On order-types of models ofarithmetic

by

Andrey I. Bovykin

A thesis submitted to the University of Birminghamfor the degree of Ph.D. in the Faculty of Science

School of Mathematics and Statistics,The University of Birmingham,

Edgbaston, Birmingham, B15 2TT.

13 April, 2000

Synopsis

In this thesis we study a range of questions related to the order structure ofmodels of first-order Peano Arithmetic.

In Chapter 1 we give necessary definitions and describe the current stateof the subject in the literature survey.

In Chapter 2 we study first properties of order-types of models of PA,give examples and place first restrictions on what the order-type of a modelof PA can be.

In Chapter 3 we study models generated by indiscernibles and prove thatthe model generated by a set of indiscernibles ordered as a dense linear order(C,<) is order-embeddable into C<Q.

In Chapter 4 we study a wide variety of questions associated with inter-pretability in a model of PA. In section 4.1 we prove that there is only onedense linear order interpreted in Ω |= PA, namely Q(Ω). In section 4.2 weexpress the order-type of all inner models in terms of the (<, ·)-structure ofthe outer model. In section 4.3 we introduce and study the notion of self-similarity. Section 4.4 briefly studies connections between models of PA andmodels of ZFC and their mutual interpretability.

Chapter 5 introduces a class of canonical orders of models of PA andmakes the first attempt to study it.

In Chapter 6 we prove that every model of cardinality less than λ has 2λ

order-types of elementary end-extensions of cardinality λ, that there are 2ω1

ω1-like models of PA and that (assuming ♦) there are 2ω1 ω1-like models ofPA whose multiplicative reducts are pairwise non-isomorphic.

Sections 7.1 and 7.3 present our first attempt to solve Friedman’s problemin the resplendent case. Section 7.2 connects this theme with the notion ofarithmetic saturation. Also, we obtain a consequence of arithmetic saturationfor automorphism groups.

The thesis concludes with a list of questions intended to guide and inspirefuture research.

Acknowledgements

I would like to thank my supervisor, Dr. Richard W. Kaye, for his guidanceand patience during my long years of studying in Birmingham.

Also, I thank Professor V.G. Kanovej and Professor H. Kotlarski for math-ematical discussions during their visits to Birmingham.

Contents

Introduction 1

1 A survey of the subject 81.1 Notation and useful facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Density, cells, cofinalities, weight 172.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Cofinalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Models generated by indiscernibles 343.1 Analysis of a single term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Separated and interpenetrating terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Embedding into (C, 0)<Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Inner models 454.1 Interpreted linear orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Order-types of inner models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Self-similar models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Inside models of ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Canonical orders 70

6 Counting 756.1 Order-types of cardinality λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 ω1-like models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7 Around Friedman’s Problem 827.1 Resplendency and coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Digression: arithmetic saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.3 Initial segments in resplendent models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Conclusion 94What is next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94List of questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 103

Introduction

Why study models of arithmetic

There are many reasons to study models of arithmetic coming from differentareas of man’s interest: Foundations of Mathematics, Philosophy, Method-ology of Science, and Mathematics itself. Let me list some of these reasonsand some historical landmarks in the development of the subject.

The history of the subject started with the discovery in the 1930s byGodel and Skolem of nonstandard models. Since then many people studiedmodels of arithmetic having different reasons in mind. For some, it was theusual mathematical curiosity to find out more about these ‘beautiful’ and‘interesting’ structures. For others, they were a tool to prove facts abouttheories of arithmetic, especially first-order Peano Arithmetic, the theorythat has been traditionally the main object of arithmetical research (andwhich is the main theory considered in this thesis). The discovery by Parisand Harrington in 1977 of a ‘mathematical’ statement (PH) independentof PA was a landmark. Other independent statements: “Hercules and theHydra”, “for all m the Goodstein sequence for m starting at 2 eventuallyhits zero” (by Kirby and Paris), and the statement (KM) (due to Kanamoriand McAloon) were discovered a few years later (for discussion see [14], pp.206-222). All these independence results were first proved using models ofPA. However, no general method of obtaining such results was developedand research in this direction has stalled. There is an old proof of Godel’sIncompleteness Theorems due to Kreisel that uses models of PA and theArithmetised Completeness Theorem [34]. Recently H. Kotlarski invented anew proof of Godel’s Theorem, also relying only on models of arithmetic.

Another boost to the subject was given by the discovery of recursive sat-uration and its interconnections with notions of resplendency, satisfactionclasses and automorphism groups, which produced a beautiful and harmo-nious theory. At the moment “Models of Peano Arithmetic” is an establishedarea of mathematical research profiting from methods developed in otherparts of Model Theory and exporting its results and its language to other

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areas.Nevertheless, there is a reason which distinguishes study of models of set

theories, models of arithmetic, etc, from study of other areas of mathematics.This is the idea of exploring the most basic and fundamental notions of thewhole mathematics appealing to man’s desire of the ‘search for eternal truth’.The intuitive collection N = |, ||, |||, ||||, . . . is given to us by Nature(or nature of human mind?) and we, all of us, are challenged to unveil itsmysteries. The most striking wonder of N is inadequacy of classical formallogic (with its “absolute infinity” quantifiers) to describe it as demonstratedby Godel’s Theorems.

At this stage, revolutionary-minded people split from the mass of mathe-maticians and tried to change the logic and the whole paradigm. Intuitionistand constructivist worlds are beautiful and complex and their set N is verydifferent from ours. More traditional mathematicians stayed behind in theirold (still exciting) world trying to answer new questions about incomplete-ness and Truth in terms of their old notions and beliefs. This thesis belongsto that old tradition. Throughout the thesis our logic is classical, our infinitesets are “actually infinite” and quantifiers in our formulas are understood inthe classical way.

Peano Arithmetic is a strong theory. On one hand, it represents all re-cursive functions hence giving rise to Incompleteness. On the other hand, itformalises the Arithmetised Completeness Theorem (see later) thus allowingus construct models of consistent theories ‘inside’ models of PA. (Example:if M |= PA + ConZFC then there is V |= ZFC strongly interpreted inside M .The whole of mathematics is expressed by one formula inside a model ofarithmetic!)

The study of PA and models of PA should inevitably become connectedto the study of its relatives: other ‘foundational’ theories, such as ZFC (andGNB), NF, KP, Vopenka’s set theory, nth order arithmetic, etc. There aretwo sorts of connection. Firstly, all foundational theories introduce the notionof a natural number. Different foundational theories may disagree aboutabstract sets and about cardinalities and about properties of their collectionsof ‘all’ subsets of natural numbers but they all agree that natural numbersshould satisfy PA. Secondly, PA is one of those foundational theories anda lot of mathematics can be developed in PA or interpreted in models ofPA. However, many people who studied PA believe that PA is more reliablethan other theories. Even if some foundational theories such as ZFC (whichdominates current research) are proved inconsistent (which would not damagemuch of mathematics though) PA will probably stay intact.

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Order-types

Why study order-types? The first reason is that, given a theory T withdefinable linear order, the question “What are the order-types of models ofT?” is one of the first natural questions to ask and can be understood by achild! Surely, studying order-types of models is more natural than studyingsome strange ad hoc structural properties.

Secondly, studying order-types enhances our geometrical vision of mod-els allowing us to “see” and “touch” them. This geometrical vision is lackingin the study of some of PA’s relatives, such as NF.

Thirdly, linearly ordered sets are structures of great importance intheir own right, not only throughout mathematics but, hopefully (in theremote future), in the rest of science. There are linear orders (which may beused for “modelling” “reality”) other than R and Q !

Finally, there are practical mathematical reasons: the order of a modelof PA is deeply interconnected with the rest of model’s structure. Goodillustrations are Pabion’s Theorem (see below) and Theorem 55 (page 84).

All countable nonstandard models of PA are order-isomorphic, so throughoutmost of the thesis our models will be uncountable. In the countable case,the possibility of enumerating the model gave great variety of proofs, resultsand techniques. Nothing similar exists so far in the uncountable case.

Friedman’s problem

Another natural question to ask would be “Do classes of order-types of non-standard models of different completions of PA coincide?” This question isthe 14th Friedman’s problem from his “One hundred and two problems inMathematical Logic” [7].

No-one seems to know what the answer to this question should be. Onone hand, we can not think of a good reason for two extensions of PA tohave models of different order-types, and the countable case confirms it. Onthe other hand, there is a theory Th(N) = ϕ | N |= ϕ that looks differentfrom other extensions of PA: only Th(N) has a model of order-type (ω,<)(the standard model), every model of Th(N) has no nonstandard definablepoints. Can it be a source of a counterexample?

Chapter 7 is devoted to my recent attempts to solve the problem. As-suming resplendency and some coding, the order-type of a model can beexpanded to a model of a given completion of PA.

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Resplendency

Resplendent models are the most popular kind of models considered in thisthesis. A model M is called resplendent if it realizes all Σ1

1 statements Φ(a)consistent with Th(M,a). Resplendent models, unlike saturated models,exist in all cardinalities. Very often we shall notice that resplendent modelsbehave similarly to countable models. That is why I believe that solvingFriedman’s problem in the resplendent case would be a realistic enterprise,though I did not succeed at the first try.

Classification

Non-structure theory was developed by Saharon Shelah in the 1970s. Theproblem of classifying all models of a given theory led to the development ofstability (and unstability) theory. Theory T is called unstable if there is aformula ϕ(x, y) and a model A |= T containing tuples ai, i < ω such that forall i < j < ω, A |= ϕ(ai, aj).

Obviously, PA has a formula x < y which defines linear order, hencePA is unstable. Many ‘non-structure’ results have been proved for unstabletheories. The main two are:

Theorem 1 (Shelah)If T is unstable, λ > |T |, then there are 2λ non-isomorphic models of T ofcardinality λ.

Theorem 2 (Shelah)Suppose T1 has infinite models, L(T ) ⊆ L(T1), T ⊂ T1, T is unstable, λ >|T1|. Then there are 2λ models of T1 of cardinality λ whose L(T )-reducts arepairwise non-isomorphic.

So, in the case T1 = PA, T = the theory of linear order, we can conclude thatthere are 2λ pairwise non-isomorphic order-types of models of PA in everyuncountable cardinality λ. This suggests non-classifiability of order-types ofuncountable models of PA.

In section 6.1. we prove related results. One of them is: every modelM |=PA, card(M) < λ has 2λ order non-isomorphic elementary end-extensions ofcardinality λ, and our proof is elementary, without requiring the delicatemachinery developed by Shelah.

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Cells

Let M |= PA. For every λ ≤ card(M), M is naturally partitioned intoλ-cells:

Cellλ(x) = y | there are at most λ points between x and y.

Questions rise about the structure of cells and whether we can make every cell‘regular’, e.g. order-isomorphic to the saturated discrete order of cardinalityλ or whether cells themselves can be ordered as the saturated dense linearorder. I call such ‘regular’ order-types canonical.

The idea arose from the desire to have more examples of concrete order-types to work with. You can find the whole story in chapters 2 and 5.

Note that I gave the most general definition of a canonical order buthave not proved the existence theorem (it turned out to be a hard problem).Nevertheless, I proved several particular cases of it, enough to make thenotion interesting.

We also ask how many models of each canonical order-type exist at all. Ispent some time trying to prove that there are many non-isomorphic modelsof the saturated order-type (a false statement) until H.Kotlarski refered meto Pabion’s Theorem.

Pabion’s Phenomenon

If the order-type of M is λ-saturated then M is λ-saturated.

What happens if we consider a limit of ω saturated elementary end-extensionsof the saturated model? Surprisingly, in contrast with Pabion’s Theorem,there will exist non-isomorphic elementarily equivalent models of PA of thisorder-type (see Chapter 5).

Interpretations

In a sense, all model theory is about interpretations. Constructing a modelof T is constructing an interpretation of a model of T inside a model of yourfavorite set theory. Also, most of Set Theory is about constructing modelsof set theories inside other models of set theories.

But not only models of set theory can act as the domain of our discoursewhere our mathematics takes place. Any theory formalising the Complete-ness Theorem is suitable for this purpose. PA is one of those theories.

Arithmetised Completeness Theorem

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Let Ω |= PA. Let T be a set of (Godel numbers of) sentences in L de-finable in Ω (possibly with parameters). Let Ω |= Con(T ), that is Ω |=¬∃xProv(x, pT → 0 = 1q). Then there is a model N |= T and two formulasdomN(x) and SatN(x, y) such that the domain of N is x ∈ Ω | domN(x)and for every ϕ(u1, . . . , un) ∈ L and x1, . . . , xn ∈ x ∈ Ω | domN(x)

N |= ϕ(x1, . . . , xn) ⇔ Ω |= SatN(pϕq, 〈x1, . . . , xn〉).

This kind of interpretation is called a strong interpretation and is studiedin Chapter 4.

If T = PA, then we obtain models of PA strongly interpreted insidemodels of PA + Con(PA) (inner models). The main theorem of Chapter 4(Theorem 29) says that there can be only one order-type of inner modelsinside a given model Ω, namely

Ω +Q(Ω) · (Ω∗ + Ω),

where Q(Ω) is the set of fractions of Ω and the order-multiplication · is lexi-cographic. Later we introduce a new class of models, self-similar models. IfΩ |= PA + Con(PA) then Ω is self-similar if and only if it is order-isomorphicto its inner models. Many previously studied models (saturated, resplendent,inner) turned out to be self-similar.

If T = ZFC then we ask which models Ω |= PA contain a model U |=ZFC strongly interpreted in Ω such that ωU

∼= Ω. This and other questionsconcerning mutual interpretability of models of PA and models of ZFC arediscussed in section 4.4.

Although we do not study interpretations of models of other theories inmodels of PA, I would like to mention them here. Let G be a group stronglyinterpreted in Ω. G is called Ω-finite if x ∈ Ω | domG(x) is bounded in Ω.

Question 1What do groups strongly interpreted in Ω look like? What do Ω-finite groupslook like? Which properties of N-finite (truely finite) groups do they inherit?

Arithmetic saturation

Since introduction of the notion of recursive saturation and the study of itsinterconnections with automorphism groups, resplendency and satisfactionrelations, there has been demand for other notions of saturation. The onlyother successful notion introduced was arithmetic saturation. I encounteredit by chance while trying to solve Friedman’s problem. Understanding themechanism of arithmetic saturation led further to constructing a nice auto-morphism of arithmetically saturated models. You can read the whole storyin sections 7.1 and 7.2.

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About this thesis

This thesis is based on my MPhil Thesis, which occupies chapters 2 and 3.The topics which I decided to exclude completely from the scope of the thesisare Presburger Arithmetic and Borel relations due to lack of material.

The thesis contains 42 Questions. Some questions (4, 25, 35, 40) may turnout to be easy, some (6, 10, 17, 26, 37) point at the major directions of futureresearch, one (33) is a relatively ungrounded conjecture, some are particularcases of others. Altogether, questions have to be read as an undetachablepart of the text.

Credits

A lot of ideas in this thesis arose during discussions with my supervisor.Retrospectively, it is usually hard to evaluate the extent of their influence ona particular given proof. Though some ideas were directly suggested and Itried them and they worked. An example is the idea to study interpretationsof linear orders in models of PA and internalise the proof of “every countabledense linear order without end-points is isomorphic to Q”, which eventuallygave rise to Chapter 4.

Originally, Chapter 3 arose from discussions with V.G. Kanovej who sug-gested to look at how Skolem terms fit together in order to investigate un-countable Borel models of arithmetic. The first versions of section 3.1. andCorollary 19 were written by V.G. Kanovej and presented to me as a gift.However, being not interested in Borel model theory at the time, I insteadproved a general theorem about embeddability of EM(C, p) into C<Q (The-orem 20) only to discover later that this was done by Christine Charrettonand Maurice Pouzet in 1983 (see [4]).

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Chapter 1

A survey of the subject

1.1 Notation and useful facts

1. Model TheoryFor definitions and basic facts about first-order logic, consistency, com-plete theories, Robinson’s joint consistency test, models, elementary di-agrams, compactness, completeness, Lowenheim-Skolem Theorem, ele-mentary chains lemma, arithmetical hierarchy of formulas, Σn-elementarychains lemma, ultrafilters, ultraproducts, types, back-and-forth method,universality, homogeneity, λ-saturation, saturation, saturated linear or-ders, the notion of interpretation of one structure in another, I referthe reader to Chang and Keisler [3] or Hodges [11].

2. Set TheoryFor basic set-theoretic knowledge: formalization of ZFC, ordinals, car-dinals, regularity, stationary sets, clubs, normal functions, Fodor’s The-orem, independence of (CH), (GCH), ♦, ♦λ see Kunen [20] or Jech [12].I also assume the standard pseudophilosophical discussion around theGodel’s incompleteness phenomenon in the context of ZFC to be knownto the reader. We shall also need the following fact ([12], p.59):

If λ is regular then λ can be partitioned into λ stationary subsets.

3. ChoiceAs always in Model Theory, we assume (AC).

4. Cardinal arithmeticI will use the following facts from cardinal arithmetic: Konig’s Theo-rem: if κ is infinite and cf(κ) ≤ λ then κλ > κ, Bukovsky–HechlerTheorem: if κ is singular and there is γ < κ such that for all δ such

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that γ ≤ δ < κ, 2δ = 2γ, then 2γ = 2κ, and a related fact: if λ isa limit cardinal, limi∈µ λi = λ then

∏i∈µ 2λi = 2λ. See Jech [12], pp.

42-52.

5. ArithmeticFor the discussion of PA, models of PA, standard coding devices, repre-sentation of recursive functions and recursive sets, formulas ConPA andProvPA(x), Godel’s Theorems, Tarski’s undefinability of Truth theoremin models of PA, Standard System, len(x), term(x), form(x), axiom(x)-formulas in PA and the rest of what we consider ‘standard’ notation,see [14].

6. DefinabilityLet M |= PA. If M |= ∀ y ∃! x ϕ(y, x, a) then we introduce a newfunction symbol t(y, a) and the axiom ∀x ∀y(x = t(y, a)↔ ϕ(y, x, a)).The term t(y, a) is called a Skolem term or a Skolem function.

An element b ∈ M is definable in M over A iff M |= (b = t(a)) forsome a ∈ A and some Skolem term t(y) of M . ClM(A) will denote theset of all elements definable in M over A. We know that ClM(A) is anelementary submodel of M (see [14], pp.91-96).

A subset A is called definable in M if there is a formula θ(x, a) witha ∈ M such that b ∈ A ⇔ M |= θ(b, a). We say that A is M-finite ifA is definable and bounded in M .

7. Standard SystemLet M |= PA. Then N is embedded in M as an initial segment. TheStandard System of M , SSy(M) is the collection of all subsets of Ndefinable in M . Another (equivalent) definition is:

SSy(M) = A ⊆ N | ∃x ∈M, pr(n)|x⇔ n ∈ A,

where pr(n) means “the nth prime”.

A Scott Set is a nonempty collection X of subsets of N closed under∩, ∪, complement and relative recursion and such that if T ∈X codesan infinite tree then there is an infinite path P ⊆ T , P ∈X .

If M |= PA then SSy(M) is a Scott Set. Also, if X is a Scott Set and|X | ≤ ω1 then there is M |= PA such that SSyM = X . For detaileddiscussion, see [14], pp. 172-192 and 261-265.

8. OverspillLet M |= PA and I be a proper cut of M (i.e. I is a proper subset of

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M, x < y ∈ I ⇒ x ∈ I and I is closed under the successor function).Suppose a ∈ M and ϕ(x, a) is a formula such that M |= ϕ(b, a) for allb ∈ I. Then there is c > I such that

M |= ∀ x ≤ c ϕ(x, a).

9. TypesLet M |= PA. A type over M is a set p(x, a) of formulas ϕ(x, a),a ∈ M , such that p(x, a) is finitely satisfied in M . If T is a theory, atype over T is a set of formulas ϕ(x) such that T + ϕ(c) | ϕ ∈ p isconsistent, where c is a tuple of new constants.

10. IndiscernibilityA type p(x) over a complete theory T ⊃ PA is called indiscernible iffor any M |= T , any x1 < x2 < · · · < xn, y1 < y2 · · · < yn in M , allsatisfying p, and any formula ϕ we have

M |= ϕ(x1, x2, . . . , xn)↔ ϕ(y1, y2, . . . , yn).

For the construction of an indiscernible type, see [13], page 82.

11. Quantifier Qxϕ(x)“M |= Qx ϕ(x)” is an abbreviation for “M |= ∀ x ∃ y > x ϕ(y)”, i.e.Qx ϕ(x) means “for unboundedly many x, ϕ(x) holds”.

12. Specker-MacDowell-Gaifman TheoremEvery model M |= PA has a proper elementary end-extension.

13. Conservative extensionsAn extensionN M is called conservative if for every formula ϕ(x, a)with a ∈ N there is another formula ψ(x, b) with b ∈M such that

M ∩ x | N |= ϕ(x, a) = x ∈M | M |= ψ(x, b).

In other words, every subset of M that is definable in N is alreadydefinable in M . We know that conservative extensions of models of PAare end-extensions. See [14], page 101.

14. Gaifman’s Splitting TheoremIf M ⊂ N are models of PA then K = x ∈ N | ∃y ∈M, x ≤ y is aninitial segment of N such that M ≺ K. See [14], page 89.

15. κ-like modelsA model is called κ-like if it has cardinality κ but all its proper initialsegments have cardinality less than κ. Existence of κ-like models forall κ follows from the Specker-MacDowell-Gaifman Theorem.

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16. Recursive saturationA type p(x, a) is called recursive if the set pϕ(x, y)q | ϕ(x, a) ∈p(x, a) is a recursive subset of N.

A model M |= PA is called recursively saturated if every recursivetype p(x, a) over M is realized in M .

A type p(x, a) over M is coded if pϕ(x, y)q | ϕ(x, a) ∈ p(x, a) ∈SSy(M).

A recursively saturated model realizes all its coded types. Also, if Mis recursively saturated then ThM is coded in M .

If M,N |= PA are countable and recursively saturated then (M ∼= N)⇔ (ThM = ThN and SSyM = SSyN).

17. LanguagesLPA = 0,+, ·, <, S, LZFC = ∈. The following fact is known asCraig’s trick. If a set T of formulas in some finite language L isrecursively enumerable then it is equivalent to a recursive set of for-mulas. (For proof see [2], Chapter 15 or [14], p.150.) For this reasona recursively saturated model realises all recursively enumerable types,not only the recursive ones.

18. ResplendencyA model M |= PA is called resplendent if for every Σ1

1-statementΦ(a), a ∈ M such that there exists a model N |= Th(M,a) + Φ(a),M |= Φ(a).

Every resplendent model is recursively saturated. Every countable re-cursively saturated model is resplendent.

19. Kleene’s TheoremLet L be a finite language. Let ϕn(x)n∈N be a recursive set of formulasof L (i.e. the set of their Godel numbers is a recursive subset of N).Then there is a Σ1

1-formula Φ(x) such that in all infinite L-structuresM ,

M |= ∀x (Φ(x)↔∧i∈N

ϕn(x)).

20. Wilmers’ TheoremLet X be a countable Scott Set, T be a consistent theory, T ∈ X .Then there is a countable recursively saturated M |= T such thatSSy(M) = X .

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21. Linear ordersLet (A,<) and (B,<) be linear orders. A+B is a linear order, whosedomain is the disjoint union of A and B and the order is defined by:(i) a1 <A+B a2 if a1, a2 ∈ A and a1 <A a2

(ii) b1 <A+B b2 if b1, b2 ∈ B and b1 <B b2(iii) a < b for all a ∈ A, b ∈ B.AB is a linear order with domain A× B and the order defined lexico-graphically (not antilexicographically as in some books!):(a1, b1) < (a2, b2) if (a1 < a2 or (a1 = a2 and b1 < b2)).If (X,<) is a linear order then (Xn, <) is the set Xn with the lexico-graphic order.A∗ is the ordered set A with the order < reversed.

It is known that there are 2λ pairwise non-embeddable linear orders ineach infinite cardinality λ (see [11]).

22. Saturated linear ordersThe saturated dense linear order of cardinality λ will be denoted byQλ. (This is different from Hausdorff’s notation Qα or “the ηα-set ofcardinality λ”, where λ = ℵα.) Remember: the subscript λ of Qλ willalways stand for the cardinality of Qλ. The main theorem about Qλ is:

λ is regular and Σµ<λ2µ = λ⇔ there exists Qλ.

Notice that Qω = Q.The saturated discrete linear order with first and without last elementof cardinality λ is N +QλZ.

23. TruthThroughout the thesis I avoid mentioning philosophy of mathematics,even in section 4.4 which comes close to it. However, the followingthree issues need special explanation:1) what do I mean by “true”?2) what is Th N then?3) what are “consistent theories”?Usually the word “true” will mean “provable in ZFC”. I understandthe ad hoc nature of this convention (other notions of a set will producedifferent “truths” and there is no mathematical reason for ZFC to begiven preference) but I’ll stick to it. As in much of current mathematics,I shall not exploit the strength of ZFC anyway.

For the sake of psychological convenience I suggest we may think in thefollowing way. Let us fix a ‘big’, ‘monster’ model M |= ZFC and ‘live in

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it’. Define N = ωM, Th N = ϕ | ωM |= “ϕ ∈ LPA” and ωM |= ϕ, etc.This is a neutral approach. Those who believe in existence of The OnlyTrue Universe can take M = The Only True Universe, those who don’tbelieve in it can choose M according to their preferences. All classicaldefinitions are easily translated into M-definitions. For example, let aset T of Godel numbers of sentences of L be definable in M. T is calledconsistent if M |= ¬Pr

T(p0 = 1q).

This paragraph was necessary to give meaning to such intuitive objectsas N and Th N. We got rid of foundational difficulties by simply pushingthem onto the level of our monster M.

1.2 Literature survey

Very little has been known in the area of order-types of models of PA so far:

1. In any model of PA, the terms 0, S0 =: 1, SS0 =: 2, SSS0 =: 3, . . .comprise an initial segment isomorphic to N. This initial segment iscalled the standard cut. Also, if c > N then there are the pointsSc, SSc, SSSc, . . . following it and the points Pc, PPc, . . . precedingit (if P (n)c = 0 for some n ∈ N then c = n ∈ N), hence there is the wholeZ-block of points around c. Hence the order-type of a nonstandardmodel is equal to N + A Z for some linear order A. Sometimes wedenote the Z-block of a point c by [c].

