Logic Colloquium 1981: Herbrand Symposium Proceedings

PROCEEDINGS OF THE HERBRAND SYMPOSIUM LOGIC COLLOQUIUM '8 1 Proceedings of the Herbrand Symposium held in Marseilles, France, July 198 1

Edited by

J. STERN

Universite de Caen Caen, France

1982

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK . OXFORD

ONORTH-HOLLAND PUBLISHING COMPANY - 1982 AN 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.

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Library of Congress Cataloging in Publication Data

Logic Colloquium (1981 : M a r s e i l l e , France) Logic Colloquium ‘81. ( S t u d i e s i n l o g i c and t h e foundations of mathematics ; v . 107) 1. Logic, Symbolic and mathematical--Congresses. I. S t e r n , J. (Jacques), 194911. Herbrand, Jacques, 1908-1931. 111. S e r i e s . QAS A l L 6 3 1981 5u.3 82-6433 ISBN 0-444-86417-2 AACR2

.

.

PRINTED IN T H E NETHERLANDS

A LA MEMOIRE

DE JACQUES HERBRAND

PREFACE

Fifty years after the death of Herbrand, about two hundred people attended the colloquium which was held in Marseilles from July 16 through July 24,1981. The conference has tried to encompass fifty years of logic; this could be considered as a very ambitious goal, but because of the enthusiasm of the invited speakers and of the audience, our aim was achieved to a large extent. During the opening ceremony, the participants could hear a short address by Professor A. Guinier who knew Herbrand personally; messages from Professor C. Chevalley and Professor J. Dieudonni, who also knew Herbrand, were read.

A large part of the program was devoted to invited lectures on the work of Herbrand and on the role of Herbrand's ideas in the subsequent development of logic, These lectures appear in the first part of the proceedings. The other invited lectures dealt with other topics of current research in mathematical logic (set theory, recursion theory, model theory, proof theory, computer science). In his opening address, Professor Guinier wondered about the theorems that Herbrand would have proved if he had not met with his tragic fate. This question cannot be answered. On the other hand, it is clear that the death of Herbrand somehow delayed the f m establishment of logic in France. For this reason, I want to thank warmly those who have chosen to write their contributions in the French language as a tribute to the memory of Herbrand. The symposium was sponsored by the CNRS (Centre National de la Recherche Scientifique) and was the summer meeting of the European branch of the Association for symbolic logic. Financial support was also given by the Soci6ti franpise de logique and both Universities in Marseilles.

I would like to close this preface by thanking all those who helped with the organization of this meeting and the preparation of these proceedings. J. STERN

MEMBERS OF THE PROGRAM COMMITTEE

S. Feferman R. Fra'isse (Chairman) H. Gaifman J .Y.Girard M. Guillaume T. Jech D. Lascar G.H. MiiUer I,. Pacholski J.B. Paris J. Stem

Stanford Marseille I Jerusalem Paris VII Clermont I1 Los Angeles Paris VII Heidelberg Wroclaw Manchester Caen

MEMBERS OF THE ORGANIZING COMMITTEE

G. Blanc M.R. Donnadieu R. F r & e M. Guillaume A. Preller (Chairman) C. Rambaud R. Smadja

Marseille - Luminy Marseille - Luminy Marseille I Clermont I1 Marseille - Luminy Marseille - Luminy Marseille Luminy

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PROCLBDINGS OF THE HERBSYWOSIW LOGIC COLLOQUILJM '81. J. Stem (editor) 0North-HolM Publishing Company, 1982

1

JACQUES HERBRAND par C. Chevalley

Jacques Herbrand Q t a i t d ' o r i g i n e belge. Son p i r e , ndgociant en tableaux anciens, v i n t s ' 6 t a b l i r B P a r i s oC Jacques Herbrand continua des dtudes secondaires, qui n'avaient pas t r o p bien commencd du

f a i t desdifficultds qu'il

rencontrait...

en

math6matiques ! F a u t - i l v o i r 12 un cas d'dclosion tres soudaine d'un t a l e n t mathdmatique jusqu' a l o r s cachd, ou p l u t 6 t un exemple du manque t o t a l de discernement d'un enseignant 1 q u i Herbrand

il s'exprimait

-

-

avec l a f r a n c h i s e p a r f o i s t e r r i b l e avec l a q u e l l e

a u r a i t f a i t p a r t des i n s u f f i s a n c e s de ses leqons.

Quoiqu'il en s o i t , il f u t r e p

-

premier, s i mes souvenirs sont e x a c t s

-1

1'Ecole

Normale Supdrieure, en 1925 ; c ' e s t 11 qu'entrd un an a p r h hi j e l e r e n c o n t r a i . Sans a l l e r s u i v r e l e s cours donnds 1 l a Sorbonne ou 1 l ' b c o l e mSme par l e directeur

-

ce q u i l u i semblait une p e r t e de temps

-

il s e consacra t o u t de s u i t e aux

questions q u i l ' i n t G r e s s a i e n t , ce q u i explique en p a r t i e q u ' i l a i t pu produire en s i peu de temps une oeuvre considbrable.

I1 ne s ' i n t s r e s s a i t pas seulement aux mathdmatiques, mais a u s s i 1 l a philosophie

e t 1 l a podsie. I1 me f i t ddcouvrir e t aimer Valdry e t s u r t o u t M a l l a r d pour l e quel il s u t m e f a i r e partager son enthousiasme. En philosophie, il d t a i t a t t i r 6 par 1'idBalisme absolu de Hamelin,

-

l a France n ' a v a i t a l o r s r i e n de mieux 1 of-

f r i r 1 un jeune e s p r i t ardent ; i l en c r i t i q u a cependant l a pensde, e t n o t m e n t l a deduction a p r i o r i des concepts de l a s p a t i a l i t b , dont il p e r p t l ' i n s u f f i s a n c e en l a comparant 1 l a mdthode axiomatique. Peut-dtre d ' a i l l e u r s d t a i t - i l moins a t t i r 6 par l'idbalisme de l a d o c t r i n e que par son c a r a c t z r e absolu. C'Btait en e f f e t un des t r a i t s de son e s p r i t que de pousser t o u t e s choses j u s q u ' 1 l e u r l i m i t e ext d m e e t de mdpriser t o u t e s l e s demi-mesures. S a p r a t i q u e du s p o r t mDme d t a i t f o l lement risqude : expBdition d'alpinisme s e u l e t sans guide par temps de b r o u i l l a r d dans l e s Pyrdndes, t r a v e r s d e de l'entrGe du p o r t de La Rochelle a l o r s q u ' i l s a v a i t

1 peine nager

... Par

une a d r e i r o n i e du d e s t i n il d e v a i t p 6 r i r dans un accident

qui ne r d s u l t a i t d'aucune imprudence de s a p a r t : un rocher s u r l e q u e l il d t a i t a s s i s s ' e s t subitement ddtachd e t l ' a e n t r a i n 6 dans s a chute. C'est le gofit de l ' a v e n t u r e i n t e l l e c t u e l l e q u i l e p o r t a v e r s l a logique. I1 b t a i t

C. CHEVALLEY

2

s e d u i t par l e c a r a c t s r e grandiose de l ' o e u v r e de H i l b e r t e t il c r u t un temps c ' 6 t a i t avant GEdel

-

-

que l e problsme de l a d e c i s i o n e t a i t s u r l e point d ' d t r e rd-

solu. Outre l a logique mathdmatique, qui f u t t o u j o u r s l a premiere de s e s prgoccupations, Jacques Herbrand s ' i n t c r e s s a 2 l a t h d o r i e des nombres alggbriques. Peut-dtre l ' a t tira-t-elle

par c e t t e double circonstance que c e t t e branche des mathgmatiques d t a i t

totalement inconnue en France e t que l a t h e o r i e du corps de c l a s s e s p a s s a i t 2 just e t i t r e pour l ' u n e des p l u s d i f f i c i l e s en mathgmatiques. La ddcouverte de l a l o i gdndrale de rGciprocit6 de Artin n'en a v a i t pas encore e c l a i r 6 l e s abords e t c ' e s t

1 t r a v e r s l e grand memoire d'exposition de Hasse q u ' i l s ' y i n i t i a . Le f a s c i c u l e du

"MGmorial des sciences math6matiques"

f u t redig6 en vue de f o u r n i r au public

mathematique de langue f r a n c a i s e un accss 3 l a t h e o r i e ; il s e s i g n a l e par une grande c l a r t 6 d'exposition p l u t c t que par des r g s u l t a t s nouveaux. Herbrand s ' i n t d r e s s a i t d ' a i l l e u r s moins aux fondements de l a t h d o r i e qu'aux a p p l i c a t i o n s qu'on pouvait en f a i r e . Dans son memoire

"Sur l e s c l a s s e s des corps c i r c u l a i r e s " , il

u t i l i s a l a t h e o r i e du corps de c l a s s e s pour redemontrer c e r t a i n s r e s u l t a t s de Kummer, pour en prouver d ' a u t r e s dont aucune d d i o n s t r a t i o n n ' a v a i t 6tC publide, e t s u r t o u t pour en o b t e n i r de nouveaux qui p r d c i s e n t considerablement ceux de Kummer; de p l u s , il r a f f i n e Qgalement l e s c r i t s r e s de v a l i d i t 6 pour c e r t a i n s exposants du th6orSme de Fermat. Son memoire "ThBorie des groupes de d6composition, d ' i n e r t i e e t de ramification" a pour but de determiner c e s groupes pour une extension g a l o i sienne

K/k

2 p a r t i r de l e u r connaissance pour un sur-corps

L

de

K , Qgalement

g a l o i s i e n en k. Ces r g s u l t a t s peuvent s ' e t e n d r e au cas non g a l o i s i e n , comme l ' o n t montrd l e s mdthodes de Krasner. Vous a l l e z entendre durant ce colloque des exposLs s u r l e s travaux de Herbrand en

logique. Les quelques phrases qui precedent n'ont pour ambition que de r a p p e l e r que Jacques Herbrand d t a i t ouvert 3 tous l e s vents de l ' e s p r i t e t s e r a i t probablement devenu non seulement un grand mathdmaticien mais un de ces hommes profond6ment c u l t i v d s dans tous l e s domaines dont s'honore l e u r s i b c l e .

Claude Chevalley

PROCEEDINGS OF THE HERBRAND SYMPOSIUM LGGICCOLLOQUIUM '81, J. Stem (editor) 0 North-Holland Publishing Company, 1982

3

JACQUES HERBRAND ET LA THEORIE DES NOMBRES

par J. Dieudonnd AcadCmie des Sciences

Entrd premier 1 1'Ecole Normale superieure 1 17 ans, Herbrand Ctait dlbve externe et frgquentait peu 1'Ecole. I1 dddaignait les cours de la Sorbonne et s'instruisait lui-mcme ; on le voyait cependant 1 certains cours du Collage de France et notamment au Sdminaire d'Hadamard. Selon une coutume assez rdpandue, il avait voulu se ddbarrasser des 4 certificats de Licence rCglementaires dSs sa premisre annde d'Ecole ; il y parvint sans peine sauf pour le certificat de Mdcanique rationnelle, qu'il avait entisrement ndgligd de prdparer, persuade qu'il ne s'agissait que h'applications triviales de thCorSmes d'Analyse ; mais 1 la session de Juillet le problame d'examen comportait un petit piSge, bien connu de tous ceux qui avaient suivi quelques travaux pratiques, mais ignore d'Herbrand, qui dut repasser en Octobre. J'ai donc trSs peu connu Herbrand 1 l'Ecole, oii j'dtais entre un an avant lui, mais sa reputation n'avait pas tard6 1 se repandre. I1 Ctait dCj1 docteur 2s sciences un an apras sa sortie de l'Ecole, alors que je commenqais 1 peine 1 m'initier 3 la recherche ; j'ignorais entiarement les domaines des mathdmatiques dans lesquels il travaillait, et qui me paraissaient alors inaccessibles ; aussi m'inspirait-il une admiration un peu craintive. AprSs sa thlse, il passa l'annde 1930-1931 en Allemagne, 02 je le rencontrai peu avant sa mort ; sans abandonner pour autant la logique mathgmatique, il consacra ses efforts pendant cette annde 1 la Thdorie des Nombres ; 1'Allemagne Qtait aiors la Mecque de cette discipline, et la profondeur et la nouveautd de ses idCes firent une grande impression sur E. Artin, Hasse et E. Noether, trois des principaux reprdsentants de cette Bcole. En trbs peu de temps, il obtint des rCsultats tras originaux dans trois directions diffdrentes : la thdorie du corps de classes, la thdorie des corps de nombres de degrd infini, et celle des corps cyclotomiques.

I. LA THEORIE DU CORPS DE CLASSES C'est le dgveloppement de conjectures dmises par Hilbert et H. Weber entre 1896 et 1902,

et qui a CtL au centre des travaux des arithmcticiens dans la premiSre

moitid du XXe sibcle. L'idBe centrale de Hilbert peut se ddcrire en disant qu'il

4

J. DIEUDONNE

interprPte le groupe des classes d'idlaux d'un corps de nombres k, B isomorphie pris, c o m e groupe de Galois d'une extension abelienne de k ; Weber, 1 la suite de rlsultats particuliers de Kronecker, avait conjecture independamrnent un rlsultat plus gdneral, associant de mGme une extension abelienne de k B des groupes de classes d'ideaux restreintes (deux ideaux d'une mSme classe au sen6 muel n'etant considlr6s c o m e Bquivalents que s'ils satisfont en outre certaines congruences). On donna B ces extensions ablliennes le nom de corps de classes. AprOs qu'en 1907 Furtwzngler, un LlPve de Hilbert, eut prouvl l'existence du corps de classes dans le cas particulier envisage par ce dernier, le mathlmaticien japonais Takagi, en 1920, est parvenu, non seulement B dlmntrer les conjectures de Weber dans le cas glnlral, mais aussi B Ctablir que toute extension abllienne de

k est un corps de classes pourungroupe de classes restreintes bien dgterminl. Ses resultats furent completes sur un point important par la loi de rdciprocite de E. Artin (1927), qui definit un isomorphisme canonique entre un groupe de classes d'idlaux restreintes et le groupe de Galois du corps de classes correspondant. La thlorie du corps de classes pouvait donc sembler achev6e ; mais les methodes de Takagi ltaient extrGmement compliqules et paraissaient trls artificielles. Aprls le mathematicien allemand F.K. Schmidt, Herbrand et Chevalley s'attaqulrent .1 la recherche de methodes plus simples et qui feraient mieux comprendre la structure de la thlorie. Aprls la mort d'Herbrand, Chevalley continua seul cette t3che, jalonnde par sa thlse de 1933 [l] et le memoire de 1940 [31, 06 il donna un essor nouveau 1 la thlorie des nombr'es alggbriques par l'introduction de la notion d ' e et l'utilisation de la topologie, devenue prlpondlrante dans les exposes actuels. I1 n'est Bvidemment pas ltonnant que les contributions de Herbrand B ce renouveau des conceptions relatives au corps de classes soient demeurles fragmentaires, mais deux d'entre elles ont une portle plus gln6rale et sont encore d'un emploi

courant : 1 ) Ce qu'on appelle le "leme de Herbrand" ; c'est un resultat technique de la

thlorie des groupes finis, que Herbrand ne semble pas avoir publib, mais qui est prlsentl sous ce nom par Chevalley dans sa thPse ( [ I ] , p. 375), c o m e une propriltl de deux endomorphismes d'un groupe fini satisfaisant B certaines conditions. De nos jours la forme de ce leme s'est transformee en un lnonce de la thlorie cohomlogique des groupes finis, qui a de nombreuses applications (1131, p. 143, prop. 9 ) . 2) Le mlmoire [5] est consacre B la theorie des groupes de ramification, definis

Herbrand et la thhrie des nombres

5

par Hilbert pour les extensions galoisiennes. Si K est une telle extension d'un corps de nombres k, et P un ideal premier de K, on associe B p une suite decroissante (Gi)i> de sous-groupes du groupe de Galois G de K sur k : Go, appeld groupe de decomposition, est form6 des u E G tels que u(P) = P ; i pour i > i, Gi est ford des "€Go tels que u(x)-x€P pour tout entier de K non dans p ; GI est le groupe d'inertie de P , et les Gi pour i 2 2 les groupes de ramification. Le probl2me que rdsoud Herbrand concerne un corps L intermddiaire entre k et K et galoisien sur k : B l'aide des Gi et

x

du groupe de Galois de L sur k, il determine les groupes d'inertie et de ramification de l'id6al premier P n L de L. Le resultat est devenu classique et d'un usage constant dans la thBorie des corps locaux ( [ 1 3 ] , p. 101, cor. 3). Un autre riisultat de Herbrand concerne les unites d'une extension galoisienne K d'un corps de nombres k [4]. Le groupe de Galois G de K sur k oplre naturellement sur le groupe abelien E des unites de K, d'o8 (en prenant les logarithmes des unites et l'espace vectoriel V sur 2 qu'ils engendrent) une representation lineaire de G dans l'espace vectoriel V Q C. Herbrand determine cette

-Q-

representation et en dLduit des propridtes du groupe E. Moins important actuellement que les deux rdsultats precedents, il a neanmoins QtE utilisd par Chevalley dans sa thlse, ainsi que plusieurs dthodes imaginees par Herbrand dans les articles [ 6 ] et [lo] en vue de simplifier les d6monstrations dans la theorie du corps de classes. 11. LES CORPS DE NOMBRES DE DEGRE INFINI Ce sont les extensions algdbriques du corps 4 des rationnels, dont le degr6 sur Q est infini. L'dtude de l'anneau des entiers d'un tel corps, et notamment de ses idgaux premiers, avait dt6 commencde dans la pdriode 1920-1930 par Stiemke et Krull. Herbrand, dans les memoires [S] et [91, se proposa d'examiner les generalisations possibles des groupes de d6composition, d'inertie et de ramification dans la situation 05 les corps k et K D k sont de degrd infini sur 2. Le principal int6rbt de ces memoires est que Herbrand y inaugure une methode nouvelle basLe sur les notions de limite inductive et de limite projective, qu'il introduit systEmatiquement pour la premiere fois en mathdmatiques, et qui sont devenues fondamentales de nos jours, aussi bien en Algibre qu'en Topologie. I1 considere un corps de nombres de degre infini comme limite inductive d'une suite de sous-corps de degre fini, et fait une etude systematique du "passage B la limite" pour diverses notions lides B ces corps. Quant 1 la notion de limite projective, elle se presente naturellement quand on considere une extension galoisienne K (de degrd infini)'d'un corps de nombres k (de degr6 fini ou non sur 3), c o m e limite inductive d'extensions galoisiennes Kn de k de degrd fini, car le groupe de

J. DIEUDONNE

6

Galois de K sur

sur k

est alors limite projective des groupes de Galois des

Kn

k.

I1 n'est peut-Stre pas sans int6rSt de rappeler que c'est en prolongeant ces travaux de Herbrand que Chevalley, pour dtendre la thdorie du corps de classes aux corps de nombres de degrd infini, introduisit la notion d'idlle [21. 111. LES CORPS CYCLOTOMIQUES Le m6moire de Herbrand sur les classes d'iddaux des corps cyclotomiques [71 est celui qui maintenant est considdrd comme la p l u s originale de ses contributions 2 la Thdorie des nombres, et qui a eu le plus de rdpercussions sur les travaux

actuels. Soit K = Q(Jtcome, depends

th,? data, t h a t is, t h e formal p r e s e n t a t i o n o f t h e proof,

and on the p a r t i c u l a r e x t r a c t i o n processes available.

Some of t h e e a r l i e r ex-

perience with unwinding proofs by hand i s incorporated i n t h e new work. Reminder: as already mentioned i n t h e l a s t s e c t i o n , on science and technology, requirements for success a r e modest here; f o r example, it i s enough i f program-

mers l e a r n t o g e t e a s i l y a formal p r e s e n t a t i o n from t h e general idea of t h e proof one has i n mind; it is not necessary t o have a theory f o r t h e passage. Functional i n t e r a r e t a t i o n s :

recursiveness, c o n t i n u i t y an5 higher types.

The

idea of 'reducing' an a r b i t r a r y a s s e r t i o n , f i r s t i l l u s t r a t e d by Herbrand's disjunctions, was extended t o a r i t h m e t i c i n t h e no-countereexample-interpretation (n.c.i.)

which, i n c o n t r a s t t o H T , introduces higher types.

If

3 fm1Ao is

Finiteness theorems in arithmetic

49

the Skolem normal form o f l A then, f o r obvious reasons, V f 3nAo

is called the n.c.i.

f t+ pn A.

i s recursive and continums i n f ( f o r the product

of

A, and the map:

topology). N.B.

The Skolem functions of

i n predicate logic.

A

t i c theorems

A

formulas

A

w and w P m , w

.

A

i s provable

cannot be generally interpreted by purely universal arithmetic

formulas with recursive constants. VfAo[n/N(f)],

need not be recursive even i f

It i s an easy excercise, i n hierarchy theory, t h a t arithme-

where

N

In contrast, t h e n.c.i.

of

A

i s a recursively continuous functional of

is f

.

For

i n the language of seco7d-order arithmetic with variables over one uses certain continuous functions of a l l f i n i t e types o v e r

Again, a simple hierarchy argunent shows t h a t t h i s s o r t of passage t o

higher types i s needed.

-

Remark. Though t h e c l a s s i f i c a t i o n by quantifier com-

plexity i s adequate f o r such simple negative r e s u l t s it i s wholly inadequate f o r analysing the gain achieved by functional interpretations:

a t the cost of a

'mild' increase i n types, one has a dramatic r e s t r i c t i o n on ( t h e higher typ-) operations, f o r example, i n place of the non-recursive Skolem functions of a narrow subclass of recursive functionals i n the n.c.i. a bit:

of

A.

,

A

Exaggerating

t h e categorization of formulas by t h e i r type, also called: l e v e l of

'abstraction', hss some of the flavor familiar from a s t r o l o g i c a l categories. S t a b i l i t y r e s u l t s f o r interpretations and other transformations of given derivations.

A s a hangover from the u n r e a l i s t i c

project of formalizing proofs or,

more generally, thoughts correctly, there was the aim of showing t h a t the intended unwinding was f a i t h t b l l y rendered by the available transformations, under t h e slogan: s t a b i l i t y of E-theorems.

But the same work has a l s o an

impxtant consequence f o r the more r e a l i s t i c technological project described above.

Since lower bounds f o r any one of these transformations a r e very large,

they a r e a l l equally, s o t o speak, systematically defective.

For p r a c t i c a l

success one has t o separate e f f i c i e n t from i n e f f i c i e n t procedures.

This obvious,

but decisive conclusion was implemented i n Goad's pruning (of redundancies).

-

Reminder: The work in question i l l u s t r a t e s two general points: first, t h e weakness of s t a b i l i t y r e s u l t s without safeguard against systematic oversights; secondly, the use of i n s t a b i l i t y or s e n s i t i v i t y a s a means f o r choice (between the good and the bad). Big e f f e c t s of small changes:

e f f e c t s on the r e s u l t of transforming different

derivations which formalize ' e s s e n t i a l l y the same key idea' of a proof.

A s soon

a s people began t o look f o r t h i s phenomenon, many i s o l a t e d f a c t s and promising problems were found, but so f a r no general quantitative meta-mathematical results.

G. KREISEL

50

For most readers o f t h i s volurne probably t h e i s s u e i s b e s t i l l u s t r a t e d by t h e following question from R a r n a j theory:

Let

be an economical proof of RT,

,

Ramsey's theorem f o r p a r t i t i o n s of t h e ( i n f i n i t e ) set of a l l n a t u r a l numbers, and

let

TTF

and

RTA

arithmetic variant

-

m8ntht?mstically t r i v i a l l y d i f f e r e n t

RT,:

Ramsey's own f i n i t e version

be t h e

ITA

arguments which derive from

-

RTF

compnctness

,

its

resp.

due t o Daris and Hsrrington.

The two proofs:

1Tc0 followed by

llF,

resp. by

d i f f e r only ' s l i g h t l y ' ,

ITA So it

but t h e unwinding processes s r e 'complicated'.

seems wide open whjsther t h e bounds obtained by unwinding t k s e two proofs d i f f e r

l i t t l e or much.

- NB.

The d e r i v a t i o n

llm s t a t e d i n an appropriate formalism,

may not bme long (though it w i l l c e r t a i n l y ' b e boring t o r e a d ) ; I believe t h a t J. Ketmen uses

< 30

l i n e s i n h i s 'proof-checker'.

More mathematically i n c l i n e d readers would p r e f e r examples from t o p o l o g i c a l dynamics, such as t h e proof of von der Waerden's theorem given by Furstenberg and Weiss and analysed by Girard, or from g'unction theory such as Landau's proof of

L( I,x ) 20 by d e r i v i n g a c o n t r a d i c t i o n from assuming t h a t L ( S , ~ )c ( s ) / c ( 2 s )

s = 1

has no pole a t

,

a n l i t s modification which uses the a n a l y t i c continuation, across L(S,*)C(S)/6(2

s

1 -L(1

Recycling t r a d i t i o n a l proof theory:

,x) c ( s ) / c ( 2 )

s

=

1

, of

-

how t o g e t e f f i c i e n c y i n a r e a s of obvious

i n t e r e s t by going beyond them, f o r example, i n geometry by going beyond dimensions

4 or 5.

The idea, o f t e n repeated and sometimes v a l i d , i s t h a t s i m p l i f i c a -

t i o n s of proofs and constructions a r e needed t o succeed a t a l l i n t h e far beyond; afterwards the improvements can be brought down t o e a r t h t o g e t e f f i c i e n t solut i o n s where they a r e r e a l l y wanted.

A promising candidate i n t h s present context

seems t o be t h e nx-malization f o r

ZF

presented i n Powell's a b s t r a c t a t t h i s meeting.

H i s own aim seems t o have been the extension from the ( a l r e a d y hopelessly ' s t r o n g ' ) theory of species t o t h e stronger system

ZF

.

But another s i d e i s t h e

use of s e t - t h e o r e t i c language, with t h e following consequence.

H i s scheme can be

s p e c i a l i z e d t o q u i t e weak subsystems (where bounds are manageable), and applied,

for example, t o proofs of

V2

theorems d i r e c t l y .

Without h i s scheme, t h e sub-

systems would have t o be embedded i n appropriate type t h e o r i e s , i n which

-

the

t r a n s l a t i o n s of-the s e t - t h e o r e t i c axioms have t o be proved, and then used as formulas of r e l a t i v e l y high complexity.

&.

When, as here, t h e growth r a t e i s high,

t h i s kind of detour may make t h e d i f f e r e n c e between failure and success.

Finiteness theorems in arithmetic on a neglected c o n t r a s t .

Remark

51

H T provide, as

The l a r g e lower bounds on

already mentioned, a simple formal r a t i f i c a t i o n of a g e n e r a l conviction which t h e p r a c t i c a l need f o r a b s t r a c t

s p e c i a l i s t s have derived from experience:

So those bounds a r e of l i m i t e d i n t e r e s t t o s p e c i a l i s t s , and of g r e a t

methods.

i n t e r e s t t o o u t s i d e r s f o r whom a simple proof o f s o t o speak a p o i n t of prinAlmost t h e opposite a p p l i e s t o recent r e s u l t s

c i p l e replaces p a i n f u l experience.

on the (high) r a t e of growth of s o l u t i o n s f o r s p e c i f i c problems i n f i n i t e combin a t o r i c s which follow e a s i l y from Ramsey's theorem, Himan's on w e l l partial

A s a matter of h i s t o r i c a l f a c t ,

orderings, Kruskal's on sequences of t r e e s .

experienced s p e c i a l i s t s have had t r o u b l e guessing even roughly j u s t which coroll a r i e s do o r do not have, say, r a p i d l y growing bounding functions or s t r a i g h t forward d i r e c t proofs. NEL The r e l a t i o n of t h i s work, on bounds, t o t r a d i t i o n a l proof theory i s d e l i c a t e . duced f o r t h e sake of proof

The work sometimes uses t h e o r d i n a l notations i n t r o -

-

t h e o r e t i c a n a l y s i s ; b u t it is rewarding j u s t

because it s e p a r a t e s them from other, i r r e l e v a n t f e a t u r e s o f t h a t analysis.

-

Bernays, vol. 2, ( i ) I n t h e f i r s t e d i t i o n of H i l b e r t i s proved f o r ( v a l i d ) prenex formulas by reduction t o ( t h e v a l i d i t y of) €?? C1-formulas. I n t h i s a u x i l i a r y s t e p new function symbols f a r e introduced

Sundry observations:

which have a very c l e a r meaning i n terms of t h e n. c. i. ; they have t o be elimina-

at

ted, most simply by s u b s t i t u t i n g new v a r i a b l e s i n order t o g e t Herbrand's own form of

EC

, which

questionably, t h e r e s t r i c t i o n t o prenex formulas A

f o r terms beginning with i s much l e s s memorable.

f

,

Un-

IEP i s t o

is essential i f

be used t o meet Gentzen's demand, mentioned earlier, f o r a ' f i n i t i s t sense' which i s r e l a t e d simply

- tacitly,

- t o the

for us

form of

A

.

This i s per-

f e c t l y c o n s i s t e n t with t h e increase i n t h e computational complexity of Herbrand's disjunctions t h a t must be expected from a semantically t r i v i a l passage t o prenex form, y e t one more instance of the b i g d i f f e r e n c e between information processing by us and by c u r r e n t computers.

(Thus we remember shapes and d e s c r i p t i o n s of

s u b s t i t u t i o n s s e p a r a t e l y as shown most c l e a r l y i n experiments on p r e a t t e n t i v e ( i i ) S e n s i t i v i t y o f the n. c. i. t o t h e choice o f representations,

perception. )

f o r example, of so-called real number generators: a neglected p o i n t i n t h e literature. and l e t ximation.

F be a ( r e c u r s i v e l y ) continuous map o f t h e c i r c l e i n t o i t s e l f ,

Let

r e p r e s e n t p o i n t s of t h e c i r c l e ; s u b s c r i p t s

E

VNE?(\F(F)-l,

l'interprdtation associe au terme epsilon une fonction

dans

2.

(&-l)-aire

g

; si

dgfinie

Pour comparer la mbthode de Hilbert 1 celle de Herbrand, supposons que nous veuillons discuter le problsme de la satisfaisabilitd de la formule

-x et y

sont les seules variables (libres ou lides) de A(x,y).La

nelle non stricte de

pour satisfaisabilitg est

Vzil~A(x,y), 00

VsA(x,f(x)).

forme fonctionDans le calcul

epsilon, l'glimination du quantificateur existentiel nous donne la formule V ~ & ( E , E A(x,y))

Y- - -

et, sgmantiquement, le terme

E

A(x,y),

r--

tout comme le terme

J. VAN HEIJENOORT

66

r(z),

fonctionnel denote, dans une interergtation quelconque, une fonction de 5 Certes, le terme epsilon a une plus grande complexit& syntactique que le terme fonctionnel de Herbrand, lequel est form6 2 l'aide d'un symbole nouvellement intrcr duit. Mais apparait bientat une diffdrence plus profonde, car, 1 l'dtape suivante, les chemins divergent. Herbrand passe 5 la forme fonctionnelle stricte pour satisA(x, _ _f(x_)), faisabilitd, _

dans laquelle 5

est ddsormais une variable libre, tan-

dis que Hilbert considsre le quantificateur universe1 comme la negation d'un quantificateur existentiel et rdpste la mCme operation, de sorte qu'il obtient la formule

A(", "-A(?,

E+(X,X)),

moins transpgrente que

E

A(x,y)).

Sgmantiquement, cette formule est d6j1

-A(x, _ f(z)),r-et pourtant notre exemple Qtait l'un

des plus

simples que nous eussions pu prendre. DGs que le nombre des quantificateurs superposgs dans une formule augmente, la traduction de la formule dans le calculepsilor

devient vite sdmantiquement opaque. En entreprenant d'gliminer, d'une dihonstration dans l'arithmetique, toutes les variables, liees ou reelles, au profit de chiffres ddtermines, Hilbert traite les deux quantificateurs de la mCme manisre, alors qu'il y a, tout au long des dcrits logiques de Herbrand, un fil rouge, qui est la faqon differente dont il traite les variables universaloides et les variables existentialoydes d'une formule. Herbrand 6tudie la logique pure, passant ensuite 1 des thdories particulisres en ajoutant des hypothsses. Le systsme initial de Hilbert est dCj1 l'arithmgtique ; ainsi, dans la liste de ses axiomes, les axiomes sur le successeur, l'addition et la multiplication prdcsdent les axiomes sur les quantificateurs. Dans

1929

Hilbert opdra un passage inverse 1 celui de Herbrand et va de l'arithmdtique 1 la logique ; il abandonne le successeur et, gcrit-il (page 8),

'ceci signifie es-

sentiellement que nous faisons abstraction du caractgre ordonnd du systsme des nombres et le considerons qomme un systbme arbitraire de choses'. L'aboutissement de cette marche en arrisre, c'est que dans la seconde moitie des anndes trente, d6j1 bien aprss la publication des travaux de Herbrand, Bernays prend comme son point de ddpart la thdorie de la quantification, pour laquelle il 6tablit les deux theorsmes epsilon. Mais, ce faisant, il abandonne la fason unique de traiter les deux quantificateurs, il introduit entre eux une dissymetrie, rejoignant ainsi Herbrand (Bernays 1936, pages 93-98 ; Hilbert et Bernays 1939, pages 149-163, 1970, pages 149-169).

ou

I1 y a entre la mdthode de Herbrand et celle de Hilbert re-

vue par Bernays une analogie assez dtroite, que la prdsentation de Bernays fait bien ressortir (bien qu'il se limite 1 des formules prdnexes). Mais, ndanmoins, Hilbert et Bernays dtaient passes de l'arithmgtique 2 la logique, alors que Herbrand dtait all6 de la logique 2 l'arithmetique. Cette difference de perspective explique, par exemple, pourquoi le premier thgorsme epsilon a, pour Hilbert et Bernays, une importance particdiere qu'il n'a pas pour Herbrand. Si l'on en vient maintenant aux applications qui concernent l'arithmetique avec quantificateurs, les

L'oeuvre iogique de Jacques Herbrand

67

diffdrences s'estompent. La ddmonstration, basde s u r l e thdorsme de Herbrand, de l a non-contradiction

de c e t t e a r i t h m e t i q u e (Scanlon 1973) e s t du mdme degrd de

complexitQ que c e l l e q u i s u i t l e s i d d e s de H i l b e r t (Ackermann

1940) ; mtme

plus,

l e s deux dzmonstrations o n t de profondes ressemblances. E n f i n , dans l e u r dBmonst r a t i o n du thBor8me de Herbrand, Dreben e t Denton ( E ) ' a d a p t e n t I l a logique pure l a n o t i o n de rBsolvante, i n t r o d u i t e p a r Ackermann dans s a ddmonstration

(1940)

de l a non-contradiction de 1'arithmBtique. Herbrand indique, parmi l e s Q c r i t s q u i l u i ont s e r v i l o r s de l a rBdaction de s a thsse, LGwenheim

1915. On

ne t r o u v e , n i dans c e t t e t h e s e n i dans s e s a u t r e s B c r i t s ,

aucune remarque sur l a n o t a t i o n s i p a r t i c d i e r e u t i l i s d e dans c e t a r t i c l e e t empruntBe, comme on s a i t , 1 SchrSder. Mais Herbrand l u t certainerrcnt l ' a r t i c l e , c a r , lorsque dans son

1931 il

s'occupe du problsrne de l a r Q d u c t i o n , pour l a t h Q o r i e de

l a q u a n t i f i c a t i o n , I des p r Q d i c a t s t o u t au p l u s b i n a i r e s , il u t i l i s e des symboles (I,',

' H-' e t

'1') i n t r o d u i t s

p a r LGwenheim dans son t r a i t e m e n t de ce problsme.

