Boffa, M.

Schwichtenberg, Helmut

English

Open Access LMU ( Ludwig-Maximilians-Univ. München)

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS V O L U M E 97 Editors J. Β A R W I S E , Madison D. K A P L A N , Los Angeles H . J . K E I S L E R , Madison P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam NORTH HOLLAND PUBLISHING COMPANY AMSTERDAM · NEW YORK · OXFORD (LOGIC COLLOQUIUM 78 Proceedings of the colloquium held in Möns, August 1978 Edited by M A U R I C E B O F F A Universite de Γ Etat ä Möns, Belgium D I R K V A N D A L E N Rijksnniversiteit Utrecht The Netherlands K E N N E T H M C A L O O N Universite Paris VII France 1979 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM · NEW YORK · OXFORD Machen* at isches InstlföS dar Universität MüöchäB © NORTH-HOLLAND PUBLISHING COMPANY, 1979 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN: 0 444 85378 2 Publishers: N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y A M S T E R D A M · N E W Y O R K · O X F O R D Sole distributors for the U.S.A. and Canada: E L S E V I E R N O R T H - H O L L A N D , INC. 52 V A N D E R B I L T A V E N U E N E W Y O R K , N Y . 10017 Library of Congress Cataloging in Publication Data Logic Colloquium, Möns, 1978. Logic Colloquium '78. (Studies i n logic and the foundations of mathemat-i c s ; v. 97) 1. Logic, Symbolic and mathematical—Congresses. I . Boffa, Maurice. I I . Dalen, Dirk van, 1932-I I I . McAloon, Kenneth. IV. Ser i e s . QA9.A1L63 1978 511».3 79-21152 ISBN O-lM-85378-2 P R I N T E D IN T H E N E T H E R L A N D S Dedicated to PAUL BERN AYS and KURT GÖDEL KURT GÖDEL (1906-1978) TABLE OF CONTENTS PREFACE v i i M. BEESON 1 Continuity in i n t u i t i o n i s t i c set theories G. CHERLIN 53 Stable algebraic theories R. DAVID 75 * Un r e s u l t a t de non contradiction r e l a t i v e au sujet de l a conjecture de Solevay 0. DEMUTH and A. RIVERA 81 Remarks on constructive mathematical analysis J. DENEF 131 The diophantine problem for polynomial rings of positive c h a r a c t e r i s t i c L. VAN DEN DRIES * 147 * Algorithms and bounds for polynomial rings S. FEFERMAN 159 Constructive theories of functions and classes P. HÄJEK 225 * On p a r t i a l l y conservative extensions of arithmetic A. HAJNAL and I . JUHÄSZ 235 Weakly separated subspaces and networks L. HARRINGTON and R. SAMI 247 * Equivalence r e l a t i o n s , projective and beyond G.F. VAN DER HOEVEN and A.S. TROELSTRA 265 Projections of lawless sequences I I R. LAVER 299 Linear order in (ω) ω under eventual dominance ix χ TABLE OF CONTENTS W. MITCHELL Hypermeasurable cardinals Z. RATAJCZYK * On the number of expansions of the model of ZFC-set theory to models of KM-theory of classes U.R. SCHMERL * A fine structure generated by ref l e c t i o n formulas over primitive recursive arithmetic H. SCHWICHTENBERG * Logic and the axoim of choice S. SHELAH On the successors of singular cardinals E. SPECKER Paul Bernays J . VAANÄNEN * Abstract logic and set theory. I . Definability V. WEISPFENNING Lattice products B. WEGLORZ Some σ-fields of subsets of reals * contributed papers - ο -The following invited lectures were presented at the Colloquium, but the text i s not reproduced in the present volume. A. DODD, The core model J. FENSTAD, General recursive theory: a unified view A. JOYAL, Sheaves in constructive mathematics G. KREISEL, Gödel's l a t e r views S. KRIPKE, Non-standard models and Gödel's theorem M. MORLEΥ, Theories with few models J. PARIS, Some problems in arithmetic 303 317 335 351 357 381 391 423 427 LOGIC COLLOQUIUM 78 Μ. Boffa, D. van Daten, Κ, McAloon (eds.) © North-Holland Publishing Company 3 1979 LOGIC AND THE AXIOM OF CHOICE H. S c h w i c h t e n b e r g ( M u n i c h ) Vie s h a l l p r o v e t h e f o l l o w i n g : ( .1) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s c o n s e r v a t i v e o v e r c l a s s i c a l ( f i r s t o r d e r ) l o g i c . ( 2 ) Vx3y φ(χ,ν) -* 3 f V x ω(χ,ίχ) i s c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c l o g i c w i t h o u t e q u a l i t y . ( 3 ) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c l o g i c w i t h d e c i d a b l e e q u a l i t y . ( 4 ) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s n o t c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c l o g i c . ( 5 ) A \ Vx,3y Vx 3y f/X\co(x. ,y. ) Λ /Ά(Χ.