iS*7(2R~J). Let ΪR^KP and let

where A is transitive (in V^). By 3.4, Agj is admissible and covers 9t. Let D^ beany admissible set which covers 9W. By 3.4 and Theorem V.3.1, there is a D^-recursive isomorphism of 9W and 91, and A is D^-r. e.

3.6 (Cov^ defined. Let 9Jlt=KP and let A^ be as in 3.5. The isomorphism ί'.yi^ 9Wextends to an isomorphism of Vgj onto V^ by :

carrying every transitive set in V^ onto a transitive set of Wm. In particular, A^is carried over to an isomorphic admissible set over 9M, say A^ =


where A' = {i(a)\aεA}. We claim that this A^ is the admissible cover of ΪR.It clearly is admissible and covers 501. Let DOT be admissible and cover $ϊt. Theisomorphism i can be defined by e-recursion in D^ and so A^ £ D . Thus A^is contained in every admissible set covering M so A^^Cov^. This provesTheorem 2.2.

3.7 The projection. It is clear from the above construction of (Cov^ that everyxeM is "denoted by" <0,x> and that every αeCov^ is denoted by

where yE is the set of "notations for" members of a. Turning this around gives thedesired projection.

We saw, early in §2, how to translate Σl formulas of L into Σ^ formulas ofL*, using the covering function. We now see how we can translate Σ! formulasof L* into "formulas" about 9JI.

3.8 Translation Lemma. Let 3yφ(x,y) be a Σ^ formula of L*, where φ is Δ0,and let ψ(x,z) be the interpretation

a formula of L Let 9Wl=KP, let a = o((Covm) and let xeCov^. Then

iff there is a β<a such that

Proof. Suppose Cov^ \= φ(x, y). Then

for some "standard ordinal" z of 50l~J. Thus, by Corollary 2.5,

for some β <α. The other half follows from 3.3—3.7. D

3.9 Proof of Theorem 2.7. A complete proof of Theorem 2.7 would include aproof of the following fact. The Σ+ inductive relations on 9W contain all Σ relationsand are closed under Λ , v , 3 and substitution by total Σ1 functions. This is provedjust as in Exercise VIA 18. But, given this, we have an easy proof of Theorem 2.7


from 3.8. Suppose R is Σi on Cov^, say

where φ is Δ0. Let θ(x) = Ordty)1 and define

Γ( U) = [x I M 1= 0(x) Λ Vy Ex U(y)} .

Then Γ is a Σ+ inductive definition over Wl and IΓ is the set of {β \ β < α =Furthermore

R(p) iff 3ze/Γ00ϊ^«0,p>,z))

so R is Σ+ inductive. The other half is trivial since any Σ+ inductive definition Γover ΪR transforms into a Σ+ inductive definition Γ over Cov^, and then, byGandy s Theorem, /f is Σ1 on Cov^. D

3.10 Proof of Theorem 2.8. Suppose Wênd9l and.aR^SR. Since aRcend9l,Covαrlcend(Cov9l so any Σ predicate true in Covîs true in Cov^,. In particular,the projections for (Cov^ and Cov^ agree on αeCov^, so we may write a forthis projection without fear of confusion. Suppose αeCov^ and

where φ is Δ0. Then there is a β<o((£ov<m) such that 91 is a model of

by 3.8. Hence 91 is a model of

(1) 3z [Ord(z)7 Λ [3χrk(3θ - z Λ φ(ά,

Since ΏK^, 9K is also a model of (1). By Lemma 3.2, 501 is a model of (Founda-tion)7 so 9Jί is a model of

3z [Ord(z) l Λ [3Xrk(y) = z Λ φ(ά, y)]7 Λ

[Vwe z -i 3y(rk(y) = w Λ φ(ά,

Pick such a "least" z. Since 9Wênd9l, this least z must be ^β in the sense of E,so it must be a standard ordinal. That is, there must be some y<o(Covw) suchthat γ = z. Thus 9K is a model of

so, by 3.8,

Thus



3.11. Prove that a relation S^JR (a model of KP) is s-Πj over 9Jί iff it is s-U\over Covggj.

3.12. Prove the following result of Aczel: Let S<^Wl (a countable model of KPU).Prove that S is s-Π\ on 9Jt iff S is Σ+ inductive on SR. [Combine 2.7, 3.9 andVII.3.1.]

3.13. Extend the construction above from models of KP to models of KPU.

4. Compactness Argumentsover Nonstandard Models of Set Theory

In this final section we want to show how the admissible cover can be used toextend results from standard to nonstandard models. We give two simple examples.

We know from Theorem VII.1.3 that no countable admissible set A is self-definable. An equivalent statement (in view of Exercise VIII.4.19(iv)) is that if Ais countable, admissible and

for some first order sentence φ(R) (possibly involving constants from A) thenthere is a proper end extension 95 of A such that

Phrased this way, the result holds for any countable model of KP, standard ornonstandard (or countable model of KPU by 3.13).

4.1 Theorem. Let 9W = <M,£> be a countable model of KP such that

for some sentence φ(R). There is a proper end extension 91 of 9M such that

Proof. Let A^A^^Cov^ and let LA be the admissible fragment given by A.Let x be a constant symbol in A used to denote x, for each xeM, and let T bethe following Σ: theory of LA :

diagram (9JI)

φ(R)

(all xeM) .

4. Compactness Arguments over Nonstandard Models of Set Theory 379

We can form the first sentences since A covers $R. We must prove that T is con-sistent. Since A is a countable admissible set, the Compactness Theorem impliesthat if T is not consistent, then there is a T0^ j, T0eA such that Γ0 is not con-sistent. By Corollary 2.4,

{xeM|x occurs in T0}

is 9Jl-finite. But then there is always some yeM left over to interpret c so T0 isconsistent. D

Our final result extends Theorem 1.1 from standard to nonstandard modelsof set theory.

4.2 Theorem. Let 9M = <M,E> be any countable model of ZF. There is an endextension 91 of Wl which is a model of ZF + V = L.

Proof. Let SR0 be the submodel of SR such that

M0 = {x e M 1 9W 1= "α is the first stable ordinal" Λ x e L(α)} .