C'est donc, on peut l e d i r e avec assurance, dans l ' a r t i c l e

de LGwenheim que

Herbrand trouva l'argument sBmantique q u i l e c o n d u i s i t I i n t r o d u i r e s e s s u i t e s de champs f i n i s . Herbrand a d r e s s e 1 LSwenheim un c e r t a i n nombre de reproches. C e r t e s , l ' a r t i c l e de LGwenheim c o n t i e n t des lacunes dans l e s d E f i n i t i o n s e t l e s ddmonstrations ( v o i r , par exemple,

van Heijenoort 1967,pages

228-232),

mais les g r i e f s que Herbrand

formule c o n t r e LGwenheim ont un t o u r p a r t i c u l i e r q u ' i l nous f a u t examiner. Ces g r i e f s peuvent s e ramener 1 deux : premierement, LGwenheirn donne 1 l a n o t i o n de v a l i d i t d 'un sens i n t u i t i f ' e t , p a r s u i t e , son thdoreme ' n ' a aucun sens p r d c i s ' ; deuxiemement, s a dBmonstration manque de r i g u e u r e t r e s t e i n s u f f i s a n t e . C e r t e s , LGwenheim n'dnonce pas de d g f i n i t i o n s pour l e s n o t i o n s sdmantiques q u ' i l u t i l i s e , mais l a fagon dont il l e s manie montre b i e n , une f o i s qu'on a m a f t r i s d son langage, q u ' i l n ' y a aucun malentendu dans son e s p r i t 1 l e u r s u j e t . Ces n o t i o n s r e s t e n t p o u r l u i , il e s t v r a i , ' i n t u i t i v e s ' , c a r il l e s emprunte b l e s . Quant

a

I une t h Q o r i e nai've des ensem-

s a dQmonstration, e l l e s o u f f r e r Q e l l e m e n t de deux i n s u f f i s a n c e s : il

f a i t un dBtour p a r l ' i n f i n i ,

c'est-1-dire

c o n s i d l r e des formules

i n f i n i m e n t lon-

gues, au l i e u de s e s e r v i r de l'axiome de choix, e t , l o r s q u ' i l s ' a g i t d ' o b t e n i r l e contre-modele i n f i n i 1 p a r t i r de l a s u i t e d ' a s s i g n a t i o n s f i n i e s , il emploie une conjonction i n f i n i e ( l e t e x t e de LFwenheim e s t t e l q u ' i l n ' e s t pas f a c i l e de d i r e

s ' i l passe 11 indQment du fin:

L l ' i n f i n i ou s ' i l a un v d r i t a b l e argument en t 6 t e ) .

Les g r i e f s de Herbrand semblent donc j u s t i f i d s s i l ' o n s ' e n t i e n t au sens l i t t d r a l des mots : il y a dans l e s n o t i o n s de LGwenheim de 1 " i n t u i t i f '

e t dans s e s dB-

monstrations des lacunes. Mais, d e r r i e r e ces mots, Herbrand a v a i t envue des reproches b i e n d i f f e r e n t s de ceux que nous venons de r a p p e l e r . Ce q u ' i l pense, ce n ' e s t nullement que LGwenhe.im a u r a i t df donner des d d f i n i t i o n s e x p l i c i t e s des n o t i o n s

J. VAN HEIJENOORT

68

s d m a n t i q u e s q u ' i l u t i l i s e , mais q u ' i l a u r a i t dQ abandonner c e s n o t i o n s mPmes ; e t non p a s combler l e s l a c u n e s d e sa d d m o n s t r a t i o n , mais p r o u v e r un thdorSme d i f f d rent. S i , au l i e u de d i r e

r

' p o u r t o u t nombre

la

r-isme

e x p a n s i o n c o n j o n c t i v e , de

e s t s a t i s f a i s a b l e ( p a r l e s t a b l e s d e v d r i t d ) ' , nous

Herbrand de l a f o r m u l e

convenons, a v e c Herbrand, de d i r e

' F e s t v r a i e d a n s un champ

i n f i n i ' , nous pou-

vons d n o n c e r ses r d s u l t a t s fondamentaux sous l a forme s u i v a n t e ( v o i r H e r b r a n d 1 9 2 9 a , page 1077, ou

1968,page

28) :

"z

Thdorsme I. S i une f o r m u l e F est dzmontrable e n t h d o r i e d e l a q u a n t i f i c a t i o n , n e p e u t d t r e v r a i e d a n s un champ i n f i n i .

F n ' e s t pas dgmontrable e n t h d o r i e de l a q u a n t i f i c a ThdorSme 11. S i une f o r m u l e -

t i o n , on p e u t c o n s t r u i r e un champ i n f i n i d a n s l e q u e l

"x e s t v r a i e .

Herbrand v o i t d a n s l e ThdorSme IT un ' r d s u l t a t a n a l o g u e '

(1930,page

118, ou

1968,

page 143) au thdorSme d e Lijwenheim s u r l ' d q u i v a l e n c e d e l a v a l i d i t d 3 l a v a l i d i t d dans un ensemble ddnombrable. Le r d s u l t a t d e LEwenheim, m i s sous une forme compar a b l e 1 c e l l e du ThdorSme 11, s e r a i t : Thdorsme II*. S i une f o r m u l e F n ' e s t p a s v a l i d e , %F - e s t s a t i s f a i s a b l e d a n s un ensem-

b l e d6nombrable.

Lowenheim n ' a n i axiomes n i r S g l e s d ' i n f d r e n c e ,

il l a i s s e d e c 8 t d l a n o t i o n d e

d d m o n s t r a b i l i t d d a n s un s y s t s m e donnd. Donc les n o t i o n s s d m a n t i q u e s a p p a r t i e n n e n t non s e u l e m e n t 1 ses a r g u m e n t s , ,mais a u s s i 1 ses r d s u l t a t s m t m e s . L ' i n t d r d t fondamental de ces r d s u l t a t s ,

c ' e s t q u ' i l s s o n t , p o u r a i n s i d i r e , une r e d u c t i o n du non-

ddnombrable a u ddnombrable, a l o r s q u e c e u x de Herbrand d t a b l i s s e n t un p o n t e n t r e

l a t h d o r i e d e l a q u a n t i f i c a t i o n e t l a l o g i q u e p r o p o s i t i o n n e l l e . La p r d m i s s e du ThdorSme I1 c o n t i e n t ' d d m o n t r a b l e '

t a n d i s q u e c e l l e du ThdorSme II* c o n t i e n t 'va-

l i d e ' , c e q u i montre b i e n q u e les deux thEorSmes a p p a r t i e n n e n t 1 d e s domaines d i f f d r e n t s de l a l o g i q u e . Mais, comme Herbrand n ' a c c o r d e p a s aux n o t i o n s s d m a n t i q u e s d e s t a t u t p r o p r e , il v o i t d a n s l a v a l i d i t d une dbauche g r o s s i s r e d e l a ddmonstrab i l i t d ; l e s deux thdorSmes o n t donc p o u r l u i l a mdme p r d m i s s e , l e p r e m i e r sous une forme e x a c t e , l e second s o u s une forme c o n f u s e . I1 p e u t 1 l a f o i s p a r l e r d ' a n a l o g i e e n t r e les deux thdorSmes e t c r i t i q u e r Lowenheim p o u r son manque d e r i g u e u r . Quant a u ThdorZme I, Herbrand r e p r o c h e 3 Lowenheim, ' e t c e r e p r o c h e e s t l e p l u s grave'

(1930,p a g e

118, ou

1968,p a g e

143), d e l e c o n s i d d r e r c o m e d v i d e n t . S i

nous f a i s o n s s u b i r a u ThdorSme I l a t r a n s p o s i t i o n q u i n o u s a p e r m i s d e p a s s e r du Thdorbme I1 a u ThCorSme II*, n o u s o b t e n o n s :

L'oeuvre logique de Jacques Herbrand

69

T h e o r h e I*. S i une f o r m u l e F e s t v a l i d e , z x n ' e s t p a s s a t i s f a i s a b l e d a n s un ensemb l e ddnombrable. Ce q u i , du p o i n t d e v u e a d o p t 6 p a r Lb'wenheim, d d c o u l e immediatement d e s d 6 f i n i t i o n s

m8mes. Mais, ce p o i n t d e vue s d m a n t i q u e , Herbrand se r e f u s e prCcis6ment 1 l a cons i d d r e r . I1 a 6 t d v i s i b l e m e n t s d d u i t p a r l ' a r g u m e n t s g m a n t i q u e q u i e s t a u c o e u r de

l a d e m o n s t r a t i o n d e Lb'wenheim. Mais, c e t argument, Herbrand l e m o d i f i e de t r o i s m a n i s r e s , l i 6 e s e n t r e e l l e s p a r l e f a i t q u ' i l r e j e t t e , d a n s les 6 t u d e s l o g i q u e s , l a n o t i o n d'ensemble i n f i n i . P r e m i i r e m e n t , il n e c o n s i d i r e p a s l ' e n s e m b l e i n f i n i dont les champs f i n i s s o n t les a p p r o x i m a t i o n s s u c c e s s i v e s ; il mentionne p l u s i e u r s f o i s l ' e x i s t e n c e d e c e t ensemble, mais il n e l e f a i t p a s f i g u r e r d a n s ses argu-

ments. Deuxiimement, a p r P s a v o i r a j o u t d a u v o c a b u l a i r e d e s o n s y s t i m e comme const a n t e s i n d i v i d u e l l e s l e s noms d e s Bldments de c e s champs ( o u c e s Bldments euxmcmes s ' i l s s o n f c e n s 6 s S t r e d 6 j 1 d e s noms), ou c o n j o n c t i o n s f i n i e s ,

il n e c o n s i d s r e q u e d e s d i s j o n c t i o n s

p a s s a n t a i n s i d u p l a n sdmantique a u p l a n s y n t a c t i q u e .

Troisismement, l ' h y p o t h i s e de v a l i d i t 6 f a i t e p a r Lb'wenheim, Herbrand l a remplace p a r c e l l e d e d d m o n s t r a b i l i t d d a n s un systPme q u ' i l s p b c i f i e . Herbrand lui-m6me c o n s i d g r a i t s o n thgorsme comme une r e c t i f i c a t i o n de c e l u i d e Lb'wenheim, une 'prCcision', dcrit-il

(1931c, - page 4 , ou

1968,page

225),

f a i s a n t u s a g e d ' u n germanis-

me. Cette c o r r e c t i o n c o n s i s t a i t 1 Q l i m i n e r l e s n o t i o n s e n s e m b l i s t e s , mais a i n s i l e s e n s m c m e du thGorPme se t r o u v e 6videmment changd. C ' e s t d ' a i l l e u r s 1 une c o n c l u s i o n a n a l o g u e q u e semble a b o u t i r Herbrand lui-&me, c a r , a p r i s a v o i r p a r 1 6 d e p r o p o s i t i o n n ' a y a n t 'aucun s e n s p r d c i s ' e t de ddmonstrat i o n ' t o t a l e m e n t i n s u f f i s a n t e ' a y a n t de 'graves l a c u n e s ' ,

118, ou

1968,page

il Lcrit

(1930,page

144) : 'Nous pouvons d i r e q u e l a d 6 m o n s t r a t i o n d e Lb'wenheim

e t a i t s u f f i s a n t e e n mathdmatiques ; mais i l n o u s a f a l l u , d a n s ce t r a v a i l , l a rend r e "mdtamathdmatique" ( v o i r 1 ' I n t r o d u c t i o n ) p o u r q u ' e l l e n o u s s o i t de q u e l q u e u t i l i t d ' . Dans c e t t e i n t r o d u c t i o n 1 l a q u e l l e il nous r G f P r e , Herbrand a v a i t e s s a y 6 de d e l i m i t e r l e s n o t i o n s e t l e s arguments u t i l i s d s e n mdtamath6matique d e ceux q u ' a c c e p t e n t l e s mathdmatiques e n gCn6ral. A l o r s q u ' i l r e j e t t e l a n o t i o n d'ensemb l e i n f i n i e n mEtamathEmatique, mathematiques

.

il

admet

l a t h e o r i e c l a s s i q u e d e s ensembles e n

Les c r i t i q u e s que Herbrand f a i t du t h d o r i m e d e Lb'wenheim e t de sa d d m o n s t r a t i o n s ' b t e n d e n t 1 l a s o l u t i o n que Lawenheim a v a i t donnde du problPme de l ' d l i m i n a t i o n , e n t h d o r i e de l a q u a n t i f i c a t i o n , d e symboles d e p r d d i c a t a y a n t p l u s d e deux arguments. Tout ce que nous v e n o m de d i r e s ' a p p l i q u e a u s s i aux r e p r o c h e s f a i t s 1 ce s u j e t . L1 a u s s i , Lawenheim e t Herbrand d t a b l i s s e n t d e s r 6 s u l t a t s d i f f d r e n t s . Pour l e p r e m i e r , l a t r a n s f o r m e e d ' u n e f o r m u l e v a l i d e e s t v a l i d e . P o u r l e s e c o n d , il

s ' a g i t d e d e m o n s t r a b i l i t d d a n s un s y s t k n e donnd. Quant aux mdthodes, Lb'wenheim s e

J. VAN HEIJENOORT

70

s e r t d ' a r g u m e n t s s e m a n t i q u e s , a l o r s q u e Herbrand a p p l i q u e son th6orPme e t p e u t a i n s i donner 1 l a r e d u c t i o n un s e n s c o n s t r u c t i f . Herbrand mentionne p l u s i e u r s f o i s d a n s ses d c r i t s Skolem 1920 e t c e r t a i n e s de s e s remarques p e u v e n t nous f a i r e p e n s e r q u ' i l l u t l ' a r t i c l e , s a n s que nous p u i s s i o n s en a v o i r l ' a s s u r a n c e c o m p l e t e ( i l n e d i t r i e n q u i s ' a p p u i e s u r a u t r e c h o s e que ce que H i l b e r t e t Ackermann

(1928)m e n t i o n n e n t ) .

Pour l e cas d'une s e u l e formule, l a

v e r s i o n du thdorsme d e Liiwenheim d o n t Skolem donne, d a n s c e t a r t i c l e de 1920, une d e m o n s t r a t i o n amendde e s t c e l l e q u i e x t r a i t un sous-modele ddnombrable du modPle q u i est suppos6 e x i s t e r . Mais, p o u r l e c a s d ' u n ensemble ddnombrable d e f o r m u l e s , l a d e m o n s t r a t i o n que Skolem e s q u i s s e t e n d 1 d t a b l i r l ' a u t r e v e r s i o n du th&orPme, c e l l e dans l a q u e l l e on o b t i e n t l e modble denombrable e n l e c o n s t r u i s a n t 1 l ' a i d e d'approximations f i n i e s s u c c e s s i v e s . C ' e s t c e t t e e s q u i s s e d e d d m o n s t r a t i o n q u e Skolem ddveloppe dans l a s e c t i o n 3 d e son a r t i c l e d e 1922. LP, nous a v o n s , d a n s l e c a s p a r t i c u l i e r d ' u n e f o r m u l e p r d nexe p o u r l a q u e l l e t o u s l e s q u a n t i f i c a t e u r s u n i v e r s e l s p r d c b d e n t les q u a n t i f i c a -

1,d e ce que Skolem 2, q u i v o n t donner l e s a p p r o x i m a t i o n s f i n i e s du

t e u r s e x i s t e n t i e l s , l a c o n s t r u c t i o n , p o u r t o u t nombre n a t u r e 1 a p p e l l e l e s s o l u t i o n s de niveau

modsle denombrable. Ces s o l u t i o n s f o m e n t , e n f a i t , une s u i t e de champs f i n i s , au s e n s d e Herbrand. L ' a r t i c l e d e Skolem d e 1922 n e semble a v o i r 6 t B g u s r e l u . I1 n e s u s c i t a , a u t a n t que j e s a c h e , q u e deux r d a c t i o n s , un compte r e n d u d e F r a e n k e l

(9 et ) une men-

, 232) ; m a i s , dans l ' u n e t l ' a u t r e c a s , c e f u r e n t t i o n p a r von Neumann (=page

l e s problemes d e l a t h d o r i e d e s ensembles q u i r e t i n r e n t l ' a t t e n t i o n , e t non l a mdthode employ6e p o u r d t a b l i r l e theorbme d e Lb'wenheim. Dans s o n compte r e n d u , F r a e n k e l , p a s s a n t e n r e v u e l ' a r t i c l e d e Skolem, n e mentionne &me

pas l ' e x i s t e n c e

de l a s e c t i o n 3. I1 semble b i e n que Herbrand n e connut p a s l ' a r t i c l e .

I1 n e connut p a s non p l u s , on p e u t l ' a f f i r m e r s a n s c r a i n t e , Skolem 1928. Cet a r t i c l e r e v b l e , de l a p a r t d e Skolem, d e s p r e o c c u p a t i o n s t o u t

P f a i t a n a l o g u e s 21 c e l -

les de Herbrand. A u s s i e s t - i l i m p o r t a n t d ' e n e x a m i n e r l e s s i m i l i t u d e s e t l e s d i f f e r e n c e s a v e c l e s t r a v a u x de Herbrand. A l a d i f f e r e n c e de c e l u i - c i ,

Skolem n e nous

p r e s e n t e p a s un s y s t s m e d e f i n i . Aprbs nous a v o i r d i t ce q u ' e s t une e x p r e s s i o n b i e n formee de l a t h e o r i e de l a q u a n t i f i c a t i o n , il d c r i t q u e l ' o n p e u t ' r e p r e s e n t e r l e s d d m o n s t r a t i o n s mathgmatiques comme d e s t r a n s f o r m a t i o n s d e t e l l e s e x p r e s s i o n s l o g i ques s e l o n c e r t a i n e s r e g l e s ' .

Ces r ' e g l e s , i l n e nous e n donne p a s une l i s t e exhaus-

t i v e , e t il m G l e a r g u m e n t s sEmantiques e t a r g u m e n t s s y n t a c t i q u e s . L e s r s g l e s q u ' i l Qnonce e x p l i c i t e m e n t l u i p e r m e t t e n t d e p a s s e r d'une f o r m u l e c l o s e q u e l c o n q u e l ' u n e d e ses formes p r d n e x e s ; c e s o n t les r ' e g l e s d e p a s s a g e d e Herbrand, a v e c l a d i f f d r e n c e t r i v i a l e q u e l a c o n j o n c t i o n e s t , chez Skolem, p r i m i t i v e a l o r s q u ' e l l e

B

L'oeuvre logique de Jacques Herbrand

71

e s t , chez Herbrand, d b f i n i e . Skolem a f f i r m e e n s u i t e l ' e q u i v a l e n c e semantique de c e t t e formule prenexe e t de s a forme f o n c t i o n n e l l e s t r i c t e pour s a t i s f a i s a b i l i t e , F l e s deux formules -ffss ' 'voulant d i r e ' l a mPme chose. Consid6rons l e s t e r m e s o b t e n u s 3 p a r t i r d'une constante i n i t i a l e 0 p a r composition avec l e s symboles f o n c t i o n n e l s de

5

veau d'un terme d t a n t

-k-1

Fffss, -

l e ni-

C(F,&)

la

s i e t seulement s i s e s arguments s o n t au p l u s de niveau

e t l ' u n au moins e s t de ce niveau ( l e niveau de

conjonction des cas p a r t i c u l i e r s de sont l e s v a r i a b l e s u n i v e r s e l l e s de F) s i b l e s , p a r des termes de niveau

5

F -ffss

sent

0

e s t 0). Soit

obtenus l o r s q u e s e s v a r i a b l e s (qui remplacees, de t o u t e s l e s manibres pos-

au p l u s .

Nous avons maintenant l ' a l t e r n a t i v e s u i v a n t e : Ou b i e n , pour un c e r t a i n n ' e s t pas s a t i s f a i s a b l e ( p a r l e s t a b l e s de v g r i t e ) , ou b i e n , pour t o u t e s t s a t i s f a i s a b l e . Dans l e premier c a s , Skolem d e c l a r e ,

ze s t refutable.

k,C(F,k) k,C(F,k) I1 ne

donne aucun argument. I1 n ' e s t peut-Ptre pas impossible d'imaginer ce q u ' b t a i t son raisonnement :

C(z,()

n ' e t a n t pas s a t i s f a i s a b l e ,

Q J ~ ( F , ( )e s t

v a l i d e , donc demon-

t r a b l e (dans l e fragment p r o p o s i t i o n n e l du systbme dbauche p l u s h a u t ) ; p u i s on peut p a s s e r d'une demonstration de

ss(F,k) 1 sEffss

gles q u i s o n t f i a b l e s , c ' e s t - & d i r e

q u i c o n d u i s e n n u n e formule demontrable 1 une

en appliquant c e r t a i n e s rb-

formule demontrable, e t q u i s o n t les r s g l e s de s u b s t i t u t i o n e t l a r s g l e de s i m p l i f i c a t i o n . Le raisonnement s u i t de prgs ce que l ' i n t u i t i o n nous d i c t e e t peut-Ptre Skolem a v a i t - i l une vue a s s e z p r e c i s e de l'argument, mais il ne nous en d i t r i e n . Dans l e second cas p r e s e n t 6 p a r l ' a l t e r n a t i v e , Skolem cherche 1 montrer que, s i

zffsse s t r e f u t a b l e ,

a l o r s , pour un c e r t a i n nombre

&, C(F,%)

n ' e s t pas s a t i s -

f s b l e . Selon 1'Qbauche de raisonnement q u ' i l nous p r e s e n t e , une d6monstration c o n d i t i o n n e l l e de

E

&

sp

en prenant

-ffss F

mee, s i l ' o n remplace l e s v a r i a b l e s l i b r e s p a r n e l l e de

E

c u l i e r s de

-C(F,$), -

oii

& s~

-ffss, F

p

s

comme hypothbse peut t t r e t r a n s f o r -

2,

e n une demonstration condition-

en prenant comme hypothsse un c e r t a i n nombre f i n i de cas p a r t i c'est-1-dire

un c e r t a i n nombres de termes de l a conjonction

t un nombre suffisamment grand.

I1 e s t c u r i e u x de n o t e r que l e s reproches que Herbrand f a i t 1 Lgwenheim, il a u r a i t pu, avec beaucoup p l u s de j u s t e s s e , l e s a d r e s s e r 1 Skolem, s ' i l a v a i t connu l e s travaux de c e l u i - c i .

Car Skolem s e donne l e mtme but que Herbrand. AprOs a v o i r ,

t o u t c o m e LGwenheirn, considere l e s expansions e t l e s champs

comme des moyens

techniques pour o b t e n i r des r d s u l t a t s sdmantiques, Skolem en e s t a r r i v e en 1928 1 v o i r 11 une methode q u i d o i t s u p p l a n t e r l e s d e r i v a t i o n s axiomatiques t e l l e s que l e s a v a i e n t consues Frege, R u s s e l l ou H i l b e r t . C ' e s t a i n s i q u ' i l d e c l a r e : 'Je c r o i s q u ' i l e s t p o s s i b l e d'aborder l e s problPmes de deduction d'une a u t r e maniPre, une manibre p l u s commode'

(1928,page

130, ou

Heijenoort

1967,page

5 1 6 ) . Tout

J. VAN HEIJENOORT

72

comme Herbrand, il montre par des exemples que cette nouvelle mdthode permet de traiter certains problemes de ddcision. Les deux avantages de Herbrand, ce sont une plus grande prdcision et une plus grande gdndralitd. I1 spdcifie exactement le systime qu'il considere, il entreprend de donner des dgmonstrations completes de ses rdsultats (mPme s'il y a une erreur involontaire dans une de ses ddmonstrations), il considere des formules quelconques, alors que Skolem laisse son systeme dans le vague, y mgle des considerations semantiques, 6bauche 1 p e k e ses raisonnements et se borne 1 considdrer des formules prdnexes. Tous les reproches que Herbrand fait 1 LEwenheim s'appliquent exactement 1 Skolem. I1 faut noter ici que Skolem avait, envers les fondements des mathdmatiques, une attitude diff6rente de celle de Herbrand. Celui-ci, 1 la suite de Hilbert, separe nettement, quant 1 leur objet et leurs mdthodes, mathematiques et metamathdmatique. Skolem ne fait pas cette distinction et pour lui les mathematiques, ce sont les mathematiques constructives. I1 se place, quant 1 cette question, B un point de vue assez semblable B celui de Brouwer. Ceci nous conduit 1 examiner l'attitude de Herbrand envers l'intuitionnisme. Les constructivistes en mathgmatiques ont toujours mis en avant la conception que la valeur d'une fonction, pour chaque suite d'arguments donnee, doit Btre effectivement calculable. Au debut des anndes vingt, les fonctions rdcursives primitives, introduites par Dedekind des 1888, Gtaient devenues le paradigme mbme de fonction calculable et en 1923 Skolem avait publid une version de l'arithmdtique basde sur l'emploi de ces fonctions. Mais il 6tait vite devenu clair, ne fCt-ce que par un argument diagonal, qu'il est des fonctions calculables qui ne sont pas recursives primitives ; en 1925 Hilbert mentionnait dGjB une telle fonction et en 1928 Ackermann en publiait une etude detaillde. En 1931 Herbrand proposa par trois fois d'introduire une classe de fonctions calculables qui fussent plus generales que les fonctions rdcursives primitives. La premiere fois, au debut de 1931, il dcrit que, selon 1"intuitionnisme'

(et

dans ce passage il entendait par ce mot le finitisme de Hilbert, qu'il avait adoptd), 'toutes les fonctions introduites devront ftre effectivement calculables pour toutes les valeurs de leurs arguments, par des opdrations ddcrites entisrement d'avance' (Herbrand 1931a, page 187, ou

1968,page

210).

La deuxieme f o b , ce fut lorsque, 1 peu pres au mfme moment, il envoya 1 Gadel une lettre dans laquelle il proposait une definition de la notion de fonction rdcursive (gdnerale), definition qui, d a m le texte des conferences que GEdel donna 1 Princeton en 1934, est reproduite comme suit : 'Si cp est une fonction inconnue et Jll....,ik sont des fonctions connues, et si les fonctions $ et cp sont substituees

-

13

L'oeuvre logique de JacquesHerbrand

les unes aux autres de toutes les manibres possibles et certaines paires des expressions ainsi obtenues sont Bgalees, alors, si l'ensemble ainsi obtenu d'hquations fonctionnelles a une solution et une seule pour cp, cp est une fonction recursive' (GSdel 1934, page 26, ou Davis 1965, page 70). A la definition de Herbrand, GSdel ajouta deux clauses, que nous examinerons dans un instant. La troisieme fois que Herbrand proposa une definition des fonctions recursives (gengrales), ce fut dans l'article qu'il termina quelques jours avant sa mort (Herbrand 1931c, page 5, ou ~-

E,pages

226-227). LL, il Bcrivait : 'On pourra

aussi introduire un nombre quelconque de fonctions theses telles que :

ii - x1x2...&. -1-

avec des hypo-

(a) -Elles ne contiennent pas de -variables apparentes ; (b) Considerees intuitionnistiquement, elles permettent de _ -faire effectivemerit le calcul de

Lzlz2...&. pour tout systhe particulier de --___ -1

nombres ; et l'on puisse demontrer intuitionnistiquement _ que _ l'on ___ _obtient un resultat bien determine.' Et 1 la premiere occurrence du mot

'intuitionnistiquement' Herbrand avait attache

une note, la note 5, qui disait : 'Cette expression signifie : traduites en langage ordinaire, considerees comme une propriete des entiers, et non comme un pur symbole'

.

Nous avons 11 trois suggestions, car il faut parler ici de suggestions plutdt que

de definitions, et elles different. Le calcul effectif de la valeur de la fonction est mentionne dans la premibre et la troisieme, mais non dans la deuxieme. Dans la premiere ce calcul doit se faire

'par des opPrations d6crites entierement d'avan-

ce'. Dans la troisisme le calcul se base sur les proprietgs intuitives des entiers, proprietes independantes, comme semblent l'indiquer les derniers mots de la note 5, de toute definition des entiers dans un systbme formel. Des deux clauses dont GGdel jugea necessaire de completer la deuxieme suggestion de Herbrand, la premiere donnait une forme canonique au cat6 gauche des equations et la seconde, plus importante, presentait la liste finie des operations admises dans le calcul de la valeur d'une fonction pour des arguments donnes. Ainsi la dbfinition de Gb'del combinait la deuxieme suggestion de Herbrand (communiquee par Herbrand 1 GGdel dans une lettre) avec la premizre (alors inconnue de GGdel), en outre realisant pour la premiere ce qui, chez Herbrand, n'gtait encore qu'un programme ; il avait bien reclam6 'des op6rations decrites entierement d'avance', mais il n'en avait pas donne de liste. Dans une lettre datPe du 23 avril 1963 1 van Heijenoort, Gzdel faisait s u r la

I. VAN HEIJENOORT

14

manilre dont la notion de fonction rdcursive avait Ctd acquise les commentaires suivants : 'Je n'ai jamais rencontrd Herbrand. Sa suggestion fut faite par lettre

-

en 1931, et elle dtait formulde exactement comme ce l'est 2 la page 26 du texte de mes confdrences [GSdel 1934, ou Davis 1965, page 701, c'est-2-dire sans qu'il fiit fait mention de la calculabilitL. Mais, comme Herbrand dtait un intuitionniste, pour lui cette ddfinition signifiait Lvidemment qu'il existe une ddmonstration constructive de l'existence et de l'unicitb de

Q.

I1 croyait probablement qu'on ne

pouvait donner une telle ddmonstration qu'en prdsentant un procddd de calcul. (Noque, au cas 03 3lcpA(cp) est acceptable intuitionnistiquement, alors la fonction i c p A ( ~ ) est rscursive gdtez que, si la thlse de Church est correcte, il est

ndrale, bien que, pour obtenir le procddd de calcul de cp, il puisse btre ndcessaire d'ajouter certaines dquations 3 celles qui sont d6j2 contenues dans A(cp). Donc je ne pense pas qu'il y ait un ddsaccord quelconque entre ses deux ddfinitions [la deuxihe et la troisibme, GSdel ignorait alors la premibre] telles qu'il les entendait. Ce qu'il n'a pas vu (ou n'a pas clairement exprimd), c'est que le calcul, pour routes les fonctions calculables, se fait selon des rggles qui restent exactement les m8mes. ---

C'est 12 le fait qui rend possible une definition prdcise de la

rdcursivitg gdndrale. Je n'ai malheureusement pas retrouvd la lettre de Herbrand dans mes papiers. Elle a dtd probablement perdue 2 Vienne pendant la Seconde Guerre mondiale, comme tant d'autres choses. Plais mon souvenir est bien net et il Qtait encore tout frais en 1934.'

On a parfois pens6 que cette deuxigrne suggestion de Herbrand, telle qu'elle est formulde par Gadel, visait une-classe de fonctions plus gdndrales que les fonctions rdcursives, la classe des fonctions hyperarithmdtiques. Telle n'ltait pas, comme on le voit, l'opinion de GGdel et, P la lumilre des deux autres suggestions de Herbrand, l'hypothlse semble peu fondde. Dans une lettre datbe du 14 aoQt 1964, GEdel bcrivait P van Heijenoort que c'btait une exagdration que de dire que Herbrand avait 'introduit' la notion de fonction rdcursive ; il fallait, selon lui, plut6t parler d'ldbauche', 'car c'est prdcisdment en spdcifiant les rbgles de calcul qu'un concept mathdmatiquement maniable et fdcond avait btL obtenu. Herbrand, de son &td,

exclut explicitement la spdcifica-

tion de rbgles formelles de calcul par la locution "considLrdes intuitionnistiquement" (et l'explication qu'il en donne dans sa note 5). La question de savoir si la conception de Herbrand [dans

m,la troisibme] est dquivalente 1 la rdcursi-

vitd gdndrale est considdrbe par Heyting, moi-mdme et d'autres comme non rdsolue. C'est mon opinion que la thlse de Church est incontestablement correcte pour la calculabilit6 mdcanique, mais peut-btre incorrecte pour la calculabilitd intuitiolr niste (comme je l'ai clairement dit dans la note ajoutLe au texte de mes confbrences de 1934 [Davis 1965, pages 7 1 - 7 3 ] ) . '

L'oeuvre logique de JacquesHerbrand

75

Nous voici donc amends l essayer de comprendre ce que Herbrand entendait par 'intuitionnisme', un mot qu'il utilise 1 partir de 1930. Le plus souvent, dans les 6crits de Herbrand, le mot denote simplement les m6thodes adoptdes par Hilbert en mdtamathiimatique et qualifiges par celui-ci de 'finit', un mot allevand peu usitd et aujourd'hui traduit par 'finitiste' ou 'finitaire'. Nous lisons par exemple dans Herbrand 1931b (Herbrand 1968, page 216) : 'Hilbert a de plus exigd, pour dchapper l la critique destructive de Brouwer, que tout raisonnement fait en m8tamathdmatique soit du type dit

"intuitionniste".' On pourrait multiplier les cita-

tions de ce genre. Herbrand, c'est bien clair, ne s'est pas plongd dans la lecture des dcrits de Brouwer, et ce qu'il dit de celui-ci, c'est ce qu'il a pu apprendre en lisant les dcrits de Hilbert ou en conversant avec von Neumann ou Bernays. Cette connaissance de seconde main prend un tour caricatural lorsqu'il en vient 1 attribuer 1 Brouwer des id6es de Hilbert explicitement rejetdes par Brouwer : 'On a le droit de se servir de ces notions interdites [c'est-8-dire des notions non-finitistes] puisque tout resultat ddmontrd en les utilisant comme intermgdiaires ne peut dtre faux. Seulement ces notions devront Stre considgrdes par Brouwer comme des dldments sans

=,

signification rdelle, des dldments iddaux, comme dit Hilbert' (Herbrand 1930a, page 252, ou

pages 163-164).

I1 est 1 noter qu'l la fin des anndes vingt et au ddbut des anndes trente il n'dtait pas rare d'entendre dire que la mgtamath6matique de Hilbert se proposait de se limiter aux mdthodes 'intuitionnistes' (voir, par exemple, von Neumann 1927, page 3, lignes 1-5).

Cela msme incita Bernays 1 souligner la distinction entre fi-

nitisme et intuitionnisme (Hilbert et Bernays 1934, pages 34 et 43, Bernays 1934, page 69, *,

pages 89-90,

1935,page

212,

1938,page

146, et

1967,page

502).

Comme nous l'avons vu plus haut, Gzdel, dansles anndes soixante, considPre encore Herbrand comme un intuitionniste. Le seul passage dans les dcrits de Herbrand qui indique que celui-ci ait pu avoir de l'intuitionnisme une conception allant audell du finitisme de Hilbert est la note 5 de

-,

que je vats citer encore une

fois. Parlant des conditions imposdes aux fonctions rdcursives, Herbrand les dit 'considdr6es intuitionnistiquement' et il ajoute en note : 'Cette expression signifie : traduites en langage ordinaire, consid6rdes comme une propridtg des entiers, et non comme un pur symbole'. La remarque est si b r h e qu'elle reste ambi-

gee.