2. A is a dense linear order without end-points.

ProofGiven two points a and b from different Z-blocks we observe that theinteger part of a+b

2is between a and b and belongs to a block other than

[a] or [b]. A has no end-points because for any a ∈M , [a2] and [2a] are

blocks different from [a].

In particular the order-type of any countable model is N+QZ becauseQ is the only countable dense linear order without end-points.

3. A cannot be the set R of all reals.(Smorynski [33] attributes this result to Klaus Potthoff.)

ProofSuppose the order-type ofM is N+R Z. Consider an arbitrary definableM -sequence aii∈M such that for all i, j ∈ N such that i < j,

[ai] < [aj]

13

(e.g. the sequence iai∈M where a ∈M r N). Let sup[ai] | i ∈ N =[b]. Since for all n ∈ N,

an < an+1 and an < b,

by overspill, there is c ∈M r N such that for all n ∈ N,

an < ac < b.

Now, all real numbers from [aj]N<j<c lie between [an]n∈N and [b]contradicting the definition of [b].

Later in the thesis (section 2.4), the above fact will follow from a moregeneral theorem.

4. Pabion’s TheoremLet M |= PA, κ be a cardinal. If (M,<) is κ-saturated then M isκ-saturated.

Proof for κ = ω1 (due to D. Richard and J.F. Pabion)First let us prove that a model of PA is ω1-saturated if and only ifall of its ω-sequences are coded. Suppose A |= PA codes all its ω-sequences. Let p(x) = ϕn(x)n∈ω be an arbitrary type (possibly withℵ0 parameters). Let xn satisfy ϕ1(x)∧ . . .∧ϕn(x). Let a ∈ A be a codeof xnn∈ω. Let

ψk(u) = ∀y(k < y ≤ u→ ϕk((a)y)).

Informally ψk(u) means ϕk((a)k+1)∧ . . .∧ϕk((a)u) (informally, becauseit may be an infinite conjunction). For all u ∈ N, A |= ψk(u), hence,by overspill, there is ck > N such that A |= ψk(ck). Let b be a code ofthe sequence ckk∈ω. Consider the definable set B = x | ∀y ≤ x x <(b)y. Obviously N ⊂ B. Hence there is a point c ∈ B r N. For everystandard n, n < c < (c)n, hence (a)c satisfies the type p(x). Hence amodel which codes all ω-sequences is ω1-saturated.

Suppose A |= PA is of ω1-saturated order-type. Let us show that thenevery ω-sequence will be coded. Let a0, a1, . . . , an, . . . be an arbitrarysequence. Translate it into a strictly increasing sequence by puttingb0 = a0, bn+1 = an+1 + bn + 1. By ω1-saturation, there is some c > bnfor all n. Put cn = c − bn, so that cnn∈ω is strictly decreasing.Consider the following conditions:

2c0 < x < 2c0+1

14

...

2c0 + · · ·+ 2cn < x < 2c0 + · · ·+ 2cn−1 + 2cn+1

...

By ω1-saturation, there is a point u ∈ A satisfying all of the aboveconditions. Now, ck is the kth exponent in the binary expansion of u.It remains to solve a triangular system to get ak, the process which canbe formally expressed in PA. Hence, akk∈ω is coded in A.

The proof for higher cardinalities is more complicated and can be foundin [21].

5. (κ,κ)-cutsIt has been proved by Shelah that every M |= PA has a cut I ⊂ Msuch that for some cardinal κ, cf(I) = κ and cf(M r I) = κ. See [29].

6. Some results of J. SchmerlA number of possibly relevant results has been obtained by J. Schmerl.Let me quote two of them.

Fact 1 ([25])If λ is regular, N |= PA is λ-saturated and λ < |N | then there is arather classless λ-saturated M N such that |M | = |N |.

Fact 2 ([24])If M is a countable recursively saturated model of PA then for everycardinal κ there is a resplendent N of cardinality κ with N ≡∞ω Mand N generated by indiscernibles.

7. Models of singular strong limit cardinalities have long sequences.The following application of an Erdos-Rado theorem is due to Hodges[10].

Fact Let T be a completion of PA, M |= T , κ be a singular stronglimit cardinal, cardM = κ. Then M contains an increasing sequenceof order-type κ.

ProofLet κ =

∑α<µ λα, cf(κ) = µ, λαα<µ be increasing, each λα be

regular. By a Theorem due to Erdos and Rado, for each α < µ, there ishα:λα →M which is either order-preserving or order-reversing. Usingsubtraction, we can assume that each hα is order-preserving. Let B =⋃

α<µ Im(hα).

15

If for each c ∈ M , |b ∈ B | b < c| < κ then cf(M) = µ. Asµ < κ, there is β < µ such that for all δ > β, λδ > µ, hence forall δ > β, Im(hδ) is bounded in M by some point cδ. Define theincreasing sequence (aα)α∈κ as follows. For all α < λβ+1, put aα =hβ+1(α). Let δ be such that β + 1 < δ < µ and suppose for all α <supγ<δ λγ, aα has been defined. As δ < µ, aαα<supγ<δ λγ is boundedin M (otherwise cofinality of M would be less than δ), say by pointb. For α ∈ [supγ<δ λγ, λδ), put aα = b + hδ(α). Obviously, aαα<λδ

isbounded by b+ cδ.

If for some c ∈ M , |b ∈ B | b < c| = κ then suppose without lossof generality that B < c. Let f :µ→M be order preserving. Then forα ∈ [λδ, λδ+1), put aα = c · f(δ) + hδ+1(α). Obviously, aαα∈κ is anincreasing κ-sequence in M .

The condition that κ is a singular strong limit cardinal is important.In Chapter 3 we shall construct a family of models of cardinality 2ω

not even having increasing ω1-sequences.

8. Models generated by indiscerniblesIn paper [4] C.Charretton and M.Pouzet give an outline of a construc-tion of the embedding of a model generated by indiscernibles orderedas a left-right symmetric linear order C into C<Q. You can read detailsof the construction and the proof in my Chapter 3. They also outlinea possible proof of the fact that if T is a countable first-order theorywith the strict order property then, for every uncountable linear orderC, there is a model M(C) |= T such that C is embeddable into C ′ ifand only if M(C) is elementarily embeddable into M(C ′). I was not atall convinced by the details and suggest questions 7 and 8 as possibleways of proving this fact.

16

Chapter 2

Density, cells, cofinalities,weight

In this chapter we give first examples of order-types of models of PA andintroduce some important notions: density, cell decomposition, cofinality,lower cofinality, weight. Some theorems proved in this chapter will playimportant role later on in the thesis.

2.1 Density

First let us introduce a class of models M having as many points betweenany two points as possible, namely, cardM .

Definition 1 Let M |= PA, κ be a cardinal. M is called κ-dense ifcardM = κ and for all a, b ∈ M such that a < b, a and b belong to dif-ferent Z-blocks,

cardx ∈M | a < x < b = κ.

Simple Fact 1 M |= PA is κ-dense if and only if every nonstandard initialsegment of M (including M itself) has cardinality κ.

ProofIf M is κ-dense and I is a nonstandard initial segment then, for any non-standard point a ∈ I, card[0, a] = κ. Now,

κ = cardM ≥ card I ≥ card[0, a] = κ,

hence card I = κ.If all nonstandard initial segments of M have cardinality κ then, given

a, b ∈M , a < b, a,b from different Z-blocks, consider the nonstandard initial

17

segment I = x | x < b− a. As card I = κ then [a, b] = a+ x | x ∈ I hascardinality κ too.

Proposition 2 For every infinite κ there is a κ-dense model of PA.

ProofAn elementary chain argument. Let M0 |= PA, cardM0 ≤ κ. Suppose i ≤ κand for all j < i, Mj has already been defined so that j1 < j2 < i impliesMj1 ≺ Mj2 . By the elementary chains lemma,

⋃j<iMj Mk for all k < i.

Denote⋃

j<iMj by Ki. The collection of sentences

ϕ(a) | Ki |= ϕ(a), a ∈ Ki ∪ n < bi < a | n ∈ N, a ∈ Ki r N

is finitely satisfied in Ki, hence, by compactness theorem, there is N Ki,which satisfies it. Let us identify the symbol bi with its interpretation inN . Put Mi = ClN(Ki ∪ bi). We observe that cardMi = cardKi andMi Ki. Define M = Mκ. The sequence b0 > b1 > · · · > bi > · · · (i ∈ κ)is unbounded below in M r N, hence M is κ-dense.

Actually, we have proved that any model of cardinality less or equal than κhas a κ-dense elementary extension.

In the sequel we shall continue to use standard model-theoretic constructionsand results (such as the elementary chains lemma, compactness theorem, etc.,and identification of a constant symbol naming an element with the elementitself) but without explicit mention.

In the rest of this section we shall study some examples of κ-dense models.

First example

Lemma 3 Let K |= PA and X ∈ SSy(K). Then for every b ∈ K r N, thereis c < b which codes X.

ProofLet a be a code of X in K. Let

ϕ(x, a, b) = ∃y < b ∀z < x (pr(z)|a↔ pr(z)|y),

where pr(x) is the xth prime number. For every n ∈ N, M |= ϕ(n, a, b),hence, by overspill, there is c ∈M such that N < c < b and for every n ∈ N,

M |= pr(n)|a↔ pr(n)|c,

hence c codes X.

18

For any non-principal ultrafilter D on ω and any countable model M , denote∏D M by Mω. (Later we shall see that, modulo (CH),

∏D M does not

depend on the ultrafilter D.)

Proposition 4 Mω is 2ℵ0-dense.

ProofLet us prove prove that SSy(Mω) = P(N). Let X = i1, i2, . . . be an arbi-trary subset of N. Define aX = [(a1, a2, a3, . . .)j∈ω], where aj =

∏k≤j pr(ik).

Observe that for every n ∈ N,

i ∈ ω | pr(n)|ai ∈ D ⇔ n ∈ X,

so aX codes X. Thus, SSy(Mω) = P(N). By Lemma 3, every nonstandardinitial segment contains a code of X, hence, by Fact 1, Mω is 2ℵ0-dense.

A κ-dense submodel of a µ-dense model

Proposition 5 If M is µ-dense then for every κ < µ there is a κ-denseelementary submodel of M .

The following proof was suggested by my supervisor.ProofLet θ be a surjective mapping

κ θ→ (i1, i2, ..., in, j) | n ∈ N, i1, i2, ...in ∈ κ, j ∈ N.

Partition κ into a disjoint union of κ sets of cardinality κ:

κ =∐i∈κ

Si.

For every i ∈ κ, let αi:Si → κ be a bijection between Si and κ. Now we aregoing to introduce κ points of M , whose Skolem closure will be κ-dense. Leta0 ∈M be arbitrary. Suppose η < κ and for all i < η, ai are already defined.As Si | i ∈ κ is a partition of κ, η ∈ Sν for some ν. Let θ(αν(η)) =(i1, i2, . . . , in, j). If i1, i2, . . . , in < η choose aη ∈ M r ai | i < η suchthat aη < tj(ai1 , ai2 , ...ain), where tj(x1, x2, . . . , xn) is the jth element in theenumeration of all nonstandard Skolem terms of Th(M). Otherwise let aη bearbitrary. Let K = ClMai | i < κ. Obviously, cardK = κ. Consider anypoint tj(ai1 , ai2 , ...ain) ∈ K. Let I =

⋃ν∈κ α

−1ν θ−1(i1, . . . , in, j). I is cofinal

in κ because card I = κ. Introduce J = aη | η ∈ I, η > i1, i2, . . . , in.We observe that card(J) = κ. For every aη ∈ J, aη < tj(ai1 , ai2 , ...ain) bydefinition of aη. Thus K is κ-dense.

19

Ultrapower construction

Proposition 6 For every cardinal λ such that λ+ = 2λ and any countablemodel M there is an ultrafilter D on λ such that

∏D M is 2λ-dense.

ProofIt suffices to prove that

∏D N is 2λ-dense, because for any nonstandard

a ∈∏

D M there is a nonstandard b ∈∏

D N →∏

D M such that b < a.Assuming the appropriate ultrafilter is already found (we shall specify itsproperties during the proof) we need to prove that∣∣∣∣∣

x ∈

(∏D

N

)r N | x < a

∣∣∣∣∣ = 2λ

where a = [(n1n2n3...nj...)j∈λ].

Suppose there are λ points ai less than a, enumerated as

ai = [(ai1ai2ai3...aij...)j∈λ] | i ∈ λ .

We are going to construct a new point b < a which is different from any ofai. Let D have a subset F such that:

1. |F | = λ;

2. for all i ∈ λ, (ni = m) implies that i belongs to not more than (m− 1)elements of F .

Let F = ei | i ∈ λ. Let j < λ and nj = m. List all ei1 , ei2 , ei3 , ..., eim−1 ∈ Fsuch that j ∈ eik for all k = 1, 2, ...,m− 1. Let bj be a natural number suchthat bj < m and bj 6= ai1j, bj 6= ai2j, ..., bj 6= aim−1j. It is possible to choosesuch bj because not more than (m− 1) points from 1, 2, ...,m are occupiedby aikj. So bj 6= aij for all j ∈ ei. Since ei ∈ F ⊂ D, b = [(bj)λ] 6= ai for alli ∈ λ. Thus, there are more than λ points below a. Assuming λ+ = 2λ thereare exactly 2λ of them. So

∏D M is 2λ-dense.

Now we only need to find an ultrafilter with the required properties.

Definition 2 An ultrafilter D on λ is called λ-excellent if it is countablyincomplete (i.e. there is a sequence C1 ⊇ C2 ⊇ C3 ⊇ · · · s.t.

⋂i∈ω Ci = ∅)

and for every Cii∈ω ⊂ D such that C1 ⊇ C2 ⊇ C3 ⊇ · · · and⋂

i∈ω Ci = ∅there is F ⊂ D such that:

1. |F | = λ;

20

2. Every element of Cn rCn+1 belongs to not more than (n− 1) elementsof F .

In the proof of Proposition 6, for our arbitrary nonstandard point a wecan introduce Cm = i ∈ λ | ni ≥ m. Observe that

⋂m∈NCm = ∅ and for

all m ∈ N, Cm ∈ D, because a is nonstandard. Thus, the only property werequire from our ultrafilter is for it to be λ-excellent.

Observation 7 Let (M,<) be a countable linearly ordered set and D an ω1-complete ultrafilter (i.e. for any countable Cii∈ω ⊂ D,

⋂i∈ω Ci ∈ D) on a

set I. Then∏

D M = M .

ProofIf there is a point a = [(a1a2a3...)] ∈

∏D M which is not equal to any

of the old points then enumerate X = x ∈ M | x < a as xii∈ω andY = y ∈M | y > a as yii∈ω. Introduce Cn = i ∈ I | xn < ai < yn. AsX < a < Y , Cn ∈ D. However, as a 6∈M ,

⋂n∈NCn = ∅ 6∈ D, contradiction.

In view of this, all ultrafilters we consider in the future will be not only

non-principal but countably incomplete as well.

Definition 3An ultrafilter on I is called λ-regular if it has a subset F such that:

1. |F | = λ;

2. Every point of I belongs to not more than finitely many elements of F .

Definition 4 An ultrafilter on I is called regular if it is card I-regular.

Thus λ-excellent ultrafilters are a kind of regular ultrafilters.

Lemma 8 (λ > ω) & (λ+ = 2λ)⇒ there exists a λ-excellent ultrafilter.

This λ-excellent ultrafilter was constructed in my MPhil Thesis. It is a longchain construction which I decided to omit here.

We do not need a λ-excellent ultrafilter for λ = ω, because, as we saw inProposition 4 ,

∏D M is 2ℵ0-dense for any ultrafilter D on ω.

Also, there exist λ-good ultrafilters providing us, modulo (GCH), withsaturated ultraproducts. If M is the saturated model of cardinality λ thenM is λ-dense as we are going to see in a moment.

Question 2 Is there an ultrafilter D on λ such that there exists a countableM |= PA with

∏D M 2λ-dense but not saturated?

21

Saturation

Fact (Keisler)[3]If D is a countably incomplete ultrafilter then

∏D M is ω1-saturated.

Corollary (CH)Let M and N be two countable models. The following are equivalent:

1. M ≡ N ;

2.∏

D M∼=∏

E N for any non-principal ultrafilters D and E on ω;

3. there are ultrafilters D and E such that∏

D M∼=∏

E N .

Proof of the Corollary(1) ⇒ (2) because both

∏D M and

∏E N are saturated and have the same

cardinality and, hence, isomorphic.(2)⇒ (3) obvious.(3) ⇒ (1) because every model is elementarily equivalent to its ultrapower.

We deduce the following theorem [3].

Theorem (CH) Mω is saturated and, for every N ≡ M and any two ultra-filters D and E on ω,

∏D M

∼=∏

E N .

As Mω is saturated, its order-type is the saturated discrete linear order withfirst and no last point, N +Qω1Z.

Proposition 9A saturated model of cardinality κ is κ-dense.

ProofIf there was a nonstandard point a such that cardx | x < a < κ then thetype p(x) = x < a∪x 6= ai | ai < a with fewer then κ parameters wouldnot be realized. This contradicts saturation.

Resplendency

Proposition 10 If M |= PA is resplendent then M is cardM-dense.

ProofLet a ∈MrN. The statement “there are cardM points below a” is expressedby the following Σ1

1 formula: ∃f ∀xy (x 6= y → f(x) 6= f(y) & f(x) < a),which is realised in Cl(a) (because x ∈ Cl(a) | x < a is countable, hence

22

there is a bijection between x ∈ Cl(a) | x < a and Cl(a)), hence is realisedin M .

Moreover, the Σ11 formula

∃f ∀xy (x < y → f(x) < f(y) < a & ∀z∃w(z < f(x)→ f(w) = z))

is realised in Cl(a) because x ∈ Cl(a) | x < [a] has order-type N + QZ, i.e.is order-isomorphic to Cl(a). Hence, for every a ∈M r N, there is an initialsegment I < a such that (I,<) ∼= (M,<).

In section 2.3, Lemma 17, we shall actually classify all order-types of initialsegments of a resplendent model.

2.2 Cells

In this section we shall divide a model into pieces that have the same density.These pieces will be called cells. The notion of a cell extends the notion ofa Z-block into larger cardinalities.

Simple Fact 11Let M be a linear order and suppose there is a ∈M such that card[0, a] = λ.Let Iλ = a | card[0, a] ≤ λ. Then card Iλ = λ or card Iλ = λ+.

ProofIf card Iλ = β ≥ λ+, let A = aii<λ+ be the first λ+ elements in theenumeration of Iλ. A cannot be bounded in Iλ, because if A < a for somea ∈ Iλ then card[0, a] ≥ λ+. Hence, aii<λ+ is cofinal in Iλ and, thus, Iλ isa union of λ+ sets [0, ai], i ∈ λ+ of cardinality λ. Hence, β = λ+ · λ = λ+.

Definition 5 Let M |= PA.For every λ such that there is a ∈ M , card[0, a] = λ and for every x ∈M , define the λ-cell of x as Cellλ(x) = x ± Iλ = y | |x − y| ∈ Iλ.Let λ1, λ2, . . . , λi . . .i<µ be the enumeration in increasing order of all λsuch that there exists a, card[0, a] = λ. For every i < µ, if card Iλi

=λi put εi = 0, otherwise (i.e if card Iλi

= λ+i ) put εi = 1. The matrix(

λ1 λ2 · · · λi · · ·ε1 ε2 · · · εi · · ·

)i<µ

is called the sort of M .

Properties of cells

Theorem 12

1. For every i < µ, x ∈M ,Cellλi

(x) = y | there are at most λi points between x and y.

23

2. For every i < µ and x ∈M , Cellλi(x) is convex.

3. For every i < µ, x, y ∈ M , either x and y are in the same λi-cell orCellλi

(x) ∩ Cellλi(y) = ∅.

Thus, for every i < µ, M is partitioned into λi-cells.

4. A κ-dense model consists of just one cell.

5. For every i < µ, Iλiis closed under addition and multiplication.

ProofIf card[0, a] ≤ λi and card[0, b] ≤ λi then card[0, a + b] = card[0, a] +carda + x | x < b ≤ λi + λi = λi and card[0, a · b] = card[0, a] +card[a, 2a] + card[2a, 3a] + · · ·+ card[(b− 1)a, ba] ≤ λi · λi = λi.

6. For i < µ, let (M/Iλi, <) be the set of all λi-cells in their natural order.

Then (M/Iλi, <) is a dense linear order.

ProofLet a, b ∈ M , Cellλi

(a) 6= Cellλi(b). Consider c = [a+b

2] ∈ M . If there

was no cell between Cellλi(a) and Cellλi

(b) then c would belong eitherto Cellλi

(a) or to Cellλi(b). Let c ∈ Cellλi

(a). Then card[a, c] = λi.Then card[c, b] = card[a, c] = λi, hence, by property (1), b ∈ Cellλi

(a)and Cellλi

(a) = Cellλi(b). Contradiction.

7. It follows from the proof above that for every i < j < µ, Iλj/Iλi

is adense linear order too.

8. Denote (Iλ1 r N)/Z by B1, (Iλjr⋃

i<j Iλi)/⋃

i<j Iλiby Bj. Then

Iλ1 = N +B1Z,Iλ2 = N +B1Z +B2B1Z,Iλ3 = N +B1Z +B2B1Z +B3B2B1Z, etc(M,<) = N +B1Z +B2B1Z + · · ·+ ( · · ·B3B2B1︸ ︷︷ ︸

µ∗

Z).

So, the problem of specifying the order-type of M is reduced to speci-fying linear orders Bj for all j < µ. Chapter 5 deals with the situationwhen the orders Bj are combinations of ordinals and saturated ordersQλ.

9. (J. Paris and G. Mills [22])If εj = 1 then Iλj

|= PA.

10. (J. Paris and G. Mills [22])If λ1 = ω, λ2 > 2ω then Iω is closed under exponentiation and this isthe fastest function under which Iω needs to be closed. More precisely:

24

if M |= PA is countable and I ⊆ M in a nonstandard initial segmentof M closed under exponentiation then for every infinite κ > ω thereis N M such that λN

1 = ω, INω = I, λN

2 = κ.

11. (GCH) If D is a λ-excellent ultrafilter and M is a model of sort(λ1 λ2 · · · λi · · ·ε1 ε2 · · · εi · · ·

)i<µ

,

where λi ≤ 2λ for all i < µ, but M is not (2λ)+-likethen

∏D M is 2λ-dense.

ProofWe observe that card

∏D M = 2λ because, since D is a regular ultrafil-

ter, card∏

D M = (2λ)λ, (see [3]), and, by Konig’s lemma, (2λ)λ = 2λ

because cf(2λ) > λ. Actually, the picture is this:(ωλ λλ

1 . . . (2λ)λ

0 0 . . . 0

)but the cells Iωλ , Iλλ

ifor λi ≤ 2λ will form one single cell because they all

have cardinality 2λ. The sets∏

D Iλifor λi ≤ 2λ are 2λ-dense because

they have cardinality 2λ and each of their initial segments contains aninitial segment of

∏D ω, which is 2λ-dense by Proposition 6.

The main theorem about cells

Theorem 13 For every matrix

(λ1 λ2 · · · λi · · ·ε1 ε2 · · · εi · · ·

)i<µ

with λ1 < λ2 <

· · · < λi < · · · and εi ∈ 0, 1 for all i < µ, there is a model of this sort.

ProofLet Mλ1 be a λ1-dense model. If ε1=1 then let Nλ1 be any λ+

1 -like end-extension of Mλ1 , otherwise put Nλ1 = Mλ1 . Suppose j < µ and we havealready defined

Nλ1 ≺ Nλ2 ≺ · · · ≺ Nλi≺ · · · (i < j).

Put M0 =⋃

i<j Nλi. M0 has the sort

(λ1 · · · λi · · ·ε1 · · · εi · · ·

), i < j. Let M1

be any proper conservative elementary extension of M0. We are going toconstruct a sequence cii∈λj

ordered as λ∗j such that M0 < ci < M1 r M0

for all i, and there are no points between M0 and cii∈λj.

25

Lemma (#)Let M |= PA, ϕ(x) be a formula such that M |= Qxϕ(x) and θ(x, y) anarbitrary formula. Then there is a third formula ψ(x) such that:

1. M |= Qxψ(x);

2. M |= ∀x (ψ(x)→ ϕ(x));

3. for all a ∈M ,either: M |= ∃y ∀x > y (ψ(x)→ θ(x, a))or: M |= ∃y ∀x > y (ψ(x)→ ¬θ(x, a)).

For proof see [14], p. 97.

Let us enumerate all formulas of LPA as θ0(x, y0), θ1(x, y1), θ2(x, y2), . . . Weconstruct a second sequence of formulas ϕ0(x), ϕ1(x), ϕ2(x), . . . such that forall i ∈ ω, M0 |= Qxϕi(x). Let ϕ0(x) be “x = x”. If ϕi(x) is given satis-fying M0 |= Qxϕi(x) we may apply Lemma (#) to obtain ϕi+1 such thatM0 |= Qxϕi+1(x), M0 |= ∀x (ϕi+1(x)→ ϕi(x)) and for all a ∈M0,either: M0 |= ∃y ∀x > y (ϕi+1(x)→ θi(x, a))or: M0 |= ∃y ∀x > y (ϕi+1(x)→ ¬θi(x, a)).This process is repeated until ϕi(x) is defined for all i ∈ N.This collection ϕi(x)i∈ω is a particular kind of definable type, as con-structed first by Gaifman [8]. We shall refer to this type in the future asa Gaifman type for M0. (Also, it is possible to make this type indis-cernible over sufficiently small parameters, more precisely: if K |= PA anda1, a2, . . . , an and b1, b2, . . . , bn are increasing sequences in K, with each ai,bj realizing p(x) and c ∈ K with ClK(c) < ai, bj for each i, j ≤ n, then foreach LPA-formula θ(x, y),

K |= ∀y < c (θ(a, y)↔ θ(b, y).

We are going to use this fact later in Theorem 50.)Let us consider the type

p1(x) = x > a | a ∈M0 ∪ x < b | b ∈M1 rM0 ∪ ϕi(x)i∈N.

Any finite subset of this set of formulas is satisfied in M1 by a sufficientlylarge element of M0 satisfying ϕi(x) for large enough i. Let the constant c1realise this type in some N M1. Define M2 = ClN(M1 ∪ c1).