=Χ. - y . = y . ) l l i r\ J η r i " i ι Ί ι ι J η ι ι , 3 J 3 f V x φ(χ,ίχ) i s c o n s e r v a t i v e o v e r c l a s s i c a l and i n t u i t i o n i s t i c l o g i c . More p r e c i s e l y : A d d i t i o n o f f i n i t e l y many i n s t a n c e s o f t h e r e s p e c ­t i v e schema w i t h a l l (number and f u n c t i o n ) p a r a m e t e r s g e n e r a l i z e d i s c o n s e r v a t i v e o v e r any f i r s t o r d e r t h e o r y i n t h e r e s p e c t i v e l o g i c . None o f t h e s e r e s u l t s i s new. ( 1 ) i s a l r e a d y i n H i l b e r t - B e r n a y s 1939 ( p . 141 i n t h e second e d i t i o n ) . ( 2 ) - ( 4 ) a r e due t o Mine 1966, 1974, 1966; n o t e t h a t ( 3 ) i s an i m m e d i a t e consequence o f ( 5 ) . As t o ( 4 ) , a s i m p l e r c o u n t e r e x a m p l e i s i n Osswald 1975 and p r o b a b l y t h e s i m p l e s t ( w h i c h i s r e p r o d u c e d h e r e ) i n S m o r y n s k i 1978. ( 5 ) i s due t o Mine and S m o r y n s k i ; i t was f i r s t a nnounced i n Mine 1977. P r o o f s a r e i n S m o r y n s k i 1978 and ( i n a g e n e r a l i z e d f o r m d e a l i n g w i t h " s i m u l t a n e o u s Skolem f u n c t o r s " ) i n L u c k h a r d t #. The p r o o f s g i v e n h e r e a r e r e l a t i v e l y s i m p l e . F o r ( 5 ) t h e p r o o f c o n s i s t s i n a p r o c e d u r e w h i c h t r a n s f o r m a d e r i v a t i o n o f a f i r s t o r d e r f o r m u l a i n v o l v i n g t h e a x i o m o f c h o i c e i n t o a d e r i v a t i o n n o t i n v o l v i n g i t . The main t e c h n i c a l t o o l i s t h e use o f a new t y p e o f 351 352 Η. SCHWICHTENBERG f u n c t i o n v a r i a b l e s : Whenever t e r m s r . , . . . , r , s „,..., s a r e i n t r o -1 η 1 η d u c e d , t h e n f ^ i s a f u n c t i o n v a r i a b l e . The i n t e n d e d meaning o f f ~ r r i s t h a t i t s h o u l d r a n g e o v e r a l l f u n c t i o n s mapping r i n s ( p r o v i d e d r , s d e t e r m i n e a f i n i t e f u n c t i o n , i . e . /ΛΑΓ. = r . -> s . = s . ) ) . ~ i»] 1 3 1 3 As a l r e a d y n o t e d , ( 3 ) and a l s o ( 1 ) a r e easy c o n s e q u e n c e s o f ( 5 ) , s i n c e t h e p r e m i s s o f t h e i m p l i c a t i o n i n ( 5 ) i s u n d e r t h e a s s u m p t i o n Vx,y ( x = y ν χ Φ y ) e q u i v a l e n t t o Vx3y t p ( x , y ) . So we s t a r t w i t h a p r o o f o f ( 2 ) , t h e n g i v e S m o r y n s k i ' s c o u n t e r e x a m p l e t o p r o v e ( 4 ) , and f i n a l l y e x t e n d t h e method f o r p r o v i n g ( 2 ) t o a p r o o f o f ( 5 ) ; o n l y t h i s l a s t s t e p i n v o l v e s t h e f u n c t i o n v a r i a b l e s f ~ . N o t e : ( 1 ) - ( 5 ) r e m a i n v a l i d - w i t h e s s e n t i a l l y t h e same p r o o f s -when a l l v a r i a b l e s x, y, f , ... a r e r e p l a c e d by f i n i t e sequences x, y^, f , ... o f v a r i a b l e s ; f & t h e n means f x , . . . , f x. However, f o r s i m p l i c i t y we o n l y d e a l w i t h s i n g l e v a r i a b l e s h e r e . N o t e : ( 2 ) , ( 4 ) and ( 5 ) a l s o h o l d f o r m i n i m a l l o g i c . T h i s i s seen e a s i l y f r o m t h e p r o o f s . P r o o f o f ( 2 ) : I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c l o g i c w i t h o u t e q u a l i t y . We s h a l l work w i t h a Gentzen s e q u e n t c a l c u l u s as d e s c r i b e d i n Kleene 1952, p. 481 ( t h e r e i t i s c a l l e d G3). For s i m p l i c i t y we m o d i f y i t t o i n c l u d e 1 ( f a l s u m ) as a p r o p o s i t i o n a l c o n s t a n t and t r e a t -φ as d e f i n e d by φ -» 1 . F i r s t n o t e t h a t ( 2 ) can be r e d u c e d t o ( 2 ) ' I f h Vx φ ( χ , ί χ ) , Δ Ψ w i t h Δ,Ψ o f f i r s t o r d e r and w i t h o u t f , t h e n I - Vx3y i p ( x , y ) , Δ -» Ψ. P r o o f o f ( 2 ) f r o m ( 2 ) ' : L e t a c u t - f r e e d e r i v a t i o n o f Γ,Δ -* Ψ be g i v e n w i t h Δ,Ψ o f f i r s t o r d e r and Γ a l i s t o f g e n e r a l i z a t i o n s o f i n s t a n c e s t h e schema Vx3y φ(χ,γ) -* 3 f V x φ(χ,ίχ). By i n d u c t i o n on t h e l e n g t h o f t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f Δ -* Ψ. I t s u f f i c e s t o c o n s i d e r an i n f e r e n c e Vx3y -» 3 f V x , Γ ' , 0 -> Vx3y φ ( χ , Υ ) 3 f V x φ ( χ , ί χ ) , Vx3y -+ 3 f V x , Γ',0 -» χ V x 3 y 3 f V X j r t s 0 -> χ F i r s t t h e l e f t m o s t 3 f i n t h e r i g h t hand s u b d e r i v a t i o n can be c a n c e l l e d by an i n v e r s i o n lemma. Then t h e o c c u r e n c e s o f Vx3y -> 3 f V x , Γ' i n t h e a n t e c e d e n t o f b o t h s u b d e r i v a t i o n s can be c a n c e l l e d by i n d u c t i o n h y p o t h e s i s . Then by ( 2 ) ' Vx φ(χ,ίχ) i n t h e a n t e c e d e n t o f LOGIC AND