Je vois deux interprgtations possibles : (a) La remarque n'est rien d'autre qu'une allusion au caractsre intuitif de la m6tamathdmatique (qui, comme l'dcrivait von Neumann

(1927,page

3),

doit stre 'un enchainement d'aperceptions intuitives immddiatement 6videntes') ; c'est simplement une glose sur le mot 'inhaltlich', si usit6

J. V A N HEIJENOORT

16

par Hilbert, et nous sommes 12 sur un terrain commun 1 Hilbert et 1 Brouwer ; (b) La remarque vise des procgdds de calcul non formalisds et peut-dtre non formalisables, et elle laisse la porte ouverte b des m6thodes intuitionnistes, non-mgcaniques. La seconde interprgtation est celle qu'adopte Gsdel, qui voit dans la note de Herbrand un refus de se cantonner dans des rbgles formelles de calcul. Fais Herbrand n'avait-il pas Ccrit quelques mois p l u s t8t que les calculs devaient se faire 'par des opdrations dCcrites entizrement d'avance'? I1 n'avait pas don&

une

liste de ces opgrations, mais peut-on dire que la note 5 exclut categoriquement la possibilitg d'une telle liste ? Cet examen de l'intuitionnisme de Herbrand se complique encore du fait qu'il introduit lui-mdme une nouvelle distinction lorsqu'il parle de l'intuitionnisme 'dans sa forme extrdme'

(m, page

187, ou

1968,page

210).

Tout ce que dit

Herbrand de cette forme extrgme s'accorde trbs bien avec le finitisme de Hilbert. Le malheur est que Herbrand ne parle jamais d'une forme non extrdme de l'intuitiorr nisme, et la seule suggestion dans ce sens, c'est, peut-dtre, la note 5 , que nous avons dGjl discutge. (page 3 , ou

Dans

1968,page

225) Herbrand dnonce son thCorPme et ajoute :

'La dgmonstration et 1'6noncg de ce theorbme sont intuitionnistes'. La phrase se comprend trbs bien si l'on prencf 'intuitionniste' dans le sens de 'finitiste'. Elle prend cependant, dans le contexte des affirmations rGp6tges de Herbrand que son thgorbme supplLe 11'imprGcision du rdsultat ensembliste de Lswenheim, une rbsonance plus profonde. Herbrand remplace la notion de satisfaisabilitg d'une formule

?

dans un ensemble par celle-ci : pour tout nombre nature1 2, la 1-ibme ex-

pansion conjonctive de

x a la valeur logique v pour une certaine interprgtation,

c'est-l-dire pour certaines valeurs logiques attributes aux formules atomiques. Il-isme expansion conjonctive de Chaque interprgtation qui vgrifie la extension d'une interprgtation qui vdrifie la k-ibme expansion, pour

est une

& 0. The o n l y e x c e p t i o n t o t h e c o n j e c t u r e i s d < 2. T h i s negat i v e r e s u l t complements t h e c o n s t r u c t i o n by K r e i s e l and Daykin o f a d i s j u n c t i o n o f polynomi a1 nonnegat i v e l y - w e i g h t e d sum-of -squares r e p r e s e n t a t m f f,y means o f Herbrand's Theorem.

('id)).

Contents :

1. H i s t o r y o f L o g i c a l Aspects o f H i l b e r t ' s 17th Problem

(a) (b) (c) (d) 2. New 3. Two (a) (b) 4. The

Geometric O r i g i n o f t h e Problem Classical Solutions C o n s t r u c t i v iz a t i o n s and Bounds Continuous S o l u t i o n s P r o o f o f D a y k i n ' s D i s j u n c t i v e Sum-of-Squares R e p r e s e n t a t i o n L o g i c a l P o i n t s Concerning t h e F i n i t e n e s s Theorem ( 2 . 2 ) I n t u i t i o n i s t i c Considerations The Terminology " F i n i t e n e s s " Negative Answer t o K r e i s e l ' s F i r s t Q u e s t i o n

1. H i s t o r y o f L o g i c a l Aspects of H i l b e r t ' s 17th Problem (a) Geometric O r i q i n o f t h e Problem' I n h i s book [1899] on t h e f o u n d a t i o n s o f geometry, H i l b e r t showed t h a t those problems i n p l a n e g e o m e t r i c a l c o n s t r u c t i o n which can be s o l v e d by means o f o n l y h i s f i v e groups o f axioms, can always be c a r r i e d o u t by t h e use of s t r a i g h t e d g e and gauge (an example o f a gauge i s a compass whose use i s r e s t r i c t e d t o t h e l a y i n g o f f o f d i s t a n c e s on a s t r a i g h t l i n e ) . He gave two a l g e b r a i c c h a r a c t e r i z a t i o n s (Theorems 41 and 44) o f t h e s e t o f p o i n t s so cons t r u c t i b l e , i n terms o f t h e i r C a r t e s i a n c o o r d i n a t e s (fl ( x ) , f 2 ( x ) ) , where t h e g i v e n p o i n t s a r e expressed as r a t i o n a l f u n c t i o n s o f t h e parameters x = (xo. Research supported i n p a r t by NSF g r a n t No. MCS8102744. 1980 Mathematics S u b j e c t C l a s s i f i c a t i o n : P r i m a r y 03F55, Secondary 01A65, 14630.

10C04, 10C10;

P r e s t e l [1978] has a l s o g i v e n a synopsis o f t h e geometric o r i g i n o f t h e prob 1em.

C.N. DELZELL

88

...

, x n ) E R"'. The second o f h i s two c h a r a c t e r i z a t i o n s was a necessary and s u f f i c i e n t c o n d i t i o n , namely t h a t f i ( x ) be a t o t a l l y r e a l a l g e b r a i c number f o r a l l x E Qntl. ( I n t h e posthumous Seventh E d i t i o n o f h i s book [1971], a minor e r r o r i n h i s f o r m u l a t i o n o f t h i s c r i t e r i o n was c o r r e c t e d , i n Supplement I V . ) The p r o o f r e q u i r e d t h e f a c t t h a t (1.1)

i f a r a t i o n a l f u n c t i o n f E Q ( X ) , where X = (Xo, minates, i s p o s i t i v e s e m i d e f i n i t e ("psd"),2 t h e n squares o f r a t i o n a l f u n c t i o n s i n Q ( X ) .

...,Xn) f

are indeterequals a sum o f

We s h a l l a b b r e v i a t e "sum(s) o f squares" as "SOS." I n t h e f i r s t e d i t i o n [1899] o f t h e book, H i l b e r t gave t h e p r o o f o f 1.1 o n l y f o r n = 0, which was enough f o r some a p p l i c a t i o n s , such as showing t h a t those r e q u l a r polyqons c o n s t r u c t i b l e by means o f a compass and s t r a i q h t e d g e can a l s o be c o n s t r u c t e d u s i n g o n l y a qauqe and a s t r a i g h t e d q e (here n o t even 1 parameter i s i n v o l v e d ) . H i l b e r t l e f t t h e c a s e n > 0 as h i s 17th problem; i n l a t e r e d i t i o n s o f t h e book, e.g. [1971], H i l b e r t mentioned A r t i n ' s s o l u t i o n [1927] (see ( b ) below). There i s an e q u i v a l e n t f o r m u l a t i o n o f t h e problem i n terms o f homogeneous r a t i o n a l f u n c t i o n s ( i . e . q u o t i e n t s o f psd forms); a l s o we can c o n v e r t back and f o r t h between a q u o t i e n t o f SOS and a SOS o f q u o t i e n t s . " A t t h e same time, i t i s d e s i r a b l e . . t o know whether t h e c o e f f i c i e n t s o f t h e forms t o b e used i n t h e expression may always be t a k e n from t h e r e a l m o f r a t i o n a l i t y g i v e n by t h e form represented." [ H i l b e r t 19001 Thus, t o s o l v e t h e problem i n i t s f u l l g e n e r a l i t y , we must a l l o w t h e c o e f f i c i e n t s t o come from any ordered f i e l d K: K i s c a l l e d ordered once we have s p e c i f i e d an o r d e r i n q , i . e . a s e t PCK ( t h e " p o s i t i v e " elements), such t h a t P t P G P , P - P & % ( - P ) = K, and P n ( - P ) = { O } . K w i l l always denote an ordered f i e l d , and we s h a l l w r i t e Kt f o r P.

.

(b) C l a s s i c a l S o l u t i o n s Besides t h e case n = 0 mentioned above, one o t h e r case had a l r e a d y been solved, namely t h e well-known case o f q u a d r a t i c forms: g i v e n a psd form f ( X ) = xa. .X.X. ( 0 < i,j < n, aij = aji - E 1J 1 J

some pi, b . . E K w i t h pi > 0. For t h e s p e c i a l case where K = R, t h e prob'3 l e m had been s o l v e d f o r t h r e e a d d i t i o n a l c l a s s e s o f forms, n a n e l y ( 1 ) psd b i n a r y forms, which a r e SOS o f (two) forms, by t h e 2-square i d e n t i t y and t h e f a c t o r i z a t i o n o f b i n a r y forms over IR (see, e.g., [Landau 19031); ( 2 ) psd t e r n a r y q u a r t i c forms, which a r e SOS o f ( q u a d r a t i c ) forms ( [ H i l b e r t 18881; see a l s o [Choi and Lam 1977bl f o r a more e l a n e n t a r y p r o o f , u s i n g "extremal forms"); and ( 3 ) t e r n a r y forms, which a r e SOS o f ( f o u r , homogeneous) r a t i o n a l f u n c t i o n s [ H i l b e r t 18931. While f o r t h e b i n a r y , t h e q u a d r a t i c , and t h e t e r n a r y q u a r t i c forms, one c o u l d choose t h e square sunmands t o . be forms over R ! ( w i t h o u t u s i n g denominators), H i l b e r t had found exanples [1888] d-o ternary s e x t i c and q u a t e r n a r y q u a r t i c forms which were n o t SOS o f forms. Work o f E l l i s o n (1968 unpublished), M o t z k i n [1967, p. 2171, and R. M. Robinson [1973] t o w a r d s i m p l e r and/or more e x p l i c i t such examples, c u l m i n a t e d i n Choi and Lam's [1977(a), ( b ) ] c o n s t r u c t i o n o f t h e psd forms X 2 Y 4 + Y2Z4

1.e. f o r a l l

x ER"'

+

Z2X4 - 3X2Y2Z2

a t which

f

and

i s defined,

f ( x ) > 0.

Representing polynomials as sums of squares w4

+

X2Y2

+

Y2Z2

+

z2x2

-

4XYZW,

It which t h e y e a s i l y showed t o be n o t r e p r e s e n t a b l e as SOS o f ( r e a l ) forms. i s n o t known i f t h e r e s u l t ( 2 ) above on t e r n a r y q u a r t i c forms s t i l l h o l d s w i t h Q i n p l a c e o f R, b u t i t i s known t h a t t h e r e s u l t ( 1 ) on b i n a r y forms does (though more t h a n t w o sunmands a r e r e q u i r e d ) : an a l g o r i t h n o f Landau [1906] transforms t h e r a t i o n a l f u n c t i o n s g i v e n b y H i l b e r t f o r n = 0 i n t o polynomia l s , s t i l l w i t h r a t i o n a l c o e f f i c i e n t s ; i t o n l y remains t o homogenize these polynomials. However by then, Landau had a l r e a d y [1903] o b t a i n e d d i r e c t l y t h i s improvement o f t h e c l a s s i c a l r e s u l t on b i n a r y forms. I n passing from R t o Q, t h e nunber o f r e q u i r e d squares i n Landau's r e p r e s e n t a t i o n o f [1903] i n creased f r o m 2 t o 2d+2, where d i s t h e degree o f t h e form. In [1904] he lowered t h i s nunber o f squares t o 5 f o r q u a d r a t i c s ( s m a l l e s t p o s s i b l e ) and < 6 f o r q u a r t i c s ; Fleck [1906] lowered t h e 6 t o 5. Using t h e 8-square i d e n t i t y , Landau [1906] f i n a l l y proved t h a t , r e g a r d l e s s o f t h e degree, 8 squares a r e enough. Pourchet [1971] extended Landau's r e s u l t by r e p l a c i n g Q w i t h any a l g e b r a i c nunber f i e l d , and s i m u l t a n e o u s l y reduced t h e nunber o f r e q u i r e d squares t o 5, t h e s m a l l e s t p o s s i b l e .

The m a i n s t e p i n t h e h i s t o r y o f t h e 17th p r o b l e n was A r t i n ' s [1927] nonc o n s t r u c t i v e p r o o f o f 1.1, and n o t o n l y f o r t h e ground f i e l d Q, b u t even i f Q i s r e p l a c e d by any u n i q u e l y o r d e r a b l e s u b f i e l d K o f R (e.g. R o r t h e r e a l a l g e b r a i c nunbers); d r o p p i n g t h e unique o r d e r a b i l i t y hypothesis, A r t i n r e p r e s e n t e d psd ( r a t i o n a l ) f u n c t i o n s f as f = Zpiri2 where pi E K+ and ri E K ( X ) . A r t i n proved t h i s u s i n g h i s r e s u l t t h a t i n any f i e l d F o f chari s " t o t a l l y p o s i t i v e w i t h respect t o t h e ora c t e r i s t i c + 2, an e l e n e n t f d e r e d s u b f i e l d k 4 I 3 i f an o n l y i f f = Zpiri2, where pi E k+ and ri E F. Thus i t remained t o show t h a t a psd f u n c t i o n f E K(X) i s t o t a l l y p o s i t i v e i n K(X) w i t h r e s p e c t t o K. For t h i s he used a s e r i e s o f " s p e c i a l i z a t i o n lemmas" u s i n g Sturm's Theorem. I n s o l v i n g t h e problem, A r t i n had r e c o g n i z e d t h a t i t had more t o do w i t h t h e a l g e b r a i c t h a n t h e a r i t h n e t i c p r o p e r t i e s o f Q and R. S p e c i f i c a l l y , he was l e d t o i n t r o d u c e t h e axioms f o r a c l o s e d f i e l d R: ( i ) R i s a form a l l y r e a l f i e l d ( i .e. -1 i s n o t a SOS i n F ) T i m a l l a E R, e i t h e r a o r -a i s a square i n R, and ( i i i ) any odd degree p o l y n o m i a l i n R[T] has a root i n R. For f u t u r e r e f e r e n c e , we m e n t i o n t h e f a c t t h a t e v e r y ordered f i e l d i s c o n t a i n e d i n a(n e s s e n t i a l l y unique) s m a l l e s t r e a l c l o s e d f i e l d c a l l e d i t s r e a l c l o s u r e . T h i s a x i o m a t i z a t i o n n o t o n l y l e d t o g r e a t e r generali t y , b u t i t t r m a d e t h e problem e a s i e r ; t h u s h i s s o l u t i o n was perhaps t h e f i r s t s p e c t a c u l a r use o f t h e a x i o m a t i c method f o r mathematical as opposed t o metamathematical purposes, such as independence r e s u l t s . A more r e c e n t e f f o r t t o g e n e r a l i z e A r t i n ' s theorem focused on h i s hypot h e s i s t h a t KGR, which i s e s s e n t i a l l y t h e h y p o t h e s i s t h a t K be Archimedean: an o r d e r e d f i e l d K i s Archimedean over t h e s u b f i e l d k i f v c E K 3d E k such t h a t c < d; i f k i s Q, we o m i t " o v e r k." It was well-known t h a t t h e Archimedean h y p o t h e s i s played a r o l e i n h i s o r i g i n a l f o r m u l a t i o n o f Q(X) h i s theoren. For e ~ a n p l e , over ~ t h e non-Archimedean ordered f i e l d ( w h e r e X - l > Q, i . e t h e i n d e t e r m i n a t e X i s i n f i n i t e s i m a l 1 small compared t o 9. and p o s i t i v e ) , t h e ( q u a r t i c ) polynomial f ( Y ) = (V2-X){ - X3 E Q ( X ) [ Y ] (Y an i n d e t e r m i n a t e ) i s psd over Q(X) b u t n o t a SOS even i n R(Y) (where R i s t h e r e a l closure o f Q(X)). Indeed, upon f a c t o r i n g over R, we see t h a t 1 . e . . nonnegative i n e v e r y o r d e r i n g o f F e x t e n d i n g t h e o r d e r on k; o f course, F need n o t have any o r d e r i n g . Wen k = Q, we say s i m p l y " t o t a l ly positive " P. 99 o f [ A r t i n and S c h r e i e r 19271.

89

C.N. DELZELL

90 f


9) i . e . t h e s m a l l e s t e x t e n s i o n c l o s e d under e x t r a c t i o n o f square r o o t s o f p o s i t i v e e l a n e n t s . Then F, b e i n Euclidean, has a u n i q u e o r d e r , r e l a t i v e t o which Dubois showed f ( X ) = ( X 3 - t ) % - t 3 E F[X] t o be ( s t r i c t l y ) d e f i n i t e ; on t h e o t h e r hand, f(t1I3) < 0, so f cannot be a SOS i n F(X). ( c ) C o n s t r u c t i v i z a t i o n s and Bounds A r t i n wondered i f a c o n s t r u c t i v e v e r s i o n o f h i s s o l u t i o n c o u l d be given, and he considered t h i s q u e s t i o n i n a seminar which he l e d between t h e wars. I n p a r t i c u l a r , he wished t o e l i m i n a t e h i s appeal t o an i n f i n i t e tower o f f i e l d extensions, and he d e s i r e d a bound on t h e nunber and degree o f sunmands i n h i s representation. H a b i c h t [1940] gave an elementary, e x p l i c i t c o n s t r u c t i o n o f a SOS-repres e n t a t i o n o f forms f s t r i c t l y d e f i n i t e o v e r R. In f a c t , t h e denominator he g i v e s i s (X; + . . . + X;)m, some m L N, and t h e n u n e r a t o r c o n t a i n s o n l y r a t i o n a l c o e f f i c i e n t s i f f does. He d e r i v e d h i s r e p r e s e n t a t i o n by combining w i t h a theot h e " R a b i n o w i t c h t r i c k " ( i . e . adding a new i n d e t e r m i n a t e rem o f P b l y a o n t h e r e p r e s e n t a t i o n o f forms which a r e p o s i t i v e when a l l X i > 0 (except when a l l Xi = O ) . 6 H a b i c h t ' s a l g o r i t h n i s f u l l y c o n s t r u c t i v e : i t can e a s i l y be made t o produce a r e p r e s e n t a t i o n c o r r e c t t o any d e s i r e d accuracy i n an e s t i m a b l e m o u n t o f t i m e .

%+I)

A. Robinson used lower p r e d i c a t e c a l c u l u s and t h e model completeness o f t o prove a nunber o f o v e r l a p p i n g r e s u l t s . To d e s c r i b e them, we f i r s t i n t r o d u c e t h e f o l l o w i n g n o t a t i o n . Let t h e o r d e r e d f i e l d K be c o n t a i n e d i n t h e r e a l c l o s e d o r d e r - e x t e n s i o n f i e l d R. Then f o r any f i n i t e subset {gi)cK[X], write7

(R

>

UIgi 1 = I x

E

Rn+'l

Ai

gi(x)

WIgi 1 = I x

E

R"'l

I\i

g i ( x ) > 01.

01, and

set,

or We s h a l l c a l l a s e t o f t h e form WIgi 1 a b a s i c closed semi-algebraic s i m p l y a "W," and s i m i l a r l y w i t h "U" and "open" i n p l a c e o f "W" and "closed." We s h a l l c a l l S GR"' a semi-algebraic i f it i s the intersection o f a U and a W. A s e t ScRn+' i s c a l l e d s%-algekrajc ( s f o r~ s h o r t ) us a a s i c s.a. s e t i s one i f i t i s a f i n i t e u n i o n o f b a s i c s.a. s e t s . w h i c h can be d e f i n e d by an elementary f o r m u l a o f t h e language o f ordered

set

~~

~~

~

~

A. Robinson [1955] gave a s i m i l a r example. See t h e second e d i t i o n [1952] o f [Hardy, f o r an e n j o y a b l e m s h v e r s i o n o f b o t h r e s u l t s .

fli dexed by

[resp. i.

Vi]

Littlewood,

and Pdlya 19341

means i t e r a t e d c o n j u n c t i o n [ r e s p . d i s j u n c t i o n ] ,

in-

Representing polynomials as sums of squares

91

f i e l d s , w i t h n + 1 f r e e v a r i a b l e s , which has no q u a n t i f i e r s , negations, o r d i s j u n c t i o n s , w h i l e an a r b i t r a r y s.a. s e t i s one which can be d e f i n e d by any elementary ( q u a n t i f i e r - f r e e 8 ) f o r m u l a o f t h e language o f o r d e r e d f i e l d s , w i t h n + 1 f r e e v a r i a b l e s . See [ B r u m f i e l 19793 f o r an e x t e n s i v e development of semi-algebraic geometry. Robinson's f i r s t r e s u l t [1955] i s t h a t i f K i s e i t h e r r e a l c l o s e d o r Archimedean, t h e n i f f ( x ) 0 V x E Kn+'nUIgi1 (where { f , g i l c K I X ] ) then f = ZcIgIrf, where cI E K+, where t h e g I a r e ( n o t n e c e s s a r i l y d i s t i n c t ) p r o d u c t s o f t h e gi , and I r I } c K ( X ) ; f u r t h e r , i f t h e o r d e r i n g on K i s u n i q u e . t h e n t h e cI a r e t o t a l l y p o s i t i v e , hence SOS i n K, so t h a t t h e CI may be a b s o r b e d i n t o t h e t h e rI; b e t t e r s t i l l , f o r K r e a l closed, he proved t h e e x i s t e n c e o f a bound on t h e nunber and degrees o f t h e sunmands which depends on I S i } and deg f b u t n o t on t h e c o e f f i c i e n t s o f f ( o r o f course, on R). In [1956] Robinson extended t h e r e a l c l o s e d case as follows: i f V f 0 i s an i r r e d u c i b l e a l g e b r a i c v a r i e t y i n Rn+' w i t h prime i d e a l P, I f , g i lc R [ X ] , and f ( x ) > 0 Vx E V n U I g i } , t h e n h 2 f = ZgIhf (mod P) f o r we s t i l l have a some { h , h I } c R I X ] , where t h e gI a r e p r o d u c t s o f t h e gi; bound on t h e number and d e g r e e s o f t h e {h,hI}. w h i c h depends o n l y on t h e Igi l and deg f, n o t on t h e c o e f f i c i e n t s o f f.

In October 1955 A r t i n asked K r e i s e l i f e x p l i c i t bounds c o u l d be found. I n Nov. 1955, somewhat b e f o r e t h e appearance o f Robinson's r e s u l t , K r e i s e l

succeeded i n o b t a i n i n g , by two p r o o f t h e o r e t i c methods, a r i m i t i v e r e c u r s i v e bound (Robinson's was o n l y general r e c u r s i v e ) . The f i r s t i m 1 9 5 7 1 , pp. 165-6 o f [1958], and [1960]) used p r o o f t h e o r e t i c a l r e s u l t s : H i l b e r t ' s f i r s t and second c-Theorems ( o r Herbrand's Theorem). The second method [1960] cons i s t e d o f e x t r a c t i n g t h e c o n s t r u c t i v e c o n t e n t o f A r t i n ' s o r i g i n a l argument, by r e p l a c i n g A r t i n ' s use o f a r e a l c l o s e d e x t e n s i o n o f an ordered f i e l d w i t h a s p e c i f i c f i n i t e e x t e n s i o n s u f f i c i e n t f o r t h e r e s u l t ; i n t h i s replacement some elegance and c l a r i t y i s l o s t , b u t some e x p l i c i t n e s s i s gained; h e r e t h e ideas b u t no theorem o f p r o o f t h e o r y f o r f i r s t o r d e r l o g i c a r e used. In [1957(1)] K r e i s e l gave a rough e s t i m a t e ( f o r n = 2) o f t h i s p r i m i t i v e r e s u r s i v e bound. A sharper e s t i m a t e i s

.ZCd

where t h e r e a r e stant.

n

2's,

where

22 , d = deg f, and where

c

i s a p o s i t i v e con-

S t i m u l a t e d by t h e s e r e s u l t s , Henkin [1960] used model t h e o r e t i c methods s i m i l a r t o Robinson's t o p r o v e what i s now accepted as t h e most n a t u r a l formu-, l a t i o n o f t h e answer t o H i l b e r t ' s question: i f f E K[X] i s psd (over R) and i f deg f < d, t h e n f = Zcirf, where ri E K(X) and c E K+ ( A r t i n had o b t a i n e d t h i s r e p r e s e n t a t i o n under t h e h y p o t h e s i s t h a t KGR and t h a t f be psd over K; f o r K E R , psd over K i s e q u i v a l e n t t o psd o v e r R and t o psd over R, s i n c e K i s t h e n dense i n R ) . Henkin a l s o showed t h a t t h e (bounded number o f ) ci and t h e (bounded nunber o f ) c o e f f i c i e n t s o f t h e ri can be t a k e n t o be f u n c t i o n s o f t h e c o e f f i c i e n t s o f f which a r e " p i e c e w i s e - r a t i o -

*

I n c l a s s i c a l mathematics we need n o t e x c l u d e q u a n t i f i e r s , s i n c e t h e y can be e l i m i n a t e d i f necessary by t h e Tarski-Seidenberg Theorem. In intuit i o n i s t i c mathematics, however, q u a n t i f i e r - e l i m i n a t i o n i s n o t g e n e r a l l y v a l i d , u n l e s s b o t h K and R a r e r e c u r s i v e (see f o o t n o t e 9), so h e r e we do exclude quant if ie r s.

C.N. DELZELL

92

('id),

n a l " over 2 , where t h e f i n i t e l y many "pieces" are s.a. subsets o f R the space o f c o e f f i c i e n t s o f f; t h e c o e f f i c i e n t s o f these r a t i o n a l f u n c t i o n s and t h e polynomials d e f i n i n g t h e f r domains are r e c u r s i v e b u t n o t n e c e s s a r i l y primi t i v e r e c u r s i v e f u n c t i o n s o f n and d. L . van den D r i e s [1977] g e n e r a l i z e d H e n k i n ' s r e s u l t s i n a c e r t a i n d i r e c t i o n , t o p o l y n o m i a l s which are "psd over good preordered r e g u l a r r i n g s ; " case d i s t i n c t i o n s were f o r m a l l y avoided, b u t a t t h e c o s t o f an a r t i f i c i a l d e f i n i t i o n o f r a t i o n a l f u n c t i o n . Robinson gave a c o r r e s p o n d i n g l y improved f o r m u l a t i o n o f h i s r e s u l t s . In f,g E K r X ] , t h a t i f f ( x ) > 0 Vx E Z{g}, t h e n h 2 f = zcih! + kg f o r some {h,hi,k}cK[X], where ci E K+; t h i s t i m e t h e bound i s o n t h e number and d e g r e e s o f h, k, and t h e hi, and i t depends on deg f and deg g, b u t n o t on K o r t h e c o e f f i c i e n t s o f f and g. I n 68.5 o f [ 1 9 6 3 ] he r e p l a c e d Z{g} above w i t h Z { g } n U { g i } (any { g i } c K [ X ] ) provided t h a t g generates t h e i d e a l o f Z{g} and t h a t g k g i ; t h e c o n c l u s i o n t h e n i s h 2 f = zIcIgIhl + kg, where t h e gI a r e products o f t h e g i . The bound no l o n g e r a p p l i e s t o deg k, and now t h e bound depends a l s o on t h e degree o f t h e

95 o f [1957] he proved f o r

Si

.

I t i s no a c c i d e n t t h a t i n t h e l o g i c a l t r e a t m e n t s o f t h e 17th problem, t h e Archimedean p r o p e r t y was rep1 aced by t h e c o n d i t i o n t h a t t h e g i v e n polynomial b e psd over t h e r e a l c l o s u r e o f t h e ordered f i e l d o f c o e f f i c i e n t s , because t h e Archimedean p r o p e r t y cannot be expressed by an elementary statement. Since Archimedean ordered f i e l d s are isomorphic t o s u b f i e l d s o f R, and a r e t h e r e f o r e dense i n t h e i r r e a l c l o s u r e s , "psd" over an Archimedean ordered f i e l d a l ready i m p l i e s "psd" over i t s r e a l c l o s u r e .

Robinson f u r t h e r proved [1957] t h a t i f p i s t o t a l l y p o s i t i v e i n a f i n i t e , f o r m a l l y r e a l e x t e n s i o n F o f K, t h e n p = z;=1 c i r ? , where c i E K+ and ri E F; what was new was t h a t r depends o n l y on [F:K], n o t on F, K, o r 9. Thus i f a l l t h e p o s i t i v e elements o f K are SOS, and i f t h e number o f r e q u i r e d squares i s bounded, then we may absorb t h e ci i n t o t h e ri i n t h e above r e p r e s e n t a t i o n , b u t make r dependent a l s o on t h i s bound; t h i s o v e r l a p s an i m p o r t a n t theorem s t a t e d by H i l b e r t ( f i r s t proved by Siege1 [1921]) t h a t i f K = I), then r = 4, independent even o f [F:K]. Here we have an i n t e r e s t i n g historical twist: w h i l e work on H i l b e r t ' s 17th problem l e d t o a r e s u l t much l i k e S i e g e l ' s theorem, S i e g e l ' s theorem helped l a y t h e f o u n d a t i o n f o r t h e 17th problem; indeed, one o f t h e f i r s t uses o f S i e g e l ' s theorem (even b e f o r e anyone had p u b l i s h e d a p r o o f ! ) was by H i l b e r t i n h i s s o l u t i o n [1899] o f t h e case n = 0 o f t h e problem. Daykin [1960] c o n s t r u c t e d a p r i m i t i v e r e c u r s i v e , p i e c e w i s e - r a t i o n a l sol u t i o n which was s u p e r i o r t o t h e Henkin-Robinson s o l u t i o n s , by working o u t K r e i s e l ' s [1960] sketch o f t h e c o n s t r u c t i v i z a t i o n o f A r t i n ' s o r i g i n a l p r o o f . A l i t t l e more n o t a t i o n a t t h i s p o i n t w i l l h e l p us d e s c r i b e D a y k i n ' s represent a t i o n (and e v e n t u a l l y o t h e r s as w e l l ) . a n ) E Nn+' be a Let a = (ao, m u l t i - i n d e x , l e t la1 = z a i , f i x an even d E #, l e t C = (Ca),al=d be i n d e t e r m i n a t e s ( i n some f i x e d o r d e r ) , l e t c = ( c a ) l a l = d be an element o f

...,

R("", ficients

l e t f L 2[C;X] C (i.e. f(C;X) Pnd = { c

E

R(

('id)

be t h e general form o f degree d i n X w i t h c o e f = z la C a p , where = X ~ o " ' X ~ n ) , and l e t

x*

n+d n lf(c;X)

i s psd ( o v e r

R) i n

XI.

Representing polynomials as sums of squares

93

Daykin showed how t o compute e f f e c t i v e l y , f r o m n and d alone, f i n i t e l y many pij E Z[C] and rij E Q(C;X) (homogeneous i n X) such t h a t

Ai (1.3)

vc

E

pnd.

vi

I\ j

f = Z. p . . r z .

J

[

pij(c)

1J 1J

and

> 0, and t h e denominator o f

does n o t v a n i s h i d e n t i c a l l y i n

rij(c;X)

X.

1

Thus, as i n most a p p l i c a t i o n s o f t k r b r a n d ' s theorem, t h e answer i s expressed as a d i s j u n c t i o n . The s u p e r i o r i t y o f t h i s r e p r e s e n t a t i o n c o n s i s t s n o t o n l y i n t h e e x p l i c i t n e s s o f t h e bound b u t a l s o i n t h e c h o i c e o f p i e c e s on which t h e r a t i o n a l f u n c t i o n s a r e d e f i n e d : t h e e a r l i e r p i e c e s were s.a., b u t Daykin's a r e b a s i c c l o s e d s.a., nanely, Wi = WIpijl. D a y k i n ' s p r o o f was l o n g and d i f f i c u l t ; i n $2 we g i v e a q u i c k p r o o f and r e f i n e m e n t o f h i s r e p r e s e n t a t i o n , u s i n g powerful r e s u l t s i n s.a. geometry. The m a i n r e s u l t i n t h e s i x t i e s was P f i s t e r ' s e l e g a n t "2n bound" [1967] on t h e number o f square sunmands r e q u i r e d t o r e p r e s e n t a homogeneous, psd f E R(X), where R i s r e a l closed. The bound i s independent o f deg f . ( H i l b e r t had proved t h i s f o r n = 2 i n [1893].) More p r e c i s e l y , P f i s t e r has shown if [1974]: 2% f = I f f i=l with

fi

E

R(X)

homogeneous o f degree

d, t h e n t h e r e i s a r e p r e s e n t a t i o n

2n f = l g f i=l

w i t h gi E R(X) depends o n l y on q u i c k l y w i t h n.

nm-l homogeneous o f d e g r e e < C(n)"-I d"m: t h e . c o n s t a n t C(n) n, and c o u l d be determined e x p l i c i t l y ; i t p r o b a b l y grows

P f i s t e r ' s p r o o f uses (1) a s p e c i a l case o f t h e Tsen-Lang Theorem: i f C i s an a l g e b r a i c a l l y c l o s e d f i e l d and F i s a f i e l d o f transcendence degree n o v e r C, t h e n e v e r y q u a d r a t i c f o r m w i t h c o e f f i c i e n t s i n F, o f dimension > Zn, h a s a n o n - t r i v i a l z e r o i n F: and ( 2 ) h i s theorem t h a t t h e non-zero e l e ments o f a f i e l d F o f c h a r a c t e r i s t i c # 2 r e p r e s e n t e d by (what i s now c a l l e d ) a " P f i s t e r form," form a subgroup o f F*. (An independent, unpublished s t u d y b y Ax i n 1966, showed t h a t 8 squares s u f f i c e when n = 3.) It i s n o t known whether P f i s t e r ' s bound a p p l i e s i n t h e case o f o r d e r e d c o e f f i c i e n t f i e l d s IC, i n particular Q; again we should a l l o w p o s i t i v e c o n s t a n t w i g h t s on t h e squares i n t h e o r d e r e d f i e l d case, s i n c e p o s i t i v e elements o f K need n o t be sums o f (even an unbounded nunber o f ) squares. F o r r e a l c l o s e d f i e l d s i t i s n o t known whether 2n i s b e s t p o s s i b l e , exc e p t f o r n < 2; Cassels, E l l i s o n , and P f i s t e r [1971] showed t h a t t h e (psd) M o t z k i n p o l y n o m i a l 1 + X 2 Y 4 + X 4 Y 2 - 3X2Y2 i s n o t a sun o f t h r e e squares i n R(X,Y): b u t t h e i r method uses t h e t h e o r y o f e l l i p t i c curves, and does n o t ext e n d t o n > 2. The o n l y known lower bound i s n + 1: Cassels [1964] showed t h a t 1 + Xf + * . ' + , ! X, i s n o t a sun o f n squares i n R(X1,.. ,Xn), by sharpening Landau's [1906] r e s u l t t h a t a SOS o f r a t i o n a l f u n c t i o n s can be r e d u c e d t o a SOS o f r a t i o n a l f u n c t i o n s i n which any one v a r i a b l e , say X,I does n o t occur i n t h e (common) denominator; Cassels d i d t h i s w i t h o u t i n c r e a s i n g t h e H s i a and Johnson [1974] have c o n j e c t u r e d t h a t a homogenunber o f sunmands.

C.N. DELZELL

94

neous, p s d f t Q ( X ) must be a sum of 2 n + 3 squares i n Q ( X ) ( t h i s i s Lagrange's Theorem (1770) f o r n = 0 and Pourchet's Theorem [1971] f o r n = 1 , b u t i t i s not known whether any bound, independent of degree, e x i s t s f o r n

> 1).

