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

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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.

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T H E

JOURNAL OF

SYMBOLIC LOGIC E D I T E D B Y

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

(B) ID , < *hTI [

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 trans-finite 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 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 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 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 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 aus-gezeichnet" 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

WELLORDERINGS OF FORMAL THEORIES 123

2. ID 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 : £

WELLORDERINGS OF F O R M A L THEORIES 125

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