THE AXIOM OF CHOICE 353 t h e r i g h t hand s u b d e r i v a t i o n can be r e p l a c e d by Vx3y c p ( x , y ) . A c u t t h e n g i v e s t h e d e s i r e d d e r i v a t i o n 1 o f Θ -* χ. P r o o f o f ( 2 ) ' : L e t a c u t - f r e e d e r i v a t i o n o f Vx c p ( x , f x ) , Δ -> Ψ w i t h Δ,Ψ o f ' f i r s t o r d e r and w i t h o u t f be g i v e n . By i n d u c t i o n on t h e l e n g t h o f t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f Vx3y c p ( x , y ) , Δ -* Ψ. I t s u f f i c e s t o c o n s i d e r Case 1 c p ( t , f t ) , Vx c p ( x , f x ) , Γ -> χ Vx φ ( χ , ί χ ) , Γ χ R e p l a c e a l l o c c u r r e n c e s o f f t i n t h i s d e r i v a t i o n by a new v a r i a b l e w. T h i s g i v e s a d e r i v a t i o n o f Ψ ( t , w ) , Vx φ ( χ , f χ ) , Γ -* χ. By i n d u c t i o n h y p o t h e s i s we o b t a i n a d e r i v a t i o n o f < p ( t , w ) , Vx3y ψ ( κ , γ ) , Γ -» χ. A p p l i c a t i o n o f (3-*) and (V-O g i v e s t h e d e s i r e d d e r i v a t i o n o f Vx3y ( p ( x , y ) , Γ -* χ. Case 2 Vx φ ( χ , ί κ ) , Γ -> χ ( r ( f 1 1 , . . . , f t ) ) Vx φ( χ , f χ ) , Γ -> 3z χ( ζ) where f t , , . . . - f t a r e a l l o u t e r m o s t o c c u r r e n c e s o f f - t e r m s i n 1 η r ( f t ^ , . . . , f t ) . R e p l a c e a g a i n a l l o u t e r m o s t o c c u r r e n c e s o f f t . , . . . , f t i n t h i s d e r i v a t i o n by new v a r i a b l e s w„,...,w . T h i s g i v e s 1 ' η J 1 ' ' η ^ a d e r i v a t i o n o f ψ( t 1 , w l ) , . . . ,φ( t ^ ,w ) , Vx φ ( χ , ί χ ) , Γ -> χ( r ( w 1 , . . . ,wn ) ) . Then a p p l y t h e i n d u c t i o n h y p o t h e s i s , (-*3) , η t i m e s (3-») and η t i m e s (V-») . P r o o f o f ( U ) : I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c l o g i c ( w i t h e q u a l i t y ) . I t s u f f i c e s t o p r o v e I- Vx3y (χΦγ) -* ν χ 1 3 Υ 1 ν χ 2 3 ν 2 ^ l ^ l A X 2 ^ y 2 A ( x ± Z X 2 ~* y l " Y 2 ) ) ' C o n s i d e r t h e K r i p k e model a„ : a ~ c b ~ d a n : a ^ b c d T h i s i s o b v i o u s l y a model o f t h e e q u a l i t y a x i o m s . F u r t h e r m o r e , a Q lh Vx3y" (χ Φ y) s i n c e a. Ih a Φ d, b Φ c f o r a l l i . But a 0 ll_ V x i 3 y i V X 2 3 y 2 ( X 1 Φ Υ 1 A X 2 Φ Y 2 A ( X 1 = X 2 ~* Y l = y 2 ) } * T ° S e e t h i s assume t h e c o n t r a r y . Choose a f o r x^. Then y^ must be d. Choose 354 Η. SCHWICHTENBERG b f o r x^ . Then must be c. Hence et ^  I ! — ( a = b -* c = d ) . But t h i s i s a c o n t r a d i c t i o n , s i n c e lh a = b, lh c = d. ( T h i s c o u n t e r e x a m p l e i s due t o S m o r y n s k i 1978) P r o o f o f ( 5 ) : We o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c l o g i c ; i t i s e a s i l y seen t h a t t h e same p r o o f a l s o a p p l i e s t o c l a s s i c a l l o g i c . F i r s t n o t e t h a t ( 5 ) can be r e d u c e d t o ( 5 ) ' I f h Vx φ ( χ , ί χ ) , Δ -* Ψ w i t h Δ,Ψ o f f i r s t o r d e r and w i t h o u t f , t h e n t h e r e i s an η such t h a t 1- Vx^y., . . .Vx 3y [/^φ( x . , y . ) Λ F c t ( x ; y ) ] , A -» Ψ ,where F c t ( x , y ) π ι I i a b b r e v i a t e s yVA\( χ. = χ. -» y. - γ.). ( 5 ) can be p r o v e d f r o m ( 5 ) ' e x a c t l y as we p r o v e d ( 2 ) f r o m ( 2 ) ' above. To p r o v e ( 5 ) ' we c a n n o t p r o c e e d as s i m p l y as i n t h e p r o o f o f ( 2 ) ' . F o r , t h e r e p l a c e m e n t o f f t by a new v a r i a b l e w i n case 1 w o u l d n o t l e a d t o a d e r i v a t i o n anymore, s i n c e an e q u a l i t y a x i o m t = s f t = f s w o u l d be t r a n s f o r m e d i n t o an u n d e r i v a b l e f o r m u l a t = s w = f s . The i d e a now i s t o r e p l a c e f by a new v a r i a b l e w i t h t h e i n t e n d e d meaning t h a t i t s h o u l d r a n g e o v e r f u n c t i o n s m apping w i n t o t . To make t h i s p r e c i s e we f i r s t e x t e n d o u r l a n g u a g e . V a r i a b l e s and t e r m s and now g e n e r a t e d s i m u l t a n e o u s l y w i t h t h e a d d i t i o n a l c l a u s e I f r l 5 . . . , r s „,..., s ( s h o r t : r , s ) a r e t e r m s , t h e n f s i s a I n I n r f u n c t i o n v a r i a b l e ( w h e r e f i s any o f t h e c o u n t a b l y many s y m b o l s r e s e r v e d f o r f u n c t i o n v a r i a b l e s ) . C o r r e s p o n d i n g t o t h e i n t e n d e d meaning o f f ^ we add t h e f o l l o w i n g a xioms t o o u r l o g i c a l f o r m a l i s m : F c t ( r ; s ) f ^ v . - s . f o r a l l i . r ι ι We now f o r m u l a t e a g e n e r a l i z a t i o n o f ( 5 ) ' i n v o l v i n g t h e s e new-f u n c t i o n v a r i a b l e s , w h i c h can t h e n be p r o v e d by i n d u c t i o n . ( 5 ) " I f I - Vx φ ( χ , ^ χ ) , F c t ( r ; s ) , Δ - Ψ w i t h Δ,Ψ o f f i r s t o r d e r and w i t h o u t f ~ , t f r e n t h e r e i s an η such t h a t I - Vx 3y ...Vx 3y r ' l J l n J n [//^\φ(χ;ί ,y.)A F c t ( r , x ; s , y ) ] , Δ -> Ψ . P r o o f o f ( 5 ) " : For s i m p l i c i t y we o n l y w r i t e o u t t h e case f o r £, ,s empty. The g e n e r a l case can be d e a l t w i t h i n e x a c t l y t h e same manner. We use i n d u c t i o n on t h e l e n g t h o f t h e g i v e n d e r i v a t i o n , w h i c h we may assume t o be c u t - f r e e . I t s u f f i c e s t o c o n s i d e r LOGIC AND THE AXIOM OF CHOICE 355 Case 1 t p ( t , f t ) , V x c p ( x , f x ) , Γ -» χ ν χ φ ( χ , ί χ , Γ -» χ We f i r s t d e s c r i b e t h e w e l l - k n o w n t e c h n i q u e o f " e x t r a c t i n g f - s u b ­t e r m s f r o m f t " . L e t f t - , . . . , f t z f t be a l l f - s u b t e r m s o f f t o r d e r e d 1 ' ' η by i n c r e a s i n g d e p t h o f n e s t i n g o f f . L e t w^,...,w^ be new v a r i a b l e s . For any s u b t e r m s o f f t d e n o t e by s* t h e r e s u l t o f r e p l a c i n g a l l o u t e r m o s t o c c u r r e n c e s o f f - s u b t e r m s f t . , . . . , f t . i n s by w. ,...,w. . U s i n g t h e new axiom s on f ~ one can p r o v e e a s i l y by 1 1 1 k £ i n d u c t i o n on s (*) F c t ( t * ; w ) -> s ( f r * ) = s*. t * d e n o t e s o f c o u r s e t t ; n o t e t . c o n t a i n s o n l y w. w i t h i < i . ~ 1 ' * η ι 3 w - Now r e p l a c e a l l o c c u r r e n c e s o f f i n t h e g i v e n d e r i v a t i o n by f W r i t i n g t ( f ) f o r t one o b t a i n s a d e r i v a t i o n o f w w w w Φ ( ΐ ( ^ ) , f ^ t ( f ^ ) ) , ν χ φ ( χ , ί ^ χ ) ,Γ -> χ U s i n g (*) t h i s d e r i v a t i o n can e a s i l y be t r a n s f o r m e d i n t o a d e r i v a t i o n o f w i p ( t * , w n ) , ν χ φ ( χ , ί ^ χ ) , F c t ( t * ; w ) , T - χ o f t h e same l e n g h t ( i t i s n e c e s s a r y h e r t o a l l o w as axioms a l l q u a s i -t a u t o l o g i c s , i . e . a i l t a u t o l o g i c a l c onsequences o f t h e e q u a l i t y s axioms i n c l u d i n g t h e new axiom s on f ~ ) . By i n d u c t i o n h y p o t h e s i s we t h e n o b t a i n a d e r i v a t i o n o f <P( t * , w n ) , V x 1 3 y 1 . . . V x m 3 y m [ c p C x ^ y ^ Λ F c t ( t * , χ ; w ,y ) ] , Γ -> χ. i -~ ~ ~ ~ Now φ ( ΐ * ^ ) can be c a n c e l l e d s i n c e i t f o l l o w s f r o m t h e second ^ η member o f t h e a n t e c e d e n t (we may assume m > 1 ) . Then u s i n g t h e r u l e s (3-*) , (V->) we o b t a i n Vu.Bw, . . .Vu 3w Vx., 3y. . . . Vx 3y [ φ ( χ . ^ . ) Λ Fct(U,X ;W ,y ) ] ,Γ -» χ. 1 1 η η 1 1 m m . ι ι ~~ · ~ - ^ ' ' Case 2 Vx φ( χ , f χ ) , Γ -* χ ( t ) Vx φ( χ , f χ ) , Γ -> 3ζχ( ζ ) 356 Η. SCHWICHTENBERG Here a g a i n we e x t r a c t t h e f - s u b t e r m s f r o m t and t h e n r e p l a c e f w by f ~* . T h i s g i v e s as above a d e r i v a t i o n o f ~ w ν χ φ ( χ , ί ^ χ ) F c t ( t * ; w ) , Γ -* χ(ΐ*) . By i n d u c t i o n h y p o t h e s i s we t h e n o b t a i n a d e r i v a t i o n o f V x 1 3 y 1 . . . V x m 3 y m L ^ i p i x - . y . ) Λ F c t ( t * , x ; w , ^ ) ,Γ - X ( t * ) . Now a p p l y ( -»3) and t h e n p r o c e e d as i n case 1 above. R e f e r e n c e s D. HILBERT and P. BERNAYS , 1939, G r u n d l a g e n d e r M a t h e m a t i k I I . B e r l i n 2 1 9 7 0 . S.C. KLEENE , 1952 , I n t r o d u c t i o n t o M e t a m a t h e m a t i c s . Amsterdam. Η. LUCKHARDT A, C o n s e r v a t i v e Skolem F u n c t o r s . U n p u b l i s h e d , 51 pp. G.E. MINC , 1966 , Skolem's Method o f E l i m i n a t i o n o f P o s i t i v e Q u a n t i f i e r s i n S e q u e n t i a l C a l c u l i . S o v i e t Math. D o k l . 7, 861-864. G.E. MINC , 1974 , H e y t i n g P r e d i c a t e C a l c u l u s w i t h E p s i l o n Symbol. J. S o v i e t Math. 8 ( 1 9 7 7 ) , 317-323. G.E. MINC , 1977 , Review o f Osswald 1975. Z e n t r a l b l a t t f ü r Math. 325, 0 2 0 2 1 . Η. OSSWALD , 1975 , Ü b e r S k o l e m - E r w e i t e r u n g e n i n d e r i n t u i t i o n i s t i c sehen L o g i k m i t G l e i c h h e i t . I n : P r o o f T h e o r y S y m p o s i o n , K i e J 1974 ( e d . J. D i l l e r , G.H. Müller). B e r l i n , 264-266. C. SMORYNSKI , 1978 , The A x i o m a t i z a t i o n P r o b l e m f o r F r a g m e n t s . A n n a i s Math. L o g i c 14, 193 - 2 2 1 . 