( d ) Continuous Solutions

The improvements found in t h e f i f t i e s t o A r t i n ' s s o l u t i o n brought only temporary s a t i s f a c t i o n , and by t h e e a r l y s i x t i e s Kreisel wondered i f one could n o t do b e t t e r . In p a r t i c u l a r , t h e piecewise c h a r a c t e r of t h e r e p r e s e n t a t i o n s meant t h a t when computing a r e p r e s e n t a t i o n from given c o e f f i c i e n t s of f , one f i r s t had t o d e t e r m i n e i n which p i e c e of Pnd t h e c o e f f i c i e n t s l a y . This amounts t o t e s t i n g various polynomial i n e q u a l i t i e s in t h e c o e f f i c i e n t s . For r e c u r s i v e ordered f i e l d s g such as Q or t h e r e a l a l g e b r a i c nunbers, the t e s t i s e f f e c t i v e . B u t such t e s t i n g i s p r e c i s e l y what we cannot do i n , say, R, an element of which m u s t be presented a s , s a y , a decimal, o r an o s c i l l a t i n g decimal used i n computer science, o r a p a i r ( ( r n ) , u ) of some kind of Cauchy sequence of r a t i o n a l s and a "modulus of convergence" function u s a t i s f y i n g V k > 0, Vn,m Y(K) [lrn-rml< l / k ] . Thus i t i s a t t h e d i s c o n t i n u i t i e s t h a t we a r e unable t o compute t h e r e p r e s e n t a t i o n . While c o n t i n u i t y i s not required by c l a s s i c a l a l g e b r a i s t s (who i m p l i c i t l y use t h e d i s c r e t e topology when doing a l gebra, a s i f r e a l nunbers were presented with i n f i n i t e p r e c i s i o n ) , t h e lack of i t i s enough o f a problem t o leave H i l b e r t ' s 17th problem s t i l l unsolved from a c o n s t r u c t i v i s t , o r , f o r t h a t m a t t e r , a t o p o l o g i c a l , point of view. Thus, by t h e e a r l y s i x t i e s , t h e following two questions were open. F i r s t Question: Are the case d i s t i n c t i o n s o f 1.2 and 1.3 unnecessary? such t h a t f = Cpjr$ and I . e . , do t h e r e e x i s t pj E Q ( C ) and r j E Q ( C ; X ) Vc E P n d , each p j ( c ) > O?

Second Question: Can a topological or "continuous" version of A r t i n ' s theorem be given? This question has two p a r t s : ( a ) can we choose represen(where by a "continuous t i n g r a t i o n a l f u n c t i o n s which a r e continuous in Rn+' r a t i o n a l function" we mean a continuously-extendible r a t i o n a l function; e . g . , t h e represent a t ion 2 2 1 = X + Y x2

+

Y2

x2

+

Y2

i s discontinuous a t the o r i g i n in R2), and ( b ) can t h e weights, and t h e coeff i c i e n t s of t h e nunerator and denominator i n each r a t i o n a l function, be chosen t o be continuous f u n c t i o n s of t h e given c o e f f i c i e n t s ( i n P n d ) ? When we r e f e r t o c o n t i n u i t y , we a r e mainly i n t e r e s t e d i n t h e usual i n t e r v a l topology on K o r R; when K = R = R, we should a l s o consider various "computational" topolo g i e s on "enrichments" of R by s p e c i f i c r e p r e s e n t a t i o n s , say Cauchy sequences of r a t i o n a l s with the topology i n h e r i t e d from the product topology on Qw. These questions appeared in p r i n t , e . g . , on p. 102 of Kreisel [1969], on p p . 115-6 of [1977a], in footnote 1 o f [1977b], and in [1978]. In view o f t h e g e o m e t r i c o r i g i n of H i l b e r t ' s 17th problem, s t r e s s e d in h i s own presentation, i t seems natural enough t o impose topological condit i o n s . K r e i s e l ' s interest i s l o g i c a l : To determine the e x t e n t t o which curr e n t mathematical notions express adequately o r b e t t e r t h e aims usually s t a t e d i n t e r n s of so-called c o n s t r u c t i v e , in p a r t i c u l a r , of i n t u i t i o n i s t i c foundations. By a recursive ordered f i e l d we mean a nunbering of i t s underlying s e t f o r which t h e f i e l d operations and t h e order (hence a l s o e q u a l i t y ) r e l a t i o n a r e recursive.

Representing polynomials as sums of squares

95

The m i n r e s u l t o f t h i s paper i s a simple, geometric proof (4.1) o f the The proof negative answer t o K r e i s e l ' s f i r s t question, except when d < 2. amounts t o showing (4.2) t h a t Pnd i s n o t a basic s.a. set, except when d < 2, when i t i s a basic closed s.a. set. When d = 2, not only i s the answer a f f i r m a t i v e , b u t the rj can be found i n Q ( C ) [ X ] , t h a t i s , they can be chosen t o be l i n e a r forms i n X (see also next page). The case d > 2 reduces t o the case n = 1, d = 4, and i t i s i n t e r e s t i n g t o compare what 4.2 says about P1,4 t o what c l a s s i c a l algebra t e x t s say about it. The standard approach i s t o reduce t o monic polynomials w i t h no X 3 term: a(X) = X 4 + qX2 t r X + s. Write L = 8qs 2q3 - 9 r 2 and l e t D be the d i s c r i m i n a n t o f a, namely

-

D = 4(4s + ") 3 Write

P;,4cR3

2 3

-

27(3qs 3

-

r2- 2 q 3 )

f o r the set o f (coefficients

q u a r t i c polynomials -8

a

2

27

.

(q,r,s)

o f ) reduced, monic

w i t h no r e a l o r n u l t i p l e roots.

By Sturm's theorem,

P1,4 = ( U { D } O W { q } ) U (U{D}nW{-L})

( c f . [Jacobson 19743, pp. 299-300).

The

c l o s u r e i n R 3 o f 7i,4 equals the s e t P;,4 o f reduced, monic q u a r t i c psd polynomials. Thus we get a d e s c r i p t i o n o f P1,4 i n terms o f basic s.a. sets. 4.2 now says t h a t t h i s d e s c r i p t i o n i s simplest possible, i n the sense t h a t no s i n g l e basic s.a. set w i l l s u f f i c e .

two

For p a r t (a) o f the second question, c o n t i n u i t y i n the variables o f the r a t i o n a l functions, K r e i s e l had found [19781 a simple proof o f the a f f i r m a t i v e answer, u s i n g S t e n g l e ' s [19741 " P o s i t i v s t e l l e n s a t z : " f o r tf,gi l c K [ X I , if Vx E W{gi 1, f ( x ) 0, then (1.4) some s E N, pjI E K+, and hjI E KLX] (j = 1,2), where the gI are pro( I n [19791, Stengle arranged f o r the hjI t o be homoged u c t s o f t h e gi neous i f f i s . ) I f we want t o transform t h i s i n t o a (nonnegatively-weighted) SDS o f r a t i o n a l functions, we j u s t m r l t i p l y the numerator and denominator o f 1.4 by t h e denominator. S i n c e WIgi In Z(denominator)_cZ{f)cRn+l, the Squeeze Theorem e a s i l y i m p l i e s t h a t each o f the r e s u l t i n g r a t i o n a l functions (and n o t merely the sum o f t h e i r squares, o f course) extends--by 0--to a funct i o n o f X which i s continuous throughout WIgi).

.

I n forthcoming papers we s h a l l prove the a f f i r m a t i v e answer t o both p a r t s o f K r e i s e l ' s second question, a t l e a s t when the ground f i e l d K i s real closed [Delzell, i n preparation (a)], and when i t i s a countable s u b f i e l d o f R [ D e l z e l l , i n preparation (b)]. Precisely, we c o n s t r u c t a f i n i t e set o f f u n c t i o n s pi : Pnd + R+ and ri: Pnd x RW1 + R t a k i n g p o i n t s w i t h coordin a t e s i n K t o p o i n t s w i t h coordinates i n K, s a t i s f y i n g , V c c Pnd, f ( c ; x ) a f u n c t i o n which i s continuous, = Zpi ( c ) r i ( c ; x ) 2, w i t h each summand p i r f r e l a t i v e t o t h e usual i n t e r v a l topology on R, hence a l s o r e l a t i v e t o the comp u t a t i o n a l topologies i n case R = R, (and "s.a." i n t h e case when K i s real c l o s e d ) s i m u l t a n e o u s l y i n c and x f o r (c;x) E Pnd x ~ " 1 , and w i t h ri homogeneous and r a t i o n a l i n X. For t h e case R = R, t h i s l a s t sentence provides the f i r s t constructive, i n p a r t i c u l a r , i n t u i t i o n i s t i c , s o l u t i o n t o

C.N. DELZELL

96

H i l b e r t ' s 17th p r o b l e m , s i n c e ( 1 ) t h e s.a. d e s c r i p t i o n s o f t h e pi and ri ( a n d a l s o o f t h e X - c o e f f i c i e n t s o f t h e r i ) are r e c u r s i v e i n n and d, and ( 2 ) w h i l e elements o f IR can be g i v e n o n l y as approximations, we can a p r o x i mate pi and ri b y approximating c, by c o n t i n u i t y . I n [1980] and [ t o appear], we a l s o g i v e c o n t i n u i t y r e s u l t s , p o s i t i v e and n e g a t i v e , f o r r e p r e s e n t i n g psd q u a d r a t i c and psd t e r n a r y q u a r t i c forms as SOS o f forms: t h e former can be r e p r e s e n t e d over K by continuous summands, a t l e a n t h e number o f summands i s increased g r e a t l y beyond n + l , w h i l e if t h e l a t t e r are represented as SOS o f q u a d r a t i c forms (even over R), c e r t a i n c o e f 2 V)2. f i c i e n t s must jump a t ( ~ + Our h i s t o r y has d e a l t m a i n l y w i t h t h e l o g i c a l aspects o f H i l b e r t ' s o r i g i n a l problem; we have t r i e d t o i n c l u d e e s p e c i a l l y t h o s e r e s u l t s which appear t o be l e s s well-known. However, a l a r g e l i t e r a t u r e on o t h e r aspects o f t h e problem has developed, and we now mention enough r e f e r e n c e s t o guide t h e i n t e r e s t e d reader i n t o these areas. Bochnak and Efroymson [1980] cover t h e c u r r e n t k n o w l e d g e o f SOS o f 6" f u n c t i o n s , Nash f u n c t i o n s ( r e a l a l g e b r a i c a n a l y t i c f u n c t i o n s ) , and r e a l a n a l y t i c f u n c t i o n s . They g e n e r a l i z e both S t e n g l e ' s and P f i s t e r ' s Theorems t o c e r t a i n s u b r i n g s o f t h e r i n g o f Nash f u n c t i o n s on open s . a . s u b s e t s o f i r r e d u c i b l e n o n s i n g u l a r a l g e b r a i c s e t s i n Rn. They c o n s i d e r b o t h g l o b a l f u n c t i o n s and germs o f f u n c t i o n s . They s i m i l a r l y g e n e r a l i z e P r o c e s i ' s [1978] r e p r e s e n t a t i o n s o f s y n m e t r i c psd f u n c t i o n s t o Nash f u n c t i o n s i n v a r i a n t under a L i e group a c t i o n on Rn. (The s t u d y o f SOS o f r e a l a l g e b r a i c f u n c t i o n s was i n i t i a t e d by A r t i n [1927].) Lam [1980] g i v e s a b i b l i o g r a p h y on t h e 17th problem, i n c l u d i n g references t o a non-commutative g e n e r a l i z a t i o n o f t h e problem, a p - a d i c analog, and a g e n e r a l i z a t i o n t o psd symmetric m a t r i c e s over polynomial r i n g s . Pfister [1976] a l s o g i v e s h i s t o r i c a l r e f e r e n c e s . We s h a l l n o t t r y t o d u p l i c a t e here t h e s e t h r e e main b i b l i o g r a p h i e s , b u t i n s t e a d conclude w i t h some r e f e r e n c e s n o t i n c l u d e d i n these b i b l i o g r a p h i e s . Berg, Christenson, and Ressel [1976] s t u d i e d p o s i t i v e d e f i n i t e f u n c t i o n s on A b e l i a n semigroups, and approximated d e f i n i t e p o l y n o m i a l s by SOS o f polynom i a l s , o f n o t n e c e s s a r i l y bounded degrees. Bose [1976] gave a l g o r i t h m s t o t e s t p o l y n o m i a l s f o r psd-ness. E l l i s o n [1969] considered a "Waring's problem" f o r forms. Oickmann [1980] c h a r a c t e r i z e d d e f i n i t e p o l y n o m i a l s over " r e a l closed r i n g s . "

2. NEW PROOF OF DAYKIN'S DISJUNCTIVE SUM-OF-SQUARES REPRESENTATION As i n 51, l e t

si

E

f

E

Z[C;X]'

be t h e general form i n

Theorem 2.1: There e x i s t f i n i t e l y many P4 ( k = 1,2) such t h a t

giJ

E

X.

a[c],

hkiJ

E

Q[C;Xl,

(2.1.1)

2) ( 2 .l. Remark: 2.1 r e f i n e s Oaykin's r e p r e s e n t a t i o n (1.2), s i n c e we can t r a n s f o r m 2.1.1 i n t o a nonnegatively-weighted SOS of r a t i o n a l f u n c t i o n s by m u l t i -

Representing polynomials as sums of squares

97

p l y i n g t h e n u n e r a t o r and denominator o f 2.1.1 by t h e denominator. The r e s u l t i n g r a t i o n a l f u n c t i o n s need n o t be homogeneous i n X, b u t s i n c e f i s , we can e x t r a c t t h e l o w e s t homogeneous components from t h e n u n e r a t o r and denominat o r and s t i l l have an i d e n t i t y , though i t s denominator may n o t have t h e spec i a l s t r u c t u r e shown i n 2.1.1. The p r o o f o f 2.1 w i l l use t h e P o s i t i v s t e l l e n s a t z (1.4)

Theorem 2.2 i s s.a., -~

GR"+~

(a)

S

(The F i n i t e n e s s Theorem,

[Delzell

and

1980 and 19813):

S

then

i s open i f and o n l y i f

=ui U{gijl.

some f i n i t e s e t

{gij1cK[X];

S = Wi Wigij),

some f i n i t e s e t

{gijIcK[X].

S

equivalently, (b)

If

S

i s closed i f and o n l y i f

K recursive, a s.a. s e t .

the

gij

a r e computable from t h e p r e s e n t a t i o n

Df

S 5

The " i f " d i r e c t i o n s a r e t r i v i a l . The e q u i v a l e n c e o f (a) and (b) f o l l o w s by t a k i n g conplements and d i s t r i b u t i n g . 2.2 i s d e c e p t i v e l y s i m p l e f o r n = 0, and d e c e p t i v e l y d i f f i c u l t f o r n > 0: f o r n = 0, we combine R o l l e ' s Theorem w i t h i n d u c t i o n on t h e degree ( o v e r K) o f t h e e n d p o i n t s o f t h e i n t e r v a l s comp r i s i n g S ; f o r n > 0 we combine t h e "Good D i r e c t i o n Lemma" w i t h a paramet r i z e d v e r s i o n o f t h e case n = 0.

.

P r o o f o f 2.1: Pnd i s a s.a. s e t i n R , s i n c e c E Pnd i f and o n l y i f t h e elementary formula V x o - * * v x n f(c;xo,-..,xn) a 0 h o l d s , and t h e Tarski-Seidenberg Theorem produces an e q u i v a l e n t q u a n t i f i e r - f r e e formula. A l so, Pnd i s c l o s e d , s i n c e a l i m i t o f psd forms i s s t i l l psd. (Furthermore, T h e r e f o r e we may a p p l y i t i s e a s y t o s e e t h a t Pnd i s even a convex cone.) t h e F i n i t e n e s s Theorem ( b ) t o Pnd: t h e r e e x i s t f i n i t e l y many g i j E 2[C] such t h a t , w r i t i n g Wi = WIgiil, , P, =Ui. Wi.. For each i we a p p l y t h e Pos.u ('id )+n+l i t i v s t e l l e n s a t z (1.4) t o f, which i s nonnegative on W i C R (we a r e now v i e w i n g {g. . l as b e i n g i n t h e l a r g e r r i n g Z[C;X]). 2.1 f o l l o w s immedi1J a t e l y , t a k i n g t h e giJ t o be p r o d u c t s o f t h e g i j . Q. E. D.

3. TWO LOGICAL POINTS CONCERNIWG THE FINITENESS THEOREM (2.2) (a) I n t u i t i o n i s t i c C o n s i d e r a t i o n s The f i r s t p o i n t i s addressed t o r e a d e r s i n t e r e s t e d i n i n t u i t i o n i s t i c mathematics. Since we do n o t assune i n 2.2 t h a t K and R a r e r e c u r s i v e , s e v e r a l s t e p s i n t h e p r o o f o f 2.2 are, i n t u i t i o n i s t i c a l l y , p r o b l e m a t i c . Spec i f i c a l l y , even i f S i s d e f i n e d by a q u a n t i f i e r - f r e e f o r m u l a (and n o t by an a r b i t r a r y f i r s t - o r d e r f o r m u l a ) i n t h e language o f o r d e r e d f i e l d s , and t h e gij are presented by terms i n t h a t language, t h e e q u i v a l e n c e ( v x ~ R ~ + ~ E) [ Sx H x U i W { g i j l ] may b e v a l i d c l a s s i c a l l y b u t n o t i n t u i t i o n i s t i c a l l y . I n broad terms: t o e a c h x E S we a r e t o f i n d i such t h a t x E W{gij}, and so, i f S happens t o be connected t h e map x * i r m s t b e constant, s i n c e i n t u i t i o n i s t i c f u n c t i o n s a r e continuous. A c o n c r e t e example r e f u t i n g 2.2(b) w i l l be g i v e n a f t e r t h e p r o o f o f 4.2 below. E v i d e n t l y , t h e s i t u a t i o n changes i f we E

C.N. DELZELL

98 replace, UI - 9 i j 1 .

i n 2.2(b),

t h e u n i o n by i t s c l a s s i c a l l y e q u i v a l e n t f o r m

R"'-

n u 1

J

These s p e c i f i c a l l y i n t u i t i o n i s t i c o r , i n a r e l a t e d c o n t e x t , s h e a f - t h e o r e t i c r e q u i r e m e n t s s h o u l d be d i s t i n g u i s h e d f r o m q u e s t i o n s about t h e ( r e c u r s i o n - t h e o r e t i c ) c o m p l e x i t y o f t h e terms ( d e f i n i n g ) gij as f u n c t i o n s o f t h e presentation o f S, e i t h e r by a f i r s t o r d e r f o r m u l a o r more s p e c i f i c a l l y a q u a n t i f i e r - f r e e one. These q u e s t i o n s s i m p l y concern t h e v a l i d i t y of 2 . 2 ( a ) and 2.2(b) i n t h e c l a s s i c a l t h e o r y . Reasonably good bounds on t h e c o m p l e x i t y can be e x t r a c t e d f r o m o u r p r o o f i n [1980] and [1981]. But i t i s worth n o t i n g t h a t mere r e c u r s i v e n e s s o f some s u i t a b l e gij, as f u n c t i o n s o f ( t h e p r e s e n t a t i o n o f ) S f o l l o w s t r i v i a l l y from t h e l o q i c a l form o f 2.2 by t h e ( r e c u r s i v e ness o f t h e ) T a r s k i - S e i d e n b e r g a l g o r i t h m m e F b y r e c u r s i v e l y enumerating a l l p r e s e n t a t i o n s o f s.a. s e t s S, and a l l p r e s e n t a t i o n s o f , t h a t i s , f o r m u l a e d e f i n i n g b a s i c open, r e s p . closed, s e t s , denoted above u s i n g qij. The l a t t e r enumeration induces enumerations o f f i n i t e u n i o n s Ui U{gij},

resp. uiW{gij},

o f such b a s i c s e t s . Now, f o r any g i v e n p r e s e n t a t i o n o f S ( w i t h parameters, h t e r p r e t e d t o d e n o t e elements E K) and any g i v e n p r e s e n t a t i o n o f a u n i o n o f i s recurb a s i c s e t s , t h e v a l i d i t y o f S = Ui U{gij}, r e s p . S =Ui W{gij}, s i v e l y d e c i d a b l e by t h e T a r s k i - S e i d e n b e r g a l g o r i t h m . I n o t h e r words, t h e r e l a t i o n , s a y R , b e t w e e n p r e s e n t a t i o n s o f S and g . . s a t i s f y i n g 2 . 2 ( a ) ,

IJ

r e s p . 2.2(b), i s r e c u r s i v e . (For t h e c o n c l u s i o n below i t would be s u f f i c i e n t i f t h a t r e l a t i o n were r e c u r s i v e l y enumerable.) Thus V S 3 { g i j } 9 (S,{gij}) is t r u e and hence t h e r e i s a r e c u r s i v e map y : S - r {gij}

s a t i s f y i n g vSR(S,y(S)).

E v i d e n t l y , c o r r e s p o n d i n g r e s u l t s a p p l y t o 1.4 and 2.1. Remark: The sane c o n s i d e r a t i o n s are used i n t h e work o f Robinson and Henkin, d i s c u s s e d i n $1, p r o v i d i n g r e c u r s i v e bounds, and more g e n e r a l l y , i n a p p l i c a t i o n s o f f i r s t - o r d e r model t h e o r y t o theorems w i t h a r e s u r s i v e l y enumerable s e t o f axioms. T h i s b r i n g s us t o o u r second l o g i c a l p o i n t : ( b ) The Terminology ' ' F i n i t e n e s s " Can g e n e r a l ( c l a s s i c a l ) model t h e o r y be used t o g i v e a simple, o r a t l e a s t new, p r o o f o f 2.2? The q u e s t i o n a r o s e - - i f f o r no o t h e r reason--because o f a pun: t h e compactness theorem o f l o g i c was o r i g i n a l l y c a l l e d t h e " F i n i t e ness Theorem" j u s t l i k e o u r 2.2. The l a t t e r i s c a l l e d " F i n i t e n e s s Theorem" because, a t l e a s t s u p e r f i c i a l l y ( t h a t i s under q u i t e g e n e r a l c o n d i t i o n s ) , i t i s t r i v i a l t h a t S i s open i f and o n l y i f i t i s a p o s s i b l y i n f i n i t e u n i o n of b a s i c open s e t s . It i s w o r t h p a u s i n g a moment t o l o o k a t t h e assumptions behind the question. set

F i r s t o f a l l , t h e " t r i v i a l " r e p r e s e n t a t i o n o f any ( n o t o n l y : s . a . ) open S by an i n f i n i t e u n i o n o f b a s i c open s e t s d e f i n e d b y use o f q i j - - t a c i t -

l y , w i t h c o e f f i c i e n t s i n K--assumes t h a t K i s dense i n R . I n t h a t case x S can be surrounded b y an ( n + l ) - b a l l i n S, and t h e l a t t e r b y a b a s i c open (The c e n t e r o f t h e ( n + l ) - b a l l need K-R-s.a. ( n + l ) - b a l l , a l s o l y i n g i n S.

E

n o t be x i t s e l f , f o r example i f x $ K"'.) But, i n g e n e r a l , t h a t i s i f K i s n o t Archimedean, K need n o t be dense i n R a t a l l , w i t h consequences ill u s t r a t e d by t h e p e n u l t i m a t e paragraph o f $ l ( b ) . More s p e c i f i c a l l y , t h e r e p r e s e n t a t i o n o f S as an i n f i n i t e u n i o n o f b a s i c open s e t s d e f i n e d by use o f parameters f o r elements E K o n l y , i s by no means t r i v i a l . What can be done--and t h i s i s i m p l i c i t i n a s u g g e s t i o n made by R. L. Vaught a t t h e c o n f e r e n c e - - i s t h i s . Suppose t h a t t h e gap mentioned i n t h e l a s t p a r a g r a p h i s f i l l e d , b u t p o s s i b l y i n a n o n - u n i f o r m way. S p e c i f i c a l l y , suppose i t i s shown t h a t , f o r each open s.a. s e t S, t h e r e i s some i n f i n i t e f a m i l y o f

Representing polynomials as sums of squares

99

such t h a t S = U i Ui, b u t p o s s i b l y a d i f f e r e n t f a m i l y f o r d i f f e r Ui = U{gijl e n t R (and t h e sane d e s c r i p t i o n o f s ) . Then t h e compactness theorem o f l o g i c ensures a l s o a u n i f o r m r e p r e s e n t a t i o n as f o l l o w s . We f i r s t expand t h e l a n g u a g e b y a d d i n g a c o n s t a n t a, and c o n s i d e r t h e c o n j u n c t i o n , f o r Ui ( b u i l t up f r o m t h e p a r a n e t e r s i n S, i n o t h e r words, i n t h e f i e l d generated by them), o f (acSAa$Ui ) V (aEUir\ a$S). T h i s i s i n c o n s i s t e n t f o r each r e a l c l o s e d f i e l d , by our h y p o t h e s i s t h a t t h e r e a r e non-uniform r e p r e s e n t a t i o n s . The compactness theorem t h e n ensures t h a t a f i n i t e subset i s i n c o n s i s t e n t , and hence uniformity. It i s e v i d e n t t h a t t h e Tarski-Seidenberg Theorem by i t s e l f does n o t prov i d e t h e e l i m i n a t i o n o f n e g a t i o n s and = and < r e l a t i o n s from q u a n t i f i e r f r e e p r e s e n t a t i o n s o f open s e t s s.

4. THE NEGATIVE ANSWER TO KREISEL'S FIRST QUESTION Q[C;X]

for -

Theorem 4.1: For d < 2, t h e r e a r e ( k = 1,2), such t h a t

d

(4.1.1)

>

2, t h e r e do n o t e x i s t f

= ZJ. p J. r ?J

pj

8

Q(C),

and

s

and

E

rj

(4.1.2)

The p r o o f i s based on a f i n e r a n a l y s i s o f F i n i t e n e s s Theorem:

{gJiCZ[C],

N,

E

Q(C;X)

VC E

Pnd

Pnd,

such t h a t

each

b a s i c s.a.

pj(C) a-0.

t h a n t h a t g i v e n by t h e

Theorem 4.2: Pd, i s n o t a b a s i c s.a. set, except when i s a b a s i c c l o s e d s.a. s e t . (4.2 i s t o be expected: sets.)

{hk3k

d < 2, when i t

s e t s a r e r a t h e r s p e c i a l among a l l s.a.

Proof o f 4.2: For t h e case d = 2 we use i n d u c t i o n on n. For n = 1, P1,2TA,C,ATB21 ( w r i t i n g f(A,B,C;X,Y) = AX2 + 2BXY + CY2). To prove P n 2 i s a s i n g l e W f o r n > 1, we may suppose, i n d u c t i v e l y , t h a t t h e c o n d i .,Xn-l t o be psd i s a c o n j u n c t i o n o f nont i o n f o r a quadratic form i n s t r i c t i n e q u a l i t i e s i n C. W r i t e

b,.,

where d e g fi = i ( i = 0,1,2). l h e n f i s psd i f and o n l y i f f,, f,, and a r e a l l p s d i n Xo,...,Xn-l; t h i s i s j u s t a conjunction o f three f,f, - f! c o n j u n c t i o n s , s i n c e t h e s e t h r e e forms a r e q u a d r a t i c ( e x c e p t t h e c o n s t a n t form f,, f o r which t h e psd p r o p e r t y i s an "improper" c o n j u n c t i o n , nanely., w i t h o n l y one c o n j u n c t ) . For t h e case d > 2, n o t e t h a t i f Pnd were a b a s i c s.a. s e t f o r d 4, t h e n by s e t t i n g some c o e f f i c i e n t s equal 0, we would have t h a t P1,4 i s a bas i c s.a. s e t . To d e r i v e a c o n t r a d i c t i o n f r o m t h i s , l e t Y be a s i n g l e i n d e -

C.N. DELZELL

100

t e r m i n a t e and c o n s i d e r fa,b(Y) = (Y2 + a)' + b E R[Y] f o r (a,b) E R2. Gn t h e one hand, t h e s e t A = {(a,b) E R 2 I f a , b ( Y ) i s psd i n Y over R} must be o f t h e form W{gi(a,b)}nU{hi(a,b)l f o r some gi, h i E Z[a,b]. On t h e o t h e r hand, A = I ( a , b ) I b > O v c b - a < 01 = { ( a , b ) I b > O v ( a > O A b > - a 2 ) ] ( s t r i p e d i n t h e f i g u r e b e l o w ) . Then some gi o r hi would have t o change s i g n across a Z a r i s k i - d e n s e subset o f t h e n e g a t i v e a-axis, hence t h e SignChanging Theorem i n [Dubois-Efroymson 19701 would t h e n i m p l y t h a t t h i s gi o r hi w o u l d have t o be d i v i s i b l e b y ( p r e c i s e l y ) an odd power o f b, which would make i t change s i g n across even t h e p o s i t i v e a-axis, t h e r e b y e x c l u d i n g p a r t o f A. Q. E. D. ib

a

Note t h a t t h e c l o s e d s e t

i s a counterexanple t o t h e F i n i t e n e s s TheoThe above p r o o f showed, i n e f f e c t , t h a t no b a s i c s.a. subset o f A can c o n t a i n a s e t o f t h e f o r m B n A , where B i s any open d i s k about t h e o r i g i n i n R2. T h e r e f o r e i f a Cauchy sequence o f p a i r s o f r a t i o n a l s (rn,sn) + (0,O) E A i s given, we w i l l n o t a l ways b e a b l e t o compute i such t h a t (rn,sn) E Wi. A

rem ( b ) , when i n t e r p r e t e d i n t u i t i o n i s t i c a l l y .

P r o o f o f 4.1: For d = 2 we j u s t combine 2.2 and 1.4, o f 2 . m t h a T g l e Wi , so t h a t we may d r o p t h e i.

as i n t h e p r o o f

F o r d > 2, suppose t h a t 4.1 were f a l s e , and w r i t e each p. = r . / s . , w i t h J J J r e l a t i v e l y prime i n Q[C]; t h e n we c o u l d conclude t h a t Pnd = W{rjsjl" Q. E. D. U { s j l ( zby 4.1.1 and c _ by 4.1.2), c o n t r a d i c t i n g 4.2.

rj, s j

The r e s u l t s o f t h i s paper appeared i n my d i s s e r t a t i o n [1980]. I an g r a t e f u l t o P r o f e s s o r Gregory B r u n f i e l , my t h e s i s a d v i s o r , and t o P r o f e s s o r Georg K r e i s e l , f o r many h e l p f u l c o n v e r s a t i o n s on t h i s s u b j e c t .

BIBLIOGRAPHY Artin, E. Uber d i e Zerlegung d e f i n i t e r Funktionen i n Quadrate, Ahhandl. Math. Sem. Hamburg 5 (1927), 100-15. A r t i n , E. and S c h r e i e r , O., A l g e b r a i s c h e K o n s t r u k t i o n e n r e e l l e r Korper, hand1 . Math. Sem. Hamburg 5 (1927), 85-99. .

&-

~

Berg, C . , Christenson, J.P.R., and Ressel, P., P o s i t i v e d e f i n i t e f u n c t i o n s on A b e l i a n semigroups, Math. Ann. 223(3), (1976), 253-74. Bochnak, J. and Efroymson, G., problem, Math. Ann. (1980). Bose,

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N.K,, New techniques and r e s u l t s i n m u l t i - d i m e n s i o n a l F r_ a n_k_ l i n_ I_ n s t . 301( 1 - 2 ) , (1976), 83-101. _

problems,

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Representing polynomials as sums of squares

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B r u n f i e l G., P a r t i a l l y Ordered Rings and Semi-Algebraic Geometry. Lecture N o t e S e r i e s o f t h e LondonMath. S o y ( C a m b r i d g e m v . Press, Cambridge, 1979) . Cassels, J.W.S., &I t h e r e p r e s e n t a t i o n o f squares, A c t a A r i t h . 9 (1964), 79-82.

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E l l i s o n , W.J., and P f i s t e r , A., On suns o f squares and on Cassels, J. W.S., e l l i p t i c curves o v e r f u n c t i o n f i e l d s , J. Number Theory 3 ( 2 ) , (1971), 125-49. Choi. M.D.. and Lam. T.-Y.. An o l d a u e s t i o n o f H i l b e r t . ~ Proc. _ Conf. _ on Oua_ . d r a t i c Forms--1976, (Queen's Papers on Pure and A p p i i e d Math., No. 4 6 3 . Orzech, ed.), Queen's Univ., Kingston, O n t a r i o , 1977), 385-405. E x t r a n a l p o s i t i v e s e m i d e f i n i t e forms, Math. Ann. 231 (1977), 1-18. Daykin, Thesis, Univ. o f Reading, 1960 ( u n p u b l i s h e d ) ; c i t e d by K r e i s e l , A surv e y of p r o o f t h e o r y , 2. S y b . Logic 33 (1968), 321-88. D e l z e l l , C . , A c o n s t r u c t i v e , c o n t i n u o u s s o l u t i o n t o H i l b e r t ' s 17th problem, and o t h e r r e s u l t s i n s e m i - a l g e b r a i c geometry, Ph.D. dissertation, S t a n f o r d Univ., 1980 (Univ. M i c r o f i l m s I n t e r n a t i o n a l , Order No. 8024640). C f . a l s o D i s s e r t a t i o n A b s t r a c t s I n t e r n a t i o n a l 41, no. 5, 1980. A f i n i t e n e s s theorem f o r open s e m i - a l g e b r a i c sets, w i t h a p p l i c a t i o n s t o H i l b e r t ' s 17th p r o b l e m , Proc. AMS Special Session on Ordered F i e l d s and Real A1 e b r a i c Geometry, Jan. 7-8, 1 9 8 1 , n F r a n c i s c o . D.W. Dubois, : de ~ e ~ a ~ ~ s ~ s -Continuous suns o f squares o f forms, t o appear i n Proc. L.E.J. Brouwer Cent e n a r y Sjmposium, June 8-13, 1981, Nordwijkerhout, H o l l a n d , A.S. m t r a and 0. van m e n , eds., North H o l l a n d . A c o n t i n u o u s , c o n s t r u c t i v e s o l u t i o n t o H i l b e r t ' s 17th problem, i n preparat i o n ( a ) . See a l s o a p r e l i m i n a r y a b s t r a c t i n A b s t r a c t s , Jan. 1981.

fi

W, i n p r e p a r a t i o n

On H i l b e r t ' s 17th p r o b l e m o v e r c o u n t a b l e s u b f i e l d s o f (b).

Sur l e s anneaux de polynbrne a c o e f f i c i e n t s dans un anneaux Dickmann, M.A., r e e l c l o s , Comptes Rendus ( P a r i s ) , S e r i e A 290 (1980), 57. D u b o i s , D.W., Note on A r t i n ' s s o l u t i o n o f H i l b e r t ' s 17th problem, B u l l . h e r . Math. SOC. 73 (1967), 540-1. -~ Dubois. D.W.. and E f r o m s o n . G.. A l a e b r a i c t h e o r v o f r e a l v a r i e t i e s . I. Studi e s and^ Essays Presented t o Y;-Why Chen on' h i s 60th Birthday; p p . 1 0 7 T35,Taiwan U n i v e r s i t y , 1937. --- - ~

E l l i s o n , W.J., A "Waring's problem" P h i l o s . SOC. 65 (1969), 663-72. ___-

f o r homogeneous forms,

Proc.

Cambridge

F l e c k , A., Zur D a r s t e l l u n g d e f i n i t e r b i n l r e Formen a l s Summen von Quadraten ganzer R a t i o n a l z a h l i g e n Formen, A r k i v der 5 & Physik 3d Ser. 10 (1906), 23-38, and 3d Ser. 16 ( 1 9 1 O r n 5 - r Habicht, W., k e r d i e L o s b a r k e i t gewissen a l g e b r a i s c h e r Gleichungssysteme, Comm. M. Helv. 12 (1940), 317-22. (An E n g l i s h v e r s i o n i s i n t h e second[1952]ion o f [Hardy, L i t t l e w o o d , and Pblya, 19341, below.)