Now we are going to show that M2 is a conservative extension of M0.Consider an arbitrary formula θ(u, b), where b = (b1, . . . , bn) and bj is defined

26

by the formula ηj(v, a, e, c1), j = 1, 2 . . . n, a ∈M0, e ∈M1 rM0.As M1 is a conservative extension of M0 we can find for the formula

δ(x, y, a, e) = ∃v

(n∧

j=1

ηj(vj, a, e, x) & θ(y, v)

)

another formula ξ(x, y, g) with a parameter g ∈M0 such that for all x, y ∈M0

M0 |= ξ(x, y, g) ⇔ M1 |= δ(x, y, a, e).

The formula ξ(x, y, y1, . . . , yk) is θi(x, y, y1, . . . , yk) for some i ∈ N.We claim that for all d ∈M0,

M2 |= θ(d, b1, b2, . . . , bn)⇔M0 |= ∃w ∀x > w (ϕi+1(x)→ ξ(x, d, g)).

(⇐) If w0 ∈M0 and M0 |= ∀x > w0 ( ϕi+1(x)→ ξ(x, d, g) )then, by our choice of ξ, for all x ∈M0, x > w0,

M1 |= ϕi+1(x)→ δ(x, d, a, e).

Then, by overspill, there is a0 ∈M1 rM0 such that

M1 |= ∀ x ( w0 < x < a0 & ϕi+1(x)→ δ(x, d, a, e)).

Then, as M2 M1,

M2 |= ∀ x ( w0 < x < a0 & ϕi+1(x)→ δ(x, d, a, e)).

Then, putting x = c1 we get M2 |= δ(c1, d, a, e),which means that M2 |= θ(d, b).(⇒) Let M2 |= θ(d, b). By the construction of ϕi+1(x),either: M0 |= ∃w ∀x > w (ϕi+1(x)→ ξ(x, d, g))or: M0 |= ∃w ∀x > w (ϕi+1(x)→ ¬ξ(x, d, g))If M0 |= ∀ x > w0 (ϕi+1(x)→ ¬ξ(x, d, g)) then,by our choice of ξ, for all x ∈M0, x > w0,

M1 |= ϕi+1(x)→ ¬δ(x, d, a, e).

Again, by overspill,

M1 |= ∀x ( w0 < x < a0 & ϕi+1(x)→ ¬δ(x, d, a, e)).

Then, as M2 M1,

M2 |= ∀x ( w0 < x < a0 & ϕi+1(x)→ ¬δ(x, d, a, x)),

27

which is not true, because

M2 |= w0 < c1 & c1 < a0 & ϕi+1(c1) & δ(c1, d, a, e).

Hence, M0 |= ∃ w ∀ x > w ( ϕi+1(x)→ ξ(x, d, g)).So, the new constant did not spoil the conservativeness of our extension M1.Again consider the set of formulas

p2(x) = x > a | a ∈M0 ∪ x < b | b ∈M2 rM0 ∪ ϕi(x)i∈N.

It is finitely satisfied in M2, so we can realise it by c2 and setM3 = Cl(M2 ∪ c2). M3 is still a conservative extension of M0.Analogously we can define Mi for all i < λj.

LetMλj=⋃

i<λjMi. Mλj

has the sort

(λ1 λ2 · · · λj

ε1 ε2 · · · 0

)because

⋃i<j Nλi

is an initial segment of Mλjand, given a point a >

⋃i<j Nλi

we can findk < λj such that a ∈ Mk and observe that there are λj distinct pointscii>k below a.If εj=1 we take as Nλj

any λ+j -like end-extension of Mλj

, otherwise putNλj

= Mλj. Having constructed Nλj

for all j < µ, we observe that⋃

j<µNλj

has the required sort.

Actually, we have proved even more:

Corollary 14 For any sort-matrix, every complete theory T ⊃ PA and anyM |= T such that cardM ≤ λ1, there is N M of this sort such that forevery η < µ, Iλη |= T .

Another corollary of the proof of Theorem 13 will be required later.

Corollary 15Let M |= PA, p(x) be a Gaifman type for M . Then for any cardinal α, thereis an end-extension N M , cardN = maxcardM,α, and a sequencecii∈α of elements of N r M ordered as α∗ such that for all i ∈ α, cisatisfies p and for any a ∈ N rM there is i ∈ α such that ci < a.

2.3 Cofinalities

Definition 6Let (A,<) be a linear order. Then Υ(A) = the least cardinal γ such that thereis a nondecreasing sequence of length γ unbounded in A.

Υ

(A) = Υ(A∗).

28

Definition 7 Let (A,<) be a linear order. Then cf(A), cofinality of A,is the least cardinal γ such that there is a set B ⊆ A unbounded in A,cardB = γ. lcf(A), lower cofinality of A, is cf(A∗).

Theorem 16

1. For any linear order A, Υ(A) = cf(A),

Υ

(A) = lcf(A).

2. For any linear order A, cf(A) and lcf(A) are regular.

3. For any regular α, β, there is a model M |= PA such that cf(M) = α,lcf(M r N) = β.

4. Let cardM ≤ κ. Then for any regular α, β ≤ κ, there is a κ-denseN M with cf(N) = α, lcf(N r N) = β.

5. If M is λ-like then Υ(M) = cf(λ).

6. If M |= PA, cardM = λ and M is saturated then λ is regular andcf(M) = λ.

7. If M |= PA is resplendent then cf(M) = lcf(M r N).

8. Let M |= PA. For any regular α and any cardinal κ ≥ α, there is aresplendent N M , cardN = κ such that cf(N) = α.

Proof

1. Firstly, Υ(A) ≥ cf(A), because an unbounded sequence is also an un-bounded set. Let B ⊆ A be unbounded in A, cardB = cf(A) andconstruct an increasing sequence bii<γ≤cf A unbounded in A. Letb0 ∈ B be arbitrary. Suppose we already have an increasing sequencebii<δ. If bii<δ is unbounded in A, put γ = δ, otherwise choosearbitrarily bδ ∈ B, bδ > bii<δ. As cardB = cf(A), γ ≤ cf(A). HenceΥ(A) ≤ cf(A).

2. If (ai)i<α is a nondecreasing unbounded sequence in A and (βj)j<cf(α)

is a nondecreasing unbounded sequence in α then (aβj)j<cf(α) is a non-

decreasing unbounded cf(α)-sequence in A.

3. The usual chain-argument.

4. The usual chain-argument.

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5. Firstly, there is an increasing cf(λ)-sequence, (ai)i<cf(λ) unbounded inM (the same proof as the proof of (1)). Suppose there is an unboundedsequence (bi)i<γ, γ < cf(λ). Then f : γ → cf(λ) defined as f(i) =max j aj < bi is unbounded in cf(λ), which contradicts regularity ofcf(λ).

6. If there was a sequence aii<γ, γ < λ unbounded in M , then the typex > aii<γ would have to be realized by saturation.

7. The statement Υ(M) =

Υ

(M) is expressed by the following Σ11 formula:

∃f ∃a∀xy > a(f(x) < a) & (x < y → f(y) < f(x)) & ∀b < a ∃x >a (f(x) < b), which is realised in all countable models.

8. A chain-argument, using the fact that any model has a resplendentelementary extension of the same cardinality.

Lemma 17 Let M |= PA be resplendent, I, J be any two initial segments ofM . Then (I,<) ∼= (J,<) ⇔ Υ(I) = Υ(J).

It follows that (I,<) ∼= (M,<) ⇔ Υ(I) = Υ(M).

ProofLet a1 and a2 belong to different Z-blocks and b1 and b2 belong to differentZ-blocks. The Σ1

1-statement

∃ f : [a1, a2] → [b1, b2], f is an order-isomorphism

is realized in Cl(a1, a2, b1, b2), hence is realized in M , hence [a1, a2] and [b1, b2]are order-isomorphic. In particular, given an arbitrary a ∈MrN, any infinitesegment [b1, b2] is order-isomorphic to [0, a].

Denote the order-type of [0, a) by (A,<). Let Υ(I) = α, (ai)i∈α be anincreasing sequence unbounded in I, a0 = 0. Then (I,<) = Σi∈α[ai, ai+1) ∼=Σi∈αA ∼= αA.

Also, we have obtained the order-type of our resplendent model M in termsof (A,<):

(M,<) ∼= Υ(M) A.

2.4 Weight

Definition 8 Let I be a linearly ordered set. An interval in I is an infiniteset of the form (u, v), where u, v ∈ I.

30

Definition 9 The weight of I, w(I), is the supremum of cardinals γ suchthat there is a disjoint collection of intervals of I, which has cardinality γ.

Examples: w(Q) = ℵ0, w(R) = ℵ0, w(R2) = 2ℵ0 .

Definition 10 Let I be a set, (A,<) a linear order, 0 ∈ A an arbitrarypoint, f : I → A. Then supp(f) = x ∈ I | f(x) 6= 0.

Definition 11 If (I,<), (A,<) are linear orders, 0 ∈ A, then(A, 0)<I = f : I → A | supp(f) 6= ∅, supp(f) is finite with order definedlexicographically: f < g if for a = mini | f(i) 6= g(i), f(a) < g(a).

Elements of I will sometimes be called coordinates or coordinate axes.The linear orders (A, 0)<I will play important role later, in Chapter 3.If (A,<) is 2-homogeneous, (i.e. for all a1, a2 ∈ A, there is g ∈ Aut(A,<)such that g(a1) = a2) or I is finite then it is easy to see that for any 01, 02 ∈ A,

(A, 01)<I ∼= (A, 02)

<I .

Also, (0, 1, 0)<ω 6∼= (0, 1, 1)<ω because (0, 1, 0)<ω has least element butno last element while (0, 1, 1)<ω has last element but no least element. Thefollowing question is relevant to the material of the next chapter.

Question 3 Are there dense linear orders (I,<), (A,<), without end-pointsand 01, 02 ∈ A such that

(A, 01)<I 6∼= (A, 02)

<I ?

Theorem 18 Let M |= PA.

1. If I is a nonstandard initial segment of M then w(I) = card(I).

2. For any i, j < µ such that i > j, w(Iλi/Iλj

) = card(Iλi/Iλj

) = card Iλi.

3. Let (X,<) be a linear order, n ∈ N. Let M be λ-dense, λ > w(X), I bea nonstandard initial segment of M . Then I is not order-embeddableinto (Xn, <).

4. With the same conditions as in (3) above, I is not embeddable into(X<ω∗ , <).

Proof

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1. If I is card I-like then, by Theorem 12 (5), I is closed under multipli-cation. Then, given any a ∈ I r N, we observe that (ia, (i + 1)a)i∈I

is a collection of cardinality card I of disjoint intervals of I.

If I is not card I-like then let us show that there is b ∈ I such thatcard[0, b] = card I and b2 ∈ I. By Theorem 12 (5),

⋃µ<card I Iµ =: K is

closed under multiplication. Let c ∈ I r K. For all x ∈ K, x2 ∈ K,in particular x2 < c. Hence, by overspill, there is b ∈ I rK such thatb2 < c.

Now (0, b), (b, 2b), . . . , ((b−1)b, b2) is a collection of disjoint intervalsof cardinality card[0, b] = card I.

2. If a ∈ Iλir Iλj

then for any k ∈ Iλi, ka and (k+1)a belong to different

λj-cells, hence (ka, (k + 1)a)k∈Iλiis a disjoint system of intervals of

Iλi/Iλj

which has cardinality card Iλi.

3. For n = 1, see part (1) above. Suppose for all nonstandard initialsegments I, I is not embeddable into (Xn, <). Let J be a nonstandardinitial segment embeddable into (Xn+1, <) and f : J → (Xn+1, <) bethe order-embedding. For every c ∈ X introduce Ac = x ∈ J | f(x) =(c, . . .). Obviously, each Ac is convex. Let us show that for all c ∈ X,cardAc ≤ w(X). Suppose to the contrary that cardAc > w(X) forsome c ∈ X. Let a ∈ Ac. Then one of the sets A1 = x ∈ Ac | x < aand A2 = x ∈ Ac | x > a has cardinality cardAc.

Let B =

x | a− x ∈ A1 if cardA1 = cardAc

x | a+ x ∈ A2 otherwise.

We observe that B is an initial segment of cardinality cardAc > w(X)embeddable into Xn. Contradiction. Hence cardAc ≤ w(X) for allc ∈ X.

As M is λ-dense, J is not card J-like, hence there is b ∈ J such thatb2 ∈ J and card[0, b] = card J = λ > w(X) ≥ cardAc. Hence, for alli, j ∈ J , i 6= j implies ib ∈ Aci

, jb ∈ Acjfor ci 6= cj and each interval

(ci, cj) is infinite. The collection (ci, ci+1)i∈[0,b] of intervals of X isdisjoint and has cardinality card J = λ > w(X). Contradiction.

4. Let f : I → (X<ω∗ , <) be an order-embedding. Let x1, x2 ∈ I be suchthat card[x1, x2] = card I. Let n ∈ ω be the first n such that f(x1) 6=f(x2). Then [f(x1), f(x2)] is embedded into Xn. This contradicts (3).

32

Notice that clauses (3) and (4) of Theorem 18 can be applied not only tomodels of PA themselves but to all factor-sets M/Z, M/Iλ as well, by thesame proof.

An easy modification of the proof of Theorem 18 produces the followingcorollary.

Corollary 19 No uncountable model of PA is embeddable in any of thefollowing linear orders: R, Rn, R<ω∗, ωω, ω∗1Q, ω∗1Rn.

Clearly, it would be nice to generalise this result to other ordinals.

Question 4 Is it true that if γ is an ordinal and cardM > w(X) then(M,<) is not embeddable into (X<γ∗ , <)?

Question 5Is there an uncountable model of PA order-embeddable into (R<ω, <)?

If there is no such model then we can suggest the following conjecture.

Question 6If (I,<) is a countable linear order then the following are equivalent:

1. For every linear order (A,<) there is an uncountable MA |= PAembeddable into A<I .

2. I contains a copy of Q.

Chapter 3 gives a partial answer to the (2⇒1)-part of this problem. Theorem18 proves that there are no such models for I finite or I = ω∗.

33

Chapter 3

Models generated byindiscernibles

Indiscernibles are one of the major methods of constructing examples ofmodels of PA developed so far. In this chapter we study order-types ofmodels of PA generated by indiscernibles.

Let T be an arbitrary completion of PA, (C,<) a linearly ordered set. Bythe Ehrenfeucht–Mostowski Theorem (see [3], or [11], section 11.2), there isa model M |= T such that:

1. (C,<) is order-embeddable into M (so, we can identify (C,<) with itsimage);

2. for all y ∈M , y = t(c1, . . . cn) for some c1, . . . cn ∈ C and some Skolemterm t;

3. M |= ϕ(ci1 , . . . , cin)↔ ϕ(cj1 , . . . , cjn) for all LPA -formulas ϕ(x1, . . . xn)and all ci1 < . . . < cin , cj1 < . . . < cjn in C.

The set p = ϕ | there are c1, . . . cn ∈ C such that c1 < . . . < cn andM |= ϕ(c1, . . . cn) is called the indiscernible type satisfied by C. C iscalled an indiscernible sequence in M . The model M is called a modelgenerated by indiscernibles ordered as C or Ehrenfeucht-Mostowskimodel and is denoted by EM(C, p). For any (C,<) and any indiscernibletype p, EM(C, p) is uniquely defined.

Theorem 20If (C,<) is a dense linear order without end-points and there is an orderpreserving f : (C,<) −→ (C∗, <) and a point 0 ∈ C such that f(0) = 0, thenfor any indiscernible type p, EM(C, p) is order-embeddable into (C, 0)<Q.

34

Notice that if C is not dense or there is no function f :C∗ → C fixing apoint then we can embed C into some dense linear order D for which there isf :D∗ → D which fixes one point. This will induce an embedding of EM(C, p)into EM(D, p) and hence, by Theorem 20, into (D, 0)<Q. Thus, every modelgenerated by indiscernibles is embeddable into (D, 0)<Q for some D.

The examples of C’s I had in mind while proving the Theorem are R, Qor, more generally, any densely ordered group. This influenced my notationfor the rest of the chapter.Notation. For any x ∈ C denote f(x) by −x. If x1, x2, x3 ∈ C, x1 <x2 < x3 are the points we are considering at the moment then x2 + ε is anypoint between x1 and x3 which is different from x2. Sometimes we requirex2 + ε ∈ (x2, x3). Let us pick any point a ∈ C, a > 0 and denote it by 1.Another symbol ‘1’ will appear in the following notation. For every x ∈ C,let 1x = x, 0x = 0, −1x = −x.

3.1 Analysis of a single term

Consider a Skolem term t(v1, . . . , vn) of T and points x1, . . . , xn ∈ C. Whenwe write t(x1, . . . , xn) we shall always implicitly assume x1 < x2 < . . . < xn.If x1, . . . , xn are ordered differently then reorder them in the right orderand consider t(v1, . . . , vn) with its variables reordered (and vi and vj suchthat xi = xj substituted by the same new variable) as a new term. Thisis done for the sake of indiscernibility as we shall see below. Now, let Vt =t(x1, . . . , xn) | x1, . . . , xn ∈ C, x1 < · · · < xn. The question for this sectionwill be: “What is the order-type of Vt in M”?

Definition 12Let 1 ≤ i ≤ n. We say that t i-increases if for any xi + ε ∈ (xi, xi+1),

t(x1, . . . , xi + ε, . . . , xn) > t(x1, . . . , xi, . . . , xn).

We define t i-decreases or is i-constant similarly.

This definition does not depend on the choice of x1, . . . , xn ∈ C, by indis-cernibility. Also, it does not depend on the choice of xi+ε because x1 < . . . <xi < xi + ε1 < xi+1 < . . . < xn and x1 < . . . < xi < xi + ε2 < xi+1 < . . . < xn

are similar distributions, hence, by indiscernibility of C,

t(x1, . . . , xi + ε1, . . . xn) > t(x1 . . . , xi, . . . , xn)←→

←→ t(x1 . . . , xi + ε2, . . . , xn) > t(x1, . . . , xi, . . . , xn).

35

Definition 13 Let 1 ≤ i 6= j ≤ n. We say that j is more valuable in tthan i and write i / j if, given x1, . . . , xn ∈ C, x1 < . . . < xn,

t(x1, . . . , xi, . . . , xj + δ, . . . , xn) > t(x1, . . . , xi + ε, . . . , xj, . . . , xn)←→

←→ t(x1, . . . , xi, . . . , xj + δ, . . . , xn) > t(x1, . . . , xi, . . . , xj, . . . , xn)

for all xj + δ ∈ (xj−1, xj+1) r xj, xi + ε ∈ (xi−1, xi+1) such that (xj, xj + δ)and (xi, xi + ε) do not overlap.

Again the definition does not depend on the choice of xi + ε and xj + δ.

LemmaLet 1 ≤ i 6= j ≤ n. Then precisely one of the following holds:

1. i / j

2. j / i

3. t is both i-constant and j-constant.

Proof If t is i-constant, but not j-constant then i / j. Without loss ofgenerality assume t is i-increasing and j-increasing. (The proof for the otherthree cases is the same.) Let xi + ε ∈ (xi, xi+1), xj + δ ∈ (xj, xj+1).

If t(x1, . . . , xi, . . . , xj + δ, . . . , xn) > t(x1, . . . , xi +ε, . . . , xj, . . . , xn) then j . i.

If t(x1, . . . , xi, . . . , xj + δ, . . . , xn) < t(x1, . . . , xi +ε, . . . , xj, . . . , xn) then j / i.

If t(x1, . . . , xi, . . . , xj + δ, . . . , xn) = t(x1, . . . , xi + ε, . . . , xj, . . . , xn) then, by

indiscernibility t(x1, . . . , xi, . . . , xj, . . . , xn) = t(x1, . . . , xi+ε, . . . , xj, . . . , xn),

hence t is i-constant. By the same argument, t is j-constant.

So, “/” is a linear order on the set i | t is not i − constant. Let us orderi | t is i− constant arbitrarily to extend “/” to a linear order on i | 1 ≤i ≤ n, and let the permutation τ of 1, . . . , n be such that τ1.τ2. · · ·.τn.

In order to illustrate the definition of “j is more valuable than i” let usstudy two examples. We shall always identify xi ∈ C with its value in M .Don’t get confused by different +’s!

1. If M |= x1 + x2 < x3 for some x1, x2, x3 ∈ C with x1 < x2, then fort1(x1, x2) = x1 +x2, 2.1, because the shift x2 → x2 + ε is equivalent tothe shift x2 → x3, which increases our term in spite of any shifts of x1

inside (−∞, x2), by indiscernibility. (More formally: by indiscernibility,

x1 + x2 < (x1 − δ) + (x2 + ε) ⇔ x1 + x2 < (x1 − δ) + x3.

The last inequality is true by our assumption x1 + x2 < x3.)

36

2. If M |= xx23 < x4 & xx3

4 > xx24 + x1 for some x1 < x2 < x3 < x4

in C, then for t2(x1, x2, x3) = xx23 + x1, 3 . 2 . 1. Here, x3 is the

main variable because (x3 + ε)x2 + x1 is a point between x3 + ε and(x3 + ε,+∞) = y ∈ C | x3 + ε < y, while both xx2+δ

3 + x1 andxx2

3 +(x1+δ) are between x3 and (x3,+∞). 2.1 because xx2+ε3 > xx2

3 +x1,and, hence, xx2+ε

3 + x1 > xx23 + (x1 + δ).

The conditions x1 + x2 < x3, xx23 < x4, x

x34 > xx2

4 + x1 in these two examplesare clearly satisfied by x1 < x2 < x3 < x4 when the indiscernible sequenceC is cofinal in M . It is unclear how the terms t1(x1, x2) and t2(x1, x2, x3)behave if C is not cofinal in M .

Denote (x1, . . . , xn) ∈ Cn | x1 < x2 < · · · < xn by Dn.Introduce the function s : 1, . . . , n → −1, 0, 1 as follows.

s(i) = si =

−1 if t is τi− decreasing

0 if t is τi− constant1 if t is τi− increasing.

Let Et = (zτ1, zτ2, . . . , zτn)| zτi = sixτi, (x1, . . . , xn) ∈ Dn.Define f :Vt → Et as

f(t(x1, x2, . . . , xn)) = (zτ1, zτ2, . . . , zτn).

Now it is easy to prove (repeatedly applying definitions 12 and 13) that thisfunction is an order-isomorphism between Vt and Et.

3.2 Separated and interpenetrating terms

Let t1(u1, u2, . . . , un) and t2(v1, v2, . . . , vm) be two Skolem terms of T in thelanguage LPA ∪ C with all their free variables shown such thatτ1 . τ2 . · · · . τn in t1 and σ1 . σ2 . · · · . σm in t2.

Definition 14 Call t1 and t2 separated if either

for all x ∈ Dn, y ∈ Dm, M |= t1(x) > t2(y)

or for all x ∈ Dn, y ∈ Dm, M |= t1(x) < t2(y)

and interpenetrating otherwise.

Notice that “being interpenetrating” is an equivalence relation on the set ofall Skolem terms of T .

37

Lemma 21 If t1 and t2 are interpenetrating and not constant then

either (xτ1 > yσ1 → t1(x) > t2(y)) & (xτ1 < yσ1 → t1(x) < t2(y))

or (xτ1 > yσ1 → t1(x) < t2(y)) & (xτ1 < yσ1 → t1(x) > t2(y)).

ProofAssume for the moment that t1 and t2 are i-increasing for all i.Suppose there are tuples (x1, . . . , xn) ∈ Dn and (y1, . . . , ym) ∈ Dm such that

x1 < y1 < . . . < xτ1 < . . . < yσ1 < . . . < xn (∗)

and t1(x1, . . . xn) > t2(y1, . . . ym) and tuples (x′1, . . . , x′n) ∈ Dn, (y′1, . . . , y

′m) ∈

Dm such that

x′1 < . . . < y′σ1 < . . . < x′τ1 < . . . < x′n (∗∗)

and still t1(x′1, . . . , x

′n) > t2(y

′1, . . . , y

′m).

We are going to show that now t1 and t2 will be separated.Let xi < yσ1 and yj be the element in the sequence (∗) immediately precedingxi. We are going to swap xi and yj preserving the inequality t1 > t2. Let usshift yj → xi+ε, yσ1 → yσ1−ε, where xi < xi+ε < xi+1, yσ1−1 < yσ1−ε < yσ1 .

s ssxi

yj yσ1

?

?

Doing these shifts we are of course using the density of C. As yσ1 is the mainvariable of t2, t2 will decrease after this shift and t1 > t2 will still be true. Bysome sequence of such shifts we can obtain any distribution of variables of t1and t2 with xτ1 < yσ1 preserving the inequality t1 > t2. The same argumentfor the distribution (∗∗) gives us that t1 > t2 for the rest of the distributions.The proof would be analogous if for some i our terms were i-decreasing.

Thus, if t1(u1, . . . , un) and t2(v1, . . . , vm) are interpenetrating and we wantto compare the values

t1(x1, . . . , xn) and t2(y1, . . . , ym)

(where (x1, . . . , xn) ∈ Dn, (y1, . . . , ym) ∈ Dm) then we simply compare theirmain variables xτ1 and yσ1. But what happens if xτ1 = yσ1 =: b? (We shall

38

denote the fact that xτ1 in t1 and yσ1 in t2 are equal to the same elementb ∈ C by writing t1(x1, . . . , xn | xτ1 = b) and t2(y1, . . . , ym | yσ1 = b).)If t1(x | xτ1 = b) and t2(y | yσ1 = b) are separated then the process ofcomparing has finished. But if t1(x | xτ1 = b) and t2(y | yσ1 = b) are stillinterpenetrating then we shall have to compare their second most valuablevariables, etc. We continue doing so until either t1(x | xτ1 = b1, . . . , xτm =bm) and t2(y | yσ1 = b1, . . . , yσm = bm) become separated at some stage m orwe run out of variables. Lemma 22 (1) and (3) proves the inductive step ofthis construction.

Lemma 22Consider t1(x | xτ1 = b1, . . . , xτm = bm) and t2(y | yσ1 = b1, . . . , yσm = bm),where b1, . . . , bm ∈ C and bi < bj ⇔ xτi < xτj. Then the following hold.

