T H E

JOURNAL OF

SYMBOLIC LOGIC EDITED BY

HERBERT ENDERTON

DAGFINN E0LLESDAL

Y . N . MOSCHOVAKIS

M a n a g i n g E d i t o r : ALFONS BORGERS

A L O N Z O C H U R C H

W I L L I A M C R A I G

C . E . M . Y A T E S

A N I L NERODE

Jiftf B E £ V Ä R A N D R Z E J B L I K L E M A R T I N D A V I S E D W A R D E . D A W S O N V E R E N A H . D Y S O N C A L V I N C . E L G O T F R E D E R K : B . F I T C H

C o n s u l t i n g Editors:

G . H A S E N J A E G E R V

H A N S H E R M E S H . J E R O M E K E I S L E R A Z R I E L L E V Y D O N A L D M O N K G E R T H . M Ü L L E R M I C H A E L O . R A B I N G E N E F . R O S E

B A R K L E Y ROSSER A R T O S A L O M A A K U R T S C H Ü T T E DANTA S C O T T JOSEPH R . SHOENEIELD J . F . T H O M S O N J E A N V A N H E I J E N Q O R T

V O L U M E 4 3 N U M B E R 1

M A R C H 1978

PUBtrISHED Q U A R T E R L V B Y T H E ASSOCIATION FOR S Y M B O L I C L O G I C , INC. W I T H

SUPPORT F R O M I C S U A N D FROM INSTITUTIONAL A N D CORPORATE M E M B E R S

v Copyright © 1978 by the Association for Symbolic Logic, I n c . Reproduction by photostat, photo-print, microfilm, or like process by permission only

?0 G

TABLE OF CONTENTS

On the inconsistency of Systems simüar to J^* . ByM. W . BUNDER and R . K . MEYER . 1

First Steps in intuitionistic model theory. By H . DE SWART...... 3 Controlling the dependence degree of.a recursively enumerable vectot Space. By

RICHARD A. SHORE...... 13 Relativized realizability in intuitionistic arithmetic of all finite types. By

NICOLAS D . GOODMAN '. 2 3 Degrees of sensible lambda theories. By HENK BARENDREGT, J A N BERGSTRA,

J A N WILLEM KLOP and HENRI VOLKER.. . . 4 5 Projective algebra and the calculus of relations. By A. R . BEDNAREK änd S. M .

U L A M . . 5 6 Saturated ideals. By KENNETH KUNEN................ .. 6 5 Separation principles and the axiom of determinateness. By ROBERT A. VAN

WESEP 7 7 The model theory of ordered differential fields. By MICHAEL F. SINGER 8 2 Rings which admit elimination of quantifiers. By BRUCE I . ROSE.. 9 2 Nöte on an induction axiom. By J . B . PARIS 1 1 3 Provable wellorderings of formal theories for transfinitely iterated inductive

definitions. By W . BUCHHOLZ and W . POHLERS.... 1 1 8 Existentially cojnplete torsion-free nilpotent groups. By D, SARACINO 126 Sets which do not have subsets of every higher degree. By STEPHEN G . SIMPSON 135 Reviews. , L.; , 139

The JOURNAL OF SYMBOLIC LOGIC is the official organ of the Association for Symbolic Logic Inc., published quarterly, in the months of March, June, September, and December. The JOURNAL also contains the! official notices of the Division of Logic, Methodology and Phüosophy of Science of the International Union of History and Phüosophy of Science.

Membership in the Association is open to any one interested in its work. Annual dues for students are $9.00. Dues for other individual members are $18.00 per year. Dues include subscription to the current volume of the JOURNAL. For their personal use, all members may purchase back issues at 25% discount.

For nonmembers, subscription to the current volume of the Journal is $48 per year. Back volumes may be purchased at $20 per volume for volumes 1-17, $25 per volume for volumes 18-36, $45 for volume 37, $53 for volume 38, and $60 per volume for volumes 39-41. Back volumes of JSL are also available on 16 mm microfilm and may be mounted on spools or in Eastman or 3M cartridges. Volumes 1-37 (1936-1972) are mounted on 5 spools and cost $325.00 for spools or $340.00 for cartidges.

Applications for membership in the Association, requests for information, all business corre-spondence and notices of change of address should be sent to T h e Association f o r Symbolic L o g i c , P . O . Box 6 2 4 8 * Providencty Rhode I s l a n d 0 2 9 4 0 . Dues and subscriptions to the JOURNAL should be sent to T h e Association f o r Symbolic L o g i c , P . O . Box 1 5 7 1 , A n n e x S t a t i o n , Providenee, Rhode I s l a n d 0 2 9 0 1 ; all Orders must be accompanied by payment.

Two copies of any article, being submitted for publication in the JOURNAL should be sent to Y i a n n i s Moschovakis, D e p a r t m e n t of M a t h e m a t i c s , U . C . L . A . , Los Angeles, C A 9 0 0 2 4 or to W i l l i a m

C o n t i n u e d on inside back cover

Second-class postage rates paid at Providence, R.I. and additional Offices.