C.N. DELZELL

102

Hardy, G.H., L i t t l e w o o d , J.E., and Pblya, G., I n e q u a l i t i e s (Cambridge u l i v . Press, Cambridge 1934); second e d i t i o n , (1952). Henkin, L., Sums o f squares, Summaries o f T a l k s Presented a t t h e Summer I n s t i t u t e o f Sjmbolic Lo i c T n T a t C o m U n i v e r s i t y - ( I n t i D e 6 5 5 i Z e Analyses, P r i n c e t o ~ 9 6 ~ , ___ ~ - ~ . H i l b e r t , D., Uber d i e D a r s t e l l u n g d e f i n i t e r Formen a l s Summe von Formenquad r a t e n , Math. Ann. 32 ( 1 8 8 8 ) , 342-50; s e e a l s o Ges. Abh. 2, (Springer, B e r l i n T 3 3 ) , 154-61. Uber t e r n a r e d e f i n i t e Formen, k t a Math. _ Ges. __ Abh. _ 2, ( S p r i n g e r , Berlin,m3),5-66.

17 (1893),

169-97;

see

also

Grundlagen der Geornetrie (Teubner, 1899); t r a n s l . by E.J. Townsend (Open C o u r t P u b E h i n g Co., La Salle, IL, 1902); t r a n s l . by L. Unger f r o m t h e t e n t h German e d i t i o n (Open Court, 1971). Mathematische Probleme,

G o t t i n q e r N a c h r i c h t e n (1900).

I.

Hsia,

28, (Amer.

-

Math.

253-97.

and A r c h i v

SOC., Providence,

J.S., and Johnson, R.P.. O n t h e r e p r e s e n t a t i o n i n suns o f squares f o r d e f i n i t e f u n c t i o n s i n one v a r i a b l e over an a l g e b r a i c nunber f i e l d , @ -~ J . Math. 9 6 ( 3 ) , (1974), 448-53.

Jacobson, N.,

Basic Algebra 1 (Freeman, San Francisco, 1974).

K r e i s e l , G., H i l b e r t ' s 17th problem, I and 11, B u l l . h e r . Math (1957), 99 and 100.

Sot.

Mathematical s i g n i f i c a n c e o f c o n s i s t e n c y proofs, J. Symb. (1958), 155-82 ( r e v i e w e d by A. Robinson, 2. Sjmb. Log= 311,28.)

63

Logic

23

Suns o f squares. Summaries o f T a l k s Presented a t t h e Summer I n s t i t u t e for S y n b o l i c ' Lo i c i n 957 ZF GESl U n i v e r s i G , ( I n T f X i X e f o r Defense Analyses, & c e 5 n n , & 0 r 3 d

Two n o t e s on t h e f o u n d a t i o n s o f s e t t h e o r y , 114.

D i a l e c t i c a 23 (1969),

93-

On t h e k i n d o f d a t a needed f o r a t h e o r y o f p r o o f s , Lo i c Colloquium 1976 and Hyland, J.M.E., eds.), N o r t h - H o h P u b l i s h i n g (candy, R.O. Amsterdan, 1977), 111-28. (E58#21397) Review o f L.E.J. Brouwer, C o l l e c t e d Works, Vol. I , P h i l o s o p h y and Foundat i o n s o f M a t h e w ( A . t k y t i n g x ) T ul T . her. Math. Soc. 83 (1977);86-93. Review o f Ershov, Z e n t r a l b l a t t 374 (1978), 18-9, 602027. Lam,

T.-Y., The t h e o r y o f o r d e r e d f i e l d s , Ring Theory and Algebra 111, B. McDonald, ed., (Marcel Bekker, New Y o r k , X O ) -

Landau, E., h e r d i e D a r s t e l l u n g d e f i n i t e r b i n a r e r Formen durch Quadrate, Math. _ _ Ann. 57 (1903), 53-65.

Representing polynomials as sums of squares

10

Uber d i e Zerlegung d e f i n i t e r Funktionen d u r c h Quadraten, Ark. f u r Math. und P h y s i c , 3d Ser. 7 (1904), 271-7. Uber d i e O a r s t e l l u n g d e f i n i t e r (1906), 272-85. Lang, S.,

Algebra (Addison-Wesley,

Functionen i n Quadraten,

Math.

Ann.

62

Reading, MA, f i r s t e d i t i o n , 1 9 6 5 ) .

Motzkin, T.S., The a r i t h n e t i c - g e o m e t r i c i n e q u a l i t y , I n e q u a l i t i e s 1, S i s a , O., ed., (Academic Press, New York, 1967), 204-24. P f i s t e r , A., Invent.

Zur D a r s t e l l u n g d e f i n i t e r Functionen a l s Summe von Quadraten, Math. 4 (1967), 229-37. (An E n g l i s h v e r s i o n i s i n [1976]

below.)-

L e t t e r t o K r e i s e l , The K r e i s e l Papers ( S t a n f o r d Univ. Archives, 1 9 7 4 ) . H i l b e r t ' s 17th p r o b l e m and r e l a t e d problems on d e f i n i t e forms, Mathematic a l Developments A r i s i n from H i l b e r t Problems (F. Browder, &, ) . de G ~ p .- i n_Pure _ M -a t h x y P m Y Math.Soc.,), 483-9.

e

Pourchet, Y . , Sur l a r e p r e s e n t a t i o n en somme de c a r r e s des polyndmes une i n d e t e r m i n e e sur un c o r p s de nombres algebriques, k t a A r i t h . 19 (1971), 89-104. P r e s t e l , A. Sums o f squares o v e r f i e l d s , Atas da 5a Escola de Algebra (Soc. B r a s i l e i r a de Matemdtica, Rio de J a n e i r 5 9 7 8 ) - -Procesi, C., 219-25.

P o s i t i v e synmetric functions,

Robinson, A., 257-71.

Ch ordered f i e l d s and d e f i n i t e forms,

F u r t h e r remarks on ordered (1956), 405-9.

fields

Pdvances

and d e f i n i t e

29 (1978),

i n Math.,

Math. Ann. 130 (1955), forms,

Math.

Ann.

130

Some problems o f d e f i n a b i l i t y i n t h e lower p r e d i c a t e c a l c u l u s , Funda. Math. 44 (1957), 309-29. I n t r o d u c t i o n t o Model Theory and t o t h e Metamathematics o f Algebra (NorthHolland &blishingCo.,ATteTdam%i1963). Robinson, R.M., Some d e f i n i t e p o l y n o m i a l s which a r e n o t suns o f squares o f r e a l polynomials. Notices h e r . Math. Soc. 16 (1969), 554; Selected Q u e s t i o n s i n Algebra and Log& (Vol. d e d i c a t e d t o t h e memory= z d a t . "Nauka." T b i r s k Otdel N o v o s i b i r s k (1973), 264-82; o r Acad. Sci.= -647). Siegel,

C.L.,

D a r s t e l l u n g t o t a l p o s i t i v e r Zahlen d u r c h Quadrate, Math. Z e i t .

11 (1921), 246-75; G e S . Abh. 1 (1966), 47-76.

Stengle, G., A N u l l s t e l l e n s a t z and a P o s i t i v s t e l l e n s a t z f o r semi-algebraic geometry, Math Ann. 207 (1974), 87-97. I n t e g r a l s o l u t i o n o f H i l b e r t ' s 17th problem, Math. Ann.

246

(1979), 33-9.

Van den D r i e s , L., A r t i n - S c h r e i e r t h e o r y f o r commutative r e g u l a r r i n g s , Math. Logic 12 (1977), 113-50.

Ann.

PROCEEDINGS OF THEHERBRAND SYMPOSILIM LOGIC COLLOQUIm ’81, J. Stem (editor) @ North-Holland F‘ublishing Company, 1982

ON L O C A L AND NON-LOCAL

105

PROPERTIES

Haim Gaifman The Hebrew University Jerusalem and University Paris VI

SO.

INTRODUCTION

The result to be presented here was motivated by questions of first-order definability within the class of finite relational structures. These questions arose in the research of suitable languages for databases (cf [AHU],

[AUI and [CHI). A stan-

dard example is the following : Can we express in first-order language the property that a graph (i.e. a binary symmetric relation) is connected : The negative answer is easily proved either by a compactness argument, or by forming the ultrapower of a sequence of connected graphe, Mi, i<w, such that for each n all Mi

almost

have diameter > n. If, however, we pose the question in the domain of

finite graphs : is there a first-order sentence cp

M bcp iff M is connected

?

such that for all

finite M,

then these easy arguments do not carry over. The

negative answer neecis another kind of proof (cf [ A U I , [ C H I )

which, though not dif-

ficult, involves a finer analysis of the situation. Since detabases are, essentially, finite relational structures, their investigation leads naturally to questions concerning finite models. Some questions which have easy solutions for infinite models become not so easy and sometimes quite difficult when transferred to the finite domain. The method to be presented is an analysis of first-order formulas in terms of local properties. We use a natural simple metric in the model and define the concept of a k-local formula where k

is any natural number. Roughly speaking, a k-local

formula is one which asserts something about some k-neighborhood around a point,

x, i.e. about the model (Vck)(x), x), where V(k)(x) is the set of all pointsof distance < k from x ; this means that all quantifiers arerelativized to V(k)(x) and

x

is the free variable of the formula. The main theorem asserts that every

first order sentence, cp, is logically equivalent to a Boolean combination of sentences that assert, each, something of the following form : There exist

s

disjoint

r-neighborhoods, each satisfying the r-local formula JI.

H. GAIFMAN

106

If cp

is a formula, one has to add to the combination r-local formulas in the

free variables of CD. The theorem is proved by quantifier elimination. The proof yields an effective translation of the formula cp

into the Boolean combination, as well as upper

bounds for the neighborhood radius, r, and the number, s , of neighborhoods, in terms of the quantifier depth of cp. If the quantifier depth speaking, the upper bound for r

is

is n then, roughly,

7n-1. By considering particular examples,

one can establish a lower bound which is

Zn-'

.

(For the exact details see § I ) .

This poses the problem of narrowing the gap. The reader not interested in these details o r in the technicalities of the proof can go on directly to 82, once the theorem is understood. The point is

to

establish a precise sense in which first

order sentencesare local and to use it in order to show that such and such properties are not characterized by first-order sentences because they are not local, Thus, to cite one of the examples given in 8 2 , within the class of finite graphs one cannot define by a first-order sentence those that are planar. In general, let

Co

and

C1

be two classes of models. Assume that for every n

one can find M ECo, MIECl such that for all

r,s

< n,

every combination of

s

disjoint r-neighboods in one model is isomorphic to a similar combination in the other. Then Co

and

C1

cannot be seperated by means of a first-order sentence.

Moreover, as proved in §2, such classes cannot be seperated in any richer logic obtained by introducing predicates which denote local properties ties of k-neighborhoods,where k

-

i.e., proper-

ranges over the natural numbers. There is no

restriction on the properties allowed,provided only that they are preserved under isomorphisms and that they are local. The examples given in 62 are all from graph theory,which seems to be the most natural domain for applying the method. The proofs are very easy in drawing prototypes of the Mo

and

shows that the same combinations of a

M,

-

for they consist

mentioned above. A glance suffices to

"small" number of

"small" neighborhoods

are realized in both. In $ 3 a different kind of application is given

-

a much shorter proof of a previous

result of the author, answering a set-theoretical question of Levy, [ G I . As stated at the end, this leads to the existence of a certain curious transitive set. A s it is,the method does not apply to models possessing a more regular mathemati-

cal structure - s u c h as a linear ordering, for then the whole model is equal to some small neighborhood. Any extension of the method to such modelswill have to use a much more sophisticated distance function.

Local and non-localproperties

107

We should mention that a result of L. Marcus [M 1 which is an immediate corollary of the theorem, can be regarded as a forerunner of it. There seems-to be no direct proof of the theorem, or the applications, from the corollary.

$ 1 . THE THEOREM

Let L be a first-order language with finitely many predicates including equality and no function symbols (and no individual constants). The restriction on function symbols is not essential, since they can be eliminated in the usual way by using predicates. Our results can be formulated in general, but it is more convenient to formulate and prove them for a language that has only predicates. "x", "y", "z"

We shall use

both for individual variables in L

as well as for

members of models. We shall also use the members of models as names of themselves

--

__ ...,_x,y,z

and occasionally substitute them for free variables in formulas. a,b, are used for tuples. If

_x,y_

=

xo,.

..

are in X. Let M

x = xo,. ..

yo,. . .

By

By

and '%€ X"

-

y = yo,.

..,yn-l

then

-

we mean that all the elements of x

"Fcj" we mean that every element in X

is also in 7 .

be a model for L, say M = (M, Ro, ...). Define a metric

d = dM

over M

as follows : d(x,x) =

Dfo

for some predicate d(x,y) Q 1 eDF

-

tuple x E M

containing both

d(x,y)

If L

= m

x

R (including equality) and some

and y. M bR(x).

wDFfor all n ,

d(x,y) $ n.

has one binary predicate besides equality then M dM

path connecting x

and

If all predicates of

y, disregarding the direction of the edges.

L are monadic or binary then d(x,y)

the formula Wi(Ri(x,y)

can be regarded as a

is the usual distance : dM(x,y) = length of the smallest

directed graph and

v Ri(y,x))

=

1

is definable by

H. GAIFMAN

108

where the Ri's

are all the binary predicates in L.

In general, d(x,y)

= 1

is definable by a formula that involves quantifiers. Here the finiteness of the number of predicates is used. Since the forthcoming result concerns single formulas and each single formula involves finitely many predicates, the result holdsfor any language provided that we define the distance using only the predicates that occur in the formula in question.

d(x,y)

Q

is definable by

n

3v0,.

..,v

n

If M'cM

> dM(x,y)

then evidently dM,(x,y)

(x= v hy = v A M d(vi,vi+l) (1). O i 2k+l].

IS'^) (x) , being

bounded by

V(k) (x) , are eli-

d(G,x) > 2k+l V(2k+l) (v) (7k+3) Consequently 7 B A C.(T) implies n ~ ) q(7k+3)(~) for some cp (v) I J j+l (7k+3) and the first disjunct can be rewritten as B. h i B A C . 6) A cp (v)

minable in favour of quantifiers bounded by

V(6k+3+k)(G).

Also

.

is expressible by a formula in which all quantifiers are bounded by

3

q.e.d.

j+l

J

lemma.

End of the proof : Since every quantifier-free formula is of the form the claim holds for n

=

quantifier depth = 0. The claim carries over, trivially,

to Boolean combinations. It suffices to prove the claim for a = 3ua'(;,u) has quantifier depth n, n depth. Let

-

v = vo

cp(o)(z),

>

,...,vm-,.

that

1 , assuming it for all formulas of smaller quantifier

Apply the claim to

a ' ( v , u ) , write the resulting

Boolean combination of formulas of forms (I) and (11) in disjunctive normal form and distribute 3u

-

around v,u

over this disjunction. Using the fact that k-local formulas

are closed under Boolean combinations we get a disjunction of formu-

las, each of the form 3U[B(k)(G,u)

A

M Ail

iE1 where the

A.'s

are sentences of form

(I) or their negations. This is equiva-

lent to (3u B C k )

( 7 , ~ )A )

M Ai.

iEI By the induction hypothesis each

.., z s,-l[Mi<s,I)'~')

(zi)

sentence, where r' Q 7n-2

and

3z0,.

variables of a'

A

Xi

is either

d(zi, z . ) > 2r'I or a negation of such a M i<j<s' J s'-l

contain infinitely many disjoint copies of each

bers of copies of the other basic models. Let ting one copy of

the

is isomorphic to

there are points of distance

tail). Now fix r > O CL, L = O,l,

in a copy of

CL and as there is no point of distance

(L+l)-neighborhood.

corresponds to

C.'s,

(where any number of isomorphic copies of each of these models

may be used). Note that for x with the copy of

{ O } and

add new members

,

der models which are disjoint unions of isomorphic copies of the

*

and

inter-

let CL be obtained from the cycle CL by

add to the binary relation the pairs and the C.'s, j > O

P

V(')

(xi)

in M'

such

are isomorphic.

(for in that case each of the

that occur in the collection in M

can be matched by a copy of

Consequently Boolean combinations of sentences from

Local and non-localproperties

u LS(i,j)

U

U

i < r 1Gj

O


by induction, using the equivalences : d(u,v) G 1

d (u,v) G 2k

R(u,v),

3w[d(u,w) G k

d(u,v)GZk+l

A

-

X.

.

This is easily established,

3w[d (u,w) G k A d (v,w) k]

and

d(v,w) G k+l].

Consequently d(u,v) > n, which is equivalent to 1 (d(u,v)Gn), needs ?og2z quantifier depth and this is also true for d(u,v) = n (which is equivalent to (d(u,v) G n A d(u,v) > n-1)). Hence, for r > 0, cr(x) can be written with rlog rl + 1

quantifier depth. From this it follows easily that, for r > 0, rl bog rl + 1 +

can be expressed in quantifier depth

s.

s

quantifier depth. Evidently,

Theorem : For each n > 2

there exists a sentence, u , of quantifier depth n

such that any equivalent Boolean combination of sentences of U must contain members form each of LS(2n-2-i, =

n-1, LS(2-',

0,s

r,s needs

From this we get the following lower bound.

~~

i

rl

..

S i

. U LS(i.j) lGj

i+l) , i = 0,. ,n-1 where, for

is to be reread as LS(0,n).

n)

This simply means that we get the lower bound by replacing in the upper bound everywhere

7"-'

by

2"-'.

(and rereading 2-1 as 0).

Problem : Narrow down the gap between the upper bound 2n-2

7"-'

and the lower bound

In the case of formulas a similar construction yields the same kind of lower bound where, in addition,

1

(7"-1)

is to be replaced by

As

2"-'.

far as the applications of the next section are concerned there is no need for estimates on the values of r and s . Here is an outline of a shorter semantic

proof of the theorem as stated :

H. GAIFMAN

118

For €;

M

k = 0,I , .

-

let the local type of a be the set of all local formulas cp(k)

..

such that M ~ ~ ( k ) ( Now ~ ) assume . that for all

-a € LS(i,j), b

=

bo,

M1

/= a

...,bmdl €

e,

M2

i> 0 , j> 1

(G),

and all

1 u . Assume furthermore that a = ao,. ..,am-1 E M 1 , and a and b have, in their respective models, the same

M2

local type. Finally, assume that M1 and M2 are u-saturated. Then for every am E M1 there exists bm € M2 such that a,a and b,b, have the same local type. that

(This holds also for m = 0). Playing a FraissG-Ehrenfeucht game we deduce

-

(Ml,a) : (M2,%).

extensions MY

If M1 and

and Mi

M2

are not w-saturated we can get elementary

that are w-saturated and from

*-

(Ml,a)

(M2,%)

5

we dedu-

(Ml,Y) ? (M2,b). Now use the theorem that if every two models (of a theory T) that satisfy the same sentences out of some class S are elementary ce again

equivalent, then every sentence is equivalent (in T) to a Boolean combination of S.

members of

12. LOCAL INSEPERABILITY

Definition : Let Co and

C1 be two classes of models for L and

a sentence. Say that cp seperates Co

and

C1 if

is true in all members of

(0

one class and false in all members of the other. C and C

ble if

0

some sentence

if no sentence in L If

P

cp

in L

and

kq,

If

4

p

is first-order within the

{MEC : M does not satisfy F 1

first-order seperable. This means that for some M

are first-order sepera-

seperates them.

IMEC : M satisfies P I

if

1

seperates them. They are first-order inseperable

is some property of models then we say that

class C

let cp be

cp

are

we have : M satisfies P



for all M E C . is any set of formulas, then

(M,ao

,...,a,-l)

5

@(M',b0

,...,bk-l)

means

Definition : ( I ) Say that Co and C1 are locally inseperable if for every natural number n there exist a pair of models Mo E Co, M1 E C1 such that, for all r,s

< n, the same (up to isomorphism) collections, of

r-neighborhoods are realized in Mo lection V(r)(xo),

.. .,V(r)(xs-,)

corresponding collection )'(V (~(~'(x~),

J = (~(~'(y~), x.)

(11) Say that

s

disjoint simple

and M1. By this we means that for every col-

of disjoint neighborhoods in Mi (yo),

yj)

CO

,

...,V(r) (ys-l) in Midi

there is

a

(also disjoint) such that

for all j<s.

and

C1 are locally first-order inseperable if

119

Local and non-localproperties there exist Ma E Co, M1 and every collection of disjoint neighborhoods

for every n and every finite set of formulas Q r,s G n

such that for all

. . .,V (r) ( x ~ - ~ )in (yo), . . . v(r) in

V(r)(xo),

Mi, (i = 0 , l )

"(r)

M1-i such that

there is a corresponding collection (V(r)(x.),x.) J

all j<s. Evidently, if

Co

and

O(V(r)(yj),yj),

f

J

for

C1 are locally inseperable they are also locally first-

order inseperable, but the converse need not hold. As a corollary of the Main Theorem we have : Theorem 2.1.

: If

C,

and

C,

are locally first-order inseperable then they are

first-order inseperable. Proof :

Let cp be any first-order sentence. By the Main Theorem cp

is equiva-

lent to a Boolean combination of sentences of the form 3G(M $(r)(vi)

A.M.

i<s

d(vi,v.)>2r).

Let

be greaterthan all the r

R

and

s

that

l<J<S

consist of all the J l ' s such that

let 0

are involved and

this combination. Let M

E Co

M1 E C 1

0 rements of the definition with respect to n

hence cp does not seperate Co

and

Jl(r)(vi)

occurs in

be the two models satisfying the requiand

0. Then

Mo

I= cp

0

M1 (= cp,

C1.

In order to show that a certain property P

q.e.d. is not first-order within

C

it

{MEC : M satisfies P} and {MEC : M does not satisfy PI are

suffices to show that

locally first-order inseperable. In most examples that we have thought of these two classes satisfy the stronger requirement of local inseperability and it is this that one proves directly. Yet there are cases in which only the first property holds and for this reason we have introduced this weaker version. If two classes Co

and

C1

are locally inseperable then they are inseperable in

the somewhat stronger logic obtained as follows : Let

{P i}iEI

be any family of properties of models of the form

is a model for L

and

aEM. There is no restriction on the

be preserved under isomorphisms : If then

(M',a')

te P F )

to

satisfies Pi. For each L

and

let L*

for L, enlarge it to a model of xEM

such that

put : M

(VCk)(x),x)

I= cp(z) 0 Df M* I=

P.

(M,a)

where M

except that they

satisfies P. and (M',a')=(M,a) P i and each kEw add a monadic predica-

(M,a)

be the language thus obtained. If M

is a model

for L* by interpreting each P F ) as the set

M*

satisfies

cp(g).

P.. If cp(F)

is a formula of

L*

H. GAIFMAN

120

We shall call the logic thus obtained a monadic arbitrary local logic (of L), or for short, a MAL-logic. The word

"arbitrary"

is no restriction on the properties

is used to indicate that there

of the neighborhoods. We could define

Pi this logic, equivalently, by adding, instead of monadic'standardlyinterpreted predicates, new quantifiers 3!k), exists x

such that

Theorem 2.2.

If

Co

such that

(V(k);x),x) and

In order to prove it we first show

____ Lemma 2.3.

If

Co

are Mo E Co,

and

y = yo

is interpreted as

and

,...,ym-l E

"there

...'I.

L*

seperates them).

:

C1 are locally inseperable then for every m > 1

MI E C1 which realize the same m-neighborhoods of

That is to say, for each

-

Pi

C1 are locally inseperable then they are inseperable

(i.e. no formula o f

in the MAL-logic

."

"3.!k)x..

satisfies

x = xo ,...,xmbl E Mi

,

i = 0,1,

there

m-tuples.

there exists

satisfying :

It is easily seen that any isomorphisms of

(V(m)(x),x)

onto (V(m)(T),y)

carries

< m. 0'' * "xjs-l 0 s- 1 Hence the condition in the lemma is a natural generalization of the condition de-

v(r)

) isomorphically onto

(Xj

V(r)(yj

)

,...,yj

for all

r,s

fining local inseperability. The,lemma asserts that this generalization is already implied by local inseperability. Proof of 2.3. Let

t

:

> 0, k >

the models MO,MI Then M

0

and M

We show the following : 1

and assume that for all

s d n,

realize the same collections of

and all s

1 realize the same t-neighborhoods of

r

< 3k-1t +

I k

~ ( 3-3)

disjoint r-neighborhoods. k-tuples.

This implies the lemma. For, given m, take M O W O , MIECl which satisfy the requirement in the definition of local inseperability for n The proof is by induction on k. If k = 1, then 3

k-l

=

Zm-l .m+$(3m-3).

1 k .t+?(3 -3) = t, hence the

condition is that both models realize the same simple t-neighborhoods, i.e. same t-neighborhoods of

1-tuples.

Assume the conditions to hold for t and

k+l and let x

=

xO,

...,xk

be a

the

Local and non-localproperties

121

(k+l)-tuple in one of the models say Mo. We have to find the corresponding y ' s j in MI. First consider the case where d(x.,x.) > 2(t+l) for all i<ja. Then 1

j

V(t+l)(xj), Since t+l

=

0,...,n

3

form a collection of

< gk+l. t + 13. (gk+I-3)

k+l

disjoint

(t+l)-neighborhoods.

the condition implies the existence of f * (V(t+')(x.),x.) (V(t+l)(y.)y.), j' J J J J are disjoint. If ~€v(~)(x.), z'€V(~)(X~,)

yo, ...,yk E MI and of isomorphisms

j

re the V(t+l)(y.), j # j'

V(t)(xj),

U f j 2. Hence V(t) (x) - is the disjoint sum of the models j = 0, k. The same argument applies to the V(t)(yj)'s. Hence is an isomorphism of (V(t)(T),:) onto (V(t) (Y) - ,Y). -

then d(z,z')

...,

d(xi,x.) G 2(t+l)

Now assume that for some i < j < k

3

; say, without loss of gene-

rality, d(xk-l, x,) < 2(t+l). Replace k + l by k and t by 3(t+l) ; since 1 k k 1 3k- 1 (3(t+l)) + 7 (3 -3) = 3 .t + 2 (gk+I 3) the condition for k + l and t

-

implies the condition for k and (with 3(t+l)

instead of t)

3(t+l).

By the induction hypothesis for k

there is a k-tuple yO,...,yk-lEMI

and an isomor-

phism :

Since xk E v(~(~+')) ( x ~ - ~ ) ,we have V(t)(x,J yk

=

Df f(x,).

that V(t)(yk)

It is easily seen that c V (3t+2)(yk-l).

function inside V(3(t+1))

(xo,.

Mo and similarly for V(t)(yk) in §l).

Hence f maps V(t)(x,)

v(~) (G) onto

yk

Put

C V (3t+2) "(2(t+l))

(yk-l) and, consequently,

Moreover, for members of V(t)(xk)

.. ,xk-1 )

the distance

is the same as the distance function in

and M I . (This follows from the argument for -(I) onto V(t)(yk).

It follows that f maps q.e.d.

vet)(7).

Proof of Theorem 2.2 : Given cp

in L* we have to show that cp does not sepeP(k) in cp. Let ? rate Co and j be obtained from L by adding these predicates and for every model M for L i = 0 , I . Let $ be the Let = {$ : MECi}, let 2 the enriched model for C1. There are alltogether finitely many

2.

Zi

2.

maximum of all k such that P!k) is in For each n, there are by lemma 2.3 J models M EC and MIECl that realize the same (n+g+l) - neighborhoods of n0 0tuples. Let x = xo, x n-l € Mo, Y = yo,. ..,Y,~ E M1 and let

...,

: v(n+g+l)(;)

Vfk)(z)

I

v(n+2+l) (y). If z€V(")(;)

onto Vtk).(f(z)).

Whether F'y)(a)

then for each k < g holds in Mi

(i

f maps

= 0,l)

depends only

H. GAIFMAN

122

on the isomorphism type of (V(k)(a),a). Hence for z€V(")(X) we have (k) 2O )= Pj( k ) ( ~ ) 0 81 I=P.J (fz). Thus f maps V(")(X) isomorphically onto V(n) (Y ), where these are considered as neighborhoods in Go and G1. Consequently to and ?1

are locally inseperable. By Theorem 2 . 1 ,

cp

does not seperate Co and C1. q.e.d.

By analogy to monadic local properties we can consider n-ary that is to say, n-ary relations, R, such that the holding of by the isomorphism type of

(V(k)(a),a),

natural number. Let the AL-logic

local properties ; R(Z)

is determined

where k = kR is an arbitrary, but fixed,

(i.e, arbitrary local logic) be obtained Ly

adding predicates denoting arbitrary local properties and let the n-AL-logic be the one in which only predicates of arity



2

one usually changes the distance and a k-neighborhood in the enri-

ched model may contain members which are not in the k-neighborhood in the original model. We do have however :

~

Theorem 2 . 4 .

If Co

and

C1 are locally inseperable and for each k, only fini-

tely many isomorphism types of the models of

Co U C 1 , then

k-neighborhoods of

Ci

and

(n-1)-tuples are realized in

C1 are inseperable in the n-AL-logic.

Actually, it suffices to have a locally inseperable pair of subclasses Ci

C

Co

,

Ci c C1 satisfying the condition of finitely many realizable isomorphism types. The second claim is trivially implied by the first. For if

Ct0 and

CI1 are

inseperable in some logic so are any pair of classes Co,C1 that include them. If, for

some m, all simple 1-neighborhoods in the models of

Ch

U Ci

haveQm

members, then also, for each k,n, the k-neighborhoods around n-tuples are uniformly bounded in size. (This situation obtains in the forthcoming examples). Then, evidently, the condition of theorem 2.4.holds ; but the theorem becomes superfluous, because within

Ch U Ci

, every

formula in the AL-logic is then equivalent to a

first-order formula. Theorems 2.2. 1-neighborhoods realized in

and 2.4. are of interest only when the sizes of

C; U Ci

are not uniformly bounded by some integer.

The proof of 2 . 4 . is given in the appendix

123

Local and non-localproperties Examples

In all cases the property in question, P, is not first-order definable within the indicated class, C. The subclasses IMEC : M satisfies P } not satisfy P }

and

IMEC : M

does

are locally inseperable (except for the last example where they

are only first-order inseperable). This is shown by pointing out a pair

Mo, MI E C r,s

tions of be

of

prototypes such that Mo

satisfies P, M1 does not and, for all

sufficiently small with respect to the size of the prototype,the same collecdisjoint simple r-neighborhoods are realized in both. The models can

s

"blown up"

in the obvious way to any desired size, while

retaining these

properties. 1.

P = Connectedness, C

=

class of finite graphs.

MO

M1 Fig 1

If on each circle we have > 2r+l points then all simple r-neighborhoods are of the form :

and if, on each circle, we have > (2r+l).s

points then s

disjoint simple

r-neighborhoods can be realized. In

[CHI

the method of Fraissc-Ehrenfeucht games is applied

to

this prototype

pair in order to show that connectedness is not first order definable in C. In principle

the method of games is applicable to the other examples. However, if

the models are not as homogoneous, a description of the strategy for n

moves canbe quite involved,

2nd

player's winning

whereas a glance may suffice to

see that the neighborhoods are the same. What is more important is that the method indicates the way in which the prototype models should be constructed. In the following examples we let the drawings speak for themselves.

H. GAIFMAN

124 2.

P

=

a - connectedness

C

=

finite (a-1)-connected graphs.

(A graph is k-connected if the removal of any k-1 edges does not disconnect it). Fig 2.2.

corresponds to the case

L = 4 . In general,

bridges between the two components of M I . For

L- 1

I,-even,

,t

is the number of is the nomber of.edges

issuing from any ordinary vertex (i.e. vertex not connected by a bridge). The case of an odd

I?

is obtained from that of

ordinary vertices in Mo

!,+I

by removing an edge between two

and the corresponding edge in one of the components in

M1.

MO

3.

P

=

Fig 2.2

being planar

C = class of finite graphs

(A-graph is planar if it is representable in the plane so that each edge is an arc and the arcs do not intersect except at their common end points. In Fig. 2.3. is planar, but

M, is not).

Fig 2.3

MO

125

Local and non-localproperties 4.

P C

=

Hamiltonian (i.e.,

=

class of finite k-regular graphs with a Hamilton path. Here k

3 . For k = 3 , 4 , 5

having a Hamilton cycle)

we can restrict C

should be

further by adding the requirement that

the graphs be planar. (A Hamilton path is a path in the graph passing through each vertex exactly once.

If such a path is also a cycle then it is a Hamilton cycle. A graph is k-regular if every vertex has degree k, where degree (x) = number of edges containing x).

MO

Fig 2 . 4

Figure 2 . 4 . is the construction for C

=

class of 4-regular planar graphs having

a Hamilton path. The construction for k>4 is obtained by replacing each vertex in Fig. 2 . 4 . by a graph

G

graph. Let

(a,b)

(a,a')

and

Hamilton cycle containing

as follows. Let (a,a')

G*

be a k-regular Hamiltonian

be two edges in G* and not containing

such that there is an (a,b)

. Get

G by removing

these two edges (without removing vertices). Now replace each vertex of 2 . 4 . by a copy of

G and use edges issuing from

rent copies. Figure 2.5.

a,a', b

in order to connect the diffe-

indicates how this is to be done. For k = 5 take G*

to be the icosahedron. Since this is planar the construction can be carried out so as to yield a planar graph. (For k>5 the graph cannot be planar, since each planar graph has a vertex of degree 4 5). We leave the case k = 3

for the reader.

H. GAIFMAN

126

Fig 2 . 5

I' (W

4.

=

a, 0

P

=

=

a', O = b. arrows show parts of a Hamilton path) C = connected finite graphs

Eulerian

(A graph is Eulerian if it contains an Euler cycle, i.e. a cycle passing through

each edge exactly once).

As is well known, a (finite) connected graph is Eulerian iff each vertex has an even degree. This is obviously a local property. Consequently, within the class of connected graphs those that are Eulerian are not locally inseperable from those that are not ; in fact

-

they are definable in the MAL-logic. Yet they are first-

order locally inseperable. This is seen by letting C* the form

consist of all graphs of

:

a

Fig 3.7

Let

CocC*

consist of those having an even number of vertices and let

Then, for GEC*, G

is Eulerian or not according as G E C o

place the binary predicate R each M E C*

let M'

or

C1=C*-C0-

GEC'l. Now re-

of our language, L, by a monadic predicate, P. For

be the model for the new language, L', obtained by interpreting

Local and non-localproperties

P as {xEM : x # a and x # b}. Then

This implies that every sentence ( P E L

127

:

is translatable into some tp'fL'

such

that, for all M E C*,

If rp were to seperate C@ C l 0 = {MI : MECo}

models of L'

from

-

C1 then

~ p '

would have seperated

from C'l = {M' : M E Cl}. But this is impossible, because for

we have d(x,y) =

for all

x # y

,

implying that each simple

neighborhood consist of one point and consequently, that

.

Cl0 and CI1 are locally

inseperable

$3.

CERTAIN TRANSITIVE MODELS

Levy's hierarchy classifies formulas in the language of set theory as follows : C -formulas are those in which all quantifiers are bounded, i.e., of the forms 0

3xEy, VxEy. The higher levels are obtained in the usual way by tacking on alternating blocks of unbounded quantifiers. We have : x = 0

x

c=>

VyEx (y#y)

is transitive

x is an ordinal

0

VyEx VzEy (ZEX)

0x

is transitive AVuEx VvEx [u=v V uEv V vEul

x

is zero or a successor

x

is a natural number

0

0

x is an ordinal A(X=OV3UEX Vvfx(v=u V veu))

x is zero or a successor

A

VyEx (y is zero or a successor).

Consequently all these notions are Xo Now in

ZFC

(i.e. expressible in ZFC by

A

z

is finite

x

is finite VZV~EX[ z is not a mapping of x- {y}

0

3z3y [y is a natural number

XI onto XI

maps y onto

x

The first is easily seen to be

C 1 , the second-lI1. Thus finiteness is, in ZFC,

a A -notion. The natural question is whether it is Co. 1

C-formulash 0

finiteness can be defined in either of the two ways :

The answer is negative in

the strongest possible sense : There is a transitive infinite set A , of rank w , such that for any

Eo-

formula

H. GAIFMAN

128

cp(v), A

there exists a finite transitive subset A ' c A ,

iff it is true for A ' . Furthermore A

te is not characterizable

by

such that in cp'

is true for

a Z o - formula in the real world.