Then by Shoenfield's Absoluteness Lemma (see § V.8)

Let A^Cov^R, A0 = CovaRo, so that Ao^A by Theorem 2.8. Let T be thetheory of LA containing

ZF

Vt;[ι;Ex<-»\/y6jeEι; = y], for all xeM.

The proof now proceeds exactly like the proof of Theorem 1.1 except that themodel of Γ0 is not 9W0 but the model 90^ where

M i = {x e M 1 9PΪ 1= "x is constructive" } .

The reason for using 3Jlί9 rather than 9Jt0> is that SO^-^SDl (parameters are notallowed in Shoenfield's Lemma) but the statement of Theorem 2.8 requires -<^One could equally well improve 2.8. D


4.3. Prove that both assumptions ΪR|=KP and $R is countable are needed forTheorem 4.1.

4.4. Show that if ZF is consistent then there is an uncountable model of ZFCwhich has no end extension satisfying ZF + V = L.

References

Aczel, P.1970 Implicit and inductive definability (abstract). J. Symbolic Logic 35, 599 (1970).

Aczel, P., Richter, W.1973 Inductive definitions and reflecting properties of admissible ordinals. In: J. E. Fenstad and

P. Hinman, eds., Generalized Recursion Theory, pp. 301—381. Amsterdam: North-Holland1973.

Barwise, J.1967 Infmitary Logic and Admissible Sets. Ph. D. Thesis. Stanford, CA: Stanford Univ. 1967.1968 The Syntax and Semantics of Infmitary Logic, editor (Lecture Notes in Math., Vol. 72).

Berlin-Heidelberg-New York: Springer 1968.1969 a Infmitary Logic and Admissible Sets. J. Symbolic Logic 34, 226—252 (1969).1969 b Applications of Strict Π} predicates to Infmitary Logic, J. Symbolic Logic 34, 409—423 (1969).1971 Infmitary Methods in the Model Theory of Set Theory, Logic Colloquium 69, pp. 53—66.

Amsterdam: North-Holland 1971.1973 a Back and Forth Through Infmitary Logic, Studies in Model Theory, pp. 5—34, edited by

M. Morley. MAA Studies in Mathematics 1973.1973 b Abstract Logics and L^, Ann. of Math. Logic 4, 309—340 (1973).1973 c A Preservation Theorem for Interpretations, Proceedings of the Cambridge Logic Conference,

pp. 618—621 (Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.1974 a Admissible Sets over Models of Set Theory, Generalized Recursion Theory, pp. 97—122.

Amsterdam: North-Holland 1974.1974b Mostowski's Collapsing Function and the Closed Unbounded Filter. Fund. Math. 82, 95—103

(1974).

Barwise, J., Fisher, E.1970 The Shoenfield Absoluteness Lemma. Israel J. Math. 8, 329—339 (1970).

Barwise, J., Gandy, R., Moschovakis, Y.1971 The next admissible set. J. Symbolic Logic 36, 108—120 (1971).

Barwise, J., Kunen, K.1971 Hanf numbers for fragments of Lnω. Israel J. Math. 10, 306—320 (1971).

Chang, C. C.1964 Some new results in definability. Bull. Amer. Math. Soc. 70, 808—813 (1964).1968 Some remarks on the model theory of infmitary languages. In: Barwise [1968], pp. 36—63.

Chang, C. C., Moschovakis, Y. N.1968 On Σ}-relations on special models. Notices Amer. Math. Soc. 15, 934 (1968).1970 The Suslin-Kleene theorem for Vκ with cofmality (κ) = ω. Pacific J. Math. 35, 565—569 (1970).

Church, A.1938 The constructive second number class. Bull. Amer. Math. Soc. 44, 224—232 (1938).

Church, A., Kleene, S. C.1937 Formal definitions in the theory of ordinal numbers. Fund. Math. 28, 11—21 (1937).

References 381

Cutland, N.1972 Σ!-compactness and ultraproducts. J. Symbolic Logic 37, 668—672 (1972).

Devlin, K. V.1973 Aspects of constructibility (Lecture Notes in Math., Vol. 354). Berlin-Heidelberg-New York:

Springer 1973.

Dickmann, M. A.1970 Model theory of infmitary languages (Aarhus University Lecture Notes, series No. 20) 1970.

Ehrenfeucht, A., Kreisel, G.1966 Strong Models of Arithmetic. Bull. Acad. Polon. Sci. 14, 107—110 (1966).

Ehrenfeucht, A., Mostowski, A.1956 Models admitting automorphisms. Fund. Math. 43, 50—63 (1956).

Enderton, H.1972 A Mathematical Introduction to Logic. New York and London: Academic Press 1972.

Engeler, E.1961 Unendliche Formeln in der Modell-Theorie. Z. Math. Logik Grundlagen Math. 7, 154—160

(1961).

Feferman, S.1965 Some applications of forcing and generic sets. Fund. Math. 56, 325—345 (1965).1968 Lectures on Proof Theory, Proceedings of the Summer School in Logic, Leeds 1967 (Lecture

Notes in Math., Vol. 70). Berlin-Heidelberg-New York: Springer 1968.1974 Some predicative set theories, Axiomatic Set Theory II. Amer. Math. Soc., Providence, R.I.

(1974).

Feferman, S., Kreisel, G.1966 Persistent and invariant formulas relative to theories of higher order. Bull. Amer. Math. Soc.

72, 480—485 (1966).

Flum, J.1972 Hanf numbers and well-ordering numbers. Arch. Math. Logik Grundlagenforsch. 15,164—178

(1972).

Friedman, H.1973 Countable models of set theories, Cambridge Summer School in Math. Logic, pp. 539—573

(Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.

Friedman, H., Jensen, R.1968 Note on admissible ordinals. In: Barwise [1968], pp. 77—79.

Gaifman, H.1970 On local arithmetical functions and their application for constructing models of Peano's

arithmetic, Mathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, ed.), pp.105—121. Amsterdam: North-Holland 1970.

Gale, D., Stewart, F. M.1953 Infinite games of perfect information. Ann. Math. Studies 28, 245—266 (1953).

Gandy, R. O.1974 Inductive definitions, Generalized Recursion Theory, pp. 265—300. Amsterdam: North-

Holland 1974.1975 Basic functions, Axiomatic Set TheoryvII. Amer. Math. Soc., Providence, R. I. (1974).