1. If t1 is τ(m+ 1)-increasing and t2 is σ(m+ 1)-decreasing thent1(x | xτi = bi, i ≤ m) and t2(y | yσi = bi, i ≤ m) are separated.

2. If xτ(m+1) and yσ(m+1) belong to different intervals (bl1 , bk1) and (bl2 , bk2)then t1(x | xτi = bi, i ≤ m) and t2(y | yσi = bi, i ≤ m) are separated.

3. If t1(x | xτi = bi, i ≤ m) and t2(y | yσi = bi, i ≤ m) are interpenetratingthen

(xτ(m+1) > yσ(m+1) → t1 > t2) & (xτ(m+1) < yσ(m+1) → t1 < t2)

iff t1 and t2 are τ(m+ 1)- and σ(m+ 1)-increasing, and

(xτ(m+1) > yσ(m+1) → t1 < t2) & (xτ(m+1) < yσ(m+1) → t1 > t2)

iff they are τ(m+ 1)- and σ(m+ 1)-decreasing.

Proof1. Let t1 < t2 for some distribution of variables xτi, yσj, (i, j ≥ m + 1)with xτ(m+1) < yσ(m+1). As t2 is σ(m + 1)-decreasing, the shift yσ(m+1) →xτ(m+1) − ε will only increase t2, and, hence, preserve the inequality t1 < t2.Now, let yσj < xτi, (i, j ≥ m + 1) be two neighbouring variables belongingto the same interval (bl, bk).If j 6= m+ 1 then the shifts yσj → xτi + ε, yσ(m+1) → yσ(m+1) − ε,and if j = m+ 1 then the shifts xτi

→ yσj − ε, xτ(m+1) → xτ(m+1) − εwill swap them preserving the inequality t1 < t2. By some composition ofsuch shifts we can obtain any distribution of xτi, yσj

(i, j ≥ m+ 1).

2. Again we can swap any two neighbouring variables xτi and yσj whichbelong to the same (bl, bk), preserving the relation between t1 and t2.

39

3. Let t1 and t2 be τ(m+ 1)- and σ(m+ 1)-decreasing respectively.(If they were both increasing the proof would be the same.)Assume xτ(m+1) < yσ(m+1) → t1 < t2, and we shall get a contradiction.As t1 is τ(m+ 1)-decreasing, the shift xτ(m+1) → x′τ(m+1) = yσ(m+1) + εdecreases t1 and, hence, preserves the inequality t1 < t2.Thus, we have two distributions of xτi, yσj, (i, j ≥ m+ 1)

x1 < . . . < bl < . . . < xτ(m+1) < . . . < yσ(m+1) < . . . < bk < . . . < ym

and x1 < . . . < bl < . . . < yσ(m+1) < . . . < x′τ(m+1) < . . . < bk < . . . < ym

such that t1 < t2 for both of them.Now, again, swapping any two xτi and yσj, (i, j ≥ m + 1), which belongtwo the same (bl1 , bk1) and preserving t1 < t2, we get that t1 < t2 for alldistributions, and, hence, t1(x | xτi = bi, i ≤ m) and t2(y | yσi = bi, i ≤ m)are separated. Contradiction.

Notice that Lemma 21 is just a particular case of Lemma 22 (3) (whenm = 0)but I had to include it for better understanding.

3.3 Embedding into (C, 0)<Q

Before describing the general construction, let us study an example. Letp ⊃ x2

1 < x2, t1(x) = 0, t2(x) = x, t3(x) = x2. We define for all i ∈ 1, 2, 3embeddings ϕi : Vti → C<Q so that the diagram

Vti

⋃3j=1 Vtj

C<Q

ϕi

in

PPPPPP

?1

q

would commute. The term t1(x) will be forever separated from all otherterms and will be less than them. For all x ∈ C, put ϕ1(t1(x)) =

( . . . 1 . . . 0 . . . 1 . . . )100 101 102

.

By this notation we mean the element f ∈ C<Q,

f(q) =

1 if q = 100 or q = 1020 if q = 1010 otherwise.

40

The term t2(x) is separated from t1(x) because for all x, y ∈ C,M |= t2(x) > t1(y). We choose a coordinate coming earlier than 100, sayd1 = 20, and put g(20) = 1 so that f < g. Define ϕ2(t2(x)) =

( . . . 1 . . . x . . . 1 . . . )20 30 31

.

The term t2(x1) is 1-increasing, hence, by the analysis of a single term, ϕ2 isan order-embedding.

The terms t3(x) and t2(x) are interpenetrating, but, once the main vari-able is fixed (by comparing the main variables, we appeal to Lemma 21),t3(x | x = b) and t2(x | x = b) become separated because b2 > b. Defineϕ3(t3(x)) =

( . . . 1 . . . x . . . 1 . . . )20 30 301

2

.

Putting the value h(3012) = 1 makes t3(x | x = b) and t2(x | x = b) separated

because 3012

comes earlier than 31.Now, let us repeat this process carefully for all Skolem terms of T . Let

t1(x1, . . . , xn1), . . . , ti(x1, . . . , xni), . . . be an enumeration of all Skolem terms

of T . Let A1 = d1 < a1 < · · · < dn1 < an1 < dn1+1 ⊂ Q. Defineϕ1(t1(x1, . . . xn1)) =

( 1 s1xτ1 1 . . . 1 sn1xτn1 1 )d1 a1 d2 . . . dn1 an1 dn1+1

.

Coordinates di will be used for separation, ai for interpenetration. Let τi ∈Sni

be the permutation such that τi1 . τi2 . · · · . τini in ti and

sij =

1 if ti is τij − increasing0 if ti is τij − constant−1 if ti is τij − decreasing.

For the sake of better readability we shall write sj instead of sij and τ1 . τ2 .

· · · . τni instead of τi1 . τi2 . · · · . τini because the index i will always bespecified in the expression ‘ti’ anyway.

Assume that for all i ≤ k we have already constructed the sets

Ai = di1 < ai

1 < di2 < · · · < di

ni< ai

ni< di

ni+1 ⊂ Q

and functionsϕi : Vti −→ CAi

41

such that the diagram

Vti

ϕi−→ CAiyin

yin⋃j≤k Vtj −→ C∪j≤kAj

commutes.Let us find Ak+1 = d1 < a1 < · · · < dnk+1

< ank+1< dnk+1+1 such that

the order-preserving mapping

ϕk+1 : Vtk+1−→ CAk+1 defined as

ϕk+1(tk+1(x1, . . . , xnk+1)) =

( 1 s1xτ1 1 · · · snk+1xτnk+1

1 )| | | | |d1 a1 d2 · · · ank+1

dnk+1+1

would make the diagram

Vtk+1

ϕk+1−→ CAk+1y y⋃i≤k+1 Vi −→ C∪iAi

commutative.If for all i ≤ k, ti and tk+1 are separated then choose Ak+1 as an arbitrary(2nk+1 + 1)-element subset of

q ∈ Q | q > Ai, ti > tk+1 ∩ q ∈ Q | q < Ai, ti < tk+1,

which is an interval in Q. So, bigger terms have earlier coordinates withpositive values.

Let tk+1 and ti be interpenetrating. By Lemma 21, first we have tocompare their main variables (on the same coordinate axis). Put d1 = di

1

and a1 = ai1. Suppose we have defined dj and aj for all j < m ≤ nk+1 + 1.

(Note that m may be equal to nk+1 + 1, i.e. only the last dnk+1+1 is notdefined yet.) If for some ti,

ti(x | xτ1 = b1, . . . , xτ(m−1) = bm−1) and tk+1(x | xτ1 = b1, . . . , xτ(m−1) = bm−1)

are interpenetrating, then, by Lemma 22 (3), we have to compare their τm-thvariables, so we put dm = di

m, am = aim. Otherwise (i.e. if for all i,

ti(x | xτ1 = b1, . . . , xτ(m−1) = bm−1) and tk+1(x | xτ1 = b1, . . . , xτ(m−1) = bm−1)

42

are separated) choose dm < am < · · · < dnk+1< ank+1

< dnk+1+1 arbitrarilyfrom the set

I = q ∈ Q | q > am−1⋂

⋂q ∈ Q | q < di

m ti(x | xτ1 = b1, . . . , xτ(m−1) = bm−1) <

< tk+1(x | xτ1 = b1, . . . , xτ(m−1) = bm−1)⋂

⋂q ∈ Q | q > di

ni+1 ti(x | xτ1 = b1, . . . , xτ(m−1) = bm−1) >

> tk+1(x | xτ1 = b1, . . . , xτ(m−1) = bm−1),

which is an interval in Q. By construction, this choice of Ak+1 and ϕk+1

makes the diagram commutative.

Notice how Theorem 20 contrasts with Theorem 18 (3) and (4): if (X,<) is adense linear order, |X| > w(X) and there is a function f :X → X∗ having afixed point then no |X|-dense model M is order-embeddable into (Xn, <) or(X<ω∗ , <) but there is a model (namely EM(X)) which is order-embeddableinto (X<Q, <).

Corollary 23 EM(R, p) is an uncountable model containing no monotonousω1- or ω∗1-sequences.

Proof Present R<Q as a union: (R<Q, <) =⋃

i∈ω(Ai, <), where Ai = Rni .Let f :ω1 → R<Q be an order-embedding. Then ω1 =

⋃i∈ω Bi, where Bi is

defined asα ∈ Bi ⇔ f(α) ∈ Ai.

Hence, by regularity, there is j ∈ ω such that Bj is unbounded in ω1, henceorder-isomorphic to ω1. Then f |Bj

:Bj → Rnj is an order-embedding whichis impossible. The proof for ω∗1 is the same.

Corollary 24For any dense linear order (C,<) with no last element, EM(C, p) is order-embeddable into (C∗ + 0+ C)<Q.

The next question asks for a generalisation of the Ehrenfeucht–Gaifmanlemma (see [16], page 305) for n ≥ 1.

Question 7Let p be an indiscernible type over T , (C,<) be a linear order, M = EM(C, p).Prove that if t(u1, . . . , un) is a Skolem term of T , x1, . . . , xn ∈ C, then eithertp(t(x1, . . . , xn)) 6= tp(x1) or t(x1, . . . , xn) = xi for some i = 1, . . . , n.

43

If Question 7 has a positive solution then we can deduce that EM(C1, p) iselementarily embeddable into EM(C2, p) if and only if C1 is embeddable intoC2. Furthermore it would follow that there are 2λ pairwise non-embeddablemodels of T in any cardinality λ.

Question 8Let λ > ω. Is there a family Cii∈2λ of dense linear orders of cardinality λwith no last element such that i 6= j implies that Ci is not embeddable into(C∗

j + 0+ Cj)<Q?

If yes, we can deduce, by Theorem 20, that there are 2λ pairwise non-embeddable order-types of models of T of cardinality λ.

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Chapter 4

Inner models

In this chapter we look at ‘inner models’ (the models obtained by means ofArithmetised Completeness Theorem) and express their order-type in termsof the (<, ·)-type of the original model. This leads to a very promising notionof self-similarity.

InterpretationLet T1, T2 be consistent theories in finite languages L1 and L2. Let A |=T1, B |= T2. A is said to be interpreted in B if there are formulas (possiblywith parameters from B) dom(x), f(x, y) for every functional symbol f ofL1 and R(x) for every relational symbol R of L1 such that

B |= ∀x (dom(x)→ ∃!y dom(y) & f(x, y)) and

(x ∈ B | dom(x), f , R) ∼= A.

Strong interpretationLet T be a consistent theory in a finite language L. Let A |= T , B |= PA.Then A is strongly interpreted in B if there are two formulas dom(x)and Sat(x, y) (possibly containing parameters from B) such that there is abijection

g:A→ x ∈ B | dom(x)

such that for every ϕ(x) ∈ L, a ∈ A,

A |= ϕ(a) ⇔ B |= Sat(pϕq, 〈g(a)〉).

Obviously, if A is strongly interpreted in B then A is interpreted in B. IfA |= PA is strongly interpreted in B |= PA we shall sometimes say that A isan inner model in B.

Let M |= PA be nonstandard, T be a set of statements (in a finite lan-guage L) such that pϕq | ϕ ∈ T is coded in M by some point t. Then

45

define Con(T ) to be the following LPA ∪ t-statement:(∀i < len(t) ∀x ¬Proof(x, p((t)0 ∧ . . . ∧ (t)len(t)−1 → ∃y y 6= yq)

).

Clearly, this definition depends on the choice of parameter t.If M = N and pϕq | ϕ ∈ T is defined in N by a formula θ(x) thenCon(T ) = ConT =

∀x(∀i < len(x) θ((x)i)→ ¬∃y Proof(y, p(x)0 ∧ . . . (x)len(x)−1 → ∃zz 6= zq)).

ConPA is the following LPA-statement:

∀x((∀i < len(x) axiomPA((x)i))→

→ ¬∃y Proof(y, p(x)0 ∧ . . . (x)len(x)−1 → ∃z z 6= zq)).

Notice that N |= Con(PA) ↔ ConPA. However, if M |= PA is nonstandardthen “M |= Con(PA)” is a weaker condition than “M |= ConPA” becauseConPA states the consistency of the collection of all (including all nonstan-dard) axioms of PA, not only the ones coded by t.

The formula PrPC(x) means “the formula with Godel number x is provablein the predicate calculus”.

Arithmetised Completeness Theorem.M |= PA + Con(T ) if and only if there is N |= T which is strongly interpretedin M , i.e. there are formulas

domN(x, a) (domain of N) and SatN(x, y, a) (satisfaction for N)

such thatx ∈ N ⇔ M |= domN(x, a),

N |= ϕ(b) ⇔ M |= SatN(pϕq, 〈b〉, a).

We shall denote this model by ACT(M,T ). (Warning: even for completetheories T , ACT(M,T ) does not have to be unique!) If M |= Qx dom(x, a)then we can without loss of generality assume M |= ∀x dom(x, a).

FactIf M,N |= PA, M is nonstandard and N is strongly interpreted in M thenN is recursively saturated.

For a more detailed discussion of the Arithmetised Completeness Theorem,see [14], pp. 186-192, 224-246 and [34].

46

4.1 Interpreted linear orders

Some results about Peano Arithmetic are obtained by “internalising” classicalconstructions, i.e. simulating their proofs inside a model of PA. E.g. theArithmetized Completeness Theorem is the internalisation of CompletenessTheorem. In this section we also prove some results by internalising classicalconstructions. Lemma 25 is the internal version of the fact “every linearorder of finite cardinality is isomorphic to 1, 2, . . . , n in natural order forsome n.” Theorem 26 is the internalisation of the fact “every countable denselinear order without end-points is isomorphic to Q”. We conclude the sectionby internalising the classical Fraısse’s construction.

Definition 15 A linear order A is called boundedly interpreted in Mif it is interpreted by the formulas ϕ(x, a) (predomain), e(x, y, a) (equality),o(x, y, a) (order) with M |= ∃x∀y(ϕ(y, a) → y < x), where a ∈ M is aparameter.

In the case of PA we can always assume that e(x, y) ↔ x = y, because theorder interpreted by ϕ′(x) ↔ (ϕ(x) & ∀y < x ¬e(x, y)), e′(x, y) ↔ x = y,o′(x, y)↔ o(x, y) is isomorphic to A.

Definition 16 A linear order A is called M-finite if there is x ∈M ,x = 〈x1, x2, x3〉, where x1 codes a bounded subset of M , x2 codes equality, x3

codes order so that A ∼= x.

From the properties of the coding function we easily observe that the abovetwo definitions are equivalent. If A is M -finite, we shall sometimes writeM |= “A is finite” (M believes that A is finite).

Notice that if the formula ψ(x) means “x codes a linear order” thenM |= Qxψ(x).

Lemma 25 If A is boundedly interpreted in M by the formulas ϕ(x, a),e(x, y, a), o(x, y, a) then A ∼= [0, b] for some b ∈ Cl(a).

ProofLet M |=“A is finite”. For every x ∈M such that M |= ϕ(x) & ∃z o(x, z) wedefine the successor function s(x). Let f : [0,maxϕ] −→ [0,maxϕ] be thefollowing function:f(0) = min z o(x, z);

f(i+ 1) =

min z o(x, z) & o(z, f(i)) if ∃z o(x, z) & o(z, f(i))f(i) otherwise.

Put s(x) = f(maxϕ). Let M |=“c is the element z of [0,maxϕ] such that∀y ≤ maxϕ (ϕ(y) & ¬e(y, z)→ o(z, y))”.

47

Let g(0) = c, g(i+1) = s(g(i)). If b = max i ∃z o(g(i), z) we observe thatg : [0, b+ 1]→ A is an order-isomorphism.

Definition 17 Let (A,+, ·, <) be an ordered ring, A+ = x ∈ A | x ≥ 0.Then (Q(A+), <) is the set

(x, y) | x, y ∈ A+, x, y 6= 0

factored out by the following equivalence relation:

(x, y) ∼ (z, w)⇔ xw = yz

with the linear order on it defined as

(x, y) < (z, w) ⇔ xw < zy.

Actually we do not use addition in this definition. Later we shall be discussingwhether there are (A,+, ·, <) |= PA, (B,+, ·, <) |= PA such that(A,+, <) ∼= (B,+, <) but (Q(A), <) 6∼= (Q(B), <).Notice that if A has no last element then Q(A) is a dense linear order.

Theorem 26If (A,<) is a dense linear order without end-points interpreted in M then

(A,<) ∼= (Q(M), <).

ProofFirstly, notice that A is M -infinite, i.e.

M |= ∀x ∃y (ϕ(y) & ∀z < x ¬e(z, y))

because, by Lemma 25, all M -finite linear orders are discrete.Q(M) is interpreted in M by ϕ(x) = ‘∃x1x2 (x = 〈x1, x2〉 & x1, x2 6= 0)’,

e(x, y) = ‘x1y2 = x2y1’, o(x, y) = ‘x1y2 < x2y1’.Let Q1 and Q2 be two dense linear orders interpreted in M . We are going

to prove that they are isomorphic. Moreover, the isomorphism can be takento be a definable function in M .

Let Q1 be interpreted by (dom1(x), x <1 y) and Q2 be interpreted by(dom2(x), x <2 y). Let f1(n) = the nth element x such that dom1(x) andf2(n) = the nth element x such that dom2(x). We are going to find a Skolemterm i(x) such that

1. M |= ∀x∃y i(y) = x;

2. M |= ∀zw(f1(z) <1 f1(w))←→ (f2(i(z)) <2 f2(i(w)))

48

i.e. f2if−11 is a definable order-isomorphism between Q1 and Q2. We define

i(x) by internalizing the ‘forth’-argument in M :i(0) = 0;i(x + 1) = min z ∀y ≤ x (f1(y) <1 f1(x + 1)) ↔ (f2(i(y)) <2 f2(z)). Suchelement z can always be found because of density:

M |= ∃zmaxw≤xf2(i(w)) | f1(w) < f1(x+1) < z <

< minw≤xf2(i(w)) | f1(x+ 1) < f1(w).

Corollary 27Neither R nor Q is interpreted in uncountable models of PA.

ProofLet M |= PA be uncountable. Q cannot be interpreted in M because Q(M)is uncountable, hence Q 6∼= Q(M). R cannot be interpreted in M because Mis order-embeddable into Q(M) but not into R, and hence R 6∼= Q(M).

The next proposition (dealing with the non-dense case) is similar to thefollowing known fact.

Fact (*)(see [14], page 192)If M,N |= PA and N is strongly interpreted in M then the function f :M →N defined as f(0) = 0N , f(x + 1) = f(x) +N 1N is a definable isomorphismbetween M and an initial segment of N .

We shall need Fact (*) in the next section.

Proposition 28 If (A,<) is a linear order with first point interpreted in Mand every element of (A,<) has a successor then

1. A is M-infinite, i.e. M |= Qxy(ϕ(x) & ϕ(y) & ¬e(x, y));

2. (M,<) is definably isomorphic to an initial segment of (A,<).

Proof2. Define f :M → A as follows:f(0) = the element w such that ∀x (ϕ(x) → e(x,w) ∨ o(w, x)), f(i + 1) =the element y such that o(f(i), y) &∀z(o(f(i), z) &¬e(y, z)→ o(y, z))).Clearly, f is definable and determines an order-isomorphism between M andan initial segment of A.1. If A was M -finite then, by Lemma 25, there would exist a definable iso-morphism g:A → [0, b] for some b ∈ M . Then g f would be a definable

49

embedding of M into [0, b] whose existence contradicts the pigeonhole prin-ciple.

Notice thatA does not have to be discrete (example: A = odds(M)+evens(M))but if a ∈ A has no predecessor then the gap between x ∈ A | x < a anda has to be big. If M |= o(b, a) then f :M → A defined as:f(0) = b;f(i+ 1) = the first w such that

ϕ(w) & o(f(i), w) & ∀y(ϕ(y) & o(f(i), y)→ e(w, y) ∨ o(w, y))

embeds M between b and a.

Fraısse limit

Another example of a dense linear order interpreted in M is the internalFraısse limit. We define an M -finite order with a ‘marked’ subset as a pointy = 〈y1, y2〉, where y2 codes a subset of the M -finite order [0, y1]. Let f :M →M be an enumeration of all linear orders with a ‘marked’ subset.

Let g(0) = f(0). Suppose for x ∈ M , g(x) = 〈x1, x2, x3〉 be alreadyconstructed, where x2 codes equality in [0, x1] and x3 codes order. (Of coursewe know from Lemma 25 that g(x) is isomorphic to some linearly ordered setof the form ([0, a], <M), but we shall not need it in our construction.) Letf(x+ 1) = 〈y1, y2〉, where y2 marks a subset B of [0, y1].

Let c = minx1 + 1, card(y2). We are going to amalgamate f(x + 1)with g(x) (x1 + 2− c) times by identifying the first c points of the markedsubset of f(x + 1) with all c-element suborders of g(x) of the form k, k +1, . . . , k + c− 1 and adjusting the order between marked points. At the ith

step, (y2)0, (y2)1, . . . , (y2)c−1 will be identified with i− 1, i, . . . , i+ c− 2.Let h(0) = g(x). For h(i) already defined, let

h(i+ 1) = 〈x1 + 1 + (i+ 1)(y1 + 1), eq, order〉,

where eq defines equality in [0, x1 + 1 + (i + 1)(y1 + 1)] and order defineslinear order. Let eq code the transitive closure of the following set of pairs:

(w,w) | w ≤ x1 + 1 + (i+ 1)(y1 + 1) ∪

∪ (w1, w2) | (w1, w2) ∈ (h(i))2

(i.e. all pairs which are already equal in h(i)) ∪

∪

(u, v) | ∃z ≤ c− 1u = the (z + i− 1)th point of g(x)

v − x1 − i(y1 + 1) = (y2)z

.

50

Let order code the transitive closure of the following set of pairs:

(u, v) | (u, v) ∈ (h(i))3 ∪

∪ (u, v) | x1 + 1 + i(y1 + 1) < u < v ≤ x1 + 1 + (i+ 1)(y1 + 1)(i.e. all points in our ith copy of [0, y1] go in their natural order) ∪

(u, v) | ∃j ≤ c− 1(u, (j + i)th point of g(x)) ∈ (h(i))3

v > (y2)j + x1 + 1 + i(y1 + 1)

(i.e. points of h(i) smaller than the (j+ i)th point of g(x) (which is identifiedwith (y2)j+1) are smaller in h(i+1) than anything greater than the (y2)j) ∪

∪

(u, v) | ∃j ≤ c− 1x1 + 1 + i(y1 + 1) < u ≤ (y2)j + x1 + 1 + i(y1 + 1)

(v, j + i− 1) ∈ (h(i))3

(i.e. the points of f(x+ 1) not greater than (y2)j are smaller than anythinggreater than the point of h(i) identified with (y2)j) ∪

(u, v) | (u, (i− 1)th point of g(x)) ∈ (h(i))3,x1 + 1 + i(y1 + 1) < v < (y2)0 + x1 + 1 + i(y1 + 1)

∪

∪

(u, v) | ((i+ c− 2)th point of g(x), u) ∈ (h(i))3,(y2)c−1 + x1 + 1 + i(y1 + 1) < v < x1 + 1 + (i+ 1)(y1 + 1)

.

Put g(x+ 1) = h(x1 + 2− c).We define the Fraısse limit, F , as follows. Predomain = M . For x, y ∈ M ,let z= the first point w such that maxx, y < (g(w))1. We define

e(x, y)↔ (x, y) ∈ (g(z))2,

o(x, y)↔ (x, y) ∈ (g(z))3.

As in the classical Fraısse construction, we observe that F is dense. AlsoM |=“F is infinite”, since otherwise it would be isomorphic to [0, a] for somea by Lemma 25.

By Theorem 26, F is definably isomorphic to Q(M) as any dense orderinterpreted in M is. Also, we can check that F is “M -homogeneous”, that is:every definable order-preserving h: [0, a]→M can be extended to a definableautomorphism of F . I am not going to write a detailed proof of that here,because it follows the lines of the proof of Theorem 26.

Question 9Prove internal versions of the uniqueness of the countable atomless booleanalgebra and the random graph. What do these structures look like?

51

4.2 Order-types of inner models

Theorem 29Let M |= PA and N |= PA be an inner model in M . Then

(N,<) ∼= M +Q(M)(M∗ +M).

ProofSinceN is strongly interpreted inM , by Fact (*) (page 49) there is f :M → Ndefinable in M which determines an isomorphism between M and an initialsegment of N . Hence, (N,<) ∼= M + A(M∗ +M) for some linear order A.

For a, b ∈ N , we define a ∼ b ⇔ M |= ∃x(a −N b = f(x)) if a > b anda ∼ b ⇔ M |= ∃x(b−N a = f(x)) if b < a. We interpret A in M by meansof the following formulas:

domA(x)←→ x ∈ N & ∀y < x ¬(y ∼ x)

o(x, y)←→ x <N y.

Let us prove that A is dense. Take a, b ∈ N, a < b, a 6∼ b. If[

b−a2

]belonged

to M (i.e. to the image of f) then so would b − a because f(M) is closedunder addition. Hence, a 6∼ a+

[b−a2

]6∼ b.

As A is a dense order interpreted in M , by Theorem 26, A ∼= Q(M).

Is the order-type of an inner model determined by the order-type of the outermodel?

Question 10 Find two models A,B |= PA such that A ≡ B,(A,<) ∼= (B,<) but (A+Q(A)(A∗ + A), <) 6∼= (B +Q(B)(B∗ +B), <).