Note that for any cp(v)ECO, with v cp' (v)ECO,

such that cp(v)

is primitive recursive. Thus being fini-

as its only free variable, there exists

all quantifiers are of the form QxEv

and we have,

in pure logic : is transitive -+ (cp(v)

v cp' (v)

is obtained by replacing every

..."

every "VxEu V'(v),

by

"VxEv (xEu

--+

cp'(v))

"3xEu.. ." by "3xEv (xEu ...)" and ...)". Furthermore, we can assume that in A

v occures only as a bound for quantifiers. For if x

ferent from v

is any variable dif-

then its occurences inany quantifier-free part are within the

scope of some QxEv, hence the occurences (i.n a quantifier- free part) of VEX and v = x

can be replaced by

ly v = v and

vEv

vely. Now let cp"

x # x , and the occurences of xEv- by

are replaceable by, say Vy€v (y=y)

x =

X.

Similar-

3yEv(y#y),respecti-

be the sentence obtained by removing the bound on the quanti-

fiers, i.e., replacing each QxEv by A

and

Qx. It is easily seen that for a transitive

:

we have

Vice versa the satisfaction of any sentence in

(A,ErA)

'

is equivalent to the

A

of some Z --formula cp(v). Hence we have to construct a tran0 sitive infinite model M such that for any sentence cp there exists a finite satisfaction by

transitive M'cM

such that

The construction to be given here is similar to the one used in

[GI.

To show

that M has the desired property a rather involved argument was used there,which relied on the generalization of Marcus's result. Here we get it at a glance by realizing that the classes {MI

and

{M'cM:M'

is finite and transitive) are

locally inseperable. Construction of M. Let

l nj+l is isomorphic to the r-neighborhood of some h(w)

at level E (n , I I ~ + ~ IThe . argument given for M works also for Mm. The essenj+l tial point is that by truncating M at level % one converts all branching points at that level into leaves and consequently neighborhoods may change from type (111) to types (11) and (I). But these can be simulated in the lower region. using the leaves at level nj+2. (It is for this reason that many

leaves).

T

should contain

H. GAIFMAN

132

Given If m

r

and

=

and

hoods in M

s

j

choose k>j+3

2 j +I 0

such that nj+l - n . > 2r+l

!?,(v)

.k+l

=

$*

?(-TREE T starts with identity if for some o ~ * < B >E T for all B y

.

5.4 The treeclasses Ha[B] a < B T i s in Ha[B1 if HI. T is [a,@[-homogeneous tree H2. T is strongly well-founded H3. T starts with identity

k

B is a limitordinal

The following trees are in HO[wl ID, SUC, DOUBLE, SQUARE, ID + 2-EXP, EXP while 2-EXP and ONE are not.

Introducing homogeneous trees

155

5.5 a-commuting operations An n-ary operation 'IT is a-commuting if 01. TI : Ha[ 81" + Ha[BI for any limitordinal B For any TI, T2, ...., Tn E Ha[B1 02. f E ( a ) J ( y , B ) * f-l('IT(T1,..,T,)) = I f ( f-1 (T1),..*f-'(Tn)). 03. 6 > B * IT(T,[~I,..,T~[~~) = 'IT(T1,..,Tn)[61 (6 limit) One of the programs in the theory of homogeneous trees is to show that the usual ordinalnotations can be given by members of Hn[ol and that operations on ordinalnotations can be given by n-commuting operations. We have for example that + and a are both 2-ary 0-commuting operations. The use of the functionclass J ( y , @ ) in 02 above comes from the dissertation of Lenz [3] In my Miinchen-lectures there is used the See [2] In that case it is not true that functionclass I ( y , B ) is commuting.

.

.

.

5.5 Iteration Two basic operations in proof-theory are the Grzegorczyk-operation in hierarchies of number-theoretic functions - the derivative of normal functions of ordinals The 2-ary 0-commuting operation iteration generalizes both. It is defined as follows:

-

1. Start with the well-founded tree S . 2. For each topmost node u of S : if u tack on 3. The result is the tree then above

Tn

.

(

Here

It(S,T)

.

u

= <s

T1 = T

,';$;T

...,

S > = Tknn

,

)

Using the theory so far we can give ordinalnotations for all predicative ordinals. We havefor example:

H.R.JERVELL

156

6. RECURSION 6.1 Restriction Let T be a node in tree T T = The restriction of T to T is given by:

.

.

T

-

....,tn>

Here we have cut off everything to the right of E T 2. -k+l* 6 T

T

is the

wk

and

y

E

[a,@[ ,

y

E

[ a B[

,

such that

6 . 3 Recursion Recursion is a powerful way of producing new homogeneous trees. It was first used by Jean-Yves Girard 111 to get h(T) from T. The other operations defined above are all predicative, while the recursion is not.

Introducing homogeneous trees

157

Assume S,T trees, T well-founded and n a unary operation on trees. We define Rec(n,T/r)S by recursion over the Brouwer-Kleene ordering o f nodes T of T. Recursionstart: If TIT does not contain any critical node, then Rec(n,T/r)S = S In this case T/T contains only nodes of the form mn , and T is among the first few nodes in the Brouwer-Kleene ordering.

.

Recursionstep: If wk is the critical node of T/T , then Rec(n,T/r)S is the tree built up from 1. 6*nRec(n, ( T / T ) /mk*6)S where wk*6 is a topmost node of T/T k 2. 6*Rec(n, (T/T)/m * 6 ) S where mk*6 is a not-topmost node of T/T

.

We then define

Rec(n,T)S

:=

.

Rec (n,'rYy< > ) S

6.4 The recursiontheorem

If n is a unary a-commuting operation, then a-commuting operation.

AS,T Rec(n,T)S.is a 2-ary

For the proof we refer to our Miinchen-lectures [ 2

or Lenz [ 3 1

.

6.5 Some applications We give a few applications to indicate the power of recursion. INDUCTIVE-0 := Ii(EXP,EXP) INDUCTIVE-n+l := Rec(lt,INDUCTIVE-n)EXP We then have that I INDUCTIVE-n I is the proof-theoretic ordinal of n times iterated inductive definitions, and in particular INDUCTIVE-I gives an ordinalnotation for the Howard ordinal. It is interesting to note that these ordinalnotations are here defined without using higher number classes. These definitions can be extended by using the unary commuting operation lnd" defined by Ind' (T) := Rec (It,T)EXP Ind"" (T) := Rec(lndn,T)EXP We have not worked out the details for how these match up with the usual proof-theoretic ordinals. Define

H.R. JERVELL

158

7. REFERENCES

[I] Girard, J.Y., n:-logic, part 1. Manuscript 1979. To appear in Annals of Mathematical Logic. (Revised version 1981) [ 2 ] Jervell, H.R., Homogenous trees. Lectures given at the Uni'versity

of Miinchen, Summer 1979. Manuscript. Troms0. 131 Lenz, U., Homogene Bsume und die Howardzahl. Dissertation.

University of Miinchen. [ 4 1 Masseron, M., Rungs and Trees. Manuscript 1980. To appear in Journal of Symbolic Logic. (Revised version 1981)

PROCEEDINGS OF THE HERBRAM) SYMPOSIUM LOCK COLLOQVIUU '81,J. Stem (editor) 0North-Holland Publishing Company, I982

159

EXPONENTIAL D I O P H A N T I N E REPRESENTATION

OF RECURSIVELY ENUMERABLE SETS J.P. Jones and Ju. V. Matijasevi? University of Calgary and Steklov Mathematical Institute of the Academy of Science, Leningrad

En 1961 M. Davis, H. Putnam et J. Robinson ont demontre que tout ensemble recursivement Qnum6rable est ddfini par une 6quation diophantienne pouvant comporter la fonction exponentielle. De fason plus precise, tout ensemble recursivement enumerable, A peut se definir par n € A

,...,x

3x1,x2

le P et Q

(P(n,x,

,...,x

) = Q(n,

si et seulement si

,..., xw)),

x1

dans laquel-

sont des fonctions d'entiers naturels obtenues 1

partir des operations d'addition de multiplication et d'exponentiation. En 1979, ce resultat fut am6liorb par Ju. V. Matijasevix qui a prouve que trois inconnues x,, x2 et x suffisent (i.e. w (uni-pli) :

=

3 3) et que la representation est univoque

c'est-2-dire, la solution x1,x2,x3 est unique

pour n € A . En anglais cette representation est appele 'singlefold', en russe

'odno-kratno'.

Dans le present travail, ce resultat est encore ameliore. Nous d6montrons que tout ensemble enumerable peut ttre re-

present6 par une inegalite

3z,y[P(n,z,y)

< Q(n,z,y)

], dans

laquelle z ety sont des entiers uniques (representation univoque), P et Q sont des fonctions sur les entiers naturels ddfinies 1 partir des operations d'addition, de multiplication et de l'exponentielle

ZX (en base 2 ) . Ceci implique une forme

forte du theorhe initial des trois inconnues ci-dessus, car ici l'exponentiation que sous la forme ZX.

xY

sous sa forme generale n'apparait

De plus, on montre que deux iterations

de l'exponentielle suffisent, i.e. ZZx. Des resultats connexes et varies sont ddmontrds : un ensemble enumerable a une representation sous la forme 3zVy[R(n,z,y)

S(n,z,y)l,

dans laquelle R et S

sont obtenus

1 partir de l'addition, de la multiplication et de l'exponen-

tielle en base 2 .

J.P. JONES and Ju.V. MATIJASEVIC On montre a u s s i qu'une r e l a t i o n glementaire de Kalmar a a u s s i

une r e e r g s e n t a t i o n , avec une v a r i a b l e (que l ' o n peut supposer bornbe), i . e . sous l e s formes 3y[P(n,y) G Q(n,y)l Vy[R(n,y)

Q

S(n,y)] ;

3y,x[P(n,y,x)

=

et

e t donc sous l e s formes

Q(n,y,x) 1 e t Vy'y3x[R(n,y,x) = S(n,y.x)

l e s q u a n t i f i c a t e u r s sont born&).

1

(03

Ces Qnonces c a r a c t e r i s e n t

l e s r e l a t i o n s elgmentaires de Kalmar. Comme l'ensemble des nombres premiers e s t elementaire de Kalmar, il s ' e n s u i t que l'ensemble des nombres premiers posssde une r e p r e s e n t a t i o n SOUS

l a forme ci-dessus.

In 1961 it was shown t h a t every r e c u r s i v e l y enumerable s e t i s exponential diophant i n e . M. Davis, H . Putnam and J u l i a Robinson [19611

proved t h a t f o r each r . e .

set

W there e x i s t s an exponential diophantine equation

P(a, z 1 solvable i n the unknowns

P

and

Q

,....zw) z l , ...,zw,

=

Q ( a , z1,...,zV)

(1)

i f , and only i f a belongs t o t h e s e t W . Here

a r e functions b u i l t up from n a t u r a l numbers and v a r i a b l e s a , z l ,

by a d d i t i o n , m u l t i p l i c a t i o n and exponentiation. The unknowns

zl,

...,z

...,zv

a r e un-

derstood t o range over n a t u r a l numbers. In t h e papers of Matijasevi: e f f e c t t h a t each r . e . i . e . one i n which

xl,

119741 and [I9761

t h i s theorem was improved t o the

s e t has a - s i n g l e f o l d exponential diophantine r e p r e s e n t a t i o n ,

...,xw, when

the parameter a such t h a t

they e x i s t , a r e unique. Thus f o r each value of

a EW, t h e r e i s one and only one s o l u t i o n

zl,.

.. ,z .

In the paper of Matijaseviz [1979] t h i s r e s u l t was f u r t h e r improved t o t h e e f f e c t t h a t t h e r e always e x i s t s a s i n g l e f o l d exponential diophantine r e p r e s e n t a t i o n i n three unknowns, i . e . every r . e .

s e t can be represented i n t h e form

Here we have suppressed mention of the parameter a . P and Q a r e functions obtained from n a t u r a l numbers and v a r i a b l e s r a t e d exponentiations of type

a,x,y,z

by a d d i t i o n , m u l t i p l i c a t i o n and i t e -

u", i n two v a r i a b l e s .

In t h e paper of Jones and Matijasevi:

[1981] t h i s r e s u l t was f u r t h e r improved t o

unary s i n g l e f o l d , t h r e e unknown, exponential diophantine r e p r e s e n t a t i o n , i . e . one based only on powers of two. We obtained, a r e p r e s e n t a t i o n of type ( 2 ) i n which and

Q

a r e functions b u i l t up from n a t u r a l numbers and v a r i a b l e s

a,x,y,z

P

using

only t h e operations of a d d i t i o n , m u l t i p l i c a t i o n and t h e r a i s i n g of 2 t o a power,

Exponential diophantine representation i . e . e x p o n e n t i a t i o n s of t y p e

2'.

Since

uv

i s a two-place

a one-place f u n c t i o n , w e c a l l o u r new b a s e 2 r e p r e s e n t a t i o n

161 f u n c t i o n and

5

is

' 2

Unlimited i t e -

r a t i o n of e x p o n e n t i a t i o n i s t o be u n d e r s t o o d h e r e b u t a c t u a l l y two l e v e l s of expon e n t i a t i o n are s u f f i c i e n t f o r Theorems I and 2 . (Terms of t h e t y p e where

2 ~ ~ a' p p~e a)r ,

i s a polynomial i n z . )

f(z)

The r e s u l t s of t h e p r e s e n t p a p e r a r e e s s e n t i a l l y t h e same as t h o s e of J o n e s Matijasevi:

[ 1 9 8 1 ] . The o n l y d i f f e r e n c e b e i n g one o f c o m p l e t e n e s s . The proof h e r e

i s s e l f c o n t a i n e d . N e c e s s a r y r e s u l t s of M a t i j a s e v i :

[19791 a r e h e r e i n c l u d e d where

needed so t h a t i t i s n o t n e c e s s a r y t o r e f e r t o t h i s e a r l i e r p a p e r t o u n d e r s t a n d t h e p r o o f s . We p r o v e t h e f o l l o w i n g theorems.

THEOREM 1 . Every r e c u r s i v e l y enumerable s e t can be r e p r e s e n t e d i n t h e form 3z3y [ P ( z , y ) G Q ( s , y ) ] where

P

and

Q

are u n a r y e x p o n e n t i a l d i o p h a n t i n e e x p r e s s i o n s . F u r t h e r m o r e t h e

r e p r e s e n t a t i o n i s s i n g l e f o l d and t h e second q u a n t i f i e r , 3y may be bounded. This i s a c t u a l l y a s t r o n g e r r e s u l t t h a n t h e t h r e e unknown theorem. Taking P(x,y,z)

=

every r . e .

Q(z,y) + x

we h a v e a s an immediate c o r o l l a r y t h e t h r e e unknown theorem,

s e t c a n be s i n g l e f o l d r e p r e s e n t e d i n t h e form ( 2 ) . A l s o

P

and

Q

are

h e r e u n a r y e x p o n e n t i a l d i o p h a n t i n e f u n c t i o n s , b u i l t up o n l y o f powers of 2 . Furt h e r , x and y may b e supposed t o be l e s s t h a n some bound, where t h e bound t a k e s t h e form of a n i t e r a t e d e x p o n e n t i a l f u n c t i o n of t h e f i r s t unknown, z . We w i l l a l s o prove THEOREM 2. E v e r y r . e . where

R

and

S

set c a n be r e p r e s e n t e d i n t h e form

3zVy [R(z,y)




suppose that any solution to (3) has the property that w1 >w2,

....

$7"

and

that each of w2, w3, w is uniquely determined by wl. (If that i s not the case, then we introduce a new variable w and consider instead of (3) the equation (w - 1 -w -(wl +w2)2 - (w, +w2 +w3) 3

..

... -

= 0,

1

v 2

2

],...,w")

(w, +... +wV) ) + D (w

which has the same solutions as (3) and which possesses the required property with w

playing the role of w 1 ' )

Now it will be necessary for us to use a number-theoretic theorem of E. Kummer r18521, afterwards restated and reproved in other papers, for example Singmaster [19741.

J.P.JONES and Ju.V. MATIJASEVIC

164

KUMMER'S THEOREM. L e t

denote t h e m u l t i p l i c i t y t o which 2 d i v i d e s

K(t)

t

and

denote t h e sum of t h e d i g i t s i n t h e binary expansion of t . Then

let u(t)

u : if

We n o t e two obvious p r o p e r t i e s of t h e f u n c t i o n o ( t 1Z S + t , ) Also i f

T

( + T2)

then

= o(t,) +o(t2).

E~

(7)

IT1 < 2' G 2 r ,

i s an i n t e g e r such t h a t

~

t2 < Z S ,

then

[ I , s+l]

if

T

Ir-s+l,rI,

if

T < 0.

0,

We introduce two f u n c t i o n s

On t h e b a s i s of If

T

(7)

*

Here and i n what follows

*

i t i s easy t o show t h e following :

IT1 < 2'

i s any i n t e g e r such t h a t

o(F (r,T))

(8)

and


o(A(z))

(zV+zv-')T(z)

+ ( R ( z ) -S(z)),

i f (3w2,.. .,wv)(D(z,w2, & (i3wl

,...,wv)(wl < z &

< (zu+zv-')(T(z)+2S(z))

...,wv) = O ) ) D(wl

&

,...,wu) = O ) ) ,

i n t h e o t h e r case.

Put

According t o (15) and ( 2 4 )

From t h i s and ( 2 3 ) we f i n d t h a t

holds i f and only i f

z

i s t h e l e a s t value of

( 3 ) . (This w i l l imply t h e uniqueness of

z

w1

i n any s o l u t i o n t o equation

i n condition ( 4 ) . I f t h i s s i n g l e f o l d

aspect of t h e r e p r e s e n t a t i o n i s unimportant t o t h e r e a d e r , then t h e construction can be s i m p l i f i e d somewhat. It i s enough t o put v- 1 M(z) = z T + R - S).

A ( z ) = B+(z,z)

Now we make u s e of t h e i d e n t i t y (6) t o r e w r i t e (26)

Thus ( 2 7 ) holds i f and only i f

z

and

i n t h e form

i s t h e l e a s t value of

w1

i n any s o l u t i o n t o

equation ( 3 ) . In order t o give a unary d e f i n i t i o n of t h e condition ( 2 7 ) , w e w i l l use t h e f o l lowing generating function f o r t h e symmetric binomial c o e f f i c i e n t ,

Exponential diophantine representation

167

This s e r i e s (28) can be derived from t h e binomial s e r i e s

by using t h e i d e n t i t y

Now r e p l a c e

x

by

l/u

i n (28) and multiply by

un

t o o b t a i n t h e following

s e r i e s ( v a l i d f o r IuI > 4)

We can use a geometric s e r i e s t o estimate t h e s i z e of t h e f r a c t i o n a l p a r t of (31).

The s e r i e s (31) a l s o gives a d i r e c t formula f o r t h e symmetric binomial c o e f f i c i e n t i n terms of t h e remainder function and t h e i n t e g e r p a r t f u n c t i o n ,

We will use t h i s i n $2. To r e t u r n t o condition ( 2 7 ) ,

we see t h a t i f

1.

[

2

M

divides u,

then ( 2 7 ) can be r e w r i t t e n i n t h e form

[ " J .J1-4/U For any

E Q

1/2, i f

(4/E)4* + 4 < u ,

(34)

then by (32) we can r e w r i t e condition (34)

i n t h e form t h a t , f o r some (unique) n a t u r a l number y

J.P. JONES and Ju.V. MATIJASEVIe

168

Taking

E

=

118, we see that since

118, condition (35) implies

E'

Q

114.

Similarly, condition (36) implies condition ( 3 5 ) with

E =

318. Hence for

32. qA + 4 < u , condition ( 3 4 ) is equivalent to condition ( 3 6 ) . Condition (36) is

Im-4/u

in turn equivalent to 2

- 2 2 4

U2A

Q

I

[-

(37)

2My.

which is in turn equivalent to the condition

4

-

2

22My2]

Q 2 2My2

,

which is in turn equivalent to the condition

-

4[u 2A+1

2 2My2 (u-4)j


0 and P r o j e c t i v e Determinacy h o l d s .

ous s e t . Thus we see a sharp c o n t r a s t between t h e n = 0 and t h e n > 0 case.

This points

o u t t o a new d i s c r e p a n c y between t h e s t r u c t u r e t h e o r y o f t h e f i r s t and t h e h i g h e r odd l e v e l s o f t h e a n a l y t i c a l h i e r a r c h y . The p r o o f o f t h i s theorem makes s u b s t a n t i a l use o f Q-theory, a l t h o u g h i t s s t a t e ment c l e a r l y has a p r i o r i n o t h i n g t o do w i t h it. An e x p o s i t i o n o f Q - t h e o r y can b e found i n [3],

and t h e r e a d e r i s assumed t o b e more o r l e s s f a m i l i a r w i t h it,

+

~

A S . KECHRIS

180

although we w i l l summarize below t h e r e l e v a n t p a r t s t h a t w i l l be needed here. Our a p p l i c a t i o n o f Q-theory i n t h i s paper i s q u i t e elementary, so i t i s hoped t h a t i t w i l l b r i n g o u t c l e a r l y some o f i t s e s s e n t i a l ideas and methods, t h a t can be basic i n g r e d i e n t s i n more e l a b o r a t e uses and a p p l i c a t i o n s o f t h i s theory. This paper i s organized as f o l l o w s : 1 discuss sow r e l a t e d r e s u l t s on A2n+l

I n $ 1 we prove t h e main theorem.

-

and Q2n+l

-

I n $ 2 we

e n c o d a b i l i t y , and i n $ 3 we

mention some open problems. I n conclusion, we would l i k e t o thank D r . I l i a s Kastanas, f o r many s t i m u l a t i n g discussions on the s u b j e c t matter o f t h i s paper. $1. Proof o f t h e main theorem. We w i l l f i r s t g i v e t h e p r o o f f o r t h e case n = 1 i n 1 .l.w i t h o u t worrying about t h e amount o f determinacy used, i n order t o make t h e key ideas i n v o l v e d more transparent. to all n

2

1 i n 1.2.,

Then we w i l l discuss t h e g e n e r a l i z a t i o n

and f i n a l l y i n 1.3. t h e t e c h n i c a l m o d i f i c a t i o n s needed t o

b r i n g down t h e l e v e l o f determinacy used. 1.1.

I n t h i s subsection we s h a l l assume f u l l AD (beyond the b a s i c theory ZF + DC) ( A c t u a l l y ADLcIR1is enough, since the theorem i s

and prove Theorem 1 f o r n = 1.

absolute between t h e r e a l w o r l d and L[IR],so

we can work e n t i r e l y w i t h i n L[IR]for

t h i s proof). The proof w i l l use Mathias f o r c i n g (see [5]) over an appropriate i n n e r model o f ZFC.

This model i s an "analog" o f L f o r t h e t h i r d l e v e l o f t h e a n a l y t i c a l h i e r a r -

chy and i s defined as f o l l o w s : For each r e a l a E ww l e t L[a] be t h e r e l a t i v i z e d t o a c o n s t r u c t i b l e universe, and consider HODLCa1,

t h e i n n e r model o f a l l HOD w i t h i n L[a] sets.

For each

let

c o n s t r u c t i b i l i t y degree d = [a],

LCdl = LCal, and consider the ultrapower M~ =

nd

HOD^[^]/,,,

where p i s t h e M a r t i n measure on c o n s t r u c t i b i l i t y degrees, 7.e. by cones.

We w i l l need t h e f o l l o w i n g f a c t s about M,

Kechris, Martin, Solovay and Woodin (1) The s e t o f r e a l s i n functions from w i n t o w.)

--

t h e one generated

(proved i n [3] and due t o

see [3] f o r r e l e v a n t references).

M3 i s 9.,

(By r e a l s we always mean members o f uw,

i.e. A quick d e f i n i t i o n o f Q, i s the f o l l o w i n g : A r e a l a beZongs to Q, i f f a i s A\ i n a countable o r d i n a l , i . e . there i s 6 < w1 1 such t h a t a e A3(w) f o r a l l ( r e a l ) codes w o f 5. More than that,for each r e a l a , ifM,[a] t a i n i n g M3 and a then ww

n

M,CaI

i s t h e smallest i n n e r model of ZFC con= Q,(a),

Effective Ramsey theorems

181

where Q3(a) i s t h e r e l a t i v i z e d t o a Q., ( 2 ) M, s a t i s f i e s a "dual Shoenfield absoluteness" theorem a t t h e t h i r d 1 1 l e v e l o f t h e a n a l y t i c a l hierarchy, i . e . f o r each 1, formula cp(a) t h e r e i s a II,

-

formula cp*(a) ( e f f e c t i v e l y constructed from c p ) such t h a t V(a)

MCd

I= cp*(a),

and s i m i l a r l y interchanging t h e r o l e s o f C: It can be shown t h a t M,

between M,

n\.

and

1 i s n o t C3-correct ( i . e . Zi formulas are not absolute

and t h e universe).

This strange "dual absoluteness" w i l l t u r n o u t t o be c r u c i a l t o the argument be1ow.

(3)

M,

I= 3

(4)

Q,

has t h e f o l l o w i n g r e f l e c t i o n p r o p e r t y :

measurable c a r d i n a l s .

3a

(5)

f

Q,P(a)

and

-

3a

f

1 I f P(a) i s II, then

1 A3P(a).

are countable ( i n t h e universe).

F i n a l l y we w i l l need t h e f o l l o w i n g d e f i n a b i l i t y estimate, strengthening ( 5 ) . 0 1 1 (6) L e t y, be t h e f i r s t n o n - t r i v i a l I13-singleton ( i . e . a n3-singleton which

1 1 1 i s n o t A, and i t i s A, i n any o t h e r such II,-singleton

-- t h i s i s a good analog o f 0 1 0 0 the Kleene Q a t t h e t h i r d l e v e l ) . Then t h e r e i s a r e a l x 8 ~ ~ ( y , )such t h a t x 0 0 0 enumerates Q,, i . e . Q, = { ( x :,) n e w } , ( x )n # ( x ), i f m # n and i f ~

1

E

P

~

C

X

E

{ Z( x 0)~, : aA ( n ) = o }

EM,

1 0 E A,(Y,)

enumerating P (P i s countable by ( 5 ) ) . This j u s t says M t h a t t h e r e i s an enumeration o f (power(ww)) which i s "A; i n y j " .

then t h e r e i s x

We proceed now t o prove o u r theorem. Let

P be Mathias' n o t i o n o f f o r c i n g i n M.,

(5,s) e

max(s) < min(S).

P consists o f a l l p a i r s

Order these conditions by (S,S)

5

(t,T) o t c_

L e t A c_ [wlube agivenn,1 set. be t h e : 1

5 A

L e t cp be a

f

A

o

rpp(X)

o

M3[X]

By a basic f a c t on Mathias' f o r c i n g (see [5]) E M,

S c_ T A s

-

t c_ T.

ni formula d e f i n i n g A and l e t cq*

formula associated t o rp by ( 2 ) , so t h a t f o r any X B [ w l w : X

(0,s)

Thus

M, where s i s a f i n i t e subset o f w, S an i n f i n i t e subset o f w and

I=

cp*(X).

t h e r e i s a Mathias c o n d i t i o n

deciding t h e sentence o f t h e f o r c i n g language r p * ( i ) ,

f o r the Mathias generic r e a l , i.e.

( 0 , S > l l ~ S)*(R) o r (~,s)llg1rp*(fi).

where

fi

i s a name

A.S. KECHRIS

182

Consider now separately t h e two cases.

(0,~)11-,lrp*(i).

Case I .

L e t then H be a Mathias generic r e a l over M3 s a t i s f y i n g Such a r e a l e x i s t s and can be found i n a ~ 1 ~ 0 ( 1 0 i m p l i e s t h a t t h e r e i s a ~ ~ ( enumeration y ~ ) o f a l l dense

H c_ S.

t h e c o n d i t i o n (0,S), i.e. . fashion y ~ ) by (6), which f o r P s e t s belongingtoM3.

Then by another basic p r o p e r t y o f Mathias' f o r c i n g every i n f i n i t e X c_ H i s a l s o Mathias generic (see [5]).

thus X

6

A.

*

as w e l l , we must have

M3CXI I= W ' (XI 9 1 0 So H i s a p 3 ( y 3 ) homogeneous s e t avoiding A , so our f i r s t a l t e r n a t i v e

holds. Case 11.

and since i t s a t i s f i e s (0,s)

*.

(0,S)llprp (HI.

Let again Ho be Mathias generic over M3.

E x a c t l y as before Ho i s homogeneous

landing i n A, i.e. VX c_ HOrp(X). 1 1 Sincerp i s 113 t h e r e i s a 113 formula JI such t h a t

Let $

*

1

be t h e C 3 formula

JI(N = V X r H d X ) . associated w i t h t by ( 2 ) ,

I=

) ( H I = M$HI

I= $*(Ho),

I n p a r t i c u l a r , M3[Ho] sentence.

so t h a t

$*(HI.

I= aHJI*(H).

so M3[Ho]

Now

0

~HJI*(H) i s a C 13

But M3[Ho]

i s a m i l d . g e n e r i c extension o f M3. which i s a model con1 t a i n i n g a measurable c a r d i n a l by ( 3 ) , so by Martin-Solovay [4], M3 i s C3-correct

I=

i n M3[HO], thus M3 U, i . e . M3 I= xH$*(H); So p i c k H1 F M3 w i t h M3 thus since M3[Hl] = M3, we have M3[H1] 9 (H1), so ,(H1) holds. Thus

I=

$*(H1);

3 H E Q3$(H) and so by the r e f l e c t i o n p r o p e r t y

(4), 3 H E A',JI(H). Thus f i n a l l y l e t 1 So we see t h a t t h e r e i s a homogeneous s e t l a n d i n g i n A, which means t h a t o u r second a l t e r n a t i v e holds, and o u r proof i s complete. ( I t i s r a t h e r obvious from t h e above argument t h a t i n case A E A 1 ~ A, has a 1 homogeneous s e t ) .

-

H B A$ be such t h a t

1.2.

$(n), i.e.

vX c_ K( Xs A ) .

We now i n d i c a t e t h e m o d i f i c a t i o n s needed t o c a r r y o u t t h i s p r o o f f o r each

odd l e v e l 2n + 1 L e t T2n-l

2 3,

s t i l l under t h e assumption o f f u l l AD.

1 be the f r e e associated t o a I12n-l-scale

Assume n 1.1 below.

1 on a complete I12n-l

s e t (see

By r e p l a c i n g L[a] by L[T2n-1 ,a] (LET1 ,a]= L[a]) and HOD by HODT~,-~ ( t h e [S]). HOD f r o m T2n-l s e t s ) i n the d e f i n i t i o n o f t h e model M3 i n 1.1, we can construct a model M2n+l

s a t i s f y i n g a l l t h e c o n d i t i o n s ( 1 ) - ( 6 ) w i t h 3 replaced by 2n

+

everywhere (see [ 31). To be precise,for

each r e a l a l e t d = [ a ]

be i t s c2,-degree

@2n

defined by

1

Effective Ramsey theorems

IB:

aeePn(p) A B E As (see [2])

d = set.

183

1 e2,,(a)l, where cZn(a) i s t h e l a r g e s t c o u n t a b l e C2,,(a)

a E e2(,B)

8

0

L[T2n-l

.p 1

t h i s n o t i o n o f degree c o i n c i d e s w i t h c o n s t r u c t i b i l i t y degree modulo T2n-l.

u2n be M a r t i n ’ s measure on t h e C2,-degrees

(i.e.

Let

a g a i n t h e one generated b y cones)

and l e t

be t h e a s s o c i a t e d u l trapower. However t h e r e i s one more t h i n g t o w o r r y about. C o n d i t i o n ( 3 ) was enough t o 1 guarantee t h e Z3-correctness o f M3 w i t h i n any M3[H], where H i s Mathias g e n e r i c over M3, b u t i t i s n o t enough t o guarantee t h e Z&+l-absoluteness

o f Mpntl

within

when n 2 2. So a t t h i s p o i n t i n t h e p r o o f we need a d i f f e r e n t argument t o handle t h e general case.

M2n+l[H],

F i r s t l e t us r e c a l l a f u r t h e r p r o p e r t y o f M2n+l

and i t s e x t e n s i o n s (see [3]) which

i s a u t o m a t i c i n t h e case n = 1, and t h i s e x p l a i n s why i t was n o t mentioned exp l i c i t l y before. (7)

1 F o r e v e r y r e a l a , M2n+l[a] i s 1 2 n - c o r r e c t , i.e. I : , , formulas a r e abso-

l u t e between

[a] and t h e r e a l w o r l d .

Now l e t us go back t o t h e argument i n 1 .l,Case 11. t h e general case e x a c t l y as b e f o r e ) .

(0,s)

e M2n+l

(Case I can b e handled i n

I t i s c l e a r t h a t g i v e n a Mathias c o n d i t i o n

i t i s enough t o f i n d a m d e l N = M2n+l[a],

a Mathias g e n e r i c o v e r M2n+l

r e a l H, and M2n+l

such t h a t t h e r e i s i n N

i s Z$n+l-correct

i n N.

(Fact (7)

i s used h e r e ) . As (power(ww))M2n+1 i s c o u n t a b l e i n t h e u n i v e r s e , = ( 2 2X0)~2n+1 < wl, so i f g : w + K i s a g e n e r i c o v e r M2n+l c o l l a p s e , and we t a k e

K

N

=

M2n+l[g]

= Mpn+,[a]

where a is,a

r e a l coding t h e p r e w e l l o r d e r i n g on w induced

by g, t h e n t h e r e i s such an H i n N, so i t i s enough t o show M2n+l

i s E:n+l-correct

i n N. S i n c e N i s a homogeneous g e n e r i c e x t e n s i o n o f M2n+l t h i s w i l l f o l l o w i f we 1 s e t i n ww x ww has a d e f i n a b l e u n i f o r m i z a t i o n . can show t h a t in N e v e r y I12n F o r each i n t e g e r N > 0 l e t ZFCN be t h e f i r s t N axioms o f ZFC. I f TN,n = ZFCN t 1 A$,-DETERMINACY be t h e a s s e r t i o n t h a t a1 1 p r o v a b l e A~~-DETERMINACY,1 e t Prov 1 TN .ni n TN,n h n g a m e s a r e determined. Then we have t h e f o l l o w i n g f u r t h e r f a c t (see 131 1

(8)

F o r each r e a l a , M2n+l[a] M2ntl

[a]

I= ldn-DETERM1NACY, ProvT

b u t f o r each N,

$,,-DETERMINACY. 1

N,n Now f r o m t h e p r o o f o f t h e T r a n s f e r Theorem f o r Scales under t h e game q u a n t i f i e r

A.S. KECHRIS

184

1 (see [6], 6E.15) i t follows t h a t assuming ProvT k2,-DETERMINACY, we can show 1 N,n t h a t every I12n s e t c a r r i e s a definable scale, sc) can be definably uniformized (here N i s a large enough i n t e g e r ) . Thus our proof i s complete. 1.3. Finally, we discuss the technical changes needed t o make t h i s proof work using PD only. The only place where full AD was really used in the preceding proof was in conyd'/p2n. B u t t h i s can be easily 2n -1 avoided ( a t the expense of some loss of c l a r i t y , which explains our presentation L[TZn -1 ,a] in terms of the model MZn+, f i r s t ) by working with HOD , f o r large T2n-l enough C2,-degree of a.

structing the ul trapower M2n+l

=

IId HODT

Instead of results ( 1 ) - ( 8 ) about M2n+l t h a t we quoted in 1.l.and 1 . 2 . we can now use the following f a c t s , a l l proved i n [3] and using only PD, from which by exactly the same proof as before we can establish our main theorem. (1')

There i s a real zo such t h a t

Moreover f o r each real a there i s a real z (depending on a ) , such t h a t a, z

E

LCT2n-l

,01 = W" n

H O O ~ [a] 2n-1

L[T2n-l ,O1

= Q2n+l ( 4 .