Gandy, R., Kriesel, G., Tait, W.1960 Set existence. Bull. Acad. Polon. Ser. Sci. Math. Astron. Phys. 8, 577—582 (1960).

Garland, S. J.1972 Generalized interpolation theorems. J. Symbolic Logic 37, 343—351 (1972).

382 References

Gόdel, K.1930 Die Vollstandigkeit der Axiome des logischen Funktionenkalkϋls. Monatsh. Math. Phys. 37,

349—360 (1930).1931 Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I.

Monatsh. Math. Phys. 38, 173—198 (1931).1939 Consistency proof for the generalized continuum hypothesis. Proc. Natl. Acad. Sci. U.S.A.

25, 220—224 (1939).1940 The consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with

the Axioms of Set Theory. Annals Math. Studies 3 (Princeton, N. J., Princeton Univ. Press).

Gordon, C.1970 Comparisons between some generalizations of recursion theory. Compositio Math. 22,

333—346 (1970).1971 Finitistically computable functions and relations on an abstract structure. J. Symbolic Logic

36, 704 (1971).

Grilliot, T.1971 Inductive definitions and computability. Trans. Amer. Math. Soc. 158, 309—317 (1971).1972 Omitting types; applications to recursion theory. J. Symbolic Logic 37, 81—89 (1972).

Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C.1958 The classical and ω-complete arithmetic. J. Symbolic Logic 23, 188—206 (1958).1961 Definability of sets in models of axiomatic theories. Bull. Acad. Polon. Sci. Ser. Sci. Math.

Astronom. Phys. 9, 163—167 (1961).

Hanf, W.1964 Incompactness in languages with infinitely long expressions. Fund. Math. LIII, 309—324

(1964).

Hanf, W., Scott, D.1961 Classifying inaccessible cardinals. Notices Amer. Math. Soc. 8, 445 (1961).

Harrison, J.

1966 Recursive pseudo-wellorderings. Ph. D.Thesis. Stanford, CA: Stanford Univ.

Henkin, L.1949 The completeness of the first-order predicate calculus. J. Symbolic Logic 14, 159—166 (1949).1954 A generalization of the concept of ω-consistency. J. Symbolic Logic 19, 183—196 (1954).1957 A generalization of the concept of ω-completeness. J. Symbolic Logic 22, 1—14 (1957).

Jech,T.1973 Some combinatorial problems concerning uncountable cardinals. Annals of Math. Logic 5,

165—198 (1973).

Jensen, R. B.1972 The fine structure of the constructible hierarchy. Annals of Math. Logic 4, 229-308 (1972).

Jensen, R., Karp, C.1971 Primitive recursive set functions, Axiomatic Set Theory. Proceedings of the set theory in-

stitute, UCLA, 1967, Amer. Math. Soc. (1971), Providence, R.I.

Karp, C.1962 Independence proofs in predicate logic with infinitely long expressions. J. Symbolic Logic

27, 171—188 (1962).1964 Languages with Expressions of Infinite Length. Amsterdam: North-Holland 1964.1965 Finite quantifier equivalence, The Theory of Models. (Edited by J. Addison, L. Henkin and

A. Tarski) pp. 407—412. Amsterdam: North-Holland 1965.1967 Nonaxiomatizability results for infmitary systems. J. Symbolic Logic 32, 367—384 (1967).1968 An algebraic proof of the Barwise compactness theorem. In: Barwise [1968] pp. 80—95.1972 From countable to cofmality ω in infmitary model theory. J. Symbolic Logic 37, 430 (1972).

References 383

Keisler, H. J.1965 Finite approximations of infinitely long formulas, The Theory of Models (edited by J. Addison,

L. Henkin and A. Tarski) pp. 158—169. Amsterdam: North-Holland 1965.1968 Formulas with linearly ordered quantifiers. In: Barwise [1968], pp. 96—130.1971 Model Theory for Infinitary Logic. Amsterdam: North-Holland.1973 Forcing and the omitting types theorem, Studies in Model Theory, MAA (1973).

Kino, A., Takeuti, G.1962 On Hierarchies of predicates of ordinal numbers. J. Math. Soc. Japan 14, 199—232 (1962).

Kleene, S. C.1938 On notations for ordinal numbers. J. Symbolic Logic 3, 150—155 (1938).1944 On the forms of the predicates in the theory of constructive ordinals. Amer. J. Math. 66,

41—58 (1944).1955 a On the forms of the predicates in the theory of constructive ordinals [second paper]. Amer.

J. Math. 77, 405—428 (1955).1955 b Arithmetical predicates and function quantifiers. Trans. Amer. Math. Soc. 79, 312—340 (1955).1955 c Hierarchies of number theoretic predicates. Bull. Amer. Math. Soc. 61, 193—213 (1955).1959 a Quantification of number theoretic functions. Compositio Math. 14, 23—40 (1959).1959 b Recursive functionals and quantifiers of finite types I. Trans. Amer. Math. Soc. 91, 1—52

(1959).

Kreisel, G.1961 Set theoretic problems suggested by the notion of potential totality. In: Infinitistic Methods,

pp. 103—140. Oxford: Pergamon 1961.1962 The axiom of choice and the class of hyperarithmetic functions. Indag. Math. 24, 307—319

(1962).1965 Model theoretic invariants: applications to recursive and hyperarithmetic operatons. In:

J. Addison et al. (eds.) Theory of Models (Proceedings of the 1963 Berkeley Symposium)pp. 190—205. Amsterdam: North-Holland 1965.

1968 Choice of infmitary language by means of definability criteria. In: Barwise [1968] pp. 139—151.1971 Some reasons for generalizing recursion theory. In: R. O. Gandy and C. E. M. Yates (eds.)

Logic Colloquium 69, pp.139—198. Amsterdam: North-Holland 1971.

Kreisel, G., Krivine, J. L.1967 Elements of mathematical logic. Amsterdam: North-Holland 1967.

Kreisel, G., Sacks, G. E.1965 Metarecursive sets. J. Symbolic Logic 30, 318—338 (1965).