T H E

JOURNAL OF

SYMBOLIC LOGIC E D I T E D B Y

A L O N Z O C H U R C H

W I L L I A M C R A I G

H E R B E R T E N D E R T O N

D A G F I N N F 0 L L E S D A L

Y . N . M O S C H O V A K I S

C . E . M . Y A T E S

A N I L N E R O D E

M a n a g i n g E d i t o r : A L F O N S B O R G E R S

J IRI B E C V Ä R A N D R Z E J B L I K L E M A R T I N D A V I S E D W A R D E . D A W S O N V E R E N A H . D Y S O N C A L V I N C . E L G O T F R E D E R I C B . F I T C H

C o n s u l t i n g E d i t o r s :

G . H A S E N J A E G E R H A N S H E R M E S H . J E R O M E K E I S L E R A Z R I E L L E V Y D O N A L D M O N K G E R T H . M Ü L L E R M I C H A E L 0 . R A B I N G E N E F . R O S E

B A R K L E Y R O S S E R A R T O S A L O M A A K U R T S C H Ü T T E D A N A S C O T T J O S E P H R . S H O E N F I E L D J . F . T H O M S O N J E A N V A N H E I J E N O O R T

V O L U M E 4 3

1978

PUBLISHED QUARTERLY BY THE ASSOCIATION FOR SYMBOLIC LOGIC, INC. WITH SUPPORT FROM I C S U AND FROM INSTITUTIONAL AND CORPORATE MEMBERS

The four numbers of Volume 43 were issued at the following dates:

Number 1, pages 1-160, June 27, 1978

Number 2, pages 161-384, August 16, 1978

Number 3, pages 385-622, August 28, 1978

Number 4, pages 623-759, February 14, 1979

Numbers 1—3 of this volume are copyrighted © 197 8 by the Association for Symbolic Logic , Inc. Number 4 of this volume is copyrighted © 1979 by the Association for Symbolic Logic, Inc. Re-production of copyrighted numbers of the Journal by photostat, photoprint, microfilm, or like process is forbidden, except by written permission, to be obtained through the Business Office of the Association for Symbolic Logic, P.O. Box 6248, Providence, Rhode Island 02940, U. !S. A.

A U T H O R INDEX

A B R A M S O N , F R E D G. and H A R R I N G T O N , L E O A Models without indiscernibles.. 5 7 2

B A R E N D R E G T , H E N K , B E R G S T R A , J A N , K L O P , J A N W I L L E M and V O L K E N , H E N R I .

Degrees of sensible lambda theories 4 5 B A R W I S E , J O N and M O S C H O V A K I S , Y I A N N I S N . Global inductive definability 5 2 1

B E D N A R E K , A . R. and U L A M , S. M . Projective algebra and the calculus of relations 5 6

B E E S O N , M I C H A E L . A type-free Gödel Interpretation 2 1 3 . . Some relations between classical and constructive mathematics 2 2 8

B E R G S T R A , J A N . See B A R E N D R E G T , H E N K .

B O T T S , T R U M A N . What is the Conference Board of the Mathematical Sciences?.. 6 2 0 B R U C E , K I M B. Ideal models and some not so ideal problems in the model

theory o f L ( ß ) 3 0 4 ^ B U C H H O L Z , W. and P O H L E R S , W. Provable wellorderings of formal theories for

transfinitely iterated inductive definitions 1 1 8 B U N D E R , M . W. and M E Y E R , R. K . On the inconsistency of Systems

similar to g r * t 1 B U N D E R , M . W. and S E L D I N , J O N A T H A N P. Some anomalies in Fitch's System

Q D 2 4 7

B U R G E S S , J O H N P. On the Hanf number of Souslin logics 5 6 8 C U T L A N D , N I G E L . Zj-compactness in languages stronger than i£A 5 0 8 D E S W A R T , H . First Steps in intuitionistic model theory 3 D I P R I S C O , C. A and H E N L E , J . On the compactness of N t and N 2 3 9 4

E P S T E I N , R See P O S N E R , D.

F E R R O , R U G G E R O . Interpolation theorems for h\*k 5 3 5

G A L V I N , F., J E C H , T. and M A G I D O R , M. An ideal game 2 8 4

G O O D M A N , N I C O L A S D. Relativized realizability in intuitionistic arithmetic of all finite types 2 3

. The nonconstructive content of sentences of arithmetic 4 9 7 G R E E N , J U D Y . K-Suslin logic 6 5 9

H A R R I N G T O N , L E O A. Analytic determinacy and 0 # 6 8 5 . See A B R A M S O N , F R E D G.

H E N L E , J . See D I P R I S C O , C . A .

H I R S C H F E L D , J O R A M . Examples in the theory of existential completeness 6 5 0 H O D E S , H A R O L D T. Uniform upper bounds on ideals of Turing degrees 6 0 1

iii

iv A U T H O R I N D E X

H O W A R D , P A U L E. , R U B I N , A R T H U R L . and R U B I N , J E A N E . Independence

results for class forms of the axiom of choice 6 7 3 J E C H , T. See G A L V I N , F .

J O C K U S C H , C A R L G., J R . and P O S N E R , D A V I D B . Double jumps of minimal

degrees 7 1 5 J O H N S O N , N A N C Y . Classifications of generalized index sets of open classes 6 9 4 J O N E S , J A M E S P. Three universal representations of recursively enumerable

sets 3 3 5 K A L A N T A R I , I R A J . Major subspaces of recursively enumerable vector Spaces 2 9 3 K E C H R I S , A L E X A N D E R S. Minimal upper bounds for sequences of -degrees... 5 0 2

. The perfect set theorem and definable wellorderings of the continuum 6 3 0

K E I S L E R , H . J E R O M E . The stability function of a theory 481 K L E E N E , S T E P H E N C. An addendum to "The work of Kurt Gödel" 6 1 3 K L O P , J A N W I L L E M . See B A R E N D R E G T , H E N K .