1 1 Given a n2n+l formula cp(a) there is a c ~ formula ~ + cp*(a) ~ (effectively constructed fromcp) such t h a t f o r any real a and any model N of ZFC with N n. @ = Q2n+l ( a ) we have (2')

cp(a) a N (=cp*(a),

and similarly interchanging the roles of Ithn+l and (3')

1 I f P(a) i s 112n+l then Pa € QZn+1 P(a)

(4')

0

3a

f

Xin+, .

1 Azn+1 p ( a ) .

There i s a real z, such t h a t

where R i s a countable s e t independent of 6. 0 1 Then there i s a Let Yzn+1 be the f i r s t non-trivial i'12n+l-singleton. (5') 0 1 0 real x e A ~ (Y?,+~ ~ +) such ~ t h a t x enumerates Q2n+l, and i f ~ ( a a) a

e 2' A{(x0),: a(n) = DI e R

Effective Ramsey theorems

185

1 1 0 then t h e r e i s x a ~ ~ ~ + ~enumerating ( y ~ ~P. + ~ ) 1 I f N i s a m d e l o f ZFC with N n wo = Q2n+l(a), then N i s Z2n-correct

(where R i s a s i n ( 4 ' ) ) ,

(6') and

I= provT

&,-DETERMINACY, N,n where a s in ( 8 ) o f 1 . 2 . , T N , n = Z F C ~+ &-DETERMINACY. N

zr

A&+l - and &2n+I - emodabiZity. Consider any notion o f r e d u c i b i l i t y among real s , l i k e f o r i n s t a n c e < (Turing r e d u c i b i l i t y ) , < (A 1 - r e d u c i b i l i t y ; -T -n n 1 ~ ~ (11, i 3o r (Q2,+, -reducibil i t y ;

82.

'gn+l

5jn+l

1. a i3 a f Q2n+1(np) A real a i s c a l l e d zr-encodabte i f f vX E [ w ] ~ aY c_ X[a zrY]. When -r < = -T < we 1 t a l k about recwsiueZy encodabZe real s , when 5, = 5 about An-encodabk real s and when = &+l about Q2n+l -encodabZe real s . 1 The following r e s u l t o f Solovay computes the r e c u r s i v e l y and A1-encodable r e a l s .

zr

Theorem (Solovay [7]). The recursively encodable r e a l s a r e e x a c t l y t h e A; 1 r e a l s a r e e x a c t l y t h e real s i n L 1 , where a11 i s t h e r e a l s . The A1-encodable a1 1 f i r s t C1-reflecting ordinal. 1 Since f o r n = 0 , 0 . Again t h e s i t u a t i o n i s d i f f e r e n t from the c a s e n = 0.

Theorem 2. Let n > 0 and assume P r o j e c t i v e Determinacy. encodable real s a r e e x a c t l y the Q2n+l -real s.

Proof. Lemma 1 .

Then t h e Q2n+l-

We will need f i r s t t h e following lemna. There i s a mdel M of ZFC such t h a t

M = L[A], A a bounded s u b s e t o f q , i) i i ) I f H i s Mathias generic over M , then M[H] i s &,-correct.

I=

1 k2,-DETERMINACY, and M[H]

(Since (power(ww))M is countable such r e a l s e x i s t ) Proof (Following ideas o f Solovay). Let T2n+l be the tree a s s o c i a t e d w i t h a 1 1 I12n+l- s c a l e on a complete It2n+l set. 1 1 Every extension o f L[T2n+l] i s C2n+2 a b s o l u t e , so i t a l s o s a t i s f i e s bnDETERMINACY. Let K be the c a r d i n a l i t y o f ww i n L[T2n+l] and a the c a r d i n a l i t y o f power(ww) in L[T2n+l], so t h a t K < 1 < w l . Let f , g E L[T2n+l] be such t h a t

I=

L[T2n+l

1

LCT2n+l

I p g:a

f:

K

1-1 w ontd w power(ww).

AS. KECHRIS

186

Then t h e r e i s a bounded subset A o f wl,

A

E

L[T2n+l],

encoding f, g, thus L[A]

contains t h e same r e a l s and .sets o f r e a l s as L[T2n+l].

Hence L[A],

L[T2n+l]

have

the same Mathias conditions, and f o r every Mathias generic r e a l H over L[A], and ww n L[A,H] = w" h2,-DETERMINACY. 1

Mathias generic over L[T2n+l] a1 so Z2n+2-correct 1 and L[A,H] C a l l a r e a l a M-encodable,

I=

n

4

L[T2n+l

,H I,

so L[A,H]

where M i s a m d e l o f ZFC w i t h (power(wo))

if

yH[H Mathias generic over M

M

H is

is

countable,

3

zX c_ H(ae M [ X l ) l . The f o l l o w i n g r e s u l t i s a special instance o f a m r e general theorem o f Solovay about M-encodability ( i t s p r o o f i s i m p l i c i t i n [7] and e x p l i c i t i n [ l ] ) . Theorem (Solovay).

L e t M = L[A],

where A i s a bounded subset o f wl.

Then

every M-encodable r e a l i s i n M. F i n a l l y we s h a l l need t h e f o l l o w i n g f a c t f r u m [3]. Lemma 2.

IfM i s a Z i n - c o r r e c t m d e l o f ZFC + .&,-DETERMINACY, i.E. p E M A a e Q2n+l ( B ) =. a E M.

then M i s

downward closed under 0 t h e

~ r ela l s .~ + ~

en-

1

93. Some open problems. The f o l l o w i n g q u e s t i o n s r e l a t e d t o t h e t o p i c s discussed i n t h i s paper a r e , t o t h e b e s t o f o u r knowledge, s t i l l open. 3.1.

There i s an o r d i n a l

7

1 I f A c_ [ w I w i s Ill t h e n e i t h e r t h e r e

< w1 such t h a t :

i s a homogeneous s e t a v o i d i n g A o r e l s e t h e r e i s a homogeneous s e t i n i n A.

3.2.

What i s t h e l e a s t such T?

L landing 7

1 What a r e b e s t p o s s i b l e e s t i m a t e s f o r homogeneous s e t s f o r C2n and

s e t s of [ w ] ~ , assuming PD?

(Added i n P r o o f :

1

sub-

T h i s problem i s s o l v e d i n

H. Woodin's paper i n t h i s volume). 3.3.

1 What i s t h e s e t o f An-encodable r e a l s f o r n

2

2?

( c f . the conjecture i n $2).

REFERENCES Ennis, G.,

Mathias f o r c i n g and e n c o d a b i l i t y , p r e p r i n t , C a l t e c h (1978).

H a r r i n g t o n , L. A. and K e c h r i s , A. S., Ann. Math. Logic, 20(1981), 109-154.

On t h e determinacy o f games on o r d i n a l s ,

Kechris, A. S., M a r t i n , D. A. and Solovay, R. M., Cabal Seminar (1979-81), t o appear. M a r t i n , D. A. and Solovay, R. M., o f Math. 89 (1969), 138-160. Mathias, A.R.D.,

I n t r o d u c t i o n t o Q-theory,

1 A b a s i s theorem f o r C3 s e t s o f r e a l s , Ann.

Happy f a m i l i e s , Ann. Math. Logic, 12 (1977), 59-111.

Moschovakis, Y. N.,

D e s c r i p t i v e S e t Theory, N o r t h H o l l a n d (1980).

Solovay, R. M., H y p e r a r i t h m e t i c a l l y encodable s e t s , Trans. Amer. Math. SOC. 239 (1978), 99-122.

'The a u t h o r i s an A. P. Sloan Foundation Fellow. NSF Grant MCS79-20465.

Research p a r t i a l l y supported b y

PROCEEDNGS OF THE HERBRAND SYMPOSIUM LOCIC COLLOQUIUM '81, J. Stem (editor) 0 North-Holland Publishing Company, 1982

FINITE HOMOGENEOUS SIMPLE DIGRAPHS A.

H. Lachlan

Department of Mathematics, Simon Fraser University Burnaby, British Columbia, Canada V5A 156

Simple directed graphs are regarded as structures

where

G

-

% is an irreflexive binary relation on a

nonempty set of vertices

IGl.

A

classification is given of

all finite simple digraphs which are homogeneous in the sense of Fraiss;.

'Ihe classification extends that given by Gardiner

of homogeneous graphs.

G =
By a digraph

%>

we mean an ordered pair whose first member is a

nonempty set and whose second member is an irreflexive binary relation on that set.

lhus, in the language of graph theory, we confine consideration t o digraphs

without loops and without multiple edges. IHI c IGI

and

E = EG n IHl*; H

induced subdigraphs. of

IGI

lhe

and is denoted

power IIGII.

We call

H

a subdigraph of

G

if

our subdigraphs are what are usually called or

order of

G

is defined to be the cardinality

Following Fraiss;

[l]

G

is called homogeneous if

any isomorphism between subdigraphs of smaller power can be extended to an automorphism of

Let

IG

In this paper we classify all finite homogeneous digraphs.

G.

denote the identity relation on

-

C Eu

is

< I G I ,\GI'

I~)>. G

EG

and antisymmetric if

(x, y)

]GI.

lhe complement of

is symmetric if

%

implies

symmetric digraphs are given in Figure 1.

P Figure 1

(x, y)

E

(y, x) 4

%

E.

G

denoted

implies

(y, x )

C

W o examples of

A.H. LACHLAN

190

In general

denotes t h e complete symmetric digraph of o r d e r

K,,

a r e digraphs then t i o n of

and

G

G x

denotes t h e i r product and

H

H.

In

[21

Gardiner shared t h a t

homogeneous digraph if and only if

G

or

G

G[H] G

n.

If

G,

H

denotes t h e composi-

i s a f i n i t e symmetric

i s isomorphic t o one o f :

W e s h a l l need t o r e f e r t o t h r e e w e l l known asymmetric digraphs

T,

C,

and

D

depicted i n Figure 2.

C

T

D

Figure 2 Note t h a t digraphs

C

and

Ho, H,,

shown i n Figures

are homogeneous b u t

D Hp 3,

T

is not.

Zhree more homogeneous

are r e q u i r e d f o r t h e statement of our r e s u l t s . 4,

and 5 r e s p e c t i v e l y .

improving my o r i g i n a l diagrams.

Figure 3

I am indebted t o M.

'Ihese a r e Dubiel f o r

Finite homogeneous simple digraphs

H1 Figure 4

H2 Figure 5

191

A.H. LACHLAN

192

In the diagram of

H2 we have omitted most of the edges in the hope that this will make the picture clearer. Zhe following rule is applied to obtain the

remaining edges. from

v

to

an edge from

w

Call vertices and from

v

to

w

w

to

v, w v,

but not from

mate to which it is double-linked.

v

double-linked if there are edges both

and say that w If

to v

V.

In

v

dominates

w

if there is

each vertex has a unique

H2

dominates (is dominated by)

is dominated by (dominates) the mate of

w

then

"his leads to the insertion of

W.

another 36 edges.

We can now present our findings.

Let

H , A, S

denote the classes of all finite

digraphs which are homogeneous, homogeneous and antisymmetric, homogeneous and symmetric, respectively. G E A

THEOREM 1.

for some

n,

THEOREM 2.

1

iff

< n
3.

be a digraph and

1

< n
1.

n

'G

'G

similar

-

Kn[D]

*GI

We now consider these

remaining cases in turn.

6.3.

-

Gu.

E

If

*v

'w) = g('v)

f('v,

(n > 1).

= Kn[DI = G'

'G

one-one,

or

Fach v '

dominates

= g('w)

' G

E ' G

= vo,

= g('w)

g('v)

w '

f

g('v)

contradiction.

if and only if

g('w),

because

and

' w

are

In the latter case, considering the vo-automorphisms of

Gu

s.

Gu.

= En[;]

G '

We conclude that Gd = ' G

Similarly, unique

E

v1

G

E

'v

'G,

Without loss suppose

'G.

dominated by

Gu.

we

Gu = G '

But then one of

Gu = G ' then

,'G

'G,

D

in

or .'G

there

Gd

in

see that

is dominated by a copy of

' G

is

g

double-linked

is one-one which tells us that

g

or

each

otherwise

It follows that either

v'

'G.

But since

g('v)

dominates a unique vertex

then

exists

should be a

singleton, contradiction.

6.4.

= 'G

' G

by some v1

v '

dominates

lhus each

joint copies of in (m

Gu,

*v

c G'

v '

(Gu)'

< n) or

-

C

'G

copy of of the

'G.

-

-

Kn-l [C]

n

C

in

in

copies of

(n > 1). Gu. C

'G

7.

--

or

THE CRSE G"

D[K2],

and

H1.

-

Kn,

- x2.

is dominated

Gu

dominates

vl,

and

9.

vu

E

Gu

and distinct

Since each

has mate in *v

-

Gu.

dominate dis-

vu E Gu

has its mate (Gu)' =

km

However, previous cases show there is no

' G

G '

=

km,

UGl > 9,

G

embeds

and dominated by .'G

C,

But if

is dominated by a vertex in

Gu

Gu

-

and

(Gu)'

as well as

.

> 1 which contradicts 'G = Kn [C]

Since Gu

-

Gc = Kn[CI

t! s

each

and similarly

v '

E G '

Gd

t! s.

dominates a unique vertex of

copies dominate different vertices. morphic to

>

'G

II"GX

Hence in

€

vu

' G

being a subdigraph of G, we have

dominating

This means that

= Kn[C] = G'

-

Kn-,[C]

(m < n).

v1

and each

-

nGUll

Gu

satisfying all of:

then a copy of

by one in

6.5.

Hence

and

has a unique mate to which it is double-

v E G

Also,

and each

v1 E Gd,

has mate in ,'G

= x,,,[Cl

G

there is a unique embeds

' G

'(Gu).

CG")'

homogeneous

€

= GC

' G

dominates a copy of

Kn-,[C].

a

Since

there exists unique

'G.

linked, and each Now each

(n > 1 ) .

= l),

206

7.1.

A.H. LACHLAN

-

G’

’G = G’.

Kn[C[F2]1.

Since

n = 1.

From 1.2(3),

iio[z21.

G = 7.2.

G’

have

’G

-- 6[F2l

H ~ . Since

or

H1.

G’ = H ,

that

v

dominates

dominates

w

vertices

G’

and

K~

’G

’G = D[K2].

and

Hence each

in

’G

*v

Also

G’.

-

and

that

G’

H,

6n we

E

’G

{x’,

we easily see that

= IG’I

1’611

=

For any vertices

dominates a unique pair

y’)

have

8

we

v, w such

dominates a unique ’G

and

{v, w)

{x’,

v’

E

G’.

y’)

G’.

of IhuS

Under this

have the same automorphisms which is impossible.

G’

THE RDlAINING QLSES.

8.

=

and so it is sufficient to con-

there is a definable one-one correspondence between correspondence

= IG’I

II’GN

there are unique vertices x,y unlinked such that

{x, y}.

unlinked

- --

= ’(G’)

(’G).

From 1.2(3)

sider the case in which

and

K2

Following the line of 6.1

or

6[ii2]

-

= ’(G’)

(’G)’

--

G”

If

(n > Z ) ,

Kn

is no longer possible.

we can proceed as in $7 except

Ihe only possibility for

G

is

io[Enl.

lhis completes the verification of the first ten lines in the Table.

Now

suppose

’(G-)

. I

Km[Kn],

-

case ’G

-

G’. in

k1

8.1.

-

G’

G”

then

(m, n

>

C[Km[T?n]].

= P

or

G’

K

-

-

or

G’

x

3

(’G)’

(n > 1).

Kn[Ho]

-

H2.

or

#(G*) =

Consider first the

c and

Notice that for any vertices

there is a unique copy of

fore each

’v

E

’G



in

C

= IIG’I

!’GI

v’.

’v

and

G

such that

v, w

dominated by both

v’ c G’

determines a unique

dominated by both

of triples

Ihe same

=

6n,

Kn[HO].

w

G’

(*G)* =

lhus

1.2(3).

contradicts

(n > 1)

Kn[Ho]

dominates

in

lhis

K3.

- -

or

G’ = C[Km[?,II.

then

2)

’G

G’ = K ~ [ H ~ ] .Since

which

Kn[Ho]

-

Km[Kn]

whence

argument works when

Let

- -

G”

-

v

such that and

W.

v

mere-

such that there is a copy of

C

It follows that there are five 3-types v

dominates

w

and

w

dominates

X.

Contradiction. 8.2.

F

-

G’ = Ho = ’G. say,

dominated by

in

G’ v

Let

v c #G,

dominated by

and

F.

then as in 8.1 there is a unique copy of V.

Also,

lhe subdigraph

F.

cannot be all of

G’

and so is just

hence determines

v-

the unique member of

contains no unlinked pair

w

least four 3-types of triples

there is a unique G’ %us G’

dominating w

being only three possible 2-types for pairs

with

is homogeneous and

determines

which dominates

dominates no vertex in

w

x

<w, x>.

E

’G.

G’.

C,

w = f(v) c Gu

F uniquely F. Since

Also,

and ’G

there are at

This contradicts there

8.3.

in

-

' G

unique

Finite homogeneous simple digraphs

'G

. I

Ho.

w = f(v)

' G

E

As in the previous subcase for each GU

such that

dominated by

v

are the only ones in

'G

-

a definable one-one ' G

' G

Gd

vertices from

v

dominates

and dominating dominating

f(v).

f(v),

v1 E Gd

F,

and

v,

'G

Gu.

v E ' G

there is a

%

and a unique copy

C

of

%

As before the vertices of

whence

dominated by

dominates a vertex in Gu,

w,

correspondence which means

there exists

vertex in

207

>

IIGuM

that

Gu

merefore

8. . I

Ho

and dominating But

f(v)

or 'G,

f

-

Ho.

is

Since

whence no

is dominated by three

mis

which makes seven vertices in all.

contradiction completes the case.

-

-

= U G ' I

=

12,

[l] Fraiss;, R., Sur l'extension aux relations de quelques propri&t;s ordres, Ann. Sci. h o l e Norm. Sup. 71 (1954), 361-388.

des

8.4. ' G

= H

2

.

m e

case

' G

m

H2.

Since

('G)'

'(G-)

D

and

I' G I

Now 1.2(3) yields a contradiction.

REFERENCES

[2] Gardiner, A., 94-1 02.

Homogeneous graphs, J. Bmbinatorial "Ieory Ser. B 20 (1976),

[3] Gardiner, A., Homogeneity conditions in graphs, J. Ser. B 24 (1978). 301-310.

Bmbinatorial lheory

[4] Henson, C. W., A family of countable homogeneous graphs, Pacific J. Math. 38 (1971), 69-83. [ S ] Henson,

C. W., Buntable homogeneous relational structures and categorical theories, J. Symbolic Logic 37 (1972). 494-500.

[61 Jonsson, B.,

Universal relational systems, Math. Scand. 4 (19561, 193-208.

[7] Jonsson, B., Homogeneous universal (1960), 137-142.

relational

systems,

[ 8 ] lachlan, A.

Math.

ti., Buntable ultrahomogeneous tournaments, her. Abstracts 1 (1980). 80FA17.

Scand.

Math.

8

Soc.

191 Lachlan, A. H., and Woodrow, R. E., Buntable ultrahomogeneous graphs, Trans. Amer. Math. SOC. 262 (1980). 51-94. [lo] Morley, M., and Vaught, R., ( 1962), 37-57.

Homogeneous universal models, Math. Scand.

[ll]

Fado, R., 331-340.

[ 121

Schmerl, J., Buntable homogeneous Universalis 9 (1979). 317-321.

11

Universal graphs and universal functions, Acta Arith. 9 (1964),

partially

ordered

sets,

Algebra

A.H. LACHLAN

208

Snoothly embeddable subgraphs, J.

[131

Sheehan, J., 212-218.

London Math. Soc. 9 (1974),

[141

Iheories with a Finite Number of Cbuntable Models and a Woodrow, R. E., Small Language (Simon Fraser University, British Columbia, Canada, 1976).

[151

Woodrow, R. E., %ere are four countable ultrahomogeneous graphs without triangles, J. Combinatorial Theory Ser. B 27 (1979). 168-179.

PROCEEDINGS OF THE HERBFAND SYMpasILoM LOCK COLLOQUIW '81. J. Stem (editor) @ North-Holland Publishing Company, 1982

209

BOREL SETS AND THE ANALYTICAL HIERARCHY A . Louveau

U n i v e r s i t e P a r i s V I and UCLA

In t h i s paper, we use Moschovakis' s t r a t e g i c b a s i s theorem ( s e e Moschovakis [19801), t o r e l a t e boldface and l i g h t f a c e p o i n t c l a s s e s i n the p r o j e c t i v e h i e r a r c h y . The 1 main r e s u l t i s t h a t , assuming p r o j e c t i v e determinacy, any Borel and A 2 n + l s e t i s 1 A ' i n a A 2 n + l r e a l . This r e s u l t , t o g e t h e r with s i m i l a r ones, f o r a l l l e v e l s of t h e 1 p r o j e c t i v e h i e r a r c h y , a r e obtained i n S e c t i o n 1 . I n S e c t i o n 2, we d e r i v e some app l i c a t i o n s to b a s i s and u n i f o r m i z a t i o n problems. Case

n = 1

of the above r e s u l t i s due t o Kechris [1978a], using s t r o n g s e t theo1 s e t s . The p r e s e n t t r e a t -

r e t i c a l h y p o t h e s i s , t o g e t h e r with deep p r o p e r t i e s of X

ment owes much t o d i s c u s s i o n s I had with him when he was v i s i t i n g a t t h e Universit y P a r i s VI1,during 1978-1979. The main r e s u l t was obtained the very day my son P i e r r e was born, and I ' d l i k e t o d e d i c a t e t h i s paper both t o him and t o my wife, Lise, §I.

THE BASIC STRUCTUXAL RESULT

Let

r

C

and

A

be two l i g h t f a c e c l a s s e s of s u b s e t s of ww, with

r?A .

= U{r(a), aEwW}be the corresponding boldface c l a s s , and d e f i n e

U{r(a),aEh),

where

i s equivalent t o t h a t any s e t i n

aEh

means

{a}EA

( f o r the

A's

we a r e i n t e r e s t e d i n , i t

a E h , a s a s u b s e t of wL). For c l o s e d enough

r(A)

i s both i n

g a t e t h e converse p r o p e r t y , i . e .

A

and i n

c. What we want

to study t h e p a i r s

We l e t

r(A) t o be

(r,A)

A's,

it i s clear

t o do i s to i n v e s t i f o r which

hn;=l'(A).

I n t h e following, such a p a i r i s r e f e r r e d t o a s a n e f f e c t i v e p a i r . The f i r s t r e s u l t concerning e f f e c t i v e p a i r s i s i n Louveau 11980al : For E<wCk 1 ' 0 1 0 1 the p a i r s and a r e e f f e c t i v e . This r e s u l t has been extended to 5 1 5' 1 a l l " l i g h t f a c e " Wadge c l a s s e s of Borel s e t s (Louveau [1981]), and by Kechris 1 1 [1978a] who proves, a s s a i d above, t h a t i s an e f f e c t i v e p a i r , using AD[L( rH)

1

1.

3

1 We s h a l l use here a d i f f e r e n t , and q u i t e simpler, approach, adapted t o '2n+1 1 s e t s , and prove, assuming -determinacy, t h a t t h e p a i r s , and 1 1 1 1 1 1 < A ~ , A ~ ~ + ~ >f ,o r k , = < I , U , g > , with (of course)

Ik} ; an ultra-diagram of type

-+

i s n e c e s s a r i l y the ultrapower

A(L)

(since

map : now

the t r i p l e

L

r

A)'

g

the

i s given by i t s value ; etc.

16. ON THE PROOF. Fact. -

Let

: T

E

-+

be an elementary f u n c t o r between pretoposes. Then

T'

E

is

an equivalence i f f (i)

E

is conservative,

(ii)

E

i s subobject-full,

(iii)

every o b j e c t of

Explanation. with

E

SubT(A)

and

T'

has a cover v i a

c o n s e r v a t i v e means t h a t

E.

1-1 map SubT(A)

induces a

E

the l a t t i c e of s u b o b j e c t s of

A

in

T.

E

+

SubT, (€A),

s u b o b j e c t - f u l l means

t h a t t h e same map i s onto. To e x p l a i n ( i i i ) , I f i r s t say t h i s .

X

A p a r t i a l (A-)cover of

:

(via E )

P

i s a pair

0,

P :

EA

"1 X

with

A E IT1

and such t h a t

Y

f i n i t e family of p a r t i a l covers

i s a monomorphism. A cover of (ai,Yi)

of

X

2 x

@.

f a c t o r s through

f

(v&

E)

is a

such t h a t the only monomorphisms

f such t h a t each

X

a r e isomorphisms,

M.MAKKAI

230

-- -

Recall t h a t the Theorem says t h a t E

=

E~

: T

T'

(Mod T, SET)

Horn

i s an equivalence f o r any small pretopos T. We show t h a t (iii).

satisfies ( i ) , ( i i ) ,

E

( i ) i s e s s e n t i a l l y che R e p r e s e n t a t i o n Theorem (and no u l t r a p r o d u c t s e n t e r ( i i ) uses t h e u l t r a p r o d u c t - s t r u c t u r e on

i n t o i t s proof).

Mod T ,

but not the

ultramorphisms, and I won't say more about t h i s . I ' d l i k e t o sketch how ultramorphisms e n t e r i n t o t h e proof of Let

X

be any o b j e c t of

T',

p l i c i t y , assume t h a t a l l

i . e . an u l t r a f u n c t o r

[X,U]

X :

(iii).

Mod T -+SET.

For s i m -

a r e i d e n t i t i e s . Unraveling t h e d e f i n i t i o n o f

' c o v e r ' , we o b t a i n the following. A partial

A-cover

EM c M(A)

of

X

i s a family

X

= < ZM : M E [Mod TI >

X(M)

x

-

such t h a t the following a r e s a t i s f i e d :

( i ) whenever then

< a , x > E ZM, and

h : M

E Z N ; EXM

( i i ) whenever on I , then ( i i i ) whenever

i

i EP,

for a l l

(1,V)

then t h e r e i s

is a morphism i n

N

< < a . > / U , <x.>/U> E ZnMilu

P c I , PE U,

U an u l t r a f i l t e r

/V>E XnMilv,

EXM.

such t h a t

Mod T ,

;

i s an u l t r a f i l t e r , and P EV

of s u b s e t s

for a l l

i EP ;

1

( i v ) E ZM

A

of

X

and

< a , x ' p EXM imply

is a f i n i t e s e t

( v ) f o r every

a EM(Ai)

{Xi : i < n } of p a r t i a l covers such t h a t

M E lMod T I , and every such

that

x = x'.

i

€ E M

.

xEX(M)

there i s

The proof of the e x i s t e n c e of a cover proceeds a s follows. every

M E IMod TI

is a support of

and

x

xEX(M) t h e r e a r e

Second, we show t h a t

and

First, we

aEM(A)

show t h a t f o r

such t h a t

(A,a)

i n t h e sense t h a t

The proof of t h i s uses t h a t phism.

A € IT1

i < n and

X

p r e s e r v e s t h e diagonal ( t h e s i m p l e s t ) ultramor-

Stone duality whenever

(A,ao)

i s a support of

a p a r t i a l cover

of

C

X

23 1 then t h e r e i s

(xoEX(Mo)),

xo

.

E X M

such t h a t

0

Note t h a t having done s o we have a class-many)

"cover"

b u t w i t h p o s s i b l y i n f i n i t e l y (proper-

p a r t i a l covers i n s t e a d of f i n i t e l y many.

This i s t h e p a r t of t h e proof t h a t uses complicated ultramorphisms. We a r e t o c o n s t r u c t a family ( v i )

=

< X M : [Mod T [ >

( i ) - (iv)

satisfying

.

E EM 0

K

F i r s t , w e do something s l i g h t l y l e s s : we s e l e c t a h

and u l t r a f i l t e r s

and

U

*

K

*

i n s t e a d of

all models, *

argue t h a t t h e o r i g i n a l t a s k a l s o can be done. Having now

*

*

= . For each

( a € M(A),xEX(M))

MEK

,



into

K

,

etc.,

-

*

ZMo.

(iii)

and

we'll

we want t o b u i l d

we throw i n more and more p a i r s

i n an e f f o r t t o s a t i s f y ( i )

s t a r t with throwing

of models M, morphisms

f o r which we ensure t h e c o n d i t i o n s ; having done

V

the c o n s t r u c t i o n f o r an a r b i t r a r y s e t E

and



( v i ) . Of course, we

A t any s t a g e , c o n d i t i o n s ( i ) and ( i i )

a r e honored by simply throwing i n necessary p o i n t s . S a t i s f y i n g ( i i i ) , however,

P EV. Repeating those s t e p s t r a n s f i n i t e l y o f t e n , in-

r e q u i r e s a choice of a s e t

eluding choices

"PEV",

we end up with a family

Z

*

s a t i s f y i n g ( i ) - ( i i i ) and

( v i ) . We might have f a i l e d t o s a t i s f y ( i v ) though. We now make the assumption that, indeed, a t a l l p o s s i b l e s e r i e s o f c h o i c e s

'PEV'

w e f a i l . Using t h i s assumption

we a r e a b l e t o b u i l d two ultramorphisms, and show t h a t the f a c t t h a t these l e a d s t o a c o n t r a d i c t i o n t o t h e assumption t h a t x

. The

(A,ao)

X

preserves

i s a support of

u l t r a g r a p h s u n d e r l y i n g these ultramorphisms a r e about t h e same s i z e a s K

*

i t s e l f , and they a l s o use an u l t r a p r o d u c t by an u l t r a f i l t e r f u r t h e r and above those considered i n

K

*

. The

u l t r a g r a p h s used code, i n a s e n s e , t h e procedure of

s e a r c h i n g f o r t h e r i g h t c h o i c e s of

'PEV'

mentioned above.

The c o n s i d e r a t i o n showing t h a t doing t h e t a s k

(2)

f o r a r b i t r a r y bounded

K*

s u f f i c e s uses t h e Keisler-Shelah Isomorphism Theorem : elementary e q u i v a l e n t mod e l s have isomorphic ultrapowers. cardinal

K,

if

*

and

1,

plies that quired i n The

Z2

*

then the f a c t t h a t

*

*

C I = C2.

both s a t i s f y the c o n d i t i o n s w i t h some s u i t a b l e

= (Z2)

(C1)

*

We show, u s i n g t h i s theorem, t h a t , f o r a fixed

M

*

M

for a l l

M

of c a r d i n a l i t y l e s s than

I t i s easy t o see t h a t now we can put t o g e t h e r a


, i.e.,

e',

N

f o r w, M E N , and

Further i f

deg(N)G$.

= (w,

There i s a g r e a t d e a l known a b o u t =

$}.

deg (M)

t o be of t h e

1

1 2 i s t h e d e g r e e of t h e b a s i c d i a -

and deg (M) G

8 , 0, @)

= (w,

M

process allows us t o pass t o

deg(M)

M

i s simply e q u a l i t y

M

w e mean t h e supremum o f t h e d e g r e e s o f {<no,nl,n2>:Mkno@n =n

I<no,nl,n2> :M k n O n l

gram of

0

,d,

t h e n t h e u s u a l Henkin

G I ) , where e q u a l i t y i n

is equality

N

i s nonstandard s o i s N .

M

Do = {,$ : t h e r e i s

M CPA

nonstandard, with

The f o l l o w i n g a r e among t h e more s t r i k i n g r e s u l t s .

1) The u s u a l Henkin argument shows t h e r e i s

2) Tennenbaum [TI showed t h a t

OBD,.

(i.e.,

$EDo

with

,$GO'.

T h e r e i s n o r e c u r s i v e non s t a n d a r d

M kPA.)

3) S h o e n f i e l d [ S ] u s e d K r e i s e l ' s b a s i s theorern t o f i n d

$EDO

such t h a t

4 ) J o c k u s c h and S o a r e [ J S ] improved S h o e n f i e l d ' s r e s u l t by e x h i b i t i n g

$'

with

4

O ( n ) l . From t h i s we a r e a b l e t o deduce Knight's r e s u l t

analogue of t h e Jockusch-Soare theorem f o r PA. Our proof uses i d e a s

Th(N)

[HI c o n s t r u c t i o n of a nonstandard

M

k

PA

with

deg(M)

< 0'

Th(N) E O ( w ) .

We would l i k e t o thank Angus Macintyre and Steve Brackin f o r numerous conversat i o n s on H a r r i n g t o n ' s theorem, and Carl Jockusch f o r b r i n g i n g Knight's c o n j e c t u r e t o our a t t e n t i o n . $2. THE MAIN THEOREM

We h e a v i l y use t h e following f a c t from degree t h e o r y . Fact I : I f f o r every Proof : See E p s t e i n

n Ew

$ > O(n),

then

O(w)


t 6 . ( s , t ' ) = u.(s). A s above, whenever p o s s i b l e we suppress s j ( s , t ) J

concentrate on

and

J A. (s,t). J

1 i n d (again, we r e a l l y f i n d an For i < s worker 1 w i l l form a s e t U.(s) r . e . index ui(s) 1 f o r Ui(s)). 1 Ul(s) w i r l contain B1(Th(N)UKi,tA,(i,s)U 2 U ~l~-~(i,s)).

...

Note t h a t , a s above, i f

X

is

r.e.

d, s o i s

in

B (X), and i n f a c t t h e func-

1

t i o n which takes an index f o r x t o an index f o r B1(X) can be computed i n d ' . 1 I f r . ( v ) i s eventually c o n s i s t e n t with t h e f u l l diagram of M, t h e r e w i l l be a constant

cEC

and a s t a g e a such t h a t i f

w i l l arrange things so t h a t

'

lim u . ( s ) s

1

exist.

1

then

s ' h ,

1

Ui(s')zi(c).

w i l l denote

UI

Again we

limUi(S).

The b a s i c ideas : Before d e t a i l i n g the c o n s t r u c t i o n , we should o u t l i n e t h e ideas behind i t . There would be no d i f f i c u l t i e s involved i f we only had t o maintain t h e consistency of

T1, T

2

and T3. Worker 1 would make s u r e

Th O(N) 3 T:

''

i s consistent for

T ' C d' and worker 2 has o r a c l e d", worker 2 could maintain conT 1 (Note t h a t as T F Z ~and Th O(N) U T: i s consistent s i s t e n c y of T1 UTh(N) UT:.

each s . A s

f o r each

Th(N)UT

s

Th(N) U T 1 U T2 U T:

1

S i m i l a r l y worker 3 could ensure 3 i s c o n s i s t e n t , while completing T

The d i f f i c u l t y a r i s e s i n ensuring t h a t worker 2 w r i t e s down co

such t h a t

-

.

i s Henkinized. For example, suppose

T3

3 x V y ~ , ( x , y , c ) ,where

Vy cp (x,y,c)

witness f o r

z4

i s consistent.)

we must ensure t h a t worker 1 has s e t a s i d e a constant 1 B T S i m i l a r l y , i f worker 3 w r i t e s down

.

3ycp(co,y,F)

3xVy3zcp(x,y,z,T), worker 2 must s e t a s i d e a f o r any

c1

i s open. In order t o provide a

U,

worker 1 must ensure

c

s.t.

O-

3z(p(co,c1,z,c)

ET

3

3yVzcp ( c o , y , z , ; )

.