Kripke, S.1964 Transfmite recursion on admissible ordinals, I, II (abstracts). J. Symbolic Logic 29, 161—162

(1964).

Krivine, J. L., McAloon, K.1973 Some true unprovable formulas for set theory. Proceedings of the Bertrand Russell Logic

Conference, Denmark 1971, pp. 332—341. Leeds 1973.

Kueker, D.1968 Definability, automorphisms, and infmitary languages. In: Barwise [1968] pp. 152—165.1970 Generalized interpolation and definability. Annals Math. Logic 1, 423—468 (1970).1972 Lόwenheim-Skolem and interpolation theorems in infmitary languages. Bull. Amer. Math.

Soc. 78, 211—215(1972).

Kunen, K.1968 Implicit definability and infmitary languages. J. Symbolic Logic 33, 446—451 (1968).

Levy, A.1963 Transfmite computability, Notices Amer. Math. Soc. 10, 286 (1963).1965 A hierarchy of formulas in set theory. Memoir Amer. Math. Soc. 57 (1965).

Lopez-Escobar, E.1965 An interpolation theorem for denumerably long sentences. Fund. Math. LVII, 253—272 (1965).1966 On definable well-orderings. Fund. Math. LVIX, 13—21 and 299—300 (1966).

384 References

Machover, M.1961 The theory of transfmite recursion. Bull. Amer. Math. Soc. 67, 575—578 (1961).

Makkai, M.1964 On a generalization of a theorem of E. W. Beth. Acta Math. Acad. Sci. Hungarica 15, 227—235

(1964).1969 An application of a method of Smullyan to logics on admissible sets. Bull, de Γ Academic

Polon. des Sci., Ser. Math. 17, 341—346 (1969).1973 Global defmibility theory in Lωιω. Bull. Amer. Math. Soc. 79, 916—921 (1973).1975 Applications of a result on weak definability theory in Lωjω (to appear).

Malitz, J.1965 Problems in the model theory of infinite languages, Ph. D. Thesis. Berkeley: Univ. of Cali-

fornia 1965.1971 Infmitary analogues of theorems from first order model theory. J. Symbolic Logic 36, 216—228

(1971).

Montague, R.1968 Recursion theory as a branch of model theory. In: B. van Rootselaar et al. (eds.) Logic,

Methodology and Philosophy of Science III (Proceedings of the 1967 Congress) pp. 63—86.Amsterdam: North-Holland 1968.

Morley, M.1965 Omitting classes of elements, The Theory of Models (J. W. Addison, L. Henkin and A. Tarski,

eds.) pp. 265—273. Amsterdam: North-Holland 1965.1967 The Hanf number for ω-logic (abstract). J. Symbolic Logic 32, 437 (1967).1970 The number of countable models. J. Symbolic Logic 35, 14—18 (1970).

Morley, M., Morley, V.1967 The Hanf number for κ-logic (abstract). Notices Amer. Math. Soc. 14, 556 (1967).

Moschovakis, Y. N.1969a Abstract first order computability I. Trans. Amer. Math. Soc. 138, 427—464 (1969).1969 b Abstract first order computability II. Trans. Amer. Math. Soc. 138, 465—504 (1969).1970 The Suslin-Kleene theorem for countable structures. Duke Math. J. 37, 341—352 (1970).1971 The game quantifier, Proc. Amer. Math. Soc. 31, 245—250 (1971).1974 Elementary Induction on Abstract Structures. Amsterdam: North-Holland 1974.

Mostowski, A.1949 An undecidable arithmetical statement. Fund. Math. 36, 143—164 (1949).1961 Formal system of analysis based on an infinitistic rule of proof, Infinitistic Methods. Warsaw

1961.

Nadel, M.1971 Model Theory in Admissible Sets, Ph. D. Thesis. Univ. of Wisconsin 1971.1972 a Some Lόwenheim-Skolem results for admissible sets. Israel J. Math. 12, 427 (1972).1972b An application of set theory to model theory. Israel J. Math. 11, 386 (1972).1974 Scott sentences and admissible sets. Annals of Math. Logic 7, 267—294 (1974).

Nerode, A.1957 General topology and partial recursive functions, Summaries of talks presented at the Summer

Institute for Symbolic Logic, pp. 247—251. Cornell Univ. 1957.

Orey, S.1956 On ω-consistency and related properties. J. Symbolic Logic 21, 246—252 (1956).

Platek, R.1966 Foundations of recursion theory, Doctoral Dissertation and Supplement. Stanford, CA:

Stanford Univ. 1966.

Reyes, G. E.1970 Local definability theory. Annals of Math. Logic 1, 95—137 (1970).

Richter, W.1971 Recursively Mahlo ordinals and inductive definitions. In: R. O. Gandy and C. E. M. Yates

(eds.) Logic Colloquium '69, pp. 273—288. Amsterdam: North-Holland 1971.

References 385

Rogers, H. Jr.1967 Theory of recursive functions and effective computability. New York: McGraw-Hill 1967.

Scott, D.1964 Invariant Borel Sets. Fund. Math. 56, 117—128 (1964).1965 Logic with denumerably long formulas and finite strings of quantifiers, The Theory of Models,

pp. 329—341. Amsterdam: North-Holland 1965.

Shoenfield, J. R.1961 The problem of predicativity, Essays on the Foundations of Mathematics, pp. 132—139.

Amsterdam: North-Holland 1961.1967 Mathematical logic. Reading, Mass.: Addison-Wesley 1967.

Simpson, S. G.1974 Degree Theory on Admissible Ordinals, Generalized Recursion Theory, pp. 165—194. Amster-

dam: North-Holland 1974.

Smullyan, R.1963 A unifying principle in quantification theory. Proc. Nat. Acad. Sci. 49, 828—832 (1963).1965 A unifying principle in quantification theory, The Theory of Models, edited by J. Addison,

L. Henkin and A. Tarski, pp. 443—434. Amsterdam: North-Holland 1965.

Spector, C.1955 Recursive wellorderings. J. Symbolic Logic 20, 151—163 (1955).1960 Hyperarithmetical quantifiers. Fund. Math. 48, 313—320 (1960).1961 Inductively defined sets of natural numbers. In: Infmitistic methods, pp. 97—102. New York:

Pergamon 1961.