Heyting Predicate Calculus with Epsilon Symbol.

Introduction to Metamathematics.

Review of Osswald

Skolem's Method of Elimination of Positive Quantifiers in Sequential Calculi.

The Axiomatization Problem for Fragments.

Über Skolem-Erweiterungen in der intuitionistic sehen Logik mit Gleichheit.

Logic and the axiom of choice

Universität München: Elektronischen Publikationen

STUDIES IN LOGIC 
AND 
THE FOUNDATIONS OF MATHEMATICS 
V O L U M E 97 
Editors 
J. Β A R W I S E , Madison 
D. K A P L A N , Los Angeles 
H . J . K E I S L E R , Madison 
P. SUPPES, Stanford 
A. S. TROELSTRA, Amsterdam 
NORTH HOLLAND PUBLISHING COMPANY 
AMSTERDAM · NEW YORK · OXFORD 
(LOGIC 
COLLOQUIUM 78 
Proceedings of the colloquium 
held in Möns, August 1978 
Edited by 
M A U R I C E B O F F A 
Universite de Γ Etat ä Möns, 
Belgium 
D I R K V A N D A L E N 
Rijksnniversiteit Utrecht 
The Netherlands 
K E N N E T H M C A L O O N 
Universite Paris VII 
France 
1979 
NORTH-HOLLAND PUBLISHING COMPANY 
AMSTERDAM · NEW YORK · OXFORD 
Machen* at isches InstlföS 
dar Universität MüöchäB 
© NORTH-HOLLAND PUBLISHING COMPANY, 1979 
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, 
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording 
or otherwise, without the prior permission of the copyright owner. 
ISBN: 0 444 85378 2 
Publishers: 
N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y 
A M S T E R D A M · N E W Y O R K · O X F O R D 
Sole distributors for the U.S.A. and Canada: 
E L S E V I E R N O R T H - H O L L A N D , INC. 
52 V A N D E R B I L T A V E N U E 
N E W Y O R K , N Y . 10017 
Library of Congress Cataloging in Publication Data 
Logic Colloquium, Möns, 1978. 
Logic Colloquium '78. 
(Studies i n logic and the foundations of mathemat-
i c s ; v. 97) 
1. Logic, Symbolic and mathematical—Congresses. 
I . Boffa, Maurice. I I . Dalen, Dirk van, 1932-
I I I . McAloon, Kenneth. IV. Ser i e s . 
QA9.A1L63 1978 511».3 79-21152 
ISBN O-lM-85378-2 
P R I N T E D IN T H E N E T H E R L A N D S 
Dedicated to 
PAUL BERN AYS 
and 
KURT GÖDEL 
KURT GÖDEL (1906-1978) 
TABLE OF CONTENTS 
PREFACE v i i 
M. BEESON 1 
Continuity in i n t u i t i o n i s t i c set theories 
G. CHERLIN 53 
Stable algebraic theories 
R. DAVID 75 
* Un r e s u l t a t de non contradiction r e l a t i v e au sujet 
de l a conjecture de Solevay 
0. DEMUTH and A. RIVERA 81 
Remarks on constructive mathematical analysis 
J. DENEF 131 
The diophantine problem for polynomial rings of 
positive c h a r a c t e r i s t i c 
L. VAN DEN DRIES * 147 
* Algorithms and bounds for polynomial rings 
S. FEFERMAN 159 
Constructive theories of functions and classes 
P. HÄJEK 225 
* On p a r t i a l l y conservative extensions of arithmetic 
A. HAJNAL and I . JUHÄSZ 235 
Weakly separated subspaces and networks 
L. HARRINGTON and R. SAMI 247 
* Equivalence r e l a t i o n s , projective and beyond 
G.F. VAN DER HOEVEN and A.S. TROELSTRA 265 
Projections of lawless sequences I I 
R. LAVER 299 
Linear order in (ω) ω under eventual dominance 
ix 
χ TABLE OF CONTENTS 
W. MITCHELL 
Hypermeasurable cardinals 
Z. RATAJCZYK 
* On the number of expansions of the model of ZFC-set 
theory to models of KM-theory of classes 
U.R. SCHMERL 
* A fine structure generated by ref l e c t i o n formulas 
over primitive recursive arithmetic 
H. SCHWICHTENBERG 
* Logic and the axoim of choice 
S. SHELAH 
On the successors of singular cardinals 
E. SPECKER 
Paul Bernays 
J . VAANÄNEN 
* Abstract logic and set theory. I . Definability 
V. WEISPFENNING 
Lattice products 
B. WEGLORZ 
Some σ-fields of subsets of reals 
* contributed papers 
- ο -
The following invited lectures were presented at the Colloquium, but the text i s 
not reproduced in the present volume. 
A. DODD, The core model 
J. FENSTAD, General recursive theory: a unified view 
A. JOYAL, Sheaves in constructive mathematics 
G. KREISEL, Gödel's l a t e r views 
S. KRIPKE, Non-standard models and Gödel's theorem 
M. MORLEΥ, Theories with few models 
J. PARIS, Some problems in arithmetic 
303 
317 
335 
351 
357 
381 
391 
423 
427 
LOGIC COLLOQUIUM 78 
Μ. Boffa, D. van Daten, Κ, McAloon (eds.) 
© North-Holland Publishing Company 3 1979 
LOGIC AND THE AXIOM OF CHOICE 
H. S c h w i c h t e n b e r g ( M u n i c h ) 
Vie s h a l l p r o v e t h e f o l l o w i n g : 
( .1) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s c o n s e r v a t i v e o v e r c l a s s i c a l 
( f i r s t o r d e r ) l o g i c . 
( 2 ) Vx3y φ(χ,ν) -* 3 f V x ω(χ,ίχ) i s c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c 
l o g i c w i t h o u t e q u a l i t y . 
( 3 ) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c 
l o g i c w i t h d e c i d a b l e e q u a l i t y . 
( 4 ) Vx3y φ(χ,γ) -» 3 f V x φ(χ,ίχ) i s n o t c o n s e r v a t i v e o v e r 