Section 6.2 may give insights into how to tackle this problem.

There is nothing special about ZFC from the point of view of ModelTheory and we can deal with models of ZFC very much as we deal withmodels of other theories, such as the theory of groups or PA. Models ofZFC formalise the Completeness Theorem, hence we can talk about modelsstrongly interpreted in models of ZFC and about inner models. Also, modelsof ZFC can be strongly interpreted in models of PA + ConZFC. Thus, a hostof questions rise about order-types (and not only order-types) of modelsstrongly interpreted in another model. Let us list some of them.

Question 11 If M |= PA + Con(ZFC) and V |= ZFC is strongly interpretedin M , what can we say about (Ordinals(V ), <) in terms of (M,<)?

52

Of course, ωV will be a model of PA strongly interpreted in M , hence, byTheorem 29 has order-type M +Q(M)(M∗ +M).

Question 12 If V |= ZFC + Con(ZFC) and U |= ZFC is an inner model inV , what can we say about (Ordinals(U), <) in terms of (Ordinals(V ), <)?

Question 13

1. How many pairwise non-isomorphic elementarily equivalent models ofZFC are strongly interpreted inside M |= PA + Con(ZFC)? (If M iscountable nonstandard then there is only one (the recursively saturatedmodel whose standard system is SSy(M)). What is the order-type ofits ordinals?)

2. How many non-isomorphic order-types of ordinals of models of ZFCinterpreted in a model of PA + Con(ZFC) are there?

3. What are those order-types? How about Friedman’s problem for modelsof ZFC from the point of view of a model of PA + Con(ZFC)?

I expect Question 11 to be easier than Question 12 because models of PA aremore restrictive: Theorem 29 shows that from the point of view of a modelof PA + Con(PA), there is only ONE order-type of models of PA, namelyM +Q(M)(M∗ +M). So, from its point of view, Friedman’s problem has atrivial solution: all order-types of models of PA are isomorphic. However, aswe shall see in Chapter 6, models of ZFC believe in existence of 2λ order-typesof models of PA in each uncountable cardinality λ.

4.3 Self-similar models

What happens if some model K is strongly interpreted in N and N is aninner model in M? Is (K,<) (which is equal to N +Q(N)(N∗ +N)) a neworder-type? No! K is also strongly interpreted in M , hence, by Theorem 29,(K,<) ∼= M +Q(M)(M∗ +M) ∼= (N,<).So, N has a nice property: its inner models have order-type (N,<).

Definition 18 A model M |= PA is called self-similar if

(M,<) ∼= M +Q(M)(M∗ +M).

53

Examples of self-similar models

Proposition 30

1. Every inner model is self-similar.

2. Every countable model is self-similar, because Q(N + QZ,+, ·, <) = Q(because if a

b< c

d, then ad < bc, then 2ad < 2ad+ 1 < 2bc then

a

b=

2ad

2bd<

2ad+ 1

2bd<

2bc

2bd=c

d

i.e. Q(N + QZ) is dense) henceN + QZ +Q(N + QZ)(QZ) = N + QZ + QQZ = N + QZ.

3. No λ-like model is self-similar, because M + Q(M)(M∗ + M) alwayscontains an initial segment of cardinality |M |.

4. Every saturated model is self-similar, because Q(N + QλZ) = Qλ. (IfE < F are two subsets of Q(M) of cardinalities < λ then consider

p(x, y) = ay < bx | ab∈ E ∪ dx < cy | c

d∈ F.

p(x, y) is a type because, given a1

b1, . . . , an

bn< c1

d1, . . . , cm

dm, by the argument

from example 2 above, there are x, y ∈M such that

maxa1

b1, . . . ,

an

bn < x

y< min c1

d1

, . . . ,cmdm

.

The pair (x, y) realising p separates E and F .) Hence

N +QλZ +Q(N +QλZ)(QλZ) = N +QλZ + (Qλ)QλZ = N +QλZ.

In particular, every inner model in a saturated model is again saturatedby Pabion’s Theorem.

Theorem 31 If M is self-similar and f :M → M + Q(M)(M∗ +M) is anorder-isomorphism then there is a proper self-similar elementary extensionN M such that |N | = |M | and the diagram

Mf−→ M +Q(M)(M∗ +M)yin

yin

Nf−→ N +Q(N)(N∗ +N)

commutes.

54

The following short proof was suggested by my supervisor and replaces myoriginal one.

ProofLet L = LPA ∪ f. If M is self-similar, let us expand it to (M, f), wheref is interpreted by an isomorphism between M and M + Q(M)(M∗ + M).Any proper elementary extension (N, f) (M, f) of cardinality |M | is asrequired.

Hence any countable model has elementary self-similar extensions of all car-dinalities.

Theorem 32 If M is resplendent then (M,<) ∼= M +Q(M)(M∗ +M).

ProofEvery resplendent model is self-similar because the statement

∃ g ( g is an isomorphism between M and M +Q(M)(M∗ +M) )

is realised in all countable models. Let us show that it is really a Σ11-

statement. For that we shall need to interpret M + Q(M)(M∗ + M) inmodel M , which is done easily:

∃D (domain) ∃f ∃S (sum) ∃E (equivalence) ∃O (order)

∃ ∼ (equality in Q(M))∃ / (order in Q(M))

∃g (isomorphism between M and M +Q(M)(M∗ +M))

[∃x ∀y (D(f(y)) & O(f(y), x)) & ∀xy (x < y → O(f(x), f(y))) &

& ∀xyz(f(x) = y & O(z, y)→ ∃w f(w) = z &

(f is an order-isomorphism between M and an initial segment of D)

& ∀xyz (S(x, y) = S(y, x) & S(x, f(0)) = x & S(x, S(y, z)) = S(S(x, y), z))&

&∀x < y (E(x, y)←→ ∃z S(x, f(z)) = y) & and E is an eqivalence relation&

(i.e. E(x, y) means y −D x ∈M)

& ∀xyz(D(x) & D(y) & D(z)→ ¬O(x, x) & O(x, y)↔ ¬O(y, x) &

(O(x, y)&O(y, z)→ O(x, z)) &

(O(x, y) is a linear order on the set D)

& ∀x1 y1 6= 0 x2 y2 6= 0 (〈x1, y1〉 ∼ 〈x2, y2〉 ↔ x1y2 = x2y1) &

55

& 〈x1, y1〉 / 〈x2, y2〉 ↔ x1y2 < x2y1 &

(definition of Q(M) with equality ∼ and order /)

& ∀xyzw D(g(〈x, y〉)) &

& 〈x, y〉 ∼ 〈z, w〉 → g(〈x, y〉) = g(〈z, w〉) &

& 〈x, y〉 / 〈z, w〉 → O(g(〈x, y〉)), g(〈z, w〉)) &

(i.e. g:Q(M) −→ (D rM)/M respects equality and order)

& ¬∃v g(〈x, y〉) = f(v) &

& E(g(〈x, y〉), g(〈z, w〉))→ 〈x, y〉 ∼ 〈z, w〉 &

& ∀x(D(x)→ (∃y(E(x, y) & ∃zw g(〈z, w〉) = y) ] .

So far we know that every inner model is recursively saturated and self-similar and every resplendent model is recursively saturated and self-similartoo. Also, if M is not λ-dense then every model of PA strongly interpreted inM will have an initial segment of cardinality < λ, hence, by Proposition 10, isnot resplendent. (This actually produces a family of examples of recursivelysaturated non-resplendent models.)

Proposition 33 If N |= PA is strongly interpreted in a resplendent M |=PA then N is resplendent.

ProofSuppose Sat(x, y, a) is the formula defining in M truth for N and M |=∀x domN(x). Suppose Φ = ∃ R χ(R, b) is consistent with Th(N, b), whereR is a tuple of relation symbols (functional symbols are easily eliminated).Let us transform it into a formula Ψ = ∃ R χ(R, b, a) which in M means“there is R such that χ(R, b) holds in N” and prove that Ψ is consistent withTh(M, b, a).

Define the translation operation ˆ as follows.If ϕ(x1, . . . , xn) = R(x1, . . . , xn) then

ϕ(x1, . . . , xn) = R(x1, . . . , xn);

if ϕ(x) = R(t1(x), . . . , tn(x)), where t1, , . . . , tn are terms in LPA then

ϕ(x) = ∃ y1 . . . yn

(n∧

i=1

Sat(pu = vq, 〈yi, ti(x)〉) & R(y1, . . . , yn)

);

56

if ϕ(x) is an atomic formula in LPA then ϕ(x) = Sat(pϕ(u)q, 〈x〉);if ϕ = ψ ∨ χ then ϕ = ψ ∨ χ;if ϕ = ¬ψ then ϕ = ¬ψ;if ϕ = ∀x ψ(x) then ϕ = ∀x ψ(x).

Let A |= Th(M, b, a) be an arbitrary countable recursively saturated(hence, resplendent) model. Let B |= Th(N, b) be the inner model in Adefined by Sat(x, y, a). B is countable and recursively saturated, hence re-splendent, hence B realises Φ, hence A realises Ψ, hence, by resplendency,M realises Ψ, hence N realises Φ. Thus, N is resplendent.

Let us list some more questions about interconnections of our notions.

Question 14

1. Is there an uncountable self-similar model which is not recursively sat-urated?

2. Is there a resplendent model which is not an inner model?

3. Is there an uncountable self-similar model which is not an inner model?

4. Give an example of two elementarily equivalent non-isomorphic innermodels in M . (Of course, M will have to be uncountable.)

4.4 Inside models of ZFC

In this section we briefly investigate different kinds of interpretations of mod-els of PA in models of ZFC, study models of PA as natural numbers of somemodel of ZFC and discuss models V |= ZFC strongly interpreted insideΩ |= PA such that ωV

∼= Ω.

Throughout the section we assume N |= ConZFC.We shall encounter many different omegas: ωZFC, the Skolem term of thelanguage of ZFC expressing “the first infinite ordinal”, ω, the ‘outer’ ‘true’ω, the natural numbers of the environment we are working in, ωU , the inter-pretation of ωZFC in the universe U |= ZFC, and sometimes the symbol ω,which is later interpreted as one of the above omegas.

For convenience let LZFC = ∈, ∅, ωZFC, ×, t, i.e. already contain thesymbols for Skolem terms defining ∅, ω, direct product and disjoint unionof sets.

57

Definition 19 Let ϕ be a closed formula of LPA. Then ϕ∗ will denote thetranslation of ϕ into the language LZFC = ∈, ∅, ωZFC, ×, t relativized toωZFC, namely ϕ∗ is ϕ with

∀ x substituted by ∀ x ∈ ωZFC

∃ x substituted by ∃ x ∈ ωZFC

0 substituted by ∅

x < y substituted by x ∈ y

x+ y substituted by card(x t y),

x · y substituted by card(x× y).

Definition 20 Let ϕ be a closed formula of LZFC = ∈, ∅, ωZFC, ×, t allof whose quantifiers are relativised to ωZFC. We build the formula ϕa in thelanguage LPA as follows. Find all instances of the Skolem terms × and t in ϕand substitute expressions card(s(x)×t(y)) by s(x) ·t(y) and card(s(x)tt(y))by s(x)+ t(y). After that substitute the rest of the quantifiers ∀x ∈ ωZFC and∃x ∈ ωZFC by ∀x and ∃x and the rest of the symbols ‘∈’ by ‘<’.

For every formula ϕ ∈ LZFC with all quantifiers relativised to ωZFC,ZFC ` (ϕa)∗ ↔ ϕ and for every ψ ∈ LPA, PA ` (ψ∗)a ↔ ψ.(Notice that translation ∗ is preserved under PA-equivalence, i.e.

PA ` ϕ↔ ψ implies ZFC ` ϕ∗ ↔ ψ∗

but translation a does not have to be preserved under ZFC-equivalence. How-ever, this is not going to affect our presentation.)

Set-like models and class-like models

There are different kinds of models of PA interpreted inside U |= ZFC. Ofcourse, there is the U -standard model ωU |= Th(ωU), where Th(ωU) is ‘thetrue arithmetic from the point of view of U ’, i.e.

Th(ωU) = ϕ ∈ LPA | U |= ϕ∗.

Also, there are plenty of ‘set-like’ models of PA inside U , i.e. models N |= PAstrongly interpreted in U such that

U |= ∃x ∀y domN(y)↔ y ∈ x.

58

As Chapter 6 shows, U will believe that there are 2λ of them in cardinalityλ ∈ U such that U |= λ > ω. Note that from the point of view of U there willbe 2ωU ‘theories’ extending (PA)U = x ∈ ωU | ωU |= axiomPA(x), while forus (looking at U ‘from the outside’) the number of theories (in the standardsense) extending PA which are intersections of a definable subset of U withN will not exceed card(SSy(U)), where

SSy(U) = A ⊂ N ⊆ ωU | A is definable in U with parameters ,

so, for example, if U is countable, then only countably-many models of PAwill exist in every cardinality. (And a lot of them may turn out to be iso-morphic, even if there is no isomorphism inside U .)

Question 15Let U |= ZFC be countable, T be a completion of PA coded in U , U |= Con(T ).How many non-isomorphic models of T are interpreted in U?

By analogy with models of PA interpreted in models of PA, we may askwhether there is a model of PA whose domain is unbounded in U . Let us callsuch models ‘class-like’. (Class-like models are not ‘huge’, you cannot say‘such model is not a set but a class’. Remember: U |= ZFC is just a model.Think of it as being countable.)

Proposition 34 Let V |= ZFC. There is a class-like model of PA in V .

ProofLet us construct M |= PA by V -transfinite induction. For every ordinalα ∈ V we define Mα |= PA with domain ℵα whose satisfaction relation isSatα(x, y). Let M0 = ωV , Sat0(pϕq, x)↔ x < ω & ϕ∗(x). As ZFC ` (everymodel of PA has an elementary end-extension of any larger cardinality), wecan define

Mα+1 = ℵα+1, Satα+1(pϕq, x)↔ (x ∈Mα → Satα(pϕq, x)

(i.e. Mα+1 is an elementary extension of Mα) &

& “ϕ = ψ ∨ χ”→ (Satα+1(pϕq, x)↔ Satα+1(pψq, x) ∨ Satα+1(pχq, x) &

& “ϕ = ¬ψ”→ ((Satα+1(pϕq, x)↔ ¬ Satα+1(pψq, x) &

& “ϕ = ∃yψ(y, x)”→ ∃b Satα+1(pψq, b ∪ x))

59

(i.e. Satα+1 is a satisfaction relation ).If β is a limit ordinal, Mβ = ℵβ, Satβ(x, y) ↔ ∃α < β Satα(x, y). Nowdefine the interpretation of M as follows:

dom(x)↔ x = x, Sat(x, y)↔ ∃z Satz(x, y).

In this construction, Th(M) = Th(ωV ). This is by no means necessary. Wecould start with any nonstandard model of any complete T ∈ SSy(V ).

Corollary 35 If V |= ZFC is countable then the linear orders Q and N+QZare interpreted in V . There are bounded (‘set-like’) and unbounded (‘class-like’) interpretations.

Note that neither Q nor N+QZ can be ‘set-like’ (boundedly) interpreted in amodel of PA because, by Lemma 25, every linear order boundedly interpretedin a model of PA will have to have a last element.

Plausible arithmetics

Definition 21TA, the Theory of Arithmetic, is the following collection of statements:

TA = ϕ ∈ LPA | ZFC ` ϕ∗.

TA is the most true (and, hence, the most important) of all extensions ofPA as it contains all arithmetical statements we shall ever be able to provefrom ZFC. TA contains (PH), (KM), ConPA, ConPA+ConPA

and all first-orderconsequences of nth order arithmetic.

Definition 22 The language of nth order arithmetic, LnPA, is

LPA ∪ S(1)(u) ∪ S(k)m (u)k≤n,m∈N ∪ ∈(k)

m k≤n,m∈N

with the intended interpretation: S(1)(x) means “x is an element, i.e. a

first-order element”, S(2)m (X) means “X is an m-ary relation on first-order

elements”, i.e. a second-order element, S(k+1)m (Y ) means “Y is an m-ary re-

lation on k-order elements”. I shall usually omit S(k)m (X) and instead simply

attach indices to a letter: X(k)(u1, . . . , um).

The formula ∈(k)m (X

(k)1 , . . . , X

(k)m , Y (k+1)) means “the m-tuple (X

(k)1 , . . . , X

(k)m )

belongs to Y (k+1)”. In writing, the formula ∈(k)m (X

(k)1 , . . . , X

(k)m , Y (k+1)) will

be shortened to Y (k+1)(Xk1 , . . . , X

(k)m ).

60

Definition 23 The theory of nth order arithmetic is the first-order theory inthe language Ln

PA with the following axioms.PA− (PA without the induction scheme) ++ the second-order Induction axiom:

∀X(2)(u) ((X(0) & ∀x (X(x)→ X(x+ 1))→ ∀x X(x))

+ the Comprehension scheme:for every nth order formula ϕ(X(k)), k < n, possibly containing parametersof any order, but having the (only) variable X(k) free,

∃Y (k+1)∀X(k) (Y (X) ↔ ϕ(X)).

It is known that nth order arithmetic for n ≥ 2 proves ConPA (see [31]).

Observation 36TA proves all first-order (LPA) consequences of nth order arithmetic.

ProofThere is an obvious translation ∗ of nth order formulas into LZFC:if ϕ is first-order, put ϕ∗ = ϕ∗ in the old sense;if ϕ contains higher-order relation symbols then substitute∀X(2)(u1, . . . , um) by ∀r (r ⊆ ωm)→and X(2)(x1, . . . , xm) by (x1, . . . , xm) ∈ r;...∀X(k+1)(u1, . . . , um) by ∀s (s ⊆ (P(k)(ω))m)→and X(k+1)(x1, . . . , xm) by (x1, . . . xm) ∈ s.

The translations of the axioms of PA− + Induction follow from ZFC, ob-viously. The translations of instances of the Comprehension scheme followfrom the comprehension scheme of ZFC. Proofs translate and transfer di-rectly in the same way.

By Godel, ConZFC 6∈ TA. Let us show that TA is recursively aximomati-zable. The set of all theorems of ZFC whose quantifiers are relativised toωZFC is recursively enumerable and the map ϕ → ϕa is recursive, henceTA = ϕ ∈ LPA | there is ψ ∈ LZFC such that ψa = ϕ is recursively enu-merable, hence, by Craig’s trick, TA is recursively axiomatizable. Hence, byGodel’s Theorem, TA has 2ω completions.

Question 16 (probably a very hard one)Find a mathematical statement in the language LPA independent of TA.

61

(By a ‘mathematical’ statement I mean something different from ConZFC orConZFC+ ConZFC

. Remember the story of (PH) being the first ‘mathematical’statement known to be independent of PA.)

Let us call consistent extensions of TA plausible arithmetics. Every‘true arithmetic’ Th(ωV ) (where V |= ZFC) is plausible, obviously. Now,inside every universe V |= ZFC, the picture looks like this:

s s s

((((((((((((

hhhhhhXXXXXXXXX

(((((((((

hhhhhhhhPPPPPPP

PA TA

plausible arithmeticsextensions of PA

weak

arithmetics· · · Th(ωV )

‘truearithmetic’

The notion of a ‘true arithmetic’ is relative and depends on the universe.Also, for every plausible arithmetic T , there is a universe U |= ZFC suchthat ωU |= T .

Proof Let T be plausible. Then ZFC +ϕ∗ | ϕ ∈ T is consistent. LetV |= ZFC +ϕ∗ | ϕ ∈ T. Then ωV |= T .

Now, the question is: is any model of any plausible arithmetic an ωV in someuniverse V |= ZFC?

Existence of an ω-standard model

If U |= ZFC +“there exists a strongly inaccessible cardinal” then obviouslyV := x ∈ U | x < the first strongly inaccessible cardinal is a model of ZFCsuch that ωU

∼= ωV . I.e., assuming “there is a strongly inaccessible cardinal”there is a model of ZFC whose natural numbers are N. Is ZFC + ConZFC

enough to construct such a model? I don’t know.

Fact (Wilmers)If U |= ZFC and ωU is nonstandard then ωU is recursively saturated.

Sketch of the proofLet p(x) = ϕi(x, a)i∈N be a recursive set of formulas such that for alln ∈ N, ωU |= ∃x

∧i<n ϕi(x, a), i.e. p(x) is a type. Then for all n ∈ N,

U |= ∃x ∈ ωU (ωU |=∧

i<n ϕi(x, a)) using Tarski’s definition of truth in U .Then

A = n ∈ ωU | U |= ∃x ∈ ωU (ωU |=∧i<n

ϕi(x, a))

is definable in U and contains N, hence contains a nonstandard point, b ∈ωU r N. Hence U |= (ωU |=

∧i<b ϕi(x, a)), hence ωU |=

∧i<b ϕi(x, a), that is

62

p(x) is realised in ωU .

Thus, if U |= ZFC is strongly interpreted in V |= ZFC then U and V wouldusually drastically disagree about ωU . Model U will think that ωU is theset of natural numbers N while model V will think that ωU is a recursivelysaturated model of PA.

Question 17 Prove that if Ω |= TA is recursively saturated then there isV |= ZFC such that ωV

∼= Ω.

Proposition 37If Ω |= TA is resplendent then there is U |= ZFC such that ωU

∼= Ω.

ProofConsider Ψ = ∃E ∃f ∃ Sat Φ(E, f, Sat) =

∃E(x, y), f(x), Sat(x, y)

“(Ω, E) |= ZFC ” &

& ∀x E(f(x), ω) & ∀y (E(y, ω)→ ∃x f(x) = y) &

& ∀xyz [x+ y = z → card(f(x) t f(y)) = f(z) &

& x · y = z → card(f(x)× f(y)) = f(z) & x < y → E(f(x), f(y))]

is a Σ11-statement (where ω, card,× and t are expressions in terms of E).

Let us show that “(Ω, E) |= ZFC ” is a Σ11-statement. As ZFC is recursively

axiomatised, there is a point z ∈ Ω coding the set of all axioms of ZFC. Weshall identify each formula in the language E with its Godel number. Theexpression “ϕ = ψ ∨χ” will mean “ϕ is the Godel number of the disjunctionof two formulas with Godel numbers ψ and χ”. Let θn(x) =

x > n & ∀i < x Sat((z)i, [ ]) &

& ∀ϕ ∀a (form(ϕ) & ϕ has len(a) free variables −→(“ϕ = ψ ∨ χ” → (Sat(ϕ, a)↔ Sat(ψ, a) ∨ Sat(χ, a))) &

(“ϕ = ¬ψ” → (Sat(ϕ, a)↔ ¬ Sat(ψ, a))) &

(“ϕ = ∃x1ψ(x1, x)” → (Sat(ϕ, a)↔ ∃b Sat(ψ, 〈b, a〉)))).As θn(x)n∈N is a recursive set of formulas, by Kleene’s Theorem, thereis a single Σ1

1-sentence Θ(x) such that in any M |= PA, M |= ∀x (Θ(x) ↔∧n∈N θn(x)). Thus, “(Ω, E) |= ZFC ” is expressed by the Σ1

1-statement ∃xΘ(x).Hence, Ψ is a Σ1

1-statement, which means “there is a model of ZFC, whosenatural numbers are Ω”. Since Th(Ω) is plausible then (as proved above)there is a model M |= Th(Ω) such that there is V |= ZFC of cardinalitycardM with ωV

∼= M .Thus Ψ is consistent with Th(Ω), hence, by resplendency, is realized.

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Models of ZFC inside models of PA

Theorem 38 If Ω |= PA + Con(ZFC) is resplendent then there is V |= ZFCstrongly interpreted in Ω such that (ωV , <) ∼= (Ω, <) and SSy(ωV ) = SSy(Ω).

ProofBy Arithmetised Completeness Theorem, there is a formula Sat(x, y), whichdefines a satisfaction class for a model V |= ZFC in Ω. Let ωV be the pointx ∈ Ω such that Ω |= Sat(pv = ωq, x). Let

Φ = ∃ f

∀x Sat(pv1 ∈ v2q, 〈f(x), ωV 〉) &

& ∀x1x2 (x1 6= x2 → f(x1) 6= f(x2)) &

& ∀y(Sat(pv ∈ ωq, y)→ ∃x f(x) = y)

(i.e. f is a bijection from Ω to ωV )

& ∀xy (x < y → Sat(pv1 ∈ v2q, 〈f(x), f(y)〉))

(i.e. f is an order-isomorphism).Φ is consistent because in any countable M ≺ Ω,

(ωVM, <) ∼= (M,<) ∼= N + QZ.

Hence, by resplendency, Φ is realised in Ω, i.e. ωV is order-isomorphic to Ω.Also, SSy(ωV ) = SSy(Ω) because Ω ≺∆0 ωV .

However, in order to obtain so desired ωV∼= Ω we shall have to use the

Reflection principles.

Definition 24Let M |= PA. Let M |=“f(a) = the ath point x such that axiomPA(x)”. Forevery a ∈M define Reflectiona(PA) to be the scheme

ϕ→ Con(∧i<a

f(i) ∧ ϕ) | ϕ ∈ LPA.

Reflection(PA) is the scheme ϕ→ ConPA(ϕ) | ϕ ∈ LPA.

Obviously,M |= PA + Reflection(PA) if and only ifM |= PA + Reflectiona(PA)for all a ∈M . It is not surprising that conditions of Theorem 38 even withM |= TA + Con(ZFC) do not imply ωV

∼= Ω due to the following fact.

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Observation 39 (modification of a result by Kaye and Kotlarski [15])Let M |= PA be recursively saturated. Then there is N |= ThM strongly in-terpreted in M if and only if there is a ∈MrN such that M |= Reflectiona(PA).

ProofSuppose there is N |= ThM strongly interpreted in M . For every n ∈ N,ϕ ∈ LPA,

M |= ϕ→ Con(∧i<n

f(i) ∧ ϕ)

because otherwise (i.e. if for some n ∈ N, ϕ ∈ LPA,M |= ϕ∧¬Con(∧

i<n f(i)∧ϕ)) there would be no model of f(0) ∧ . . . ∧ f(n) ∧ ϕ strongly interpreted inM , in particular no model of ThM . Hence

p(x) = x > nn∈N ∪ ϕ→ Con(∧i<x

f(i) ∧ ϕ) | ϕ ∈ LPA

is a recursive type, hence, by recursive saturation, is realised by, say, a ∈M r N. Then M |= Reflectiona(PA).