K N I G H T , J U L I A F. An inelastic model with indiscernibles 3 3 1 . Prime and atomic models 3 8 5

K U N E N , K E N N E T H . Saturated ideals 65

L E G G E T T , A N N E . a-Degrees of maximal a-r.e. sets 4 5 6 L I H , K O - W E I . Type two partial degrees 6 2 3

^ M A A S S , W O L F G A N G . The uniform regulär set theorm in a-recursion theory 2 7 0 M A G I D O R , M , See G A L V I N , F.

M E Y E R , R . K . See B U N D E R , M . W .

^ M I L L S , G E O R G E , A model of Peano arithmetic with no elementary end extension 5 6 3

M O S C H O V A K I S , Y I A N N I S N . See B A R W I S E , J O N .

N O R M A N N , D A G . A continuous functional with noncollapsing hierachy 4 8 7 P A R I S , J . B . Note on an induction axiom 113

. Some independence results for Peano arithmetic 7 2 5 / T I L L A Y , A N A N D . Number of countable models 4 9 2

P O H L E R S , W. Ordinals connected with formal theories for transfinitely iterated inductive definitions 161

. See B U C H H O L Z , W .

P O S N E R , D A V I D B . See J O C K U S C H , C A R L G., J R .

P O S N E R , D. and E P S T E I N , R . Diagonalization in degree constructions 2 8 0 R E M M E L , J . A r-maximal vector space not contained in any maximal vector

Space 4 3 0 R E T Z L A F F , A L L E N . Simple and hyperhypersimple vector Spaces 2 6 0 R O S E , A L A N . Formalisations of further N0-valued Lukasiewicz propositional

calculi 2 0 7

A U T H O R I N D E X V

R O S E , B R U C E I. Rings which admit elimination of quantifiers 9 2 . The ttj-categoricity of strictly upper triangulär matrix rings over

algebraically closed fields 2 5 0 R U B I N , A R T H U R L. See H O W A R D , P A U L E.

R U B I N , J E A N E. See H O W A R D , P A U L E.

S A R A C I N O , D . Existentially complete torsion-free nilpotent groups 126

S C H L I P F , J O H N S T E W A R T . Toward model theory through recursive Saturation 183 ^ S C H Ü M M , G E O R G E F . An incomplete nonnormal extension of S3 2 1 1

S E L D I N , J O N A T H A N P. A sequent calculus formulation of type assignment with equality rules for the 50-calculus 6 4 3

. See B U N D E R , M. W.

S H E L A H , S A H A R O N . On the number of minimal models 4 7 5 . End extensions and numbers of countable models 5 5 0

S H O R E , R I C H A R D A Controlling the dependence degree of a recursively enumerable vector Space 13

. Nowhere simple sets and the lattice of recursively enumerable sets 3 2 2 S I M P S O N , S T E P H E N G. Sets which do not have subsets of every higher degree 1 3 5

S I N G E R , M I C H A E L F . The model theory of ordered differential fields 8 2 S O L O M O N , M A R T I N K . Some results on measure independent Gödel

speed-ups 6 6 7 S T E P A N E K , P E T R . Cardinal collapsing and ordinal definability 6 3 5 U L A M , S. M. See B E D N A R E K , A. R.

V A N W E S E P , R O B E R T A. Separation principles and the axiom of determinateness 7 7 V O L K E N , H E N R I . See B A R E N D R E G T , H E N K .

W H E E L E R , W I L L I A M H . A characterization of companionable, universal theories 4 0 2 W O O D R O W , R O B E R T E. Theories with a finite number of countable models 4 4 2 Meeting of the Association for Symbolic Logic, Campinas, Brazil, 1976 . By A Y D A

I. A R R U D A , F R A N C I S C O M I R O Q U E S A D A , N E W T O N C. A. D A C O S T A and

R O L A N D O C H U A Q U I 3 5 2

Abstracts of papers 3 5 2 Annual meeting of the Association for Symbolic Logic, Saint Louis, 1977 . By

J O N B A R W I S E , K E N N E T H K U N E N and J O S E P H U L L I A N 3 6 5

Abstracts of papers 3 6 5 Meeting of the Association for Symbolic Logic, Chicago, 1977 . By C A R L G.

J O C K U S C H , JR . , R O B E R T I. S O A R E , W I L L I A M T A I T and G A I S I T A K E U T I 6 1 4

Abstracts of papers 6 1 4 Reviews 1 3 9 Reviews 3 7 3 List of officers and members of the Association for Symbolic Logic 7 3 2

Notice of meeting 7 5 9

THE J O U R N A L O F SYMBOLIC LOGIC Volume 43, Number 1, March 1978

P R O V A B L E W E L L O R D E R I N G S O F F O R M A L T H E O R I E S F O R T R A N S F I N I T E L Y I T E R A T E D

I N D U C T I V E D E F I N I T I O N S

Introduction

By [12] we know that transfinite induction up to ©en^iO is not provable in I D N , the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ©en^iO is provable in IDjy, the intuitionistic version of I D N , and extend this result to theories for transfinitely iterated inductive definitions.