i n t u i t i o n i s t i c l o g i c . 
( 5 ) A \ Vx,3y Vx 3y f/X\co(x. ,y. ) Λ /Ά(Χ.=Χ. - y . = y . ) l l i r\ J η r i " i ι Ί ι ι J η ι ι , 3 J 
3 f V x φ(χ,ίχ) 
i s c o n s e r v a t i v e o v e r c l a s s i c a l and i n t u i t i o n i s t i c l o g i c . 
More p r e c i s e l y : A d d i t i o n o f f i n i t e l y many i n s t a n c e s o f t h e r e s p e c ­
t i v e schema w i t h a l l (number and f u n c t i o n ) p a r a m e t e r s g e n e r a l i z e d 
i s c o n s e r v a t i v e o v e r any f i r s t o r d e r t h e o r y i n t h e r e s p e c t i v e l o g i c . 
None o f t h e s e r e s u l t s i s new. ( 1 ) i s a l r e a d y i n H i l b e r t - B e r n a y s 
1939 ( p . 141 i n t h e second e d i t i o n ) . ( 2 ) - ( 4 ) a r e due t o Mine 1966, 
1974, 1966; n o t e t h a t ( 3 ) i s an i m m e d i a t e consequence o f ( 5 ) . As t o 
( 4 ) , a s i m p l e r c o u n t e r e x a m p l e i s i n Osswald 1975 and p r o b a b l y t h e 
s i m p l e s t ( w h i c h i s r e p r o d u c e d h e r e ) i n S m o r y n s k i 1978. ( 5 ) i s due 
t o Mine and S m o r y n s k i ; i t was f i r s t a nnounced i n Mine 1977. P r o o f s 
a r e i n S m o r y n s k i 1978 and ( i n a g e n e r a l i z e d f o r m d e a l i n g w i t h 
" s i m u l t a n e o u s Skolem f u n c t o r s " ) i n L u c k h a r d t #. 
The p r o o f s g i v e n h e r e a r e r e l a t i v e l y s i m p l e . F o r ( 5 ) t h e p r o o f 
c o n s i s t s i n a p r o c e d u r e w h i c h t r a n s f o r m a d e r i v a t i o n o f a f i r s t 
o r d e r f o r m u l a i n v o l v i n g t h e a x i o m o f c h o i c e i n t o a d e r i v a t i o n n o t 
i n v o l v i n g i t . The main t e c h n i c a l t o o l i s t h e use o f a new t y p e o f 
351 
352 Η. SCHWICHTENBERG 
f u n c t i o n v a r i a b l e s : Whenever t e r m s r . , . . . , r , s „,..., s a r e i n t r o -
1 η 1 η d u c e d , t h e n f ^ i s a f u n c t i o n v a r i a b l e . The i n t e n d e d meaning o f f ~ r r 
i s t h a t i t s h o u l d r a n g e o v e r a l l f u n c t i o n s mapping r i n s ( p r o v i d e d 
r , s d e t e r m i n e a f i n i t e f u n c t i o n , i . e . /ΛΑΓ. = r . -> s . = s . ) ) . 
~ i»] 1 3 1 3 
As a l r e a d y n o t e d , ( 3 ) and a l s o ( 1 ) a r e easy c o n s e q u e n c e s o f ( 5 ) , 
s i n c e t h e p r e m i s s o f t h e i m p l i c a t i o n i n ( 5 ) i s u n d e r t h e a s s u m p t i o n 
Vx,y ( x = y ν χ Φ y ) e q u i v a l e n t t o Vx3y t p ( x , y ) . So we s t a r t w i t h a 
p r o o f o f ( 2 ) , t h e n g i v e S m o r y n s k i ' s c o u n t e r e x a m p l e t o p r o v e ( 4 ) , and 
f i n a l l y e x t e n d t h e method f o r p r o v i n g ( 2 ) t o a p r o o f o f ( 5 ) ; o n l y 
t h i s l a s t s t e p i n v o l v e s t h e f u n c t i o n v a r i a b l e s f ~ . 
N o t e : ( 1 ) - ( 5 ) r e m a i n v a l i d - w i t h e s s e n t i a l l y t h e same p r o o f s -
when a l l v a r i a b l e s x, y, f , ... a r e r e p l a c e d by f i n i t e sequences 
x, y^, f , ... o f v a r i a b l e s ; f & t h e n means f x , . . . , f x. However, f o r 
s i m p l i c i t y we o n l y d e a l w i t h s i n g l e v a r i a b l e s h e r e . 
N o t e : ( 2 ) , ( 4 ) and ( 5 ) a l s o h o l d f o r m i n i m a l l o g i c . T h i s i s seen 
e a s i l y f r o m t h e p r o o f s . 
P r o o f o f ( 2 ) : 
I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c 
l o g i c w i t h o u t e q u a l i t y . We s h a l l work w i t h a Gentzen s e q u e n t 
c a l c u l u s as d e s c r i b e d i n Kleene 1952, p. 481 ( t h e r e i t i s c a l l e d G3). 
For s i m p l i c i t y we m o d i f y i t t o i n c l u d e 1 ( f a l s u m ) as a p r o p o s i t i o n a l 
c o n s t a n t and t r e a t -φ as d e f i n e d by φ -» 1 . F i r s t n o t e t h a t ( 2 ) can 
be r e d u c e d t o 
( 2 ) ' I f h Vx φ ( χ , ί χ ) , Δ Ψ w i t h Δ,Ψ o f f i r s t o r d e r and w i t h o u t f , 
t h e n I - Vx3y i p ( x , y ) , Δ -» Ψ. 
P r o o f o f ( 2 ) f r o m ( 2 ) ' : L e t a c u t - f r e e d e r i v a t i o n o f Γ,Δ -* Ψ be 
g i v e n w i t h Δ,Ψ o f f i r s t o r d e r and Γ a l i s t o f g e n e r a l i z a t i o n s o f 
i n s t a n c e s t h e schema Vx3y φ(χ,γ) -* 3 f V x φ(χ,ίχ). By i n d u c t i o n on t h e 
l e n g t h o f t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f Δ -* Ψ. I t 
s u f f i c e s t o c o n s i d e r an i n f e r e n c e 
Vx3y -» 3 f V x , Γ ' , 0 -> Vx3y φ ( χ , Υ ) 3 f V x φ ( χ , ί χ ) , Vx3y -+ 3 f V x , Γ',0 -» χ 
V x 3 y 3 f V X j r t s 0 -> χ 
F i r s t t h e l e f t m o s t 3 f i n t h e r i g h t hand s u b d e r i v a t i o n can be 
c a n c e l l e d by an i n v e r s i o n lemma. Then t h e o c c u r e n c e s o f Vx3y -> 3 f V x , 
Γ' i n t h e a n t e c e d e n t o f b o t h s u b d e r i v a t i o n s can be c a n c e l l e d by 