Suzuki, Y., Wilmers, G.1973 Nonstandard models for set theory, Proceedings of the Bertrand Russell Memorial Logic

Conference, Denmark 1971, pp. 278—314. Leeds 1973.

Svenonius, L.1965 On the denumerable models of theories with extra predicates, The Theory of Models, pp.

376—389. Amsterdam: North-Holland 1965.

Takeuti, G.1960 On the recursive functions of ordinal numbers. J. Math. Soc. Japan 12, 119—128 (1960).1965 Recursive functions and arithmetic functions of ordinal numbers. In: Y. Bar-Hillel (ed.)

Logic, Methodology and Philosophy of Science II (Proceedings of the 1964 Congress) pp.179—196. Amsterdam: North-Holland 1965.

Tague, T.1959 Predicates recursive in a type-2 object and Kleene hierarchies. Comment. Math. Univ. St.

PaulS, 97—117(1959).1964 On the partial recursive functions of ordinal numbers. J. Math. Soc. Japan 16, 1—31 (1964).

Vaught, R.1973 Descriptive set theory in Lωιω, Cambridge Summer School in Math. Logic, pp. 574—598

(Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.

Ville, F.1974 More on set existence, Generalized Recursion Theory, pp. 195—208. Amsterdam: North-

Holland 1974.

Wilmers, G.1973 An N\-standard model for ZF which is an element of the minimal model, Proceedings of the

Bertrand Russell Memorial Logic Conference, Denmark 1971, pp. 215—326. Leeds 1973.

Index of Notation

Chapter I

nolst(x), 2nd(x)

TC(α)sp(α)clpse(b)

Chapter II

A, IBo(A)

O(M)Wf

Chapter III

sub(φ)

the successor of α = αu {a} 13the first and second coordinatesof x 14φ relativized to w 15transitive closure of a 24support of a 29the collapse of b 31end extension relation 34

the universe of sets on M 42typical admissible sets 43the least ordinal not in A 45hereditarily finite sets on $R 46{aeWM\Carά(ΊC(a)}<κ} 52Gδdel's operations plus some58, 68

Chapter IV

HYP(A)

58sets constructed from a by stage α58smallest admissible above 901 60the ordinal of 901well founded part 73

subformulas of φ 81true in all ΪR-structures 88provable using 2R-rule 89class of proper infmitary formulas81Lαo ω nA 97union of countable LA 127

smallest admissible B with AelB115

Π j , Σ j

Δ}

Chapter V

Σ-Satπ

<L

τβ

α*σβ

-<!Σj, U2,

Chapter VI

Γ, Γφ

/Γ, Iφ

II/ΊIΣ+

universal and existential secondorder 116both Π} and Σ} 116

satisfaction for n-ary Σ formulas155rank functions 161the canonical well ordering of L162least nonrecursive ordinal 173βih admissible ordinal 174projection of α 174βth stable ordinal 178Σ! submodel relation 177second order quantifier forms189

inductive definitions 197, 200smallest fixed point 198, 200closure ordinal of Γ 200R-positive Σ formulas 205Moschovakis closure ordinalof <m 231

Chapter VII

F3 x φ Skolem function symbol 263Tskoiem Skolem theory 263-< (LA) elementary submodel for LA

268/ιΣ(A), /ι(A) least ordinals not pinned down

2713a(ιc) Card(Fα(M)) if Card(M) = κ

276[X]n n-element subsets of X 282= p partially isomorphic 292

Index of Notation 387

= α L^ equivalent up to α 296 Chapter VIIIovp, canonical Scott sentence for 3ft s-Π}, s-Σj, s-Δ} strict second order forms

297,301 316—317σyjι,5 α characteristic of s in ΪR 298 K!,K2,K3 Konig principles 321—326sr(ΪR) Scott rank of W 300 ^ the real power set 346

Subject Index

Absolute 34, 35Absoluteness 151— Lemma, Shoenfield-Levy 189

, Shoenfield's 195— Principle, Levy 76, 77, 243Absolute version of a predicate 243Abstract Kleene Theorem 231, 241Acceptable structure 229

, almost 229Aczel, P. 230,380— [1970] 230

Aczel-Richter [1973] 187Richter-Aczel [1974] 205

Admissible fragment 97— ordinal 45

, nonprojectible 174, projectible 174, recursively hyperinaccessible, — inaccessible 176,— Mahlo 187

— set above 902 43, original definition of 11

over 9ft 43, pure 44, recursively listed 161, 164, resolvable 163, 329, j-Δ} resolvable 332, self-definable 257, 328, —, strongly 257, Σ! compact 257, Σ! complete 260, validity 260

Admit (K, λ), to 277A-fmite 153

a.i.d. 333

Almost all 353Analysis 143—, model of 143A-r.e. 153A-recursive 153Assignment 82Axioms A1-A7 92

Back and forth property 292Barwise, J. 116, 126, 380— [1967] 4,102,291— [1968] 333,352— [1969] 75, 102, 105, 109,187— [1969 a] 4— [1969 b] 4, 333,355— [1971] 365,366— [1973] 102,105, 303— [1974] 9, 33, 365,366

Barwise-Fisher [1970] 196Barwise-Gandy-Moschovakis [1971] 116,126,220, 321Barwise-Kunen [1971] 270, 276, 291

Beta 39Beth's Theorem 104, 129Borges,J.L. 364

185 Boundedness Theorem 234Branch 311Brouwer-Konig Infinity Lemma 314

Canonical Scott Sentence 297, 301Theory 300

— structure 86Cardinal, α- 187—, inaccessible 347—, Mahlo 355—, Σ! compact 357—, strongly compact 364—, weakly compact 356Cartesian product 12Chang, C. C. 127, 303, 380— [1964] 127— [1968] 262, 303

Chang-Keisler [1973] 283, 350Chang-Makkai-Reyes Theorem 127, 131Chang-Moschovakis [1970] 241

Characteristic of s in 9ft, α- 298Church, A. 380— [1938] 2

Church-Kleene [1937] 2Church's thesis 153

Subject Index 389

Closed in K 352Closure ordinal, Moschovakis 231

ofΣ + 210— Theorem 231Coding scheme 229Cohn, P. M. 303Coinductive 212— ΦΓ 204Collapsing 39— function 29, 30— Lemma 32,41,53,54Collection, Δ0 10, 11—, full 39—,Σ 17Combination Lemma 217Compactness Theorem 101