In [14] Schütte proves the wellordering of his notational Systems X ( N ) using predicates 93 k(a) : e ( a £ M k A {X E M k : x < a } is wellordered) with M k : = {x E 2(N): ^ ( K o x ) A * • • A Sk-iCKfc-jx)}1 and 0 < k < N. Obviously the predicates 93o,.. .,93N-I are definable in ID ' N with the defining axioms:

where P r o g [ M k , X ] means that X is progressive with respect to M k , i.e.

Prog [ M k , X ] : ~ V x E M k ( V y E M k ( y < x -> X(y ) ) -> X ( x ) ) .

The crucial point in Schütte 's wellordering proof is Lemma 19 [14, p. 130] which can be modified to

(I) T I [ M k + 1 , a ] , Sb = k, 93k(fc)=> 93k((a,fc)), for 0 < k < N - l ,

where T I [ M k + i , a ] is the scheme of transfinite induction over M k + i up to a2. Checking the proof of (I) it turns out that besides (93 kl) and (23k2) (0 < k < N - 1) only finitary methods (including mathematical induction) are used. Since the proof uses "excluded middle" only for decidable formulas it is formalizable in ID' N . Following the proof of Lemma 17 in [14] one gets

(II) IDiv h 930(1) A • • • A ® N - i ( n N - , ) and

(III) I D J v h T I [ M N , a N ] .

From (III) one derives in the well-known way (due to Gentzen [5])

(IV) ID'„ h TI [ M N , cn ] for each n £ J V ,

Received January 5, 1976. 1 K n x is a finite set of subterms of x . 93„(K„x) means Vy € X„x(93„(y)). 2 For an exact definition see notational Convention (5), page 3 of the present paper.

W. B U C H H O L Z and W. P O H L E R S

(83*1)

(93k2)

Prog[M k ,93 k ] ,

Prog [ M k , m - > V x (93k (x) - + g[x ]),

118 © 1978, Association for Symbolic Logic

WELLORDERINGS OF FORMAL THEORIES 119

where C 0 :={1N, c n + 1 : = ( l , c„ ) . By (I), (II), (IV) and the facts that M 0 = S ( N ) and, 93 fc(a) implies T I [ M k , a ] one gets

(V) I D ' N h TI [2(N), ft[cn, 0]] for each n E JV,

where ft[c„, 0] :=((•• • (c„, n N_i), . . . , Hj), 1). Since s u p „ e N fi[c„, 0] = n [ ( l # l , f t N ) , 0 ] and the order type of {JC E £ ( N ) : X <f t [ ( l # l,nN),0]} is 0£n^iO 3 , t r a n s f i n i t e i n d u c t i o n up to any o r d i n a l less than 0£n^ iO is p r o v a b l e i n IDjs/, which we will abbreviate by I D N I - T I [ < ©en^iO].

Similar considerations apply to the wellordering proof of the System @({g}) given in [2]. We will prove the following results:

(A) I D > T I [ < ©erv-iO] for any countable v E @({g})\

(B) ID ,

< *hTI [<0en n i + iO] 5 ,

where IDt and ID<. are the intuitionistic versions of the theories I D V and ID<. de-fined in [4, pp. 307-308] (see also the last paragraph of page 3 of the present paper).

ID<„ is defined to be the theory U € < V I D { . By ID^we mean the theory of autonomously iterated inductive definitions (i.e. if I D h T I [ y ] thert.ID„ C I D ) . For a theory Th we take | T h | := sup { £ E O n : T h h T I [ £ ] } . There are the following ordinal theoretic relations:

(1) ©£n a i +i0 = © £ n n | + i 0 and @en,+i0 = @€n,+i0 for v<@aill0. (So the above derived results on transfinite induction in ID ' N ( N < a>) are special cases of (A).)

(2) eSlv0 = sups<v Öen^iO for limit v < @a f t l0. (3) 01X0 = v for v = © a n i o . (4) @ ü n i 0 = s u p „ e N ^ n with v 0 : = l , ivn := 0ft»* 0. By (A), and [13] (cf. footnote 4) we get the equations:

(AI) | I D , | = | IDt | = ee n ^i0 for v < @fl n i 0.

(A2) | ID<„ | = | ID!c„| = 0^,0 for limit v < ©ft^O.

(A3) _ _ HD l&n^ol = l I D ( 0 l = ®nn,0 and I D ( 0 has the same theorems as IDi&n^o.

Preliminaries. In the sequel we assume an arithmetization of the notational System @({g}), such that all relevant ordinal sets, functions and relations of [2] (as S£, K u a , S, + , 0, < , etc.) become primitive recursive6. We will identify ordinal notations and their arithmetizations.

Though we presume some familiarity with [2], we will give a short d e s c r i p t i o n of the System 0({g}). ©({g}) is a set ST of ordinal notations ordered by a r e l a t i o n < . Each element of £ has the shape 0, a + b, @ab or gab with

3 Cf. [3]. Note that the System 2(N) in [3] is a slight modification of that in [14]. In [3] the first element of X(N) is 0 instead of 1.