B T2 and

This d i f f i c u l t y i s overcome by our approximation procedures. I f worker 2 w r i t e s down

-

3xVycp(x,y,c), t h e r e w i l l be a l a t e r s t a g e where worker 1 b e l i e v e s t h a t

worker 2 wants a witness t o

V y c p ( x , y , a . A t t h i s point worker 1 w i l l f i n d a

such t h a t none of t h e workers could have considered co

c

by t h i s p o i n t and s e t

co a s i d e a s a witness ( i - e . , worker 1 w i l l n o t w r i t e down

-

3y rp(c,,y,c)),

0

As

worker 2 w i l l r e a l i z e t h a t worker 1 has done t h i s , worker 2 may a t some point w r i t e down

-

Vycp(co,y,c). From t h i s p o i n t on everyone i s committed t o t h i s choice.

Providing a witness f o r a

2:

-sentence

3x$(x)

n>2, i s a b i t more com-

where

plex. F i r s t , Worker 2 must provide a witness f o r t h e

Z

0

2

-consequences of $ ( x ) .

Models of true arithmetic

237

Secondly this witness must also have been provided by worker 1 as a witness to the Z

0

1

-consequences of

To ensure this occurs workers 1 and 2 attempt to par-

$(x).

tially saturate the model. Namely ; if they believe it is consistent with the actions of higher level workers, they will set aside a witness for the type T ! ( v ) . Witnessing

r!,

is given priority over witnessing Ti

for

i s 1 1 1 1 u i ( s ' ) = u . ( s ) . The c o n d i t i o n V s ' T u . ( s ' ) = u . ( s ) i s r e c u r s i v e i n d",

2 may c a l c t l a t e i n d i c i e s f o r

U;

,...,: A dl .

.

so worker

d"' and c a l c u l a t e s K? As worker 3 i s maintaining 1,s t o ensure t h e c o n s i s t e n c y of c o n s i s t e n c y , worker 2 may enumerate enough t o d"' 1 1 1 2 U Th(N) f o r i G s . T U Ul U V Us-l U Tsml U K?

Worker 2 enumerates more of

...

1,s

Case I : Worker 2 i s i n t h e a c t i v e mode.

Models of true arithmetic 2

Worker 2 consider the next Tf = T:-,

2

... U Us-l

denote TI U Ut U

u xT u

Case 1 : $s 2

x!

2

2

2

b) If U.(s-I) 2

contains no realization of

r.(v), 2

U;(s-I)

U

In d"

we may calculate k . Worker 2 sets U . ( s )

This is r.e. in d J

x:

U ri(v)

is consistent, but

we calculate B ~ ( T ~ urf(v) -~

and thus is

2

into the waiting mode to find a witness for Case 2 : JiS U

2 =B2(x; UUi(s-l)).

ri(v), then we set U:(s)

contains a witness for

2

... u ~ 1~ - ~ ( s )Let .

is consistent.

a) If Ui(s-l)

... U U:-l(s-l)).

u l1 ( s ) II

U Ti:

u ui(s-l)

ri(v)

u

K:,,

denote

U K;,s

j as possible.

i<j consistent for aslargeas

U.(s) j<s. Let J 1

We now must define

... U U:-,

U T 1 U Tf U Ui U

to keep Th(N)

2

U UI(s-l)U...UUi-l(s-l),

$s

which has not been attended to, and sets

(pi

or Ti =

A Q :

2

239

:

=

r .

XI

B (x?)

2

1

2

for some II Ew.

Ta.(v)

for i g j < s

2 U Ui(s-l)

U ri(v)

u

and goes

is inconsis-

tent. In the case we act just as in Ib). Case 3 :

tjS

u

x: u

X I L'

2

T~(v) is inconsistent. 2 2 u.(s) = B ~ ( X u ~ui(s-l)).

2

a)

I L ~u

b)

Otherwise set

~ ~ ( s - 1 )is consistent. Let 2

U. ( s )

=

B2(x;). 1

Case I1 : Player 2, is waiting for a witness to stage If

which was demanded at

ra.(v)

so.

sO(w)

and

$'>O

, is

t h e r e an

: such

that

:*"$?

nEw

d,>O(n)

(w),

then

,$ED1.

A positive

s o l u t i o n t o question 1 would imply t h i s conjecture. Question 2 :

Is t h e r e

,$EDI

s.t.

d,' ? O(w)

?

Knight's conjecture would imply t h e above conjecture and a p o s i t i v e s o l u t i o n t o question 2 .

D. MARKER

242

REFERENCES :

R. Epstein. Degrees of Unsolvability : Structure and Theory. Lecture Notes in Mathematics no. 759, Springer-Verlag, Berlin, 1979. S . Feferman, Arithmetically definable models of formalizable arithmetic, Notices AMS 5 (1958).

L. Harrington, Building arithmetical models of PA, handwritten notes. 0

C. Jockusch-R. Soare, X 1 -classes and degrees of theories, Transactions AMS 173 (1972). J. Knight, A nonstandard model of arithmetic of degree less than that of true arithmetic, handwritten notes. A. Macintyre, The complexity of types in field theory, in Logic Year 1979-1980 (ed. M. Lerman, J. Schmerl and R. Soare), Lecture Notes in Mathematics no. 859, Springer-Verlag, Berlin, 1981.

J. Shoenfield, Degress of Models, JSL 25 (1960) S . Simpson, Degrees of unsolvability : a survey of results, in Handbook of Mathematical Logic (ed, J. Barwise), North Holland, Amsterdam, 1978. S.

Tennenbaum. Non-archimedean models for arithmetic, Notices AMS 6 (1959).

PROCEEDINGS OF THE HERBRAND SYMposILrM LOGIC COLLOQLJIW '81, J. Stem (editor) 0 North-Holland Publishing Company, 1982

243

FIFTY YEARS OF DEDUCTI@N THEOREMS by Jean P o r t e U n i v e r s i t g des Sciences e t de l a Technologie d'Alger

I . Jacques Herbrand gave i n h i s t h e s i s

( f 2 1 1 , s e e [22] pp. 90-91)

the f i r s t

known s t a t e m e n t , w i t h p r o o f , of t h e c l a s s i c a l deduction theorem f o r an axiomatiz a t i o n of t h e f i r s t - o r d e r p r e d i c a t e c a l c u l u s . By keeping only t h e p r o p o s i t i o n a l p a r t of t h e theorem and of i t s p r o o f , we o b t a i n a v e r s i o n of them which i s p r a t i c a l l y i d e n t i c a l with what we can f i n d i n modern textbooks. I t must n o t be f o r g o t t e n t h a t Tarski a l r e a d y knew t h e r e s u l t f o r the c l a s s i c a l

p r o p o s i t i o n a l c a l c u l u s (PC), s i n c e he used i t a s in

[47]

a

( s e e a l s o t h e d i s c u s s i o n i n t h e n o t e s of

well used t h e deduction theorem as a kind of

primitive notion i n

[ 4 8 ] and

"rule"

in

[ 4 6 1 and

[ 4 9 ] ) . JaSkowski a s

[ 2 4 ] (written several

years before i t s publication). The theorem can, i n modern terms, be s t a t e d a s follows :

where

n>O,

xl,

... x n ,

y, z

system under c o n s i d e r a t i o n , r e l a t i o n of

PC, and

=)

a r e formulas, PC

,... X n , Y

X'Y

t

i s the d e d u c i b i l i t y

(1)

:

t Y + z PC

t-z-xl,"'xn PC

i s e q u i v a l e n t t o t h e detachment p r o p e r t y o f XI

is t h e i m p l i c a t i o n of t h e formal

i s a metamathematical i m p l i c a t i o n .

I t i s t o be remarked t h a t the converse o f x1

+

i s t h i s formal system,

+,

namely

Y

( 3)

The theorem i s n o t t r u e , i n form

(l),

f o r the most u s u a l a x i o m a t i z a t i o n s of t h e

p r e d i c a t e c a l c u l u s (with t h e r u l e of g e n e r a l i z a t i o n ) , when hypotheses may c o n t a i n f r e e v a r i a b l e s . See f o r i n s t a n c e s But i n t h i s s h o r t survey

I

[ 2 3 ] and [ 2 8 ] .

intend only to s t r e s s the chief l i n e s of research

which have l e d t o g e n e r a l i z a t i o n s of the c l a s s i c a l deduction theorem i n v a r i o u s propositional calculi.

J. F'ORTE

244

2 . I t i s immediate t h a t we can r e p l a c e i n

(1)

PC by the i n t u i t i o n i s t i c proposi-

t i o n a l calculus (IC), o r even (with a small modification of Herbrand's proof) by the implicational p a r t o f I C . I n 1968, Witold A . Pogorzelski found the minimal subsystem of i m p l i c a t i o n a l

IC

i n which the c l a s s i c a l deduction theorem holds ; see [ 3 6 ] . I t must be remarked t h a t the minimal system i n which both (1) and ( 2 ) hold i s t h e i m p l i c a t i o n a l I C itself. For o t h e r p r o p o s i t i o n a l c a l c u l i , the ways of g e n e r a l i z i n g (1) a r e l e s s simple ;

there a r e c h i e f l y two : (I).

To replace t h e d e d u c i b i l i t y r e l a t i o n

+ ( i d e n t i c a l with

the "consequence"

operation of Tarski [461 by a weaker r e l a t i o n . (11). To r e p l a c e and

y-w

by a more complex function,

d ( y , z ) , of t h e formulas

y

Z.

3 . Generalizations of type pendently by Church

[9]

(I) and

have been found by More space -kwei [291, and inde[ I l l (see a l s o [ 1 5 ] ) . Their importance i s i n the

f a c t t h a t they a r e one of the sources of the r e l e v a n t l o g i c s studied i n I l l . In ( l ) , what stands a t t h e l e f t of the

"turnstile",

I-- , i s a sequence of formu-

l a s ; but i n t h e ordinary treatment of d e d u c i b i l i t y ( f o r only the

set of

PC

or

I C , a s i n [181)

those formulas p l a y s any r o l e , and moreover t h a t s e t may be in-

creased with formulas n o t a c t u a l l y used i n t h e formal deduction. Those f e a t u r e s a r e changed i n order t o achieve the notion of system

of

,R

"relevant"

d e d u c i b i l i t y . I n the

[ I ] ( i d e n t i c a l with the system of Church [ 9 1 and [ l l l ) , the turn-

s t i l e means t h a t every formula w r i t t e n i n the sequence placed on the l e f t i s act u a l l y used i n order t o c o n s t r u c t a deduction of t h e formula placed on t h e r i g h t . Thus, i n

R+

we have, p and q being d i f f e r e n t p r o p o s i t i o n a l v a r i a b l e s ,

P. q / F P

(4)

P* P , P P

(5)

and even

Such r e s t r i c t i o n s of d e d u c i b i l i t y occur i n a l l the r e l e v a n t l o g i c , with implicat i o n weaker than i n

R+.

Even the semi-relevant l o g i c

RM, which accepts the dedu-

c i b i l i t y statements of t h e form

x.x

I-x

( t h i s i s often called

(6) "the Mingle property"), r e j e c t s

p,q

k p .

Various form of the deduction theorem f o r s e v e r a l r e l e v a n t l o g i c s a r e studied itt

Deduction theorems

[ l ] . They a r e complex, formulas

(my) + z

and

245

x + ( y +z)

being non-equivalent

i n these l o g i c s . 4 . Generalizations of type

(11)

have been introduced i n

[37].

It i s e a s i l y

proved t h a t a s u f f i c i e n t condition f o r x1

,... xn, y

kz

*

xl,

... xn

/-

d(y,z)

(7)

i s the conjunction of the following statements :

k-

d(x,x) d(x,y)

Y

d(x,A1),

.. .d(x.%)

d(x,B)

For every postulated r u l e of the form A1,

... 4, 1 B .

It follows t h a t , f o r the modal systems

54

and

5 5 , axiomatized with the r u l e s of

m a t e r i a l detachment and of g e n e r a l i z a t i o n , we can take

d

as

dl

defined, L

being n e c e s s i t y , by dl(y.z) = L y + z

(11)

while f o r lukasiewicz's three-valued

l o g i c w e can take

d

as

d3

d3(y.z) = Y + ( Y + z )

defined by (12)

Those r e s u l t s were independently rediscovered and generalized by p o l i s h researchers: W.A.

Pogorzelski

[33]

rediscovered (12) and generalized i t t o h k a s i e w i c z ' s n-

valued l o g i c s , while iarnecka-Biaay [ 5 2 I , [53]

rediscovered the r e s u l t s concer-

ning S4 and S5 and found s e v e r a l v a r i a n t s o f them. I t may be remarked t h a t f o r

5 4 and S5

we can use a s w e l l

d2

d2(y,z) = Ly+Lz Moreover the r e s u l t s with

defined by (13)

dl

and

known modal systems : d l and d2 f o r S4 and S5 have we the converse of

d2

can be extended p a r t i a l l y t o c e r t a i n l e s s

E4

and

E5,

(7),

i.e.

the detachment property f o r

d2 f o r

K4 and K5. But only i n d l and

d 2 . This f a c t allows t o transform every statement of d e d u c i b i l i t y i n t o an equiva-

lent

"thesishood" statement, t h e r e s u l t being t h a t the t r u t h of a statement of

d e d u c i b i l i t y i n 5 4 o r i n 55 i s a decidable problem ; see [381. In

[51],

Tokarz has found a deduction theorem f o r t h e

[ l ] , with a

dq

defined by

"mingle"

system,

RM, of

J. PORTE

246

But i t i s t o be remarked t h a t Tokarz's r e s u l t is s t a t e d f o r a d e d u c i b i l i t y i n the sense of Tarski's consequence, which i s d i f f e r e n t from t h e ty",

even i n the case of

which i s r e j e c t e d by the

p,q /-p,

" r e l e v a n t deducibili-

RM, s i n c e Tokarz's ( T a r s k i ' s ) d e d u c i b i l i t y accepts

"semi-relevant

deducibility"

of

RM.

5. The foregoing b r i e f sketch does n o t , obviously, exhaust the s u b j e c t . Other lines of r e s e r a r c h have been foollowed i n s e v e r a l papers l i s t e d i n t h e bibliography below. Among the c h i e f unsolved problems, I may s t a t e the following one : How t o prove t h a t no s o l u t i o n t o

(8)-(10) e x i s t s ? ( b a r r i n g of course u n i t e r e s t i n g t r i v i a l

s q l u t i o n l i k e d(x,y) = ( x y ) +(x-+y), which

d

and p a r t i c u l a r l y focusing on s o l u t i o n f o r

has the detachment property). I t may be conjectured t h a t such a case,

without n o n - t r i v i a l s o l u t i o n i s the modal system

T

axiomatized with r u l e s of

material detachment and n e c e s s i t a t i o n .

REFERENCES (The deduction Theorem i s mentioned i n most textbooks of logic. The books l i s t e d here a r e only those which contain'some novel views about the Theorem). and N.D.

Anderson, A.R.,

Belnap, Entailment, Princeton University Press,

Princeton, 1975. Barcan, R . C . ,

"The deduction theorem i n a f u n c t i o n a l c a l c u l u s of f i r s t order

based on s t r i c t implication", The Journal of Symbolic l o g i c , 1 1 (1946), 115-1 18. Barcan Marcus, R.C.,

" S t r i c t implication, d e d u c i b i l i t y and the deduction

theorem", The Journal of Symbolic Logic, 18 (1953). 234-236. Bunder, M.W., 1969

-

S e t Theory. based on combinatory Logic, Dissertation,Amsterdam,

see [40].

Bunder, M.W.,

"Alternative forms of p r o p o s i t i o n a l calculus f o r a given deduc-

t i o n theorem", Notre Dame Journal of Formal Logic, 20 (1979), 613-619. Bunder, M.W.,

"Deduction theorem f o r s i g n i f i c a n c e logics", Notre Dame Journal

of Formal Logic, 20 (1979), 695-700. Burgess, J.P.,

"Quick completeness proof f o r some l o g i c of conditionals",

Notre Dame Journal of Formal Logic, 22 (1981), 76-84.

Deduction theorems

247

Church, A . , Review of Quine, "A s h o r t course i n logic", The Journal of Symb o l i c Logic, 12 (1947), 60-62. Church, A . ,

"The weak p o s i t i v e i m p l i c a t i o n a l p r o p o s i t i o n a l calculus"

( a b s t r a c t ) , The Journal of Symbolic Logic, 16 (1951), 238. Church, A., "Minimal logic" ( a b s t r a c t ) , The Journal of Symbolic Logic, 16 (1951), 239. Church, A . ,

"The weak theory of implication", K o n t r o l l i e r t e s Denken.

Untersuchungen zum logikkalkiil und zur logik der Einzelwissenschaften, e d i t e d by A . Menne, A. Wihelmy, H. Angst1 (Festgabe zum 60. Geburstag von Prof. W. Britzelmayr), r o t a p r i n t , Kommissions-Verlag Karl Alber, Munich, 1951, pp. 22-37. Church, A . ,

- see [15].

Introduction t o Mathematical Logic, Princeton University Press,

Princeton, 1956. Curry, H . B . ,

"Generalizations of the deduction theorem", Proceedings I n t e r -

n a t i o n a l Congress o f Mathematicians, Amsterdam 1954, v o l . 2 , pp. 399-400. Curry, H . B . ,

"The i n t e r p r e t a t i o n of formalized implication", Theoria, 25

(1959), 1-26. [14 b i s l Curry, H.B., ted generality", [14 t e r

1

Curry, H . B . ,

"The deduction theorem i n the combinatory theory of r e s t r i c Logique e t Analyse, 3 (1960), 15-39. "Basic v e r i f i a b i l i t y i n the combinatory l o g i c of r e s t r i c t e d

generality". Essays on the Foundations of Mathematics (dedicated t o Prof. A.H.

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N o t r e Dame J o u r n a l

PROCEEDINGS OF THE HERBRAMD SYMPOSILIM LOGIC COLLOQUIUM '81, J. Stem (editor) 0 North-HolIandPublishing Company, 1982

25 1

BOUNDING GENERALIZED RECURSIVE FUNCTIONS OF ORDINALS BY EFFECTIVE FUNCTORS ; A COMPLEMENT TO THE GIRARD THEOREM J.P. Ressayre * C.N.R.S., U.E.R. de Maths, Tour 45-55, 5e Ctage, Universite de Paris 7 , 2 place Jussieu 75231 Paris Cedex 05

We give a new proof, and some extensions, of Girard's theorem on the growth of primitive recursive functors from ordinals to ordinals.

INTRODUCTION Functors from ordinals to ordinals are a basic tool of Girard's

lIi - Logic. We first recall the most basic facts concerning these

functors. A detailed exposition is made in (Gl, Ch. I).

Denote by OL be the category of linear orderings and order preserving maps ; let ON be the restriction of OL to ordinals, and ONEw the restriction to integers. The fact that OL is closed under directed limits, and that every linear ordering is a directed limit of finite orderings, allows to extend every functor F : O N F o + OL to a functor F : O L + O L , commuting to directed limits : 1.1. Definition of F - a - In order to define F(x), where x ~ O L ,one chooses any directed system (xi, fij)i<jcl of finite xi' OL, whose limit is x (such a system exists, because x is the union of its finite suborderings ! ) . Since the xi's are finite, one can assume they are integers ; then since F is a functor over ONEw, (F(xi), F(fij) )i<jc~ is defined and is a directed system in OL. One then puts p(X) =df l i", (F(xi), F(f..) )i<jcI 13

If f is a morphism of OL, that is an order preserving map : x + x ' with x, x'€OL, then one defines F(f) : F(x) + F(x') in a similar way, by choosing a directed system Sf in ONI-w, such that lim Sf and putting + F(f) =df

li",

=,

f,

F(Sf).

- 11 - It is easy to-see that the functor F commutes to l a , that is F ( lim S) = lim F ( S ) , for every directed family S. Thisproperty d

4

*An essential step of the work exposed here was done not by the expositor, but by L. Harrington -see the last remark o f § 11. The author is also indebted in J. Van de Wiele, who pointed out an error, and provided the result which allows to correct it -see th. 111-4. And the overall debt of this work to Girard's R: - Logic will be obvious, but in addition the author wants to thank Girard far his patience and for stimulating conversations.

J.P. RESSAYRE

252

c h a r a c t e r i z e s t h e f u n c t o r T up t o c a n o n i c a l isomorphjsm : i f G i s any o t h e r f u n c t o r : O L + O L s u c h t h a t G commutes t o 3 and G I ( 0 N r w ) = F , t h e n t h e r e i s a u n i q u e and c a n o n i c a l isomorphism b e t ween F a n f G . T h i s s t a r t s t o e x p l a i n why s u c h f u n c t o r s a r e a p o w e r f u l t o o l : a l t h o u g h t h e y o p e r a t e on t h e whole c l a s s O L , t h e y a r e d e t e r m i n e d by t h e i r r e s t r i c t i o n t o O N r w - h e n c e by a f i n i t e amount o f i n f o r m a t i o n , i f t h i s r e s t r i c t i o n c a n be coded i n a r e c u r s i v e way.

- c - Given G = O L + O L , h e r e i s t h e p r a c t i c a l c r i t e r i o n t o d e c i d e I f xCOL and z c G ( x ) , we s a y t h a t z w h e t h e r G commutes t o h a s a f i n i t e s u p p o r t i f t h e r e e x i s t s p t o and a n o r d e r morphism z E r a n g e o f G ( f ) ; and we s a y t h a t G h a s t h e f : p + x such t h a t f i n i t e s u p p o r t p r o p e r t y i f VxeOL VzeG(x), z has a f i n i t e support ( t h i s r o p e r t y i s t e c a t e g o r i c a l a n a l o g o f t h e p r o p e r t y G(x) = G ( x ) , :hen G i s any f u n c t i o n from s e t s t o s e t s ) . Then X finite c x G commutes t o l i m i f f G has t h e f i n i t e support p r o p e r t y .

.

4

L e t u s now c o n s i d e r F : O N P w + O N , from O N t o O N ( i n s t e a d o f OL ) :

and t r y t o e x t e n d F t o a f u n c t o r

1.2. Remarks - a - I f F(5) i s w e l l o r d e r e d f o r e v e r y w e l l o r d e r i n g x , then t h e r e s t r i c t i o n of F t o w e l l - o r d e r i n g s i s isomorphic t o a F and which w e s t i l l d e n o t e by f u n c t o r from O N t o ON,which e x t e n d s F : f o r x € On, F ( x ) i s d e f i n e d b y F ( x ) = \ F ( x ) 1 (where 1 R 1 d e n o t e s t h e t y p e o f t h e o r d e r i n g R , h e n c e I R I Con f o r e a c h w e l l o r d e r i n g R ) . And F ( f ) i s d e f i n e d i n a s i m i l a r way. S i n c e F i s e s s e n t i a l l y i s o m o r p h i c t o a r e s t r i c t i o n o f F , F a l s o commutes t o l i m

+-

-

b - If F = O N P o + O N has ~ ~ m e x t e n s i o n t o a f u n c t o r G : ON + O N ( n o t n e c e s s a r i l y commuting t o l i m ) t h e n i t i s e a s y t o see t h a t ( F ( x ) 1 6 G(x) f o r e v e r y x€On ;i+ stable ordinal, uo. I n a s e n s e , t h e f i r s t &' - r e f l e c t i n g o r d i n a l s o ( s e e R . I . 7 . b ) , and t h e . f i r s t stable ordinai u , a r e quite large countable ordinals. So t h e c o n c l u s i o n o f G i r a r 8 ' s b o u n d e d n e s s t h e o r e m h o l d s e v e r y where on a l a r g e i n i t i a l segment so o f H I , and t h e n h o l d s c o f i n a l l y i n a s t i l l much l a r g e r i n i t i a l segment u o .

Bounding generalized recursive functions of ordinals

255

1.10. Notations - a - Let u s call total functors all functors F : O N + O N , and partial functors allfunctors F : O N + O L ; for a t On, we then s a m F(a) is undefined if F ( a ) is not well ordered ; and F ( a ) is defined if I F ( a ) t o n , in which case we identify F ( a ) and IF(a)! . (These definitiois are inspired by analogy with total and partial recursive functions). - b - For a in On, let PAD(a) denote s u p { F(a) : F is a total p.r. using functor 1 (this notation was introduced by Girard, ( G . 4 . ) . , the initials of the words "prochain admissible"). And let PAD'(a) denote sup {F(a) : F is a partial p.r. functor, such that F(a) is defined I .

It is natural to investigate the relations between properties involving total functors, and corresponding properties and constructions involving partial functors. To begin with, Girard asked the following questions : -1If F is any partial p.r. functor, is F always bounded by a total p.r. functor, or by the sup of these functors ? -2What arethepossible relations between PAD(a), PAD'(a) and a+ ?

We shall answer negatively the first question, see IV.7. And we shall give a complete answer to the secund question (see IV.6 and IV.8). These results are exposed in S IV ; in 5 111, we gather the facts and results on definability of ordinals that we need. And in S V, we examine the results which one obtains when one does not restrict oneself to functors F such that Fb(ONfw) is p . r . .

J.P. RESSAYRE

256

§

I1

-

A PROOF OF GIRARD'S BOUNDENESS THEOREM

We let be a recursive language containing 6 and the distinguished constant C ; for y 6 On, we call y-model of c 8 any model M of &e such that (CM,b)'(c CM) 2 (y, r f y ) , where CM denotes {ad M : MI. a t C } . If T is a theory and (I, a sentence in $, we write T ky (I, if (I, is true in all y-models of T , and T I- (I, if a. L e t s o d e n o t e t h e f i r s t non

6-definable o r d i n a l : it i s

t h e f i r s t a f o r which G i r a r d ' s b o u n d e n e s s t h e o r e m 1 . 6 d o e s n o t h o l d ;

t o s e e t h i s , we a r e g o i n g t o s t u d y s o . Fact 1 - I f a < so, then every o r d i n a l y < a i s i n s i d e La+

.

Proof

-

By 1 . 6 . , we know t h a t a

t o r : ON-tON}.

=

sup

C - d e f i n a b l e from a

F ( a ) : F t o t a l p . r . func-

By Th. 1 . 5 , t h i s i m p l i e s t h a t a = s u p { 6 : 6 i s implies Fact 1 .

1 - d e f i n a b l e from a i n L a + 1 ; w h i c h by 1 1 1 . 4 F-a c t 2 - s o i s l i m i t o f a d m i s s i b l e o r d i n a l s . -

Proof - I f n o t , l e t a b e t h e l a s t o r d i n a l < s o which i s a d m i s s i b l e o r l i m i t o f a d m i s s i b l e ; s o a < s < a+, and a i s 6 - d e f i n a b l e . I t i s e a s y t o s e e t h a t i f a i s 6 - d e f i n a b l e , t h e n a+ a l s o i s 6 d e f i n a b l e ; s o so # a+, and a < s o < a+. Then combining a C I d e f i n i t i o n o f s o from a (which e x i s t s i n L a + , by F a c t l ) , and a 6 - d e f i n i t i o n o f a , one o b t a i n s a 6 - d e f i n i t i o n o f s o . T h i s b e i n g absurd, Fact 2 i s proved. 111.10 Theorem

( G i r a r d e x c e p t (d) )

- Let

s o be t h e f i r s t non

- a - s o i s t h e f i r s t non B - - d e f i n a b l e 6-definable ordinal ordinal. - b - so i s t h e f i r s t o r d i n a l a s u c h t h a t f o r e v e r y

p a r t i a l p . r . f u n c t o r F : O N - t O N , a C dom. F + a U { a l C dom. F . - c - s o i s t h e f i r s t c o u n t e r example t o G i r a r d ' s b o u n d e d n e s s property 1.6. Proof

-

- d

-

so i s t h e f i r s t

C:

-reflecting ordinal.

By F a c t 2 and 1 1 1 . 8 , s o i s n o t 6 - - d e f i n a b l e ,

f o r i t would

then be 6-definable ; t h i s implies ( a ) ; and ( b ) i s e q u i v a l e n t , by 1 1 1 . 7 . The c o n d i t i o n , f o r any p . r . f u n c t o r F , t h a t F " s o C s o ,

J.P. RESSAYRE

268

is2,

i n s i d e Ls+

0

; t h e n t h e a d m i s s i b i l i t y o f Ls:,

so+ > s u p { F ( s o )

: F p.r.

f u n c t o r s u c h t h a t F" s o t s o

1

i s t r u e f o r every

and s i n c e s o i s l i m i t o f a d m i s s i b l e s , F" s 0 c s o total p.r.

implies

F , hence

IF(s~)

so+ > s u p

functor : O N + O N 1

: F total p.r.

showing ( c ) . To s e e ( d ) , n o t e t h a t s o i s non B - - d e f i n a b l e i f f ( f o r e v e r y C: s e n t e n c e 3 R O ( R ) , s o b 3R0(R) 3 y < so, y 3Re(R) ), i n o t h e r words i f f s o i s C : r e f l e c t i n g . We h a v e t o show t h a t s o i s -+

C'

-1

- r e f l e c t i n g , which i s t h e same p r o p e r t y a s s e r t e d f o r f o r m u l a s

3R0(R) which may c o n t a i n p a r a m e t e r s < s o . But h e r e t h e p a r a m e t e r s , b e i n g @ - d e f i n a b l e , a r e e a s y t o e l i m i n a t e ; h e n c e s o i s .Zi - r e f l e c t i n g . And i f a < s o , a i s B - d e f i n a b l e , a f o r t i o n B - - d e f i n a b l e , non C : - r e f l e c t i n g , a f o r t i o n n o n ; : - r e f l e c t i n g . 111.11 Remark t h e o r y KP + C

.

hence

so is B*-definable : i t i s t h e l a s t y such t h a t t h e i s a d m i s s i b l e + tia < C (a i s @ - - d e f i n a b l e ) h a s a

y -mode 1

111.12 Theorem - I f t h e r e i s a w e l l o r d e r i n g p c u 2 that I p l i s a JI: s i n g l e t o n , -a & B*-definable.

-

Proof

Suppose t h a t

p

on w s u c h t h a t < w , p , Then i t i s known t h a t

@

(Aw

A

diagramm o f

t h e proof of 111. 5 .

V

p t

IpI

a and p i s t h e o n l y r e l a t i o n

=

s a t i s f i e s t h e IIi s e n t e n c e

(w)>

VSY(RS).

( s e e (R2) ) ; h e n c e p i s Z d e f i n a b l e

La+

a s t h e only p such t h a t

i n La+

where

c uz,

o f type a, such

Cp (a) Y
, v (because V can be expanded

c ~( a ) 1 < v

to an a-model of T, hence of Cp by the Claim, hence of by choice of -3 v I ( u , v ) A I F c p ( a ) I < v I cp 1.

-

1 2 a

& B-definable, then B

IV.3

Theorem

Proof

- Similar to that of

IV.4

Theorem

e s y

-

=

PAD(&)

=

PAD'(a).

IV.2 and of 1.6.

The following conditions are equivalent for

a :

B-definable and B = a+ @--definable and 6 = a+ ( 1 ) PAD(a) = a+(1)' PAD'(@) = a+ (2) F o r every tqtal a+-recursive function f, there is a total p.r. function F such that f(y) < F ( y ) for all y C [cC,a*t. (0)

(0)'

a a

Note that if we drop conditions

(0)'

dans ( l ) ' , then by

IV.1 this result is equivalent to Th. 1.8.

Proof - ( 0 ) (2) - Let f be a total a+-recursive function ; By ( 0 ) and IV.l, every ordinal < a is C -definable from a inside L a + . Then by Th. 111.2, we can assume that a is the only paramater in the 2 definition of f. Let 3 R O ( R ) be a C : definition of a among all ordinals, let T be K P + 8 (R) (a) + a < C + 7 3 y C I a,C 1 (y is an admissible ordinal) ; and choose a sentence Y s u c h that +

TI-

9

+-+

1

IFq)(C)l

< f(C)

where the symbol a replaces a in the 6 definition of f. Arguing as in the proof of Th. 1.6, one shows that F q is a total p.r. functor

Bounding generalized recursive functions of ordinals

such t h a t f ( y ) i

(2)

(1)

+

F q (y)

for a l l y

271

e [a,a+I

i s obvious.

( 1 ) + ( 0 ) - Assume PAD(a) = a+ ; t h e n 8 = a+ t h e r e remains t o s e e t h a t a i s B-definable. Let

is clear, so

a=-min I y : t h e r e i s a t c t a l p . r . f u n c t o r F s u c h t h a t F ( y ) ? 1 By t h e c h o i c e o f a l , t h e r e i s a t o t a l p . r . f u n c t o r F1 s u c h t h a t

F1 (a,)& a ; and a 1 i s c l o s e d u n d e r t o t a l p.r. f u n c t o r s ( f o r o t h e r w i s e , h a v i n g F ( y ) > a 1 f o r some y < a l and some t o t a l p . r . f u n c t o r F , ( c o n t r a d i c t i n g t h e c h o i c e o f al) s i n c e we would h a v e F1 o F ( y ) & a , functors is a t o t a l p . r .

t h e c o m p o s i t e F 1 o F o f two t o t a l p . r . functor. Claim

-

G(al)

Proof G"

And

-

T h e r e i s a p a r t i a l f u n c t o r G s u c h t h a t G" a l c a1 a& i s not defined ( i n o t h e r words, 6 ( a , ) i s n o t a w e l l ordering). O t h e r w i s e , G ( a l ) t On, f o r e v e r y p a r t i a l p . r .

a1c a l

.

G such t h a t

+

a > s u p { G ( a l ) : G p a r t i a l p . r . f u n c t o r s u c h t h a t G" a l C a,) >/ s u p { F ( a l ) : F t o t a l p . r . f u n c t o r ) 5 sup { F o F l ( a , ) : F t o t a l p . r . f u n c t o r ) &

sup { F ( a )

: F total p.r.

and t h i s c o n t r a d i c t s a+

=

functor)

=

PAD(a)

PAD(a).

A0

(The > r e l a t i o n h o l d s b e c a u s e t h e c o n d i t i o n G"alC a 1 on G i s i n s i d e La+ ; t h e o t h e r r e l a t i o n s a r e e a s i l y c h e c k e d ) . The f u n c t o r G o f t h e c l a i m a l l o w s t o p r o v e t h a t a 1 i s B - d e f i n a b l e , by t h e same argument a s i n t h e p r o o f o f Th. 1 1 1 . 8 . And by t h e c h o i c e o f al, implies PAD(al) = PAD(a), s o PAD(al) = ti1+ = 8 1 , w h i c h by I V . l t h a t a i s C d e f i n a b l e from a 1 i n s i d e L + = L a + Then t h e B - d e f i n a b i l i t y of a l e a s i l y i m p l i e s t h a t o f a . ( o ) + ( o ) ~ i s obvious,

Clearly (1)

(1)*+ (0) -

Claim 1

-

and ( o ) l

If

&

=

1

a+, t h e n

-+

implies y+

=

a+

(1)' =

.

a1

f o l l o w s from Th. I V . 2 .

a+.

implies

= y

+

+ = a , f o r every

ordinal y.

-_ Proof -

I f y+ = a

al' PAD(a)

=

PAD' ( a ) .

IV.6 Theorem -

-

Proof

If a is

@*-definable,

Assume t h a t a i s @ * - d e f i n a b l e

then

PAD(a) = P A D ' ( a ) .

: there is a recursive

t h e o r y To s u c h t h a t a i s t h e l a s t o r d i n a l f o r w h i c h To h a s a n a - m o d e l . We c a n assume t h a t t h e symbol t i s r e p l a c e d i n To by a n o t h e r symbol E . Given a p . r . f u n c t o r F s u c h t h a t I F ( a ) I t On, such t h a t we l e t T be t h e t h e o r y KP + T o , and J, a s e n t e n c e o f TI-

@

+ +

-

Claim

7 IFJ, (C)