i n d u c t i o n h y p o t h e s i s . Then by ( 2 ) ' Vx φ(χ,ίχ) i n t h e a n t e c e d e n t o f 
LOGIC AND THE AXIOM OF CHOICE 353 
t h e r i g h t hand s u b d e r i v a t i o n can be r e p l a c e d by Vx3y c p ( x , y ) . A c u t 
t h e n g i v e s t h e d e s i r e d d e r i v a t i o n 1 o f Θ -* χ. 
P r o o f o f ( 2 ) ' : L e t a c u t - f r e e d e r i v a t i o n o f Vx c p ( x , f x ) , Δ -> Ψ w i t h 
Δ,Ψ o f ' f i r s t o r d e r and w i t h o u t f be g i v e n . By i n d u c t i o n on t h e l e n g t h 
o f t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f Vx3y c p ( x , y ) , Δ -* Ψ. 
I t s u f f i c e s t o c o n s i d e r 
Case 1 c p ( t , f t ) , Vx c p ( x , f x ) , Γ -> χ 
Vx φ ( χ , ί χ ) , Γ χ 
R e p l a c e a l l o c c u r r e n c e s o f f t i n t h i s d e r i v a t i o n by a new 
v a r i a b l e w. T h i s g i v e s a d e r i v a t i o n o f Ψ ( t , w ) , Vx φ ( χ , f χ ) , Γ -* χ. By 
i n d u c t i o n h y p o t h e s i s we o b t a i n a d e r i v a t i o n o f < p ( t , w ) , 
Vx3y ψ ( κ , γ ) , Γ -» χ. A p p l i c a t i o n o f (3-*) and (V-O g i v e s t h e d e s i r e d 
d e r i v a t i o n o f Vx3y ( p ( x , y ) , Γ -* χ. 
Case 2 Vx φ ( χ , ί κ ) , Γ -> χ ( r ( f 1 1 , . . . , f t ) ) 
Vx φ( χ , f χ ) , Γ -> 3z χ( ζ) 
where f t , , . . . - f t a r e a l l o u t e r m o s t o c c u r r e n c e s o f f - t e r m s i n 1 η 
r ( f t ^ , . . . , f t ) . R e p l a c e a g a i n a l l o u t e r m o s t o c c u r r e n c e s o f 
f t . , . . . , f t i n t h i s d e r i v a t i o n by new v a r i a b l e s w„,...,w . T h i s g i v e s 1 ' η J 1 ' ' η ^ 
a d e r i v a t i o n o f 
ψ( t 1 , w l ) , . . . ,φ( t ^ ,w ) , Vx φ ( χ , ί χ ) , Γ -> χ( r ( w 1 , . . . ,wn ) ) . Then a p p l y 
t h e i n d u c t i o n h y p o t h e s i s , (-*3) , η t i m e s (3-») and η t i m e s (V-») . 
P r o o f o f ( U ) : 
I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c 
l o g i c ( w i t h e q u a l i t y ) . I t s u f f i c e s t o p r o v e 
I- Vx3y (χΦγ) -* ν χ 1 3 Υ 1 ν χ 2 3 ν 2 ^ l ^ l A X 2 ^ y 2 A ( x ± Z X 2 ~* y l " Y 2 ) ) ' 
C o n s i d e r t h e K r i p k e model 
a„ : a ~ c b ~ d a n : a ^ b c d 
T h i s i s o b v i o u s l y a model o f t h e e q u a l i t y a x i o m s . F u r t h e r m o r e , 
a Q lh Vx3y" (χ Φ y) s i n c e a. Ih a Φ d, b Φ c f o r a l l i . But 
a 0 ll_ V x i 3 y i V X 2 3 y 2 ( X 1 Φ Υ 1 A X 2 Φ Y 2 A ( X 1 = X 2 ~* Y l = y 2 ) } * T ° S e e 
t h i s assume t h e c o n t r a r y . Choose a f o r x^. Then y^ must be d. Choose 
354 Η. SCHWICHTENBERG 
b f o r x^ . Then must be c. Hence et ̂  I ! — ( a = b -* c = d ) . But t h i s i s 
a c o n t r a d i c t i o n , s i n c e lh a = b, lh c = d. ( T h i s c o u n t e r e x a m p l e 
i s due t o S m o r y n s k i 1978) 
P r o o f o f ( 5 ) : 
We o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c l o g i c ; i t i s e a s i l y 
seen t h a t t h e same p r o o f a l s o a p p l i e s t o c l a s s i c a l l o g i c . F i r s t n o t e 
t h a t ( 5 ) can be r e d u c e d t o 
( 5 ) ' I f h Vx φ ( χ , ί χ ) , Δ -* Ψ w i t h Δ,Ψ o f f i r s t o r d e r and w i t h o u t f , 
t h e n t h e r e i s an η such t h a t 
1- Vx^y., . . .Vx 3y [/^φ( x . , y . ) Λ F c t ( x ; y ) ] , A -» Ψ ,where F c t ( x , y ) π ι I i 
a b b r e v i a t e s yVA\( χ. = χ. -» y. - γ.). 
( 5 ) can be p r o v e d f r o m ( 5 ) ' e x a c t l y as we p r o v e d ( 2 ) f r o m ( 2 ) ' above. 
To p r o v e ( 5 ) ' we c a n n o t p r o c e e d as s i m p l y as i n t h e p r o o f o f ( 2 ) ' . 
F o r , t h e r e p l a c e m e n t o f f t by a new v a r i a b l e w i n case 1 w o u l d n o t 
l e a d t o a d e r i v a t i o n anymore, s i n c e an e q u a l i t y a x i o m t = s f t = f s 
w o u l d be t r a n s f o r m e d i n t o an u n d e r i v a b l e f o r m u l a t = s w = f s . The i d e a 
now i s t o r e p l a c e f by a new v a r i a b l e w i t h t h e i n t e n d e d meaning 
t h a t i t s h o u l d r a n g e o v e r f u n c t i o n s m apping w i n t o t . To make t h i s 
p r e c i s e we f i r s t e x t e n d o u r l a n g u a g e . V a r i a b l e s and t e r m s and now 
g e n e r a t e d s i m u l t a n e o u s l y w i t h t h e a d d i t i o n a l c l a u s e 
I f r l 5 . . . , r s „,..., s ( s h o r t : r , s ) a r e t e r m s , t h e n f s i s a I n I n r 
f u n c t i o n v a r i a b l e ( w h e r e f i s any o f t h e c o u n t a b l y many s y m b o l s 
r e s e r v e d f o r f u n c t i o n v a r i a b l e s ) . 
C o r r e s p o n d i n g t o t h e i n t e n d e d meaning o f f ^ we add t h e f o l l o w i n g 
a xioms t o o u r l o g i c a l f o r m a l i s m : 
F c t ( r ; s ) f ^ v . - s . f o r a l l i . r ι ι 
We now f o r m u l a t e a g e n e r a l i z a t i o n o f ( 5 ) ' i n v o l v i n g t h e s e new-