4Recently the second author [13] was able to show I D „ / T I [ Ö e n „ + i 0 ] . 5Kino's wellordering proof for her ordinal diagrams Od(J) [8] is formalizable in ID'<.. Hence

IOlc-hTI[||Od(/), <-||]- But as remarked in [2] ||Od(/),<~||<Oa n |(r + 1)<0(n n , + 1)0< 0e ^ , 0 for || / | | = 1 + r < e(nn, + 1)0.

6 For a Subsystem of 0({g}) such an arithmetization will be carried out in [15].

120 W. B U C H H O L Z A N D W. P O H L E R S

a, b E ST. The Symbols + (ordinal sum), 0 and g denote 2-place ordinal functions. So each term a E St canonically represents an ordinal | a |, for example |©00 | = 1, | © 0 1 | = a>, | ©101 = e0. For the order relation < we have a < b * * \ a \ E \ b \ . Terms of the shape & a b or g a b a r e called main terms; they represent ordinals closed under + . Ä is the set of all terms g a b E St\ the elements of Si represent initial ordinals > o). There is a primitive recursive order isomorphism a »-» C l a from 2T onto Ä 0 := {0} U Ä with f l 0 = 0 and | fta | = f l | a | for a ^ 0. For each a E ST there is exactly one x E St with C l x < a < H x + ] ; we dehne Sa: = C l x and call it the level (Stufe) of a . For all a , b E St we have S@ab = Sfc, h e n c e 0 a O < f t , . We dehne M 0 :={x E S T : Sx =0} = {x E S~: x < fli}. M 0 represents a Segment of the countable ordinals, i.e.

(1) { £ E O n : £ < |a |} = {|x|: x E M 0 A X < a ) for a E M 0 . For a E S" and w E Ä 0 , is a finite set of main terms with levels < u. The

sets K u a have the following properties: (2) If Sa < u, then iCMa is the set of components of a 7. (3) K u b C K u ( a + b ) C K u a \ J K u b . (4) w < ü A c e K v a -» K w c C X w a . (5) v < ( l a - * K v a , a C K v a C K v n a U { l } . We fix t h e f o l l o w i n g n o t a t i o n a l C o n v e n t i o n s : (1) a, b, c, x, y, z denote elements of 3T. (2) M, u, w denote elements of Ä 0 . (3) SR,3E serve as syntactical variables for sets {x: $[x ] }cS", where $[x] is a

formula of the theory considered. (4) X D a : = { x G X : x < a } , X n w + := {x E X : Sx < w}. (5) Prog [SR, X] abbreviates the formula V x E »(SR n x C £ -> x E £ ) .

TI[SR,£, a] abbreviates the fromula a E SR A (Prog [9t, £]-»SR D a C X \ and TI[SR, a] denotes the scheme {TI[SR, a ] } * expressing the principle of trans­finite induction over SR up to a .

T r a n s f i n i t e i n d u c t i o n s p r o v a b l e i n ID'„ a n d ID<*. Our main tool in proving transfinite inductions will be the concept of the a c c e s s i b l e p a r t W[SR] of a set SR, usually defined by W[SR] = {x E SR: SR H x is wellordered}, which is a second-order definition. This definition however can be replaced by an inductive definition, which is expressible in a first-order language by the infinite liSt of axioms:

(i) Prog [SR, W[SR]] and (ii) Prog [SR, £ ] - > W [SR ]C£ for each X . The t h e o r i e s IDj, ( w i t h v E M 0 ) a n d ID<- are formal theories for iterations of

such inductive definitions. They are first-order extensions of Heyting's arithme­tic, where ID'„ allows iteration of monotone inductive definitions along the segment M 0 f l v, while ID<» allows iteration along the accessible part W 0 : = W [ M 0 ] of M 0 . Besides the axioms for iteration of inductive definitions (cf. [4, p. 307, (i), (ii)]) there are the axioms:

(Tl . ) Prog [Mo, X ) M 0 f l v C X for each 3E, in ID . ,

7 For each a^0 there are uniquely determined main terms a x > ••• > a„ ( n > 1) such that a = ÜI + • • • + a n . We call a u . . . , a n the components of a . 0 is defined to have no components.

W E L L O R D E R I N G S O F F O R M A L T H E O R I E S 121

asserting the wellordering of M 0 ,

(W 01) Prog [Mo, Wo] and

(W 02) Prog [Mo, £]-» W o C l for each 3E, in IDÜ-,

defining the accessible part W 0 of M 0 . In order to treat I D l and ID<* simultaneously as far as possible, we refer to

both as ID* and define A to be the set { C l x : x < v ) in the case of I D l and the set { i l x : x E Wo} in the case of ID<*. Then A is a segment of Ä 0 H (In, with 0 E A .

I n t h e sequel w, v a r e r e s e r v e d t o denote elements of A ! Define 2I[X, Y , x , y ] to be the formula g[x] A V x 0 < x ( g [ x 0 ] - » x 0 E X ) ,

where g[x] Stands for Sx < ü y A V Z , < y({(z 0 , z,}: ZoE K < i z x } C Y ) . Then 2l[X, Y, x, y] is an arithmetic formula such that each occurrence of X is positive. To 21 corresponds a set constant P % (cf. [4, p. 307]). We define

W a y : = {x : (x ,y>EP 9 1 } and

M u : = { x : S x < w A V V < u ( K v x C W V ) } .