Conversely, assume M |= Reflectiona(PA) for some a > N. Let e ∈ Mcode ThM . By reflection, for every standard n, M |= Con(

∧i≤n(e)i), hence,

by overspill, M |= Con(∧

i≤α(e)i) for some nonstandard α < len(e). Now,by Arithmetised Completeness Theorem, there is a model N |=

∧i≤α(e)i

strongly interpreted in M . As α > N, N |= ThM .

It is possible to weaken the assumption of recursive saturation (we only needthat ThM is coded and p(x) is realised) but in view of Wilmer’s Theoremabove, we consider only recursively saturated models anyway. So, clearly,Ω |= TA + Reflectiona(PA) for some a > N is a necessary condition forωV∼= Ω.

Question 18 Prove that if Ω |= TA + Reflection(PA) is resplendent thenthere is U |= ZFC strongly interpreted in Ω such that ωU

∼= Ω.

Let us discuss a possible solution to this question.

Definition 25 Let Ω |= PA be nonstandard, T be a set of first-order sen-tences in any language coded by t ∈ Ω. Then TΩ

t is the set of all (standard)Ω-consequences of (slightly overspilled) T , i.e. statements ϕ ∈ LT such thatΩ |= PrPC(

∧i<len(t)(t)i → ϕ).

This definition is very sensitive to the choice of parameter t (different pa-rameters t in general lead to different sets TΩ

t ). But once t is fixed we shallomit it from our notation and write TΩ.

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Let M |= PA be nonstandard, z ∈M r N code the set of all theorems ofZFC (again, this is possible because the set of all theorems of ZFC is recur-sively enumerable) so that M |= ∀i < len(z) formZFC((z)i) (this is possibleby overspill). From the point of view of M there are at least three variantsof TA:

1. True TA = ϕ ∈ LPA | N |= PrZFC(ϕ∗) =: TAN .TAN is recursively enumerable, hence is definable in M . Let

M |= e = the least element x such that

∀i < len(z) ( (z)i is relativised to ω → ∃j < len(x) ((z)i)a = (x)j)) &

&∀j < len(x)∃i < len(z)((z)i is relativised to ω and ((z)i)a = (x)j.

Obviously, e codes TAN.

2. TAM = ϕ ∈ LPA | M |= PrPC(∧

i<len(e)(e)i → ϕ) - the set of all

M -consequences of (slightly overspilled) TAN.

3. (ZFCM)a = ϕ ∈ LPA | M |= PrPC(∧

i<len(z)(z)i → ϕ∗) - the set of

all arithmetical sentences which are M -consequences of (slightly over-spilled) ZFC.

(There are also other (non-definable) versions, such as⋃

n∈ωϕ ∈ LPA |M |=PrPC(

∧i<n(e)i → ϕ) which reflect M -proofs more accurately and may be

useful in the future, but I am not going to mention them again.)Obviously, TA ⊆ TAM ⊆ (ZFCM)a. Also, in N all three notions coincide:TA = TAN = (ZFCN)a, by the definition of TA.

Observation 40 If M |= PA +Π1 Th N then for any z ∈M coding ZFC ande chosen as above, TA = TAM = (ZFCM)a.

ProofLet M |= ∃xψ(x) where ψ(x) ∈ ∆0. If N |= ¬∃xψ(x) then ∀x¬ψ(x) ∈Π1 Th N then M |= ∀x¬ψ(x), contradiction. Hence, N |= Σ1 ThM .

Let ϕ(x) be formPA(x) & PrZFC(x∗). If M |= n ∈ (ZFCM)a then M |=ϕ(n) (if n follows from an M -finite fragment of ZFC then it follows from fullZFC) then, as N |= Σ1 ThM , N |= ϕ(n), that is n ∈ TA.

Question 19 Is it true that in every M |= PA there is z coding ZFC suchthat TAM = (ZFCM)a?

Theorem 41 Let Ω |= (ZFCΩ)a be resplendent. Then there is U |= ZFCstrongly interpreted in Ω such that ωU

∼= Ω.

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Note that any resplendent model of TA +Π1 Th N satisfies the conditions ofTheorem 41.

ProofLet ZFC be coded by z ∈ ΩrN, each (z)i denoted by ψi. Let Th Ω be codedby a ∈ ΩrN, each (a)i denoted by ϕi. Let us show that for arbitrary n ∈ N,

Ω |= Con(ψ1 ∧ . . . ∧ ψn ∧ ϕ∗1 ∧ . . . ∧ ϕ∗n).

Suppose Ω |= PrPC(ψ1∧ . . .∧ψn → ¬(ϕ∗1∧ . . .∧ϕ∗n)). Then, ¬(ϕ∗1∧ . . .∧ϕ∗n) ∈ZFCΩ, hence, as Ω |= (ZFCΩ)a, ¬(ϕ1∧. . .∧ϕn) ∈ Th Ω, contradiction. Hence,Ω |= Con(ψ1 ∧ . . . ∧ ψn ∧ ϕ∗1 ∧ . . . ∧ ϕ∗n).

Now, by overspill, there is α ∈ Ω such that

N < α < minlen(a), len(z)

and Ω |= Con(∧

i<α ψi ∧∧

i<α ϕ∗i ). By the Arithmetised Completeness The-

orem, there is U |= ZFC +(Th Ω)∗ strongly interpreted in Ω by means of asatisfaction relation Sat(x, y). (Notice that since we have just proved thatΩ |= (ZFCΩ)a implies Ω |= Con(Th Ω), and hence, by Observation 39 there isα > N such that Ω |= Reflectionα(PA), the condition “Ω |= (ZFCΩ)a” shouldbe regarded as one from the family of Reflection principles.)

As in Theorems 37 and 38 above, let

Ψ = ∃ f ( ∀x Sat(pv1 ∈ v2q, 〈f(x), ωU〉) &

& ∀x1x2 (x1 6= x2 → f(x1) 6= f(x2)) &

& ∀y (Sat(pv ∈ ωq, y)→ ∃x f(x) = y)

(i.e. f is a bijection from Ω to ωU) &

& ∀xyz (x+ y = z → Sat(pu t v ∼= wq, 〈f(x), f(y), f(z)〉)) &

& ∀xyz (x · y = z → Sat(pu× v ∼= wq, 〈f(x), f(y), f(z)〉)) ) .

Let us prove that Ψ is consistent. Let M be any countable recursively sat-urated elementary extension of ClΩ(a, z) and let V |= ZFC +(Th Ω)∗ be themodel defined by Sat(x, y). By construction, Th(ωV ) = Th Ω. Also, ωV isstrongly interpreted in M , hence SSy(ωV ) = SSy(M) and ωV is recursivelysaturated. Hence ωV

∼= M . Thus, Ψ is consistent, hence by resplendency, isrealised. Hence, ωU

∼= Ω.

But life would become much easier if Question 19 were settled positively, i.e.if TAM = (ZFCM)a in every model M .

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Observation 42If Ω |= TA + Reflection(PA) is resplendent and TAΩ = (ZFCΩ)a then thereis U |= ZFC strongly interpreted in Ω such that ωU

∼= Ω.

(Thus, the positive solution of Question 19 would provide a characterisationof the theory of such models: there is a model Ω of T ⊃ PA strongly interpret-ing some U |= ZFC with ωU

∼= Ω if and only if T ⊇ TA + Reflection(PA).)

ProofBy reflection, Ω |= Con(Th Ω). As Th Ω is complete and Ω |= Con(Th Ω),(Th Ω)Ω = Th Ω. TA ⊂ Th Ω, hence TAΩ ⊂ (Th Ω)Ω = Th Ω. As (ZFCΩ)a =TAΩ, Ω |= (ZFCΩ)a and we find ourselves in the conditions of Theorem 41.

Comparison

Can methods developed for models of PA benefit the study of models of ZFC?I hope so. On the superficial level, models of both theories are very similar:

1. their ‘ordinals’ are linearly ordered, though a model of PA always co-incides with its ordinals;

2. every non-empty definable subset has the least element;

3. ω is always an initial segment;

4. Standard System can be defined, and is always a Scott Set;

5. completeness is formalised inside a model.

Can a unified theory of such models be developed?However, at some point differences occur as well. A conservative extension

of a model of PA is always an end-extension. A conservative extension of amodel of ZFC is always cofinal (i.e. does not add ordinals above all existingordinals)[6]. Every model of PA has an elementary end-extension. There aremodels of ZFC having no elementary end-extensions (i.e. extensions with allnew ordinals being greater than the existing ones) [6].

Important notice: most theorems in this section would hold if we replacedZFC by any other foundational theory. We only assumed consistency of ZFCand the fact that ZFC `“ω ` PA”.

Before closing this section, let us list some open problems which are simplyreformulations of established results or established open problems for the thecase of ZFC.

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Question 20 What are order-types of ordinals of countable models of ZFC?How many of them are there? Can there be some classification result?

General problem:

Question 21 What are the order-types of ordinals of models of ZFC? Howdoes the order-type of ordinals of a saturated model look like? How manymodels with the saturated order-type are there?

Question 22 Do classes of order-types of ordinals of models of differentextensions of ZFC coincide?

Problem 21 asks for a version of Pabion’s Theorem, Problem 22 is a refor-mulation of Friedman’s Problem for ZFC.

This area (interconnections between models of PA and ZFC) is very richand profound. It deals with the very nature of mathematical objects and iswaiting to be explored.

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Chapter 5

Canonical orders

In this chapter we consider the simplest possible order-types of models ofPA: the combinations of Qλs and ordinals.

This chapter is a relative failure. It is short, and is called a “chapter”because I spent more time and effort on it than on any other two chapters.Also, I believe that the notion introduced here will play an important rolein the future. To guarantee existence of Qλs for regular λs I assume (GCH)throughout the chapter.We shall use λ-good ultrafilters. For the definition, see [3].

Fact 1 For every α, there is an α+-good ultrafilter over α.Fact 2 Ultraproducts over λ-good ultrafilters are λ-saturated.

Definition

Definition 26 The class of canonical orders is the smallest collection oflinear orders such that

1. N is a canonical order;

2. if A is a canonical order then for any regular λ ≥ |A|, any µ ≤ λ+,A+ µQλ(A

∗ + A) is a canonical order.

Let us list a first few canonical orders:N,N + QZ (A = N, λ = ω, µ < ω or µ = λ),N +QλZ (A = N, λ = λ, µ < ω or µ = λ),N + ω1QZ (A = N, λ = ω, µ = ω1),N + QZ +QλQZ (A = N + ω1QZ, λ = λ, µ < ω or µ = λ),N + ω1QZ +Qλ(ω

∗1 + ω1)QZ (A = N + ω1QZ, λ = λ, µ < ω or µ = λ),

N + ωQλZ (A = N, λ = λ, µ = ω),

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N + ω1QλZ,. . .N + λ+QλZ.

Note that sometimes the same canonical order can be obtained by dif-ferent means. If A = N + QλZ and B = N then A + Qλ(A

∗ + A) ∼=B +Qλ(B

∗ +B). Also, N + ωQℵω+1Z ∼= N + ℵωQℵω+1Z.The notion of canonical order arises naturally when we start thinking

about cells and saturation. I believe that for every canonical A and anyT ⊇ PA there is M |= T such that (M,<) = A. Let us prove two particularcases of it.

First example

Proposition 43 If T ⊇ PA is complete and λ1, λ2, . . . , λn are successorcardinals such that 2ω ≤ λ1 < λ2 < . . . < λn then there is M |= T oforder-type

N +Qλ1Z +Qλ2Qλ1Z + · · ·+Qλn . . . Qλ2Qλ1Z.

ProofLet M0 be the model of T of order-type N + QλnZ (unique by Pabion’sTheorem). Suppose 0 ≤ i < n and Mi is constructed having order-type

N +Qλn−iZ +Qλn−i+1

Qλn−iZ + · · ·+Qλn . . . Qλn−i+1

Qλn−iZ.

As λn−i−1 is a successor, there is a λn−i−1-good ultrafilter D on λ−n−i−1 (whereλ−n−i−1 is the predecessor of λn−i−1). Put Mi+1 =

∏D Mi.

Mi+1 is λn−i−1-saturated because D is λn−i−1-good.∏

D N is an initialsegment of Mi+1 because, as λn−i is regular, no sequence of λ−n−i−1 points ofMi r N can reach the bottom of Mi r N. Also, (

∏D N, <) = N + Qλn−i−1

Zbecause it is saturated. Also,∏

D

(Mi r N)/∏D

N = Qλn−i+ · · ·+Qλn . . . Qλn−i

because if A,B ⊂ Qλn−i+j. . . Qλn−i

, |A|, |B| < λn−i, A < B, then, as λn−i

is regular, A and B can be separated by a point of Qλn−i+j. . . Qλn−i

. ThusMi+1 has order-type

N +Qλn−i−1Z +Qλn−i

Qλn−i−1Z + · · ·+Qλn . . . Qλn−i

Qλn−i−1Z.

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Second example

Proposition 44 Let T ⊇ PA. Then for every regular λ and every µ ≤ λ+

there is a model of T of order-type N + µQλZ.

ProofLet us first prove that if M∗ is the saturated model of cardinality λ and I isa proper initial segment of M∗ then there is a proper initial segment K suchthat

I ⊂ K ∼= M∗.

Let N be any elementary end-extension of M∗, a ∈M∗ r I. By universality,there is an elementary embedding f :N → M∗ such that f(a) = a. Observethat f(M∗) ≺M∗ because M∗ ≺ N . Let

K = x ∈M∗ | ∃y > x, y ∈ f(M∗).

By Gaifman’s Splitting Theorem, K is an elementary submodel of M∗. Also,K is a proper initial segment of M∗ because it is bounded by f(b) whereb ∈ N r M∗. I ⊂ K because a ∈ K. By Pabion’s Theorem, as (K,<) ∼=N +QλZ, K is saturated, hence isomorphic to M∗.

Let M0 be the saturated model embedded as an initial segment into M∗.Suppose at stage α < λ+ we already have an initial segmentMα |= T of order-type N + αQλZ. If Mα is a proper initial segment let Mα+1 be a saturatedelementary end-extension of Mα that is again a proper initial segment. IfMα = M∗, let M∗ M∗ be a saturated end-extension of M∗ and chooseMα+1 as before.

Now, Mµ =⋃

α<µMα is a model of order-type N + µQλZ as required.

Pabion’s Theorem and the above proof may lead us to the suggestion thatthe model of order-type N + µQλZ is unique for all µ ≤ λ+.

Counterexamples for µ = ω

Proposition 45 For any regular λ there are three pairwise non-isomorphicelementarily equivalent models of order-type N + ωQλZ.

ProofLet T be a completion of PA, M |= T have order-type N +QλZ. Let [Cl ∅]be the convex closure of ClM ∅ in M . By Gaifman’s Theorem, [Cl ∅] ≺ M .As cf([Cl ∅]) = ω and lcf([Cl ∅] r N) = λ, ([Cl ∅], <) ∼= N + ωQλZ. Also,[Cl ∅] is not isomorphic to Mω from Proposition 44 because Cl ∅ is boundedin Mω and unbounded in [Cl ∅].

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Let a ∈ M , a > Cl ∅ and [Cl(a)] be the convex closure of ClM(a) in M .Then [Cl(a)] is not isomorphic to Mω because for every c ∈ Mω, ClMω(c)is bounded in Mω. Also, [Cl(a)] is not isomorphic to [Cl ∅] because Cl ∅ isbounded in [Cl(a)].

I believe that by varying the type of a we can obtain many more pairwise non-isomorphic models of the form [Cl(a)] (and, hence, of order-type N+ωQλZ).

Question 23 How many models of order-type N + ωQλZ are there?

Question 24 Are there non-isomorphic models of order-typeN + ω2Qω1Z? What happens in the general case?

A spectrum-like situation will appear in the general case. For a canonicalorder A, define i(A) = the number of non-isomorphic models of T ⊇ PA oforder-type A. We already know: i(N) = 1, i(N + QZ) = 2ω, i(N +QλZ) =1, i(N + ωQλZ) ≥ 3. Theorem 50 will show that i(N + ω1QZ) = 2ω1 .

In order to construct examples of non-isomorphic models of order-typeN + µQλZ we may need to solve the following problem.

Question 25 Find four models, M1, M1, M2, M2, all copies of the satu-rated model, such that M1 is an elementary end-extension of M1, M2 is anelementary end-extension of M2, but there is no isomorphism f : M1 → M2

such that f(M1) = M2.

Suggestion: try different lower cofinalities of M1 rM1 and M2 rM2.

Future research

Question 26Prove that there is a model of PA of every canonical order-type.

The success in proving Propositions 43 and 44 is due to the fact that wecould start with a saturated model and build the model of desired order-typearound it. However, the situation is very different if our canonical order-typehas a countable initial segment. We can not start with a saturated modelbecause if M is saturated and N ∆0 M then SSy(N) = P(ω), hence, byLemma 3 each initial segment of N is uncountable.

The first two obstacles we encounter are order-types N + QZ + Qω1QZand N + ω1QZ +Qω1(ω

∗1 + ω1)QZ.

Question 27Prove that there is a model of PA of order-type N + QZ +Qω1QZ.

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Question 28 Prove that there is a model of PA of order-typeN + ω1QZ +Qω1(ω

∗1 + ω1)QZ.

If we attempt to construct models of these order-types as chains, we mayneed to solve the following problem.

Question 29Prove that there is a countable M |= PA such that for every countable end-extension N M and subsets A,B ⊂ N/M , A < B, there is K N suchthat there is c ∈ K such that A < c < B and x ∈ K | ∃y ∈M, y ≥ x = M .

Can arithmetic saturation (see Chapter 7) in the case ThM = Th N helpconstruct by back-and-forth an elementary embedding f :N → N fixing Msuch that there is a point x ∈ N such that f(A) < x < f(B)? This is afuture project.

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Chapter 6

Counting

6.1 Order-types of cardinality λ

Theorem 46 Let λ be an uncountable cardinal. For any consistent theoryT ⊇ PA there are 2λ order-types of models of T of cardinality λ.

Lemma 47 If M |= PA then for every uncountable regular λ ≥ cardM ,there are 2λ order-types of elementary end-extensions of M of cardinality λ.

ProofWe know from Corollary 15 that every model M has an end-extension N1 M such that

Υ

(N1 r M) = ω, cardN1 = cardM and an end-extensionN2 M such that

Υ

(N2 r M) = ω1 and cardN2 = maxcardM,ω1. LetAα | α < λ be a partition of λ into λ stationary subsets. For every setW ⊆ λ, define the model MW as the limit of the following elementary chain.Let MW

0 = M . Suppose for all γ < α, MWγ are already defined. Denote⋃

γ<αMWγ by Kα. Let

MWα =

an end-extension N Kα such that

Υ

(N rKα) = ωand cardN = cardKα, if α ∈

⋃β∈W Aβ

an end-extension N Kα such that

Υ

(N rKα) = ω1

and cardN = maxcardKα, ω1, if α 6∈⋃

β∈W Aβ.

Now we are going to prove that if W and U are different subsets of λthen (MW , <) 6∼= (MU , <). Suppose f : (MW , <) → (MU , <) is an order-isomorphism. Introduce the functions i, j : λ→ λ as

i(α) = minγ ∈ λ | f(KWα ) ⊆ KU

γ

j(β) = minδ ∈ λ | f−1(KUβ ) ⊆ KW

δ .

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Let Fix(i) and Fix(j) be the sets of all fixed points of i and j respectively.As for every limit ordinal µ,

⋃α<µK

Wα = KW

µ and, hence⋃

α<µ f(KWα ) =

f(⋃

α<µKWα ) = f(KW

µ ), thus supα<µ i(α) = i(µ), i.e. i is continuous. Also, iis unbounded, hence normal, hence Fix(i) is a club. By the same argument,Fix(j) is a club.

Let γ ∈ λ be such that γ ∈ W ⇔ γ 6∈ U . As Aγ is stationary, there is apoint ξ ∈ λ such that

ξ ∈ Fix(i) ∩ Fix(j) ∩ Aγ.

As ξ ∈ Fix(i) ∩ Fix(j), f |KWξ

: (KWξ , <) ∼= (KU

ξ , <) is an order-isomorphism.

However, by the construction of MWξ and MU

ξ , neither (MWξ r KW

ξ ) canbe mapped onto an initial segment of (MU

ξ r KUξ ) nor (MU

ξ r KUξ ) can be

embedded into an initial segment of (MWξ r KW

ξ ). Contradiction. Hence,(MW , <) 6∼= (MU , <).

Theorem 48 Let M |= PA. Then for every λ > maxcard(M), ω1, thereare 2λ order-types of λ-like elementary end-extensions of M .

ProofCase 1: λ is regular is given by Lemma 47. Notice that at every stage α,cardMW

α = cardKWα , hence

⋃α<λM

Wα is λ-like.

Case 2: λ is singular, cf(λ) = µ.If there is γ < λ such that for all δ such that γ ≤ δ < λ,

2δ = 2γ

then, by Bukovsky–Hechler theorem (see page 8), 2γ = 2λ. Now, given aregular δ such that card(M) < γ ≤ δ < λ, by Lemma 47, we have 2δ = 2λ

order-types of elementary end-extensions of M of cardinality δ. Extend themto λ-like models. No pair of their order-types will be isomorphic becauseinitial δ-cells have to map onto initial δ-cells.Now, suppose for every γ < λ there is δ such that γ < δ < λ and 2γ < 2δ <2λ. Then there is an increasing sequence λii<µ such that: λ =

∑i<µ λi,

λ0 = ∅, for all i, λi is regular and

2Σj<iλj < 2λi < 2λ.

For every g ∈∏

i<µ 2λi = g:µ → 2λ | g(i) ∈ 2λi, we shall define M g

as an elementary chain so that f 6= g implies (M f , <) 6∼= (M g, <). Ascard(

∏i<µ 2λi) = 2λ, we shall have 2λ of those models M g.

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Let M∅ = M . Suppose for i ∈ µ and for all f ∈∏

j<i 2λj , M f are

defined. Denote limj<i λj by δ. Notice that card(∏

j<i 2λj) = 2δ < 2λi .

Enumerate all elements of∏

j<i 2λj as fk | k ∈ 2δ.

Let ` < 2λi and suppose that for all k ∈ 2δ, all j < `, M fk,j has beendefined so that

(M fk,j, <) 6∼= (M fn,m, <) if k 6= n or j 6= m.

Define M fk,`k∈2δ to be a collection of λi-like models such that for everyk ∈ 2δ, M fk,` is an elementary end-extension of M fk and

(M fk,j, <) 6∼= (M fn,`, <) if k 6= n or j 6= `.

It can be done by Lemma 47 because∣∣M fk,j | j < `, k ∈ 2δ∣∣ = max2δ, ` < 2λi .

Hence, we have defined M g for all g ∈∏

j<i 2λj × 2λi =

∏j≤i 2

λj .

For every f ∈∏

i∈µ 2λi , define

M f =⋃i∈µ

M fi.

Suppose γ < µ, f(γ) 6= g(γ) and h: (M f , <)→ (M g, <) is an order-isomorphism.Then

⋃i≤γ M

fi ∼=⋃

i≤γ Mgi. Contradiction.

Corollary 49 Let λ be uncountable.

1. If λ is regular then there are 2λ order-types of λ-dense models.

2. Let λ be regular and (λ1 λ2 · · · λε1 ε2 · · · εµ

)be a sort. If εµ = 0 then there are 2λ order-types of models of this sort.If εµ = 1 then there are 2(λ+) order-types of models of this sort.

3. Let (λ1 λ2 · · ·ε1 ε2 · · ·

)be a λ-like sort, λ > ω1. Then there are 2λ order-types of models ofthis sort.

4. Let T ⊇ PA be consistent. Then there are 2λ order-types of self-similarmodels of T of cardinality λ.

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Proof

1. Take M0 to be λ-dense and apply Lemma 47.

2. Consider any model of the sort(λ1 λ2 · · · λε1 ε2 · · · 0

)(it exists by Theorem 13). Consider 2λ order-types of its elementaryend-extensions of cardinality λ (use Lemma 47) if εµ = 0 and 2(λ+)

order-types of its λ+-like end-extensions if εµ = 1.

3. Follows the lines of the proof of Theorem 48.

4. Consider the new language L ∪ a. The theory S = PA + “a codesT” + Con(T ) in the language L ∪ a is consistent. Consider 2λ ordernon-isomorphic λ-like models of S. By applying the Arithmetised Com-pleteness Theorem to each of them, we obtain 2λ inner models (hence,self-similar) of T of which no pair can be order-isomorphic because theunions of their initial δ-cells for δ < λ are not order-isomorphic.

Question 30 Extend Corollary 49 (1) and (2) to the singular case.

Note, that our use of Corollary 15 in this section was essential. Lemma47, Theorem 48 and Corollary 49 are not true for models of ZFC becausethere are models of ZFC without elementary end-extensions [6].

Question 31 Prove that there are 2λ order-types of resplendent models ofPA of cardinality λ.

For this you may want to use Shelah’s methods of constructing many non-isomorphic structures in a pseudoelementary class and the following theorem:

Theorem (J. Schmerl [24])Let T ⊇ PA. Then there is an indiscernible type Σ such that wheneverN |= T is generated by a set I of indiscernibles having indiscernible-type Σand having no last element then N is resplendent.

6.2 ω1-like models

We already know that all ω1-like models of PA have the same order-typeN + ω1QZ.

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Theorem 50For any T ⊇ PA there are 2ω1 models of T of order-type N + ω1QZ.

ProofWe can easily obtain 2ω ω1-like models by taking 2ω countable models non-embeddable in each other and considering their end-extensions of order-typeN + ω1QZ. Let us find 2ω1 of them.

Firstly, by Corollary 15, if N |= PA and p(x) is a Gaifman type then thereis an end-extension K1 N such that x ∈ K1 | p(x) is unbounded below inK1 rN . Let us prove that there is an end-extension K2 N such that x ∈K2 | p(x) is not unbounded below in K2 rN . Let K2 be an end-extensionobtained from N by means of a Gaifman type indiscernible over N , (see proofof Theorem 13 or [14], pages 97-101), i.e. K2 = Cl(N ∪c), where c satisfiessuch a type p(x). Let t(c, a) ∈ K2 rN , t(c, a) < c, a ∈ N . Let us show thatthen there is a formula ϕ(x, a) such that K2 |= ϕ(c, a) ∧ ¬ϕ(t(c, a), a).

Define t(u)(x, a) as the uth iteration of t, i.e.

K2 |= t(0)(x, a) = x ∧ ∀ u t(u+1)(x, a) = t(t(u)(x, a), a).

Put g(x, a) = min u (t(u+1)(x, a) ≥ t(u)(x, a)). Obviously,

t(x, a) < x→ g(x, a) = g(t(x, a), a) + 1.

Hence, g(c, a) = g(t(c, a), a) + 1, which implies that one of them is even andthe other is odd. Hence c and t(c, a) realise different types. Thus, K2 is anelementary end-extension of N such that x ∈ K2 | p(x) is not unboundedbelow in K2 rN .

Let Aγ | γ < ω1 be a partition of ω1 into ω1 stationary subsets. LetW ⊆ ω1, M

W0 |= T, cardMW

0 = ω, p(x) be a Gaifman type. Suppose for alli < α, MW

i is already defined. Denote⋃

i<αMWi by KW

α . Define

MWα =

an end-extension N KWα such that N is countable and

x ∈ N rKWα | p(x) is unbounded below in N rKW

α

if α ∈⋃

γ∈W Aγ

an end-extension N KWα such that N is countable and

x ∈ N rKα | p(x) is not unbounded below in N rKWα

if α 6∈⋃

γ∈W Aγ.

Put MW =⋃

α<ω1MW

α . Let W 6= U , γ ∈ W ⇔ γ 6∈ U . Suppose f : MW −→MU is an isomorphism. Define i(α) = minγ ∈ ω1 | f(KW

α ) ⊆ KUγ , j(β) =

minδ ∈ ω1 | f−1(KUβ ) ⊆ KW

δ . Again, i and j are normal functions, henceFix(i) and Fix(j) are clubs, so

C = Fix(i) ∩ Fix(j) ∩ Aγ 6= ∅.

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Let β ∈ C.We observe that f :KWβ∼= KU

β , but f cannot be extended further,because in only one of MW

β r KWβ and MU

β r KUβ , x | p(x) is unbounded

below.

It has been proved by Harnik [9] that all models of PA of order-type N +ω1QZ have the same (M,<,+)-isomorphism type. The 2ω1 models in ourTheorem 50 all have different (M,<,+, ·)-isomorphism types (they are non-isomorphic); however can some of them be (M, ·)-isomorphic?

Question 32 Prove that there are 2ω1 models Mi |= PA, i < 2ω1 of order-type N + ω1QZ such that (Mi, ·) 6∼= (Mj, ·) for i 6= j.

The following Theorem solves this Question for the universes satisfying (♦).The proof generalises Harnik’s proof in [9], where he constructs two suchmodels. This is an example of the common situation in model theory whereyou can get 2λ models for the price of just two.

Diamonds

Theorem 51 (♦) There are 2ω1 models Mi |= PA, i < 2ω1 of order-typeN + ω1QZ such that (Mi, ·) 6∼= (Mj, ·) for i 6= j.

We shall use the following formulation of (♦) which is equivalent to the usualone (see [20], page 92 or [12], page 229):

(♦): There is a sequence fα | α < ω1 of functions fα: (1+α)ω → (1+α)ωsuch that for all F : (1 + ω1)ω → (1 + ω1)ω the set α | F (1+α)ω= fα isstationary.

If M |= PA, let p(M) be the set of all M -primes, N be an end-extension ofM . We say that X ⊆ p(M) is coded in N if for some a ∈ N ,

x ∈ p(M) | N |= x|a = X.

Also, if K is an end-extension of N then the same subsets of p(M) are codedin K as in N . Let a ∈ K rN code X ⊆ p(M), c ∈ N rM . Define

b =∏

x<c, pr(x)|a

pr(x)

where pr(x) means ‘the xth prime’. As b < (pr(c))!, b ∈ N . Also b codes X.The following fact from [23] or [32] will be used in the proof:

Proposition (#)If M |= PA is countable then there are continuum many X ⊂ p(M) that are

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codable in some end-extension N M .

Proof of Theorem 51For every f :ω1 → 2, we construct M f as a union of elementary chain ofcountable models M fαα<ω1 .

Let M∅ be an arbitrary countable model with domain ω. Suppose forall β < α, f ∈ 2ω1 , M fβ are already constructed having domains (1 + β)ω.If α is a limit ordinal, let M fα =

⋃β<αM

fβ . Obviously, M fα has domain(1 + α)ω.

Let α = δ + 1. As cardf :α → 2 ≤ 2ω =♦ ω1, we can enumerate allfunctions f :α→ 2 as gi:α→ 2 | i < ω1. Consider M g0δ . Let M g0 be anyelementary end-extension of M g0δ with domain (1 + δ + 1)ω.

Suppose for all j < i, M gj is already constructed. i is a countable ordinal,and each M gj , j < i codes only countably many subsets of p(M gjδ), hence,by Proposition (#), there is Y ⊆ p(M giδ) such that f−1

α (Y ) is not coded inany M gj , j < i. Let M gi be any elementary end-extension of M giδ whichcodes Y and has domain (1 + δ + 1)ω. Thus define M gi for all i < ω1. Now,for h ∈ 2ω1 , let Mh =

⋃α<ω1

Mhα .Let f, h ∈ 2ω1 , γ < ω1, f(γ) 6= h(γ). Suppose there is F : (1 + ω1)ω →

(1 + ω1)ω, which is an isomorphism between (M f , ·) and (Mh, ·). Thenα | F α= fα is stationary, hence unbounded. Hence, for some α > γ,F α: (1+α)ω → (1+α)ω is an isomorphism between (M fα , ·) and (Mhα , ·).Let at stage (α+ 1) of our construction, M fα+1 = M gj , Mhα+1 = M gi withj < i. Let Y ⊆ p(Mhα) be the set from the definition of M gi , coded inMhα+1 by the point a. Then f−1

α (Y ) is coded in M f by F−1(a), hence isalready coded in M fα+1 , which contradicts our definition of Y .

We believe that (♦) is not necessary for the conclusion of Theorem 51 to betrue and propose a more general Problem.

Question 33 If M |= PA has cardinality ω1 and is not saturated then (M,<)has 2ω1 (M, ·)-non-isomorphic expansions to a model of PA.

This hard problem is deeply connected to the material of Chapter 7, whichalso deals with non-isomorphic (and even not elementarily equivalent) ex-pansions of linear orders to models of PA.

Question 34Try tackling Question 10 in the universes satisfying ♦.

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Chapter 7

Around Friedman’s Problem

In this chapter we present our first two attempts to solve Friedman’s prob-lem in the resplendent case and investigate the connection with the notionof arithmetic saturation we encountered. Also, section 7.2 gives two otherapplications of arithmetic saturation.

Definition 27 Let A ⊆ N be arbitrary. Let L = LPA ∪ A(x), whereA(x) is a new predicate symbol which will later be interpreted as x ∈ A. Asconstructed in [14], chapter 9, there is a Σ1-formula SatΣ1(x, y) in L suchthat for every Σ1-formula θ(v1, . . . , vn) in L,

(N, A) |= ∀x SatΣ1(pθ(v1, . . . , vn)q, 〈(x)1, . . . , (x)n〉) ↔ θ((x)1, . . . , (x)n).

Define A′ = n ∈ N | (N, A) |= SatΣ1((n)1, (n)2).

From this definition we can prove that if SSy(M) is closed under jump andθ(x) is a formula of LPA then n ∈ N | N |= θ(n) is coded inM . In particular,Πn Th N is coded in M for every n because Πn Th N is defined in N by theformula “x is a Godel number of a closed Πn-formula” & SatΠn(x, [ ]). (Here,SatΠn(x, y) is the Πn-satisfaction predicate in LPA (see [14], pp 126-128).)

Fact (Kirby and Paris)Let M |= PA be recursively saturated. Then the following conditions areequivalent.

1. For any f ∈ M coding a function f : N→ M , there is c ∈ M r N suchthat for all n ∈ N, f(n) > N ⇔ f(n) > c.

2. SSy(M) is closed under jump.

A model satisfying the above conditions is called arithmetically saturated.Arithmetic saturation is discussed in [16], pp. 253-256. Also, in this chapter

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we shall need the following variation of the Friedman’s embedding theorem.

Fact Let M,N |= PA, M be countable. Then for all n ∈ ω, the followingconditions are equivalent.

1. There is an embedding h:M → N such that h(M) ≺Σn N .

2. SSy(M) ⊆ SSy(N) and N |= Σn+1 ThM .

7.1 Resplendency and coding

We already know from Chapter 4 that if M |= PA is self-similar, T ⊇ PAis coded in M and M |= Con(T ) then (ACT(M,T ), <) ∼= (M,<). In theresplendent case we do not need to assume M |= Con(T ) as the followingtheorem shows.

Theorem 52 If M |= PA is resplendent and c ∈M codes a consistent theoryT ⊇ PA then (M,<) can be expanded to a model of T .

ProofThe following Σ1

1-statement is realised in any countable elementary sub-model of M containing c (since all countable models are order-isomorphic),hence is realized:

∃⊕,⊗,,O,S, Sat(∀xy (x y ↔ x < y)

∧“(⊕,⊗,,O,S) |= T”

).

Let us write “(⊕,⊗,,O,S) |= T” formally. Introduce new Godel numbersfor the symbols ⊕,⊗,,O,S and extend this numeration to all formulasin the language ⊕,⊗,,O,S in some standard way. From now on weshall identify each formula ϕ in the language ⊕,⊗,,O,S with its Godelnumber. Let θn(x, c) =

(x > n) & (∀i < x Sat((c)i, [ ])

& ∀ϕ ∀a (form(ϕ) & ϕ has len(a) free variables −→(“ϕ = ψ1 ∨ ψ2” → (Sat(ϕ, a)↔ Sat(ψ1, a) ∨ Sat(ψ2, a))) &

(“ϕ = ¬ψ” → (Sat(ϕ, a)↔ ¬ Sat(ψ, a))) &

(“ϕ = ∃x1ψ(x1, x)” → (Sat(ϕ, a)↔ ∃b Sat(ψ, 〈b, a〉)))).As θn(x, c)n∈N is a recursive set of formulas, by Kleene’s Theorem, there isa single Σ1

1-sentence Θ(x, c) such that in any M |= PA, M |= ∀x (Θ(x, c)↔∧n∈N θn(x, c)). Thus, “(⊕,⊗,,O,S) |= T” is expressed by the Σ1

1-statement∃ xΘ(x, c).

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Corollary 53If M is resplendent and SSy(M) = P(N) then for any consistent T ⊇ PA,(M,<) can be expanded to a model of T .

Actually, what this Corollary requires from SSy(M) is that it contains allcompletions of PA. The following argument (by my supervisor) shows thatthis will imply SSy(M) = P(N).

Fact(Kaye) For every X ∈P(N) there is a complete T ∗ ⊃ PA such that Xis recursive in T ∗.

Proof By Rosser’s Theorem, given a recursive consistent T ⊇ PA (as usualwe identify each formula of T with its Godel number, so T is a recursive subsetof N), there are canonical sentences LT and RT = ¬LT such that T +LT andT + RT are both consistent, and this function T → LT is computable. LetT0 = PA, Tn+1 = Tn ∪ LTn if n ∈ X and Tn+1 = Tn ∪ RTn otherwise.Let T ∗ be any completion of

⋃n∈N Tn. To determine if n ∈ X first determine

whether i ∈ X for each i < n, thus obtaining Tn. Compute LTn . Ask ifLTn ∈ T ∗. Hence X is recursive in T ∗.

Corollary 54 If M is resplendent and ω1-saturated then (M,<) can be ex-panded to a model of any consistent extension of PA.

(We also know that, by Pabion’s Theorem, this expansion of M will haveto be ω1-saturated because it has an ω1-saturated order-type. Can we alsomake it resplendent?)

Theorem 55If M is resplendent and cf(M) > ω then for all n ∈ ω, Πn Th N is coded in M .

(Hence, as PA is recursive, by Theorem 52, for every n ∈ N, (M,<) is ex-pandable to a model of PA∪ Πn Th N, i.e. Th N can be “approximated” asclosely as you want.)

Proof (Resembles the process of fishing.)For any n ∈ ω, let us introduce Σn Def = the set of all nonstandard definablepoints of M defined by a Σn-formula.

As cf(M) > ω, there is a > N such that Σ1 Def > a. Define A1 =p∀xϕ(x)q | ϕ ∈ ∆0, M |= ∀x < a ϕ(x). Now, A1 ⊆ Π1 Th N becauseN ≺∆0 M . Also, Π1 Th N ⊆ A1 because if for some ϕ ∈ ∆0 such that N |=∀xϕ(x) there existed x < a ¬ϕ(x) then minx¬ϕ(x) would be a nonstandardΣ1-definable point less than a. Hence, A1 = Π1 Th N. A1 is definable, hencecoded in M .

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Suppose at stage n we already know that Πn Th N ∈ SSy(M). Let b ∈Mcode Πn Th N. Consider the statement

Φn(b) = ∃⊕n,⊗n,n,On,Sn

∀xy (xn y ←→ x < y)∧

∧“(⊕n,⊗n,n,On,Sn) |= PA∪ Πn Th N”

∧∧

“ SSy(M,⊕n,⊗n,n,On,Sn) ⊆ SSy(M)”.

Let us show that the last line is expressible by a Σ11-sentence. Let ϕm(x) =

(x > m) & ∀ z ∃ y ∀ i < x ((z))i = (y)i, where ((z))i means (z)i in thelanguage ⊕n,⊗n,n,On,Sn. ϕm(x) is a recursive set of formulas, hence,by Kleene’s Theorem, there is a Σ1

1-sentence Θ(x) such that in any K |=PA, K |= ∀ x (

∧m∈N ϕm(x) ↔ Θ(x)). Then SSy(M,⊕n,⊗n,n,On,Sn) ⊆

SSy(M) is implied by the Σ11-sentence ∃xΘ(x). Hence, Φn(b) is a Σ1

1-sentence.Φn(b) is consistent because, by Wilmers’ Theorem, as (PA∪ Πn Th N) ∈

SSy(M), there is a countable model

N |= PA ∪ Πn Th N, SSy(N) = SSy(ClM(b)).

Hence, by resplendency, Φn(b) is already realised in M .Denote the model (M,⊕n,⊗n,n,On,Sn) byMn. By construction,Mn |=

PA∪ Πn Th N, (Mn, <) ∼= (M,<), SSy(Mn) ⊆ SSy(M).Let (Σn Def)Mn > a > N. Consider

An+1 = p∀xϕ(x)q | ϕ ∈ Σn, Mn |= ∀x < a ϕ(x).

An+1 ⊆ Πn+1 Th N because if Mn |= ∀x < a ϕ(x) but N |= ∃x ¬ϕ(x) then forsome k ∈ N, N |= ¬ϕ(k), which is a Πn-statement. Hence, as Mn |= Πn Th N,Mn |= ¬ϕ(k), contradiction.

Πn+1 Th N ⊆ An+1. Let N |= ∀xϕ(x), where ϕ(x) ∈ Σn. If Mn |= ∃x <a ¬ϕ(x) then c =: min x¬ϕ(x) is a Σn-definable point less than a. If c ∈N then Mn |= ¬ϕ(c), which is a Πn-statement not belonging to Πn Th N.Contradiction with Mn |= Πn Th N. Hence N < c < a, which contradicts theassumption that (Σn Def) > a.

Therefore Πn+1 Th N = An+1, which is coded in Mn. As SSy(Mn) ⊆SSy(M), Πn+1 Th N is also coded in M .

At this point my supervisor noticed that I actually proved nothing new,because resplendency and uncountable cofinality imply arithmetic saturation,and any arithmetically saturated model codes all Πn Th N.

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Indeed, resplendency implies recursive saturation and for any f : N −→M there is a ∈ M such that ∀n ∈ N (f(n) > N ⇒ f(n) > a) becausecf(M) > ω.

In the next section we shall investigate whether recursive saturation anduncountable lower cofinality give us more information about coding thanjust arithmetic saturation. The answer will be “No”. But first let us studya corollary.

A consistent theory T is called arithmetic if it has an axiomatization Ssuch that S = n ∈ N | N |= θ(n) for some formula θ(x) ∈ LPA. Recur-sive extensions of PA are examples of arithmetic theories. Also, there arecomplete arithmetic theories. See [14], Chapter 13.

Proposition 56For any arithmetic theory T ⊇ PA, if M |= PA is resplendent and SSy(M)is closed under jump then there is N |= T such that (N,<) ∼= (M,<).

Proof Let T = n ∈ N | N |= θ(n). As SSy(M) is closed under jump, T iscoded in M . Hence, as M is resplendent, (M,<) is expandable to a modelof T , by Theorem 52.

7.2 Digression: arithmetic saturation

Lemma 57 Let M |= PA be recursively saturated. Then M is arithmeticallysaturated if and only if for all a ∈M , Cl(a)rN is bounded below in M rN.

ProofSuppose, for all a ∈ M , Cl(a) r N is bounded below. Let f ∈ M code afunction. For every n ∈ N, f(n) ∈ Cl(f). If Cl(f) > b then for all n ∈ N,(f(n) > N⇔ f(n) > b).

Let M be arithmetically saturated, c > N. The type which says: F ∈Mcodes a function F : [0, c] −→ M with F (pθq) = tθ(a) (where θ ranges overall formulas of LPA with two variables and tθ is the Skolem term defined byθ) is recursive, hence realized. But if Cl(a) r N is unbounded below thenF (pθq) ∩ (M r N) is not separated from N, which contradicts arithmeticsaturation.

Let E = x ∈M | there are no nonstandard definable points below x.If a ∈M r Cl ∅, define

Ea = x ∈M | for all c ∈ Cl ∅, c < x↔ c < a .By Lemma 57, E 6= ∅ and for any a such that N < a < Cl ∅ r N, Ea = E.The following lemma establishes some homogeneity properties of Ea whichwill be important in the rest of this section.

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Lemma 58 Let M be recursively saturated, a ∈M r Cl ∅.

1. If p(x, b) is realized by c ∈ Ea, c > Cl(b)∩Ea then for all x ∈ Ea thereis y > x such that p(y, b).

2. If p(x, b) is realized by c ∈ Ea, c < Cl(b)∩Ea then for all x ∈ Ea thereis y < x such that p(y, b).

Proof1. Let Aupper = x ∈ M | ∃ y ∈ Cl ∅, a < y < x. For an arbitrarye ∈ Ea, let us find d > e such that p(d, b). Let us show that for all θ(x, b) ∈p(x, b), M |= θ(y, b) for unboundedly-many y ∈ Ea. Consider the two cases.If A = x ∈ Aupper | M |= θ(x, b) is unbounded below then M |= θ(y, b) forunboundedly-many y ∈ Ea by overspill. Otherwise, let k ∈ Cl ∅, a < k < A.Define g = maxx < k θ(x, b). We observe that g ∈ Cl(b), while c ≤ g <Aupper, which is a contradiction.

Thus for any e ∈ E, p(x, b) ∪ x > e is finitely satisfied. By recursivesaturation, p(x, b) ∪ x > e is coded, hence realized.2. Analogous proof, left to the reader.

Lemma 59 Let M |= PA be a countable arithmetically saturated model,N < e < Cl ∅ r N. Then there is an elementary embedding h : M →M suchthat for all x > N, h(x) > e.

ProofA forth-argument. Let us enumerate M as a1, a2, . . . , ai, . . .i<ω and buildinductively a sequence b1, b2, . . . , bi, . . .i<ω with tp(b1, . . . , bi) = tp(a1, . . . , ai)and bi > e ⇔ bi > N for all i and define h(ai) = bi.

Suppose at stage i we already have tp(a1, . . . , ai) = tp(b1, . . . bi), e <Cl(b1, . . . , bi)rN. By Lemma 57, Cl(a1, . . . , ai+1)rN is bounded below. Letc < Cl(a1, . . . , ai+1) r N. By Lemma 58 (2),

p(x, a1, . . . , ai, c) = θ(a1, . . . , ai, x) | M |= θ(b1, . . . , bi, e) ∪ x < c

is satisfied, say, by e∗ ∈ E.As tp(a1, . . . , ai, e

∗) = tp(b1, . . . , bi, e), by recursive saturation, there is anelementary embedding (actually, an automorphism) h : M → M such thath(a1) = b1, . . . , h(ai) = bi, h(e

∗) = e. Put bi+1 = h(ai+1). By construction,e < Cl(b1, . . . , bi+1).

Hence, M has an elementary extension N M , N ∼= M and there is a ∈N r M such that N < a < M . Since a union of an elementary chain

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of recursively saturated models is recursively saturated, we can repeat thisextension ω1 times to obtain the following Theorem, which was promisedearlier.

Theorem 60Let X be a countable Scott Set. Then X is closed under jump if and onlyif there is a recursively saturated M |= PA, cf(M) > ω, SSy(M) = X .

Later I discovered that this fact is also known to J. Schmerl [26].

The countability assumption cannot be dropped yet because for Scott Sets Xwith card X > ω1, the existence of a modelM |= PA such that SSy(M) = Xis still an open problem.

Question 35Modify the proof of Lemma 59 to include the case card X = ω1. (Start witha countable Scott Set Y0 ⊂ X and construct M as an elementary union ofMi |= PA, SSy(Mi) = Yi, card Yi = ω, Yi ⊆ Yi+1,

⋃Yi = X .)

Automorphisms moving all non-definable points

Now, as we are discussing arithmetic saturation, let us turn for the rest ofthis section to the rapidly developing area of automorphism groups wherearithmetic saturation has profound consequences.

Fact 1 (Kaye, Kossak, Kotlarski)[16]If M |= PA is countable and arithmetically saturated then M has an auto-morphism which moves every nondefinable point.

Fact 2 (Kossak)[19]If M |= Th N is countable and arithmetically saturated then there existsh ∈ Aut(M) such that for all x > N, h(x) > x, i.e. h moves every non-standard point up.

For some reason this automorphism is becoming known in the literature as‘Kossak’s bump’. We are going to prove a theorem which generalises both ofthe above results. In general, if Th(M) 6= Th N, there exists no h ∈ Aut(M)such that for all x 6∈ Cl ∅, h(x) > x.

Proof Let a < b < e, e ∈ Cl ∅ r N, h(a) = b. Then e − a > e − b, henceh(e− a) > h(e− b), hence e− b > h(e− b).

But what we can expect is the following Theorem.

Theorem 61If M |= PA is countable and arithmetically saturated then there is h ∈

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Aut(M) such that for all x ∈ E, h(x) > x and h moves every nondefin-able point.

Since in the case of Th(M) = Th N, E = MrN, this Theorem generalizesKossak’s result. The proof uses Kossak’s method and the following twolemmas.

Lemma (#) (Kaye, Kotlarski)If M is arithmetically saturated, tp(a) = tp(b) and for any Skolem term t,

t(a) = t(b) ⇒ t(a) ∈ Cl ∅

then for any c ∈M there is d such that tp(a, c) = tp(b, d) and for any Skolemterm t,

t(a, c) = t(b, d) ⇒ t(a, c) ∈ Cl ∅.

Notice that Fact 1 follows from Lemma (#) by a back-and-forth argument.

Lemma 62 Let M |= PA be recursively saturated.

1. If c < Cl(a) r N then for any b there is b′ such that tp(a, b) = tp(a, b′),c < Cl(a, b′) r N.

2. If c > Cl(a) ∩E then for any b there is b′ such that tp(a, b) = tp(a, b′),c > Cl(a, b′) ∩ E.

Proof1) By Lemma 58, (1), there is d < Cl(a, b) such that tp(a, c) = tp(a, d). Byrecursive saturation, there is h : M → M, h(a) = a, h(d) = c. Denote h(b)by b′. As h is elementary, tp(a, b) = tp(a, b′). As d < Cl(a, b), c < Cl(a, b′).2) Similar proof.

ProofWe shall construct a string of points dii∈Z unbounded above and belowin E such that our future automorphism h takes di to di+1 which will guar-antee that each point of E moves upwards: if a ∈ (di, di+1) then h(a) ∈(hdi, hdi+1) = (di+1, di+2). Also, it obviously follows that there will be noh-fixed initial segment in E other than supE and N.

By Lemma (#) there are c0, c1 ∈ E such that tp(c0) = tp(c1) and (t(c0) =t(c1) ⇒ t(c0) ∈ Cl ∅), hence, considering the type ϕ(x) | M |= ϕ(c1) ∪t(x) 6= t(c1) | t(c1) 6∈ Cl ∅, we deduce, using Lemma 58, that there ared0, d1 ∈ E such that

tp(d0) = tp(d1),

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t(d0) = t(d1)⇒ t(d0) ∈ Cl ∅,

Cl(d0) ∩ E < d1,

Cl(d1) r N > d0.

Let sii∈ω be an enumeration of the whole of MrCl ∅. By stage n we shallalready have:

a = a0, . . . , a2n−1,

b = b0, . . . , b2n−1,

d = d−n, d−n+1, . . . , dn, dn+1

satisfying the following conditions:

tp(a, d−n, . . . , dn) = tp(b, d−n+1, . . . , dn+1),

d−n < Cl(b, d−n+1, . . . , dn+1) ∩ E,dn+1 > Cl(a, d−n, . . . , dn) ∩ E,

t(a, d−n, . . . , dn) = t(b, d−n+1, . . . , dn+1) ⇒ t(a, d−n, . . . , dn) ∈ Cl ∅.

(At stage n = 0, a and b are empty.)BackLet b2n = sn. Let e < Cl(b2n, b, d−n, . . . , dn+1). By Lemma (#) (applied tothe tuples (b, d−n+1, . . . , dn+1) and (a, d−n, . . . , dn) and the new point d−n),the set of formulas

p(x) = ϕ(a, x, d−n, . . . , dn) | M |= ϕ(b, d−n, d−n+1, . . . , dn+1) ∪

∪ t(a, x, d−n, . . . , dn) 6= t(b, d−n, d−n+1, . . . , dn+1) || t(b, d−n, d−n+1, . . . , dn+1) 6∈ Cl ∅

is realized, hence, by Lemma 58 (2), is realized by a point less than e, hence,by Lemma 62 (2), is realized by a point d−n−1 < e such that

Cl(a, d−n−1, . . . , dn) ∩ E < dn+1.

Let q(x) = ϕ(a, d−n−1, . . . , dn, x) | M |= ϕ(b, d−n, . . . , dn+1, b2n ∪

∪ t(a, d−n−1, . . . , dn, x) 6= t(b, d−n, . . . , dn+1, b2n) || t(b, d−n, . . . , dn+1, b2n) 6∈ Cl ∅.

By Lemma (#), q(x) is realized, hence, by Lemma 62 (2), is realized by somepoint a2n such that

Cl(a, d−n−1, . . . , dn, a2n) ∩ E < dn+1.

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By construction,

t(a, d−n−1, . . . , dn, a2n) = t(b, d−n, . . . , dn+1, b2n)⇒⇒ t(b, d−n, . . . , dn+1, b2n) ∈ Cl ∅,

i.e., every nondefinable point of Cl(b, d−n, . . . , dn+1, b2n) moves. Let us showthat if t(b, d−n, . . . , dn+1, b2n) ∈ E then

t(a, d−n−1, . . . , dn, a2n) < t(b, d−n, . . . , dn+1, b2n).

If t(b, d−n, . . . , dn+1, b2n) > dn+1 then t(a, d−n−1, . . . , dn, a2n) ∈ (dn, dn+1)because Cl(a, d−n−1, . . . , dn, a2n) ∩ E < dn+1. If t(b, d−n, . . . , dn+1, b2n) ∈(di, di+1), i = −n, . . . , n, then t(a, d−n−1, . . . , dn, a2n) ∈ (di−1, di).If t(b, d−n, . . . , dn+1, b2n) < d−n then, by construction of d−n−1,t(b, d−n, . . . , dn+1, b2n) ∈ (d−n−1, d−n), hence t(a, d−n−1, . . . , dn, a2n) < d−n−1.ForthLet a2n+1 = sn. Using Lemmas (#), 58 (1), 62 (1), we choose dn+2 such that

tp(b, b2n, d−n, . . . , dn+2) = tp(a, a2n, d−n−1, . . . , dn+1),

t(b, b2n, d−n, . . . , dn+2) = t(a, a2n, d−n−1, . . . , dn+1)⇒⇒ t(a, a2n, d−n−1, . . . , dn+1 ∈ Cl ∅,

dn+2 > Cl(a, a2n, a2n+1, d−n−1, . . . , dn+1) ∩ E,d−n−1 < Cl(b, b2n, d−n, . . . , dn+2) ∩ E.

Now, using Lemmas (#) and 62 (1), we choose b2n+1 such that

tp(b, b2n, b2n+1, d−n, . . . , dn+2) = tp(a, a2n, a2n+1, d−n−1, . . . , dn+1),

t(b, b2n, b2n+1, d−n, . . . , dn+2) = t(a, a2n, a2n+1, d−n−1, . . . , dn+1)⇒⇒ t(a, a2n, a2n+1, d−n−1, . . . , dn+1) ∈ Cl ∅,

and d−n−1 < Cl(b, b2n, b2n+1, d−n, . . . , dn+2).

Having obtained the points ai, bi for all i ∈ ω, we observe that h:M →Mdefined as h(ai) = bi for all i ∈ ω is an elementary isomorphism posessingthe required properties.

I was told that the automorphism constructed above in Theorem 61 maybe useful in the study of NF, in particular the problem of consistency of NF.For an up-to-date discussion of related automorphisms and their connectionto models of NF and models of PA occuring as natural numbers of models ofNF, see [18].

The following problem has been suggested by my supervisor:

Question 36 Let M |= Th N be countable and arithmetically saturated.Find g, h ∈ Aut(M) such that Aut(M) is generated by g and h.

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7.3 Initial segments in resplendent models

In this section we present our second attempt to solve Friedman’s problemfor resplendent models of PA.

Theorem 63Let T be a completion of PA. Suppose M is resplendent, cf(M) = ω and Mcodes ΣnT for all n. Then (M,<) is expandable to a model of T .

ProofLet b ∈M be a code of PA∪ Σ1T in M . Consider the following statement:

Φ(b) = ∃⊕,⊗,,O,S(∀xy (x y ←→ x < y)

∧“(⊕,⊗,,O,S) |= PA∪Σ1T”

∧∧

SSy(M,⊕,⊗,,O,S) = SSy(M)).

The last line SSy(M,⊕,⊗,,O,S) = SSy(M) can be written as a Σ11-

sentence because if

ϕn(x) = (x > n) & ∀ v ∃ y∀ i < x ((v))i = (y)i

∀ z ∃ w ∀ i < x (z)i = ((w))i

then ϕn(x)n∈N is a recursive set of sentences, hence, by Kleene’s Theorem,there is a Σ1

1-sentence Ψ(x) such that in any A |= PA, A |= ∀x(∧

n∈N ϕn(x)↔Ψ(x)). Thus, the Σ1

1-sentence ∃ x Ψ(x) implies SSy(M,⊕,⊗,,O,S) =SSy(M). Hence, Φ(b) is a Σ1

1-sentence. Again, it is realised in Cl(b) byWilmers’ Theorem, hence realised.

Define K = (M,⊕,⊗,,O,S) such that

K |= PA ∪ Σ1T,

(K,<) ∼= (M,<),

SSy(K) = SSy(M).

Let us construct a countable N |= T with SSy(N) ⊆ SSy(K). Let X ⊆SSy(K) be a countable Scott Set containing all ΣnT . As (PA ∪ Σ1T ) ∈X ,there is a model N1 |= PA∪ Σ1T, SSy(N1) = X . Suppose we already have

Ni |= PA ∪ ΣiT, SSy(Ni) = X .

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As (PA∪Σi+1T ) ∈X , there is a countable

Ni+1 |= PA ∪ Σi+1T, SSy(Ni+1) = X .

By Friedman’s Theorem, as Ni |= Σi ThNi,

Ni ≺Σi−1Ni+1.

Define N =⋃

i∈ω Ni. Obviously, SSy(N) = X . By the Σn-elementary chainslemma, for all n ∈ ω, N |= ΣnT , hence N |= T .

Now, by the variation of Friedman’s Theorem, as K |= Σ1T , there is anembedding h:N → K such that h(N) ≺∆0 K. Consider

N∗ = x ∈ K | ∃y ∈ h(N), y ≥ x.

By Gaifman’s splitting theorem, N∗ h(N), hence N∗ is an initial segmentof K satisfying T . As h(N) is cofinal in N∗, Υ(N∗) = ω, hence, by Lemma17, (N∗, <) ∼= (M,<).

Question 37 If M is resplendent and codes ΣnT for all n, is (M,<) ex-pandable to a model of T?

The obvious attempt to generalise Theorem 63 to higher cofinalities byproving that if I |= T is an initial segment of M 6|= T then there is an initialsegment J I fails, because given I0 = I we define Ij =

⋃i<j Ii if j is a

limit ordinal, Ii+1 = an initial segment J Ii if Ii 6= M , which gives us anelementary chain of some length γ. Now, M =

⋃i<γ Ii |= T , contradiction.

(The initial segments satisfying T exist as far up as one wants (by the vari-ation of Friedman’s Theorem) but they are not elementary substructures ofeach other.)

However, in the case T = Th N we may hope that arithmetic saturationgives us extra information:

Question 38 If M is resplendent and Υ(M) > ω, is (M,<) expandable toa model of Th N?

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Conclusion

What is next

I believe that in every subject, self-awareness and understanding of its moti-vation and higher goals should not be lost and replaced by indulging in accu-mulating knowledge about “structural properties” and feeding basic mathe-matical curiosity.

The ultimate question of Foundations of Mathematics is to try to under-stand the mystery of Arithmetical Truth. Even Set Theory can be assumedto be taking place in N (in a model of ZFC strongly interpreted in N), so,in this sense, set-theoretic questions are again questions about ArithmeticalTruth.

Models of PA are one of the approaches to Arithmetical Truth (other ap-proaches include Constructivism, Proof Theory, Intuitionism), the one whichis trying to answer the question “How do structures satisfying some axioms,which we believe should hold for the intuitive set N look like?” As Proposi-tion 37 shows, some of these structures are so indistinguishable from N thatthere are models of ZFC that belive that they are N.

Also, models of PA promise to be mathematically rich because they standon the crossroads of many different areas of mathematics: Set Theory (be-cause uncountable models always require some infinitary combinatorics andbecause the initial segment ωU of every universe U |= ZFC is a model of PA),Proof Theory (because of Godel’s Incompleteness, importance of reflectionprinciples, independence results like (PH) and various uses of Godel numbersof proofs in connection with Standard System (e.g. see [14], p. 177) and withrecursive saturation), Recursion Theory (needed in the study of Scott Setsand jump-operator needed in the study of arithmetic saturation, etc.) and, ofcourse, mainstream Model Theory. I propose the following seven directionsof future research. (The rest can be found in the List of Questions.)

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1. Models of TA as ωU for U |= ZFC

Theorems 37-42 are just the first results with the rest of excitement to fol-low. To characterise all models of PA which occur as ωU of some U |= ZFC(see discussion on pages 60-63 and Question 17) is a question of great im-portance for foundations of mathematics and I am unaware whether anyonehas attempted to answer it before.

2. Friedman’s Problem

What we know so far:

1. Every countable nonstandard model has order-type N + QZ.Every ω1-like model has order-type N + ω1QZ.

2. The saturated model of cardinality λ has order-type N +QλZ.

3. (GCH) For any T ⊇ PA and any µ < λ+ there is N |= T of order-typeN + µQλZ.(GCH) For any T ⊇ PA and any successor λ1 < λ2 < · · · < λn, thereis K |= T of order-type N +Qλ1Z + · · ·+QλnQλn−1 . . . Qλ1Z.

4. For any T ⊇ PA and any dense linear order (C,<), EM(C) is embed-dable into (C∗ + 0+ C)<Q.

5. If M |= PA + Con(PA) is self-similar then for any inner model N =ACT(M,PA), (N,<) ∼= (M,<).

6. If M is resplendent and T ⊇ PA is coded in M then (M,<) is expand-able to a model of T .

7. The order-type of any ω1-saturated resplendent model can be expandedto a model of any consistent extension of PA.

8. If T is an arithmetic theory, M |= PA is resplendent and cf(M) > ωthen there is N |= T such that (N,<) ∼= (M,<).

9. If M is resplendent, cf(M) = ω and M codes ΣnT for all n, then (M,<)is expandable to a model of T .

10. If M is resplendent, cf(M) = ω and SSy(M) is closed under jump then(M,<) is expandable to a model of Th N.

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Still now, I cannot even guess what the answer to the Friedman’s problemwill be. We may assume different degrees of saturation, resplendency, cod-ing and their combinations and prove that order-types of models satisfyingthose conditions can be expanded to models of other theories (sometimes “alltheories”) but again and again we shall keep returning to the same problem:what to do with different Standard Systems?

Question 39 Let T ⊃ PA, M |= T be resplendent. Let S ⊃ PA be consis-tent, S 6∈ SSy(M). Can (M,<) be expanded to a model of S?

A solution of this problem gives a solution of Friedman’s problem in theresplendent case.

A new approach, different from the two approaches we tried (resplendency& coding and resplendency & initial segments) would be to make a stepbackwards and first study resplendent linear orders and resplendent denselinear orders instead of resplendent models of PA.

Observation 64

1. If (A,<) is a resplendent discrete linear order with first and no lastelement then (A,<) is expandable to a model of PA.

2. If M |= PA is resplendent then (M,<) is resplendent.

Proof1. Let ϕii∈ω be the set of all axioms of PA in the language ⊕,⊗,,O,S.By Kleene’s Theorem,

∧i∈ω ϕi is equivalent to a Σ1

1-statement Ψ. Let Φ =∃ ⊕,⊗,,O,S (Ψ & ∀x, y(x < y ↔ x y)). Φ is consistent with thetheory of discrete linear order with first and no last element because Φ isrealised in the order-type of any model of PA. Hence, by resplendency, Φ isrealised in (A,<).2. Let ∃R f Φ(R, f, a) be consistent with Th(M,<, a). Then, as R andf are new symbols not occuring in LPA, ∃R f Φ(R, f, a) is consistent withTh(M,+, ·, <, 0, a) (by Robinson’s joint consistency test), hence realised.

Question 40

1. Prove that if (A,<) is a resplendent dense linear order then N +AZ isa resplendent discrete linear order with first and no last element.

2. Can every resplendent discrete linear order be expanded to a resplendentmodel of PA?

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3. Canonical orders and their spectrum

The main two problems here are to prove the general case of the existencetheorem and to find the number of non-isomorphic models of each canonicalorder. Sophisticated methods seem to be required in both cases because allobvious approaches failed.

4. Diamonds etc

Diamonds and other combinatorial statements have been knocking at modeltheorists’ door since 1970s. There exists a broad program to investigate theinfluence of infinitary combinatorics on construction of models and a numberof methods have beed developed in this area. Also, there are constructionsthat use V=L (‘morasses’). The plan is to learn existing methods havingmodels of PA in mind. We can start off with an attempt to generalise theproof of Theorem 51.

5. Interpretations

A myriad of questions is rising about interpretations: to develop a theory ofhow models of strong theories (i.e. the theories described on page 68) interpreteach other and what they think about each other, very much in the spirit ofsections 4.2 and 4.4, in particular to develop the theory of inner models ofPA thus investigating interpretational strength of models of PA, to state andsolve problems similar to Questions 1 and 9, giving and studying examplesof mathematical structures interpreted in models of PA and internalisingclassical results in the spirit of section 4.1.

6. Arithmetic Saturation

Arithmetic saturation is a relatively unexplored notion promising excitingresults and examples (see Questions 29 and 36). Also, there is still demandfor new useful notions of saturation that would give consequences for auto-morphism groups, satisfaction classes and the Standard System.

Question 41 Investigate the following notion of saturation: “M is recur-sively saturated and SSy(M) is closed under ω-jump: A→ A(ω)”.

By the properties of ω-jump, it implies that SSy(M) is closed under jump,hence M is arithmetically saturated. Also, Th N will have to be coded in M .

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7. Embeddability

A problem wider than just “What are the order-types of models of PA?”would be to introduce different kinds of linear orders and study their mutualembeddability with models of PA as we did in parts of chapters 2 and 3. Forexample, many questions rise about embeddability of linear orders κ and κ∗

into models of PA. Another possibility is to study the following family oflinear orders.

Definition 28 Let (A,<), (I,<) be linear orders, 0 ∈ A.(A, 0)/I = f : I → A | supp f is well-founded , with order defined lexico-graphically: f < g if for a = minsupp f, supp g, f(a) < g(a).

Question 42Study mutual embeddability of orders (A, 0)/I and models of PA.

List of questions

Question 1Let Ω |= PA. What do groups strongly interpreted in Ω look like? What doΩ-finite groups look like? Which properties of N-finite (truely finite) groupsdo they inherit?

Question 2 Is there an ultrafilter D on λ such that there exists a countableM |= PA with

∏D M 2λ-dense but not saturated?

Question 3 Are there linear orders (I,<), (A,<), dense, without end-pointsand 01, 02 ∈ A such that

(A, 01)<I 6∼= (A, 02)

<I ?

Question 4 Is it true that if γ is an ordinal and cardM > w(X) then(M,<) is not embeddable into (X<γ∗ , <)?

Question 5Is there an uncountable model of PA order-embeddable into (R<ω, <)?

Question 6If (I,<) is a countable linear order then the following are equivalent:

1. For every linear order (A,<) there is an uncountable MA |= PAembeddable into A<I .

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2. I contains a copy of Q.

Question 7Let p be an indiscernible type over T , (C,<) be a linear order, M = EM(C, p).Prove that if t(u1, . . . , un) is a Skolem term of T , x1, . . . , xn ∈ C then eithertp(t(x1, . . . , xn)) 6= tp(x1) or t(x1, . . . , xn) = xi for some i = 1, . . . , n. Deducethat EM(C1, p) is elementarily embeddable into EM(C2, p) if and only if C1

is embeddable into C2. Deduce that there are 2λ pairwise non-embeddablemodels of T in any cardinality λ.

Question 8 Let λ > ω. Is there a family Cii∈2λ of dense linear orders ofcardinality λ with no last element such that i 6= j implies Ci is not embeddableinto (C∗

j + 0 + Cj)<Q? If yes, deduce, by Theorem 20 that there are 2λ

pairwise non-embeddable order-types of models of T of cardinality λ.

Question 9Prove internal versions of the uniqueness of the countable atomless booleanalgebra, the random graph etc. How do these structures look like?

Question 10 Find two models A,B |= PA such that A ≡ B,(A,<) ∼= (B,<) but (A+Q(A)(A∗ + A), <) 6∼= (B +Q(B)(B∗ +B), <).

Question 11 If M |= PA + Con(ZFC) and V |= ZFC is interpreted in M ,what can we say about (Ordinals(V ), <) in terms of (M,<)?

Question 12 If V |= ZFC + Con(ZFC) and U |= ZFC is an inner model inV , what can we say about (Ordinals(U), <) in terms of (Ordinals(V ), <)?

Question 13

1. How many pairwise non-isomorphic elementarily equivalent models ofZFC are interpreted inside M |= PA + Con(ZFC)? (If M is countableand nonstandard then there is only one (the countable recursively satu-rated model with standard system SSy(M)). What is the order-type ofits ordinals?)

2. How many non-isomorphic order-types of ordinals of models of ZFCinterpreted in a model of PA + Con(ZFC) are there?

3. What are those order-types? How about Friedman’s problem for modelsof ZFC from the point of view of a model of PA + Con(ZFC)?

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Question 14

1. Is there an uncountable self-similar model which is not recursively sat-urated?

2. Is there a resplendent model which is not an inner model?

3. Is there an uncountable self-similar model which is not an inner model?

4. Give an example of two elementarily equivalent non-isomorphic innermodels in M . (Of course, M will have to be uncountable.)

Question 15Let U |= ZFC be countable, T be a completion of PA coded in U , U |= Con(T ).How many non-isomorphic models of T are interpreted in U?

Question 16 (probably a very hard one)Find a mathematical statement in the language LPA independent of TA.

Question 17Is it true that if Ω |= TA is recursively saturated then there is V |= ZFC suchthat ωV = Ω.

Question 18 Prove that if Ω |= TA + Reflection(PA) is resplendent thenthere is U |= ZFC strongly interpreted in Ω such that ωU

∼= Ω.

Question 19 Is it true that in every M |= PA there is z coding ZFC suchthat TAM = (ZFCM)a?

Question 20 What are order-types of ordinals of countable models of ZFC?How many of them are there? Can there be some classification result?

Question 21 What are the order-types of ordinals of models of ZFC? Howdoes the order-type of ordinals of a saturated model look like? How manymodels with the saturated order-type are there?

Question 22 Do classes of order-types of ordinals of models of differentextensions of ZFC coincide?

Question 23 How many models of order-type N + ωQλZ are there?

Question 24 For every canonical order of the form N + µQλZ, (µ ≤ λ+),find the number of pairwise non-isomorphic models of PA of this order-type.What happens for arbitrary canonical orders?

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Question 25 Find four models, M1, M1, M2, M2, all copies of the satu-rated model, such that M1 is an elementary end-extension of M1, M2 is anelementary end-extension of M2, but there is no isomorphism f : M1 → M2

such that f(M1) = M2.

Question 26Prove that there is a model of PA of every canonical order-type.

Question 27Prove that there is a model of PA of order-type N + QZ +Qω1QZ.

Question 28 Prove that there is a model of PA of order-typeN + ω1QZ +Qω1(ω

∗1 + ω1)QZ.

Question 29Prove that there is a countable M |= PA such that for every countable end-extension N M and subsets A,B ⊂ N/M , A < B, there is K N suchthat there is c ∈ K such that A < c < B and x ∈ K | ∃y ∈M, y ≥ x = M .

Question 30 Prove that if λ is singular, |M | ≤ λ, then there are 2λ λ-denseelementary extensions of M of cardinality λ.

Question 31 Prove that there are 2λ order-types of resplendent models ofPA of cardinality λ.

Question 32 Prove without ♦ that there are 2ω1 models Mi |= PA, i < 2ω1

of order-type N + ω1QZ such that (Mi, ·) 6∼= (Mj, ·) for i 6= j.

Question 33 If M |= PA has cardinality ω1 and is not saturated then (M,<)has 2ω1 (M, ·)-non-isomorphic expansions to a model of PA.

Question 34Try tackling Question 10 in the universes satisfying ♦.

Question 35Modify the proof of Lemma 59 to include the case card X = ω1. (Start witha countable Scott Set Y0 ⊂ X and construct M as an elementary union ofMi |= PA, SSy(Mi) = Yi, card Yi = ω, Yi ⊆ Yi+1,

⋃Yi = X .)

Question 36 Let M |= Th N be countable and arithmetically saturated.Find g, h ∈ Aut(M) such that Aut(M) is generated by g and h.

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Question 37 If M is resplendent and codes ΣnT for all n, then(M,<) is expandable to a model of T .

Question 38 If M is resplendent and Υ(M) > ω then (M,<) is expandableto a model of Th N.

Question 39 Let T ⊃ PA, M |= T be resplendent. Let S ⊃ PA be consis-tent, S 6∈ SSy(M). Can (M,<) be expanded to a model of S?

Question 40

1. Prove that if (A,<) is a resplendent dense linear order then N +AZ isa resplendent discrete linear order with first and no last element.

2. Find an example of a resplendent model of PA whose order-type is notresplendent.

Question 41 Investigate the following notion of saturation: “M is recur-sively saturated and SSy(M) is closed under ω-jump: A→ A(ω)”.

Question 42Study mutual embeddability of orders (A, 0)/I and models of PA.

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Bibliography

[1] Handbook of Mathematical Logic. (1977). J. Barwise, ed. North-Holland,Amsterdam.

[2] Boolos, G. and Jeffrey, R. (1988). Computability and Logic. CambridgeUniversity Press, Cambridge.

[3] Chang, C.C. and Keisler, H.J. (1973). Model theory. North-Holland,Amsterdam.

[4] Charretton, C. and Pouzet, M. (1983). Comparaison des structuresengendres par des chaine. Proc. Dietrichshagen Easter Conference onModel Theory 1983. pp. 17-27. Berlin: Humboldt Univ.

[5] Charretton, C. and Pouzet, M. (1983). Chains in Ehrenfeucht-Mostowskimodels. Fundamenta Mathematicae, CXVIII, pp. 109-122.

[6] Enayat, A. (1986). Conservative extensions of models of set theory andgeneralizations. Journal of Symbolic Logic, 51, pp. 1005-1021.

[7] Friedman H. (1975). One hundred and two problems in mathematicallogic. Journal of Symbolic Logic, 40, pp. 113-129.

[8] Gaifman, H. (1976). Models and types of Peano’s arithmetic. Annals ofMathematical Logic, 9, pp. 223-306.

[9] Harnik V. (1986). ω1-like models of Presburger arithmetic. Journal ofSymbolic Logic, 51, pp. 421-429.

[10] Hodges, W. (1973). Models in which all long indiscernible sequences areindiscernible sets. Fundamenta Mathematicae, LXXVIII, pp. 1-6.

[11] Hodges, W. (1993). Model Theory. Cambridge University Press.

[12] Jech, T.J. (1978). Set theory. Academic Press.

103

[13] Kaye, R., Kossak, R., Kotlarski, H. (1991). Automorphisms of recur-sively saturated models of arithmetic. Annals of Pure and Applied Logic,55, pp. 67-99.

[14] Kaye, R.W. (1991). Models of Peano Arithmetic. Oxford UniversityPress.

[15] Kaye, R. and Kotlarski, H. (2000). On models constructed by meansof the Arithmetized Completeness Theorem. To appear in Math. LogicQuaterly.

[16] Automorphisms of First-Order Structures. (1994). Ed. Kaye andMacpherson, Oxford University Press.

[17] Kirby, L.A.S. and Paris, J.B. (1977). Initial segments of models ofPeano’s axioms. Set Theory and Hierarchy Theory, V. Bierutowice,Poland.

[18] Korner, F. (1998). Automorphisms moving all non-algebraic points, andan application to NF. Journal of Symbolic Logic, 63, pp. 815-830.

[19] Kossak, R. (1991). Exercises in ‘back-and-forth’. Proceedings of theNineth Easter Conference on Model Theory, Gosen.

[20] Kunen, K. (1980). Set theory. North-Holland, Amsterdam.

[21] Pabion, J.F. (1982). Saturated models of Peano Arithmetic. Journal ofSymbolic Logic, 47, pp. 625-637.

[22] Paris, J.B. and Mills G. (1979). Closure properties of countable non-standard integers. Fundamenta Mathematicae, 103, pp. 205-215.

[23] Schmerl, J. (1973). Peano models with many generic classes. PacificJournal of Mathematics, 46, pp. 523-526; erratum, ibid., 92 (1981),pp.31-55.

[24] Schmerl, J. (1989). Large resplendent models generated by indis-cernibles. Journal of Symbolic Logic, 54, pp. 1382-1388.

[25] Schmerl, J. (1998). Rather classless, highly saturated models of Peanoarithmetic. Logic Colloquium ’96 (San Sebastian), Lecture Notes inLogic, 12, pp. 237-246.

[26] Schmerl, J. (1999). Automorphism groups of models of Peano Arith-metic. As yet unpublished.

104

[27] Shelah, S. (1971). The number of non-isomorphic models of an unstablefirst-order theory. Israel Journal of Mathematics, 9, pp. 473-487.

[28] Shelah, S. (1978). Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam.

[29] Shelah, S. (1994). Vive la Difference II - the Ax-Kochen isomorphismtheorem. Israel Journal of Mathematics, 85, pp 351-390.

[30] Shelah, S. (to appear in 2000). Non-structure Theory.

[31] Shoenfield, J.R. (1967). Mathematical logic. Addison-Wesley, Reading,Massachusetts.

[32] Simpson, S.G. (1974). Forcing and models of arithmetic. Proceedings ofthe American Mathematical Society , 43, pp. 193-194.

[33] Smorynski, C. A. (1984). Lectures on non-standard models of arithmetic.Logic Colloquium ’82, pp. 1-70. North-Holland, Amsterdam.

[34] Smorynski, C.A. (1977). Godel’s incompleteness theorems, in Handbookof Mathematical Logic. J. Barwise, ed. North-Holland, Amsterdam.

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