, Barwise 99, 102, 144, Kreisel 2, stable 187

Compact, Σt 328Completeness Theorem, Barwise 99, 102

, extended 100for Arbitrary Skolem Fragments, weak

266for countable fragments, weak 95for Lωιω, karp 95

,SR- 89,ω- 87,92

Complete Σx set 328— strict- Π} set 328— theory 110Conjunction 81— rule 97Consistency machine 134— property 85, 109Constructible 58— from 58— sets 3, 29, 57

with urelements 57Cotype, α-recursive 236Countable 14Cover of a model 367

model, admissible 367Craig, W. 1, 103— [1957] 105c.u.b. filter 353c.u.b. in K 352Conjunction rule 93Cutland, N. 381

Decidable structure 111Definability operator, general Δ 335Definable, invariantly 147—, semi-invariantly 147Definition, good Σί 61— Σ! 61Devlin, K. V. 381— [1973] 187

Dickmann, M. A. 381Disjunction 81Divisible part of a group 117, 204Downward Lδwenheim-Skolem-Tarski

Theorem 269

Ehrenfeucht, A. 381Ehrenfeucht-Kreisel [1966] 310Ehrenfeucht-Mostowski 278

Elementary substructure, LA- 268Enderton, H. 54, 381— [1972] 54End extension 34Engeler, E. 381Erdόs-Rado Theorem 282, 285, 286Essentially uncountable 259Extension 34Extensionality 10

Feferman, S. 381— [1968] 50— [1974] 37

Feferman-Kreisel [1966] 35Field 14Finite 14— approximations 251—, notions of 174Fisher, E. 380

Barwise-Fisher [1970] 196Fixed point 212

of an inductive definition,largest 204

, Φ- 203, Σ+ 205, ί(ίjf) 205

Flum, J. 381Forcing 4, 146Formula, V3- 191—, α-fmite 235—, atomic 79—, finite 79—, game, closed 245—, —, open 242—,—, —,ofan 251—, - -, —, recursive 242—, infinitary 81—, orderly 64—,Π 15—, proper infinitary 81—, R-monotone 200—, R-positive 156— Σ 15

'-,Σι 15—, strict-11} 316—, strict-Σj 316—, termed- 64

390 Subject Index

Formulas, Δ0 10—, first order, coextended 50—, , extended 50Foundation 10Fragment 84Friedman, H. 107, 381— [1973] 109, 137, 365

Friedman-Jensen [1968] 144Function 14— symbol, Σ 21

Henkin argument 86Hereditarily finite 46Holmes, O. W. 5Hyperarithmetic 2, 60— sets 149Hyper elementary 212—*, see extended hyperelementary 214—, extended 214— Selection Theorem 240— substitution 221

Gaifman, H. 381Gale, D. 381

Gale-Stewart 246Gale-Stewart Theorem 246Game, infinite two-person 244Gandy, R. O. 72,116,126,211,381— [1974] 211— [1975] 58, 72

Barwise-Gandy-Moschovakis [1971] 220, 321Gandy-Kreisel-Tait Theorem 116Candy's Theorem 208, 211, 377

, second half of 210Garland, S. J. 381Generalization 93Gόdel, K. 1,3, 8, 54, 57, 105, 382— [1939] 3, 62— [1940] 62Godel numbers 154GodeΓs operations 63Goofang 364Gordon, C. 382— [1970] 50, 51Grilliot, T. 382— [1972] 116Group, Krfree abelian 303— P 297—, reduced abelian p- 297Grzegorczyk, A. 382

Grzegorczyk, Mostowski and Ryll-Nardzew-ski [1959] 2Grzegorczyk, Mostowski and Ryll-Nardzew-ski[1961] 149

Hanf, W. 291,382— [1964] 262, 364

Hanf-Scott [1961] 364Hanf number 276

for Σ! theories 291for single sentences 290, 291of second order logic 351

Harrison,J. 112, 127,382Heatherton Rock Cakes 69Henkin, L. 87, 382Henkin [1949] 269— [1954] 92— [1957] 92

i.i.d. 333Implicit definition of an ordinal 314Indescribable, Πj 358—, strict- π; 315,358Indiscernibles 279—, /c-variable 283— over U 279

C7, /c-variable 283Induction over e, proof by 24

TC, proof by 26Inductive 212—*, see extended inductive 214— definition 197, 210

, absoluteness of 207, αth-iterate of an 198, closure ordinal of an 200, — ordinal of Σ + 218, extended 214, —, closure ordinal of an 215, first order positive 211, fixed point of an 197, — point of an, smallest 197

given by a formula 200, nonmonotonic 205

— — on an essentially uncountable admissibleset 262

, picture of an 199— definitions, closure properties of 221

, non-monotonic 4—, extended 214—,Φ- 203— relation, Σ+ 205

, Σ(t#) 205Infinitary proof 96Infinity 38Initial substructure 34Inner submodel 56Internal set 113, 115Interpolation Theorem 103, 129, 253, 261Interpretation 54, 56— , transitive e- 56, 57, 59Invariant definability 333— relation 301

Jech, T. 33, 382— [1973] 33

Subject Index 391

Jensen, R. B. 72, 186, 187, 381, 382— [1972] 58, 62, 72

Friedman-Jensen [1968] 144Jensen-Karp [1972] 196

Karp, C. 382— [1965] 293— [1967] 262— [1968] 9, 352

Jensen-Karp [1972] 196Karp's Theorem 294Keisler, H. J. 84, 383— [1965] 250— [1971] 84, 86, 87, 91, 92, 103, 270, 277, 283,

291—[1973] 365

Chang-Keisler [1973] 283, 350Kino, A. 383

Takeuti-Kino [1962] 196Kleene, S. C. 1, 3, 49, 201, 380, 383— [1938] 2— [1955] 2

Church-Kleene [1937] 2Kleene's Theorem 2— T-predicate 166Kleene structure, uniform 241Kδnig Infinity Lemma 311— Principle, first 321