Then the axioms (i) and (ii) of [4, p. 307] become

(Wl) Vw E A ( P r o g [ M u , Wu]) and

(W2) V K 6 A ( P r o g [ M M , . J ? ] - > Wu C X ) for each £ .

(Wl) and (W2) assert that Wu i s t h e a c c e s s i b l e p a r t of M u . Clearly for u =0, M u

coincides with the previously defined set M 0 = iT Pl C l u and in the case of ID<* the set Wo defined by (Wl), (W2) coincides with the set W 0 defined by (W 01), (W 02). A s immediate consequences of (Wl), (W2) the following formulas are provable in I D ' :

(6) Vx E W u (x E M u A M u D x = Wu H x ) , i.e. Wu is a segment of M u . (7) ö G W U ^ T I [ M U , J , a ] .

By (3) and the definition of M u we get (8) a,b£Mu-*a + b E M u and a + fcEMu—>fcE M u . The following lemmata 1-3 are straightforward modifications of correspond-

ing lemmata in [9], [10], [11] and [14]. L E M M A 1. (a) a, b E Wu—> a + b E WM arcd

(b) Sa<w A i£Ma C WM—» a E W u are p r o v a b l e i n ID' . P R O O F . By (6) and (8). a, E Wu A V X E M u f l b(a + x E WM)-> a + 6 E

M u A M„ n ( a + fc)C W u . Hence by (Wl) , a E W u - > P r o g [ M u , { x : x + a E Wu}] and thence by (W2), a, E Wu—> a + 6 E W u . Part (b) is an immediate consequence of (a) and (2).

L E M M A 2. (a) a E Wu—> Kva C Wv and (b) v < u ^ > W v = W u r \ v + a r e p r o v a b l e i n ID' . P R O O F . Suppose a E W u A v = H x . Using (TI P ) or (W 02) resp. we prove

KuöCW, by transfinite induction on x. For u < u we have K„flCW„ by a E . W u C M u . From w < i ; we get M v C \ u + C M u , hence by (Wl), Prog [M M , {x: x E M„ -* x E WJ] and thence by (W2), Wu C{x: x E M v -» x E WJ, i.e. W u nMu C W„. By the induction hypothesis we have K w a C W w for all

122 W. BUCHHOLZ AND W. POHLERS

w < v and hence a E Wu n M v C W^. By (2), (4) and (6) we then get K v a CM„ f l (a + 1 ) C W , Part (b) follows from (a) by Lemma l(b).

LEMMA 3. wE W u is p r o v a b l e i n I D \ PROOF. We have {x: £lx E A } C W 0 , which is trivial for I D < « and proved by

( T L ) and ( W l ) in ID '„ . So for u = ( l x we get x E Wo and thence by (5) and Lemma 2(a), K v u C K v x C Wv for all v < w, which implies u E M « . Now suppose a E M u n w, then Sa E A D w, X S a a C W„ and by the lemmata l(b), 2(b), a E W t t . Hence M u D u C W u and by (Wl), u E W u .

DEFINITION. Q : = { X : 3 U ( I G W u ) } = U u e A W u ; M : = { x : V v ( K v x C Wv)}. Consequences. 1. Obviously Sa E A for all a E Q. By Lemma 3 we then

have Vx(x E Q - » Sx E Q ) and Q H Ä 0 = A . Hence by Lemma 2(b) for all u E O

(9) O H M + = W U

and M U = { x : Sx < w A V W ( W E Q n u -> K w x C Q)}. That means the set Q is "ausgezeichnet" in the sense of [2, p. 18] with M ? H M + = M u and W ? = w [ M ? n M

+ ] = w u . 2. By (6) and (9) P r o g [ Q , £ ] - > Prog [ M u , { x : x E Wu—» x E £}] and thence

by (W2) (10) P r o g [ 0 , ^ ] ^ Q c £ for each £ ,

which is the first-order formulation of the fact that Q is w e l l o r d e r e d . 3. Since Q is "ausgezeichnet" (provable in I D ' ) we may follow the proof of

Theorem 15(b) in [2, p. 19] and get the formula

a E M A V x G M n a ( Q C R x ) ^ > Prog[Q, R a ] \

where R a : = { y : @ay E iT—>©ay E Q } . Here besides the premise " Q aus­gezeichnet" only methods formalizable in Heyting's arithmetic are used. By (10) it follows

(11) P r o g [ M , { x : V y E Q(@xy E 5T-^@xy E (?)}]. 4. From outside we know that ®({g}) = ( & , < ) is wellordered and hence

Wu = M U = {x: Sx < u } and M = ST which implies Q = M D C l ^ er defined by: DEFINITION.

v, in the case of I D l ,

Hi, in the case of ID<..

Of course W U = M U = {x: Sx < w} is not provable in ID' , but the weaker assertion Q = M C\ f l a is provable as the following theorem shows.

THEOREM 1. C l ^ E M a n d Q = M f l C l a a r e p r o v a b l e i n ID' . PROOF. B y Lemma 2(a) we have Q C M D H a G M and Sa E A we get

K S a CL CWSa and by Lemma l(b), a E W S a C Q . So we just have to prove a E M D Sa E A and O f f e M . The proofs differ for I D l , I D U

1. I D l Then A = { w : w <Ü<r} and trivially aEMnQ,a-*SaEA holds. By ( T L ) and ( W l ) we get a = v E W 0 . Hence by Lemma 2(a) and (5) V v i K v S l t r C K u O - C Wv) which means E M .