f u n c t i o n v a r i a b l e s , w h i c h can t h e n be p r o v e d by i n d u c t i o n . 
( 5 ) " I f I - Vx φ ( χ , ^ χ ) , F c t ( r ; s ) , Δ - Ψ w i t h Δ,Ψ o f f i r s t o r d e r and 
w i t h o u t f ~ , t f r e n t h e r e i s an η such t h a t I - Vx 3y ...Vx 3y r ' l J l n J n 
[//^\φ(χ;ί ,y.)A F c t ( r , x ; s , y ) ] , Δ -> Ψ . 
P r o o f o f ( 5 ) " : For s i m p l i c i t y we o n l y w r i t e o u t t h e case f o r £, ,s 
empty. The g e n e r a l case can be d e a l t w i t h i n e x a c t l y t h e same manner. 
We use i n d u c t i o n on t h e l e n g t h o f t h e g i v e n d e r i v a t i o n , w h i c h we may 
assume t o be c u t - f r e e . I t s u f f i c e s t o c o n s i d e r 
LOGIC AND THE AXIOM OF CHOICE 355 
Case 1 t p ( t , f t ) , V x c p ( x , f x ) , Γ -» χ 
ν χ φ ( χ , ί χ , Γ -» χ 
We f i r s t d e s c r i b e t h e w e l l - k n o w n t e c h n i q u e o f " e x t r a c t i n g f - s u b ­
t e r m s f r o m f t " . L e t f t - , . . . , f t z f t be a l l f - s u b t e r m s o f f t o r d e r e d 
1 ' ' η 
by i n c r e a s i n g d e p t h o f n e s t i n g o f f . L e t w^,...,w^ be new v a r i a b l e s . 
For any s u b t e r m s o f f t d e n o t e by s* t h e r e s u l t o f r e p l a c i n g a l l 
o u t e r m o s t o c c u r r e n c e s o f f - s u b t e r m s f t . , . . . , f t . i n s by 
w. ,...,w. . U s i n g t h e new axiom s on f ~ one can p r o v e e a s i l y by 
1 1 1 k £ 
i n d u c t i o n on s 
(*) F c t ( t * ; w ) -> s ( f r * ) = s*. 
t * d e n o t e s o f c o u r s e t t ; n o t e t . c o n t a i n s o n l y w. w i t h i < i . ~ 1 ' * η ι 3 w 
- Now r e p l a c e a l l o c c u r r e n c e s o f f i n t h e g i v e n d e r i v a t i o n by f 
W r i t i n g t ( f ) f o r t one o b t a i n s a d e r i v a t i o n o f 
w w w w 
Φ ( ΐ ( ^ ) , f ^ t ( f ^ ) ) , ν χ φ ( χ , ί ^ χ ) ,Γ -> χ 
U s i n g (*) t h i s d e r i v a t i o n can e a s i l y be t r a n s f o r m e d i n t o a d e r i v a t i o n 
o f 
w 
i p ( t * , w n ) , ν χ φ ( χ , ί ^ χ ) , F c t ( t * ; w ) , T - χ 
o f t h e same l e n g h t ( i t i s n e c e s s a r y h e r t o a l l o w as axioms a l l q u a s i -
t a u t o l o g i c s , i . e . a i l t a u t o l o g i c a l c onsequences o f t h e e q u a l i t y 
s 
axioms i n c l u d i n g t h e new axiom s on f ~ ) . 
By i n d u c t i o n h y p o t h e s i s we t h e n o b t a i n a d e r i v a t i o n o f 
<P( t * , w n ) , V x 1 3 y 1 . . . V x m 3 y m [ c p C x ^ y ^ Λ F c t ( t * , χ ; w ,y ) ] , Γ -> χ. 
i -~ ~ ~ ~ 
Now φ ( ΐ * ^ ) can be c a n c e l l e d s i n c e i t f o l l o w s f r o m t h e second ^ η 
member o f t h e a n t e c e d e n t (we may assume m > 1 ) . Then u s i n g t h e 
r u l e s (3-*) , (V->) we o b t a i n 
Vu.Bw, . . .Vu 3w Vx., 3y. . . . Vx 3y [ φ ( χ . ^ . ) Λ Fct(U,X ;W ,y ) ] ,Γ -» χ. 
1 1 η η 1 1 m m . ι ι ~~ · ~ - ̂ ' ' 
Case 2 Vx φ( χ , f χ ) , Γ -* χ ( t ) 
Vx φ( χ , f χ ) , Γ -> 3ζχ( ζ ) 
356 Η. SCHWICHTENBERG 
Here a g a i n we e x t r a c t t h e f - s u b t e r m s f r o m t and t h e n r e p l a c e f 
w 
by f ~* . T h i s g i v e s as above a d e r i v a t i o n o f 
~ w 
ν χ φ ( χ , ί ^ χ ) F c t ( t * ; w ) , Γ -* χ(ΐ*) . 
By i n d u c t i o n h y p o t h e s i s we t h e n o b t a i n a d e r i v a t i o n o f 
V x 1 3 y 1 . . . V x m 3 y m L ^ i p i x - . y . ) Λ F c t ( t * , x ; w , ^ ) ,Γ - X ( t * ) . 
Now a p p l y ( -»3) and t h e n p r o c e e d as i n case 1 above. 
R e f e r e n c e s 
D. HILBERT and P. BERNAYS , 1939, G r u n d l a g e n d e r M a t h e m a t i k I I . 
B e r l i n 2 1 9 7 0 . 
S.C. KLEENE , 1952 , I n t r o d u c t i o n t o M e t a m a t h e m a t i c s . Amsterdam. 
Η. LUCKHARDT A, C o n s e r v a t i v e Skolem F u n c t o r s . U n p u b l i s h e d , 51 pp. 
G.E. MINC , 1966 , Skolem's Method o f E l i m i n a t i o n o f P o s i t i v e 
Q u a n t i f i e r s i n S e q u e n t i a l C a l c u l i . S o v i e t Math. D o k l . 7, 
861-864. 
G.E. MINC , 1974 , H e y t i n g P r e d i c a t e C a l c u l u s w i t h E p s i l o n Symbol. 
J. S o v i e t Math. 8 ( 1 9 7 7 ) , 317-323. 
G.E. MINC , 1977 , Review o f Osswald 1975. Z e n t r a l b l a t t f ü r Math. 
325, 0 2 0 2 1 . 
Η. OSSWALD , 1975 , Ü b e r S k o l e m - E r w e i t e r u n g e n i n d e r i n t u i t i o n i s t i c 
sehen L o g i k m i t G l e i c h h e i t . I n : P r o o f T h e o r y S y m p o s i o n , 
K i e J 1974 ( e d . J. D i l l e r , G.H. Müller). B e r l i n , 264-266. 
C. SMORYNSKI , 1978 , The A x i o m a t i z a t i o n P r o b l e m f o r F r a g m e n t s . 
A n n a i s Math. L o g i c 14, 193 - 2 2 1 . 


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Logic and the axiom of choice

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