, second 322, third 323

— Principles 311KP 3,8,11,239KPU 3,8,239KPU+ 11KPU, axioms of 10,11—, intuitive set theory in 11—, nonstandard model of 72Kreisel, G. 116,255,381,383— [1959] 11

— [1965] 11

— [1968] 262

— [1971] 9

Ehrenfeucht-Kreisel [1966] 310Feferman-Kreisel [1966] 35Kreisel-Sacks [1965] 2, 168

Kreisel Basis Theorem 315— Compactness Theorem 2Kripke, S. 3, 8, 37, 54, 126, 173, 177, 187, 383— [1963] 196— [1964] 3, 11Krivine, J. L. 383

Krivine-McAloon [1973] 365Kueker, D. 33, 303, 383— [1968] 127, 303— [1972] 33Kunen, K. 380, 383— [1968] 262, 333, 364

Barwise-Kunen [1971] 276,291Kunen's example 121, 228, 229

Language 79Levy, A. 383— [1965] 10, 11, 53, 54, 72, 77, 196Lδwenheim-Skolem Theorem, upward 276,277Logic 5—, axioms of 901- 88— 9N- 88, 241—, ω- 88Lopez-Escobar, E. 383— [1965] 105— [1966] 109, 276Lyndon Interpolation Theorem 203

Machover, M. 384aW-admissible 45Mahlo cardinal 360Makkai, M. 127, 129, 143, 241, 384— [1964] 127— [1973] 129, 254— [1975] 310Malitz, J. 384— [1971] 262McAloon, K. 383

Krivine-McAloon [1973] 365Metarecursion theory 2, 168Metatheory 76Model, see structure 138— Existence Theorem 84, 86, 95, 109, 269

, extended 87, 90, 93, weak 266

Module, Noetherian 325Modus Ponens 93Monotonic operator 197Montague, R. 49, 384— [1968] 49de Morgan, A. 158Morley, M. 384— [1965] 109, 291— [1967] 291Morley, V. 384Moschovakis, Y. N. 49, 116, 126, 173, 212,

221,230,242,253,380,384— [1969a] 49— [1971] 242, 253— [1974] 173, 187, 203, 217, 221, 224, 229,

230, 232, 239, 240, 241, 242, 253— [1975] 205

Barwise-Gandy-Moschovakis [1971]220, 321Chang-Moschovakis [1970] 241

Mostowski, A. 30, 381, 382, 384— [1949] 33, 41— [1961] 41

Ehrenfeucht-Mostowski 278

392 Subject Index

Grzegorczyk, Mostowski and Ryll-Nard-zewski [1959] 2Grzegorczyk, Mostowski and Ryll-Nard-zewski [1961] 149

Nadel, M. 310, 384— [1971] 303, 310— [1974] 303, 310NadeΓs Basis Theorem 306Natural number 13Nerode, A. 384Nerode's Theorem 334Norm 232—, inductive 232Normal Form Lemma 318— function 354Notation system 2, 168, 368

, domain of a 168for IHFsR 223, 227

, univalent 172Nyberg, A. 230, 241, 334, 351

Omitting Types Theorem 91Operation, Σ 23— symbol, substitutable 70Operator, general weak metarecursive 337Ordered n -tuples 13— pair 12Ordinal 13— addition 29—, admissible 60—, least nonrecursive 60— multiplication 29—, Π implicit 323, 339—,5-Σj implicit 323,341Orey, S. 87, 384— [1956] 92

Pair 10,11Pairing function 220Parametrization 154— of extended inductive relations 214

first oder definable relations 171inductive relations 213, 235projections 171the class of A-r.e. relations 154

Partial isomorphism 292Partially isomorphic structures 292Peano arithmetic 117, 126, 130, 137, 143, 144,

146, 158, 239Perfect set argument 110, 133, 137Persistent 34, 35Pinning down ordinals 105, 270Πj reflection 187Platek, R. 3, 8, 11, 54, 126, 173, 177, 187, 384— [1965] 3, 196— [1966] 11Power set axiom 40

Predicate, absolute 33—, co-extended Σ} 117— Δ 21—, extended Πl 117— of functions, r.e. 312

integers, strict-Πj 313—, persistent 33Predicates, Δ0 14Prerequisites 1Prewellordering 164— Theorem 232Principle of parsimony 8,13Projectible 168Projectum 174—, admissibility of 184Proof, LA 97Pure part 44— set 44PZF 37

Quantifier rank of a formula 296

Rank function 29? <_ 161

Recipe 69Recursion along well-founded Relations 158—, definition by Σ 26—, Δ predicates defined by 28— Theorem, second 156, 157, 159

, ordinary 49Recursive ordinal 2Recursively saturated 74

structure 138— Σ} saturated 348Reducibility, Δ definable 335—, , truth table 335—, truth table 334—, Turing 334—, weak metarecursive 337Reduction Theorem for A-r.e. sets 165

inductive sets 240Πj sets 167, 168

Reflection Lemma, Π2 185—, Π{ 358—,Σ 11,16,17— 5-ΠJ 210—, strict- ΠJ 322,328Relation 14— symbol, Δ 19Relativization 15Replacement, Σ 17—, strong Σ 18Representability 149Representable, strongly 146, 147, 148—, weakly 146, 147, 148Resolution 163— of Πj sets 167, 168

Subject Index 393

Ressayre, J.-P. 143Retraction, e 370Reyes, G. E. 384— [1968] 127Richter, W. 384

Aczel-Richter [1973] 187Richter-Aczel [1974] 205

Rigid structure 302Rogers Jr., H. 385— [1967] 336Rule, $R- 89—, ω- 88Rules R1-R3 93Ryll-Nardzewski, C. 382

Grzegorczyk, Mostowski and Ryll-Nard-zewski [1959] 2Grzegorczyk, Mostowski and Ryll-Nard-zewski [1961] 149

Sacks, G. E. 144, 151,383Kreisel-Sacks [1965] 2, 168

Sacks school 338Satisfaction 82Schlipf, J. 139, 143, 241Scott, D. 303, 382, 385— [1964] 303— [1965] 303

Hanf-Scott [1961] 364Scott rank of a structure 300Scott's Theorem 301Search computable 49, 50