8 In [2] this formula is proved with Q in place of M, but an analysis of the proof shows that it is enough to have the premise a E M A V X G M fl a ( Q C R x ) .

WELLORDERINGS OF FORMAL THEORIES 123

2. ID<*. Suppose a E M H O r . Then K 0 a C W 0 and Sa = O for some x <0 = (7. By (5) K 0 x CK0£lx U{1}. Further K 0 S a C K 0 a and 1 E Wo. We therefore get by Lemma l(b) x E W 0 and thence Sa E A . For 0 < u, 2C 0ft n i

= 0 and K u f t a i = Obviously ftG A and hence O E W n , by Lemma 3. By Lemma 2(b) it follows V u ( K u S l a i C W u ), which means fl«, = ftn,E M

Now by (10) and Theorem 1 we get (12) TI [M ,0 , ] . Hence by (11) Vy E Q(0O,y E JT-> © f ^ y E Q ) . Since 0 E Q A 00,0 E

ST H ft, we obtain 00,0 E Q n O , Hence by (9) and (7) T I [ M 0 , 00,0]. This means we are able to c o l l a p s e the wellordering M fl O , to the provable ordinal ©OrO 9. This is a special case of the following theorem, which is proved by the above considerations with c in place of (!<,.

T H E O R E M 2 ( C O L L A P S I N G PROPERTY) . I f TI [ M , c ] is provable in I D ' and

0cOE5T, then TI [M 0 ,@cO] i s p r o v a b l e i n ID 1 . Starting from (12) we now prove T I [ M , c ] for each c E M f l ö l O r using

Gentzen's [5] method for proving T I [ < s 0 ] in number theory. DEFINITION. X : = { x : 01O, < x v V y ( M n y C l ^ M f l (y +_©0jt)C£)}. L E M M A 4. Vy ( M 0 y C X - » M fl (y + O,) C £)-H> Prog [ M , #] is p r o v a b l e

i n ID' . P R O O F . We have to prove M H (6 + ©Oa) Cäf under the assumptions

(1) V y ( M H y C l ^ M f l ( y +n . )Cf ) , (2) M f l a C l , (3) a < © 1 0 „ (4) M H c l

By (1) and (4) we get M fl (ft + O , • n) C X for all n E N using mathematical induction. Hence M PI (fr 4- ©OOr) C.3f because of sup n e J vOr • n = ©Oft,,.. Sup-posez G M n ( H ©Oa). We may assume b + ©00, < z. By (3) z < + ©Oa < b + © l O , . Hence z = b + ©Oa,- n + z, with 1 < n E AT, O , < a, < a, z, < ©Oa,. By Vu(u < Or = Sa () and the definition of Kv—it is the case that K v a x = J^ÖOa, C K v z . So we get a x E M P i a since z G M i s assumed. By (2), (3), (4) we get M fl (fr + ©Oai • (n + l))C.3f using mathematical induction. Hence Z E X .

DEFINITION. C 0 : = C l ^ c„+i: = @0c„. One easily proves cn E M , cn < ©lOr = s u p k e N c k and @c„0 E 2". T H E O R E M 3. TI [M,c„] is p r o v a b l e i n ID ' f o r each n E N . P R O O F . We prove the theorem by 'metainduction' on n . By (3)

it follows that a + b E M ^ b E M . Hence Prog [ M , £ ] A M H a C J-H>Prog[M,{jt: a + x E M - + a + x E X } ] and thence by (12)

(*) Prog [ M , X] V y ( M fl y C X - > M PI (y + O,) C £ ) .

By (*) and c 0 = Or E M we have T I [ M , c 0]. For n > 0 we have the induction hypothesis T I [ M , c„_,]. Hence Prog[M, X] —> cn_i E which implies Prog [M, l ] - ^ M D c n C if. By (*) and Lemma 4 we get Prog[M, J ] ^ P r o g [ M , X ] and therefore P r o g f M , X ] - * M fl c„ CX. Hence T I [ M , c ] .

T H E O R E M 4. I n ID', T I [ M 0 , a] is p r o v a b l e f o r each a < ©(01O,)O.

9Remember that by (1) M on0nC TO represents the ordinal ©("UO.

124 W. BUCHHOLZ AND W. POHLERS

P R O O F . For a < ©(©l f t ^O there is an n G JV with a < ©c„0. By Theorem 3 we have T I [ M , c„], which can be collapsed to T I [ M 0 , @c„0] by Theorem 2. So T I [ M 0 , a ] holds._

Since M 0 f l © ( © l f l ^ O represents the segment { £ G O n : £<0£n<,+iO}, the results (A) and (B) stated in the introduction follow from Theorem 4.

F I N A L R E M A R K S . The above proof of transfinite induction admits the following generalization. Let Th be a theory containing Heyting's arithmetic and axioms for iterations of inductive definitions along a provably wellordered subset A of Ä 0 . Then the sets Wu:= W [ M U ] , M u : = { x : S x < u A V U G A n « ( ^ C ^ ) } ( H £ A ) , Q : = { X : 3U G A ( X G W U ) } and

M : = {x: Vw G A ( K u x C Wu)} are definable in Th , and if the formula V u G A ( u G Wu A Vx G Wu(5x G A ) ) is provable in Th , one gets:

I. Q is w e l l o r d e r e d , i.e.

II. C o l l a p s i n g p r o p e r t y .

Th h T I [ M , c] ^> Th h T I [ M o , 0 c O ] (for ©cO G 3T).