, semi- 49, 50, 51Second order arithmetic 143Section 203Semantics 78— ofL^ 82Separation, Δ 17—,Δ0 10,11—, full 38— Σ! 38, 39, 41— Theorem for co-A-r.e. sets 165

coinductive sets 240Πj sets 167, 168

Shoenfield, J. R. 1, 54, 196, 385— [1961] 196— [1967] 7, 48, 54, 56, 116, 314, 315s.i.i.d. 333Simpson, S. G. 177, 339, 385— [1974] 177Skolem fragment 263

with constants 263— function symbol 263— V3 normal form 192— structure 263— theory 263Smullyan, R. 385

Special form 268— set of sentences 132Spector, C. 385— [1959] 201— [1961] 230Spector class 4Splitting 337Stable 3— ordinal 178

, β- 179, the first 189

Stavi, J. 310Stewart, F. M. 381

Gale-Stewart 246Strategy 244Stretching Theorem 279Structure 81— for L* 10— W- "88—, resplendent 351Subformula 81Substitutable function, see substitutable opera-

tion symbol 70— operation, see substitutable operation

symbol 70Subtree 311Superstable 353Supertransitive 346Supervalidity property 265Support function 24, 29Suzuki, Y. 385

Suzuki-Wilmers [1973] 365Svenonius, L. 248, 253, 385— [1965] 242Svenonius' Theorem 248, 352s.v.p., see supervalidity property 265Syntax 78—, axioms on 79— ofLw ω 81

Table 1 14— 2 22— 3 23— 4 29— 5 254Tague, T. 385— [1964] 3Tait, W. 116,381Takeuti, G. 3, 54, 383, 385— [1960] 3, 187— [1961] 3

Takeuti-Kino [1962] 196Tarski 262— criterion for -<j 180Term 79— basic 84ί-formula, see termed-formula 64Theorem of LA 94

394 Subject Index

Torsion part of a group 117Transitive closure 24Translation Lemma 376Tree 343— A- 344— argument, see perfect set argument 110—, branch thru a 344—, complete binary 110—, full binary 311— ic- 361—, path thru a 344— property 361Truncation Lemma 73, 75Two cardinal model of type (K, λ) 277

models 288Theorem 277

, Morley's 277Type, α-recursive 235

UCLA Logic year 211Unbounded in K 352Uniformization Theorem for A-r.e. sets 165

Πj sets 167Uniformly equivalent 330Union 10,11

— of chain lemma 267, 268Urelement 7, 10, 69

Validity property 92, 93, smallest 93

Variables, convention on 10, 13, 16, 157Vaught, R. 385— [1973] 253Ville, F. 75,385— [1974] 116

Weakly compact cardinal 357Weak second-order logic 51Well founded 39

part 73Well-ordering 41— definable 105— ofL 162Wilmers, G. 385— [1973] 365

Suzuki-Wilmers [1973] 365

Zermelo 9ZF 7,8,239

Perspectivesin Mathematical Logic

In recent years interconnections between differentlines of research in mathematical logic and linkswith other branches of mathematics haveproliferated. The subject is now both rich andvaried. This series, organized by the Ω-Group,aims to provide, as it were, maps or guides to thiscomplex terrain as seen from various angles. Thegroup is not committed to any particularphilosophical program. Nevertheless, the criticaldiscussion which each planned book undergoesensures that it will represent a coherent line ofthought; and that, by developing certain themes,it will be of greater interest than a mere assemblageof results and techniques.

The books in the series differ in level: some areintroductory, some highly specialized. They alsodiffer in scope, some offering a wide view of anarea while others present more specialized topics.Each book is, at its own level, reasonably self-contained. Although no book depends on anotheras prerequisite, authors are encouraged to fit theirbook in with other planned volumes—sometimesdeliberately seeking coverage of the same materialfrom different points of view.

Among the next volumes to appear will be:

P. Hinman, Inductive Definitions and Higher TypesD.S. Scott and P.Kraus, Languages and StructureA. Levy, Basic Set Theory.

Some Lecture Notes in Logic

Lecture Notes in Mathematics

6. H. Hermes, Term Logic with Choice Operator70. Proceedings of the Summer School in Logic, Leeds, 1967. Ed: M.H. Lob72. The Syntax and Semantics of Infmitary Languages. Editor: J. Barwise95. A. S. Troelstra, Principles of Intuitionism

120. O. Siefkes, Buchi's Monadic Second Order Successor Arithmetic212. B. Scarpellini, Proof Theory and Intuitionistic Systems217. T. J. Jech, Lectures in Set Theory. With Particular Emphasis on the

Method of Forcing223. U. Feigner, Models of ZF-Set Theory255. Conference in Mathematical Logic—London '70. Editor: W. Hodges306. H. Luckhardt, Extensional Gόdel Functional Interpretation.

A Consistency Proof of Classical Analysis328. J.R. Bύchi, D. Siefkes, The Monadic Second Order Theory of All

Countable Ordinals. Decidable Theories II337. Cambridge Summer School in Mathematical Logic. Eds: A.R.D. Mathias,

H. Rogers344. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis.

Editor: A.S. Troelstra354. K.J. Devlin, Aspects of Constructibility405. K.J. Devlin, H. Johnsbraten, The Souslin Problem447. S. A. Toledo, Tableau Systems for First Order Number Theory and

Certain Higher Order Theories450. Algebra und Logic. Editor: J.N. Crossley453. Logic Colloquium. Editor: R. Parikh454. J. Hirschfeld, W.H. Wheeler, Forcing, Arithmetic, and Division Rings492. D.W. Kueker, Infmitary Logic: In Memoriam Carol Karp498. Model Theory and Algebra. A Memorial Tribute to Abraham Robinson.

Edited by D.H. Saracino, V.B. Weispfenning499. Logic Conference, Kiel 1974. Edited by G.H. Mύller, A. Oberschelp,

K. Potthoff500. Proof Theory Symposium, Kiel 1974. Edited by J. Diller, G.H. Mύller

Lecture Notes in Computer Science

37. C. Bόhm, λ-Calculus and Computer Science Theory

3-540-07451-10-387-07451-1

perspectives in mathematical logic - 01 - admissible sets and structures. an approach to...

Documents