III. Extension t o the next e-number.

ThhTI [M , n a ] ^> T h h T I [ M , c ] for each c G M f l 01O,.

From I, II, III it follows: I V .

T h h f t f l G M A M n f t f l C O ThhTI[<0(0in a)O]

for 0(010, )0G£: A s an example we regard the following definition by transfinite recursion on

j / G ^ r n o :

A(0):=0, A o : = 0 .

\(v + 1) := ß A ( „ ) + „ A „ + 1 := A v U { O : A(v) < x G W[M"]}, with M v : = { x : x < \ ( v + 1) A V W G A V ( K U X C W H )} and Wu defined as above by iteration of inductive definitions along A v .

A (*>): = sup^< i,A(^)10, A v : = U ^ < V A ^ for limit ordinals v.

Let I D ! be the theory, which allows to define A v and to iterate inductive definitions along A v (ID* for example is ID<.). Then by the above consider-ations we get:

I D t h T I [ < 0(010 )̂0].

B I B L I O G R A P H Y

[1] J . B R I D G E , A s i m p l i f i c a t i o n of the B a c h m a n n method f o r g e n e r a t i n g l a r g e c o u n t a b l e o r d i n a l s , this J O U R N A L , vol. 40 (1975), pp. 171-185.

For v = o>(l + a ) it is ftA(p) = \ ( v ) = g l a .

WELLORDERINGS OF F O R M A L THEORIES 125

[2] W . B U C H H O L Z , N o r m a l f u n k t i o n e n und konstruktive Systeme von O r d i n a l z a h l e n , Proof Theory Symposium, Kiel, 1974, Lecture Notes in M a t h e m a t i c s , no. 500, Springer-Verlag, Berlin and New York, 1975, pp. 4-25.

[3] W. B U C H H O L Z and K. S C H Ü T T E , D i e Beziehungen zwischen den Ordinalzahlsystemen X und Ö(o>), Archiv für Mathematische L o g i k und Grundlagenforschung, vol. 17 (1975), pp. 179-190.

[4] S. F E F E R M A N , F o r m a l theories for t r a n s f i n i t e iterations of generalized inductive definitions and some substystems of analysis, I n t u i t i o n i s m and p r o o f theory (Kino, Myhill and Vesley, Editors), North-Holland, Amsterdam, 1970, pp. 303-326.

[5] G . G E N T Z E N , Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der t r a n s f i n i t e n I n d u k ­tion i n der r e i n e n Z a h l e n t h e o r i e , Mathematische A n n a l e n , vol. 119 (1943), pp. 140-161.

[6] H . G E R B E R , Brouwer's bar theorem and a System of o r d i n a l notations, I n t u i t i o n i s m and p r o o f theory (Kino, Myhill and Vesley, Editors), North-Holland, Amsterdam, 1970, pp. 327-338.

[7] W. A . H O W A R D , A System of abstract constructive ordinals, this J O U R N A L , vol. 37 (1972), pp. 355-374.

[8] A . KINO, O n o r d i n a l diagrams, J o u r n a l of the M a t h e m a t i c a l Society of J a p a n , vol. 13 (1961), pp. 346-356.

[9] H . PFEIFFER, E i n Bezeichnungssystem für O r d i n a l z a h l e n , Archiv für Mathematische L o g i k und Grundlagenforschung, vol. 12 (1969), pp. 12-17.

[10] , E i n Bezeichnungssystem für O r d i n a l z a h l e n , Archiv für Mathematische L o g i k und Grundlagenforschung, vol. 13 (1970), pp. 74-90.

[11] , Bezeichnungssysteme für O r d i n a l z a h l e n , Communications of the Mathematics I n s t i t u t e of Rijksuniversiteit, Utrecht, 1973.

[12] W . POHLERS, Upper boundsfor the provability of t r a n s f i n i t e induction i n Systems with N-times iterated inductive definitions, Proof Theory Symposium, Kiel, 1974, Lecture Notes in M a t h e m a t i c s , no. 500, Springer-Verlag, Berlin and New York, 1975, pp. 271-289.

[13] , O r d i n a l s connected with f o r m a l theories of transfinitely iterated inductive definitions, this J O U R N A L , (to appear).

[14] K. S C H Ü T T E , E i n konstruktives System von O r d i n a l z a h l e n , Archiv für Mathematische L o g i k und Grundlagenforschung, vol. 11 (1968), pp. 126-137 and vol. 12 (1969), pp. 3-11.

[15] , P r o o f theory, 2nd edition, .Springer-Verlag, Berlin, Heidelberg, New York, 1977. [16] J . I. Z U C K E R , I t e r a t e d inductive definitions, trees and ordinals, M e t a m a t h e m a t i c a l investiga-

tion of intuitionistic arithmetic and analysis (A. S. Troelstra, Editor), Lecture Notes i n M a t h e m a t i c s , no. 344, Springer-Verlag, Berlin and New York, 1973, pp. 392-453.

MATHEMATISCHES INSTITUT DER LUDWIG-MAXIMILIANS-UNIVERSITÄT

MÜNCHEN, FEDERAL REPUBLIC OF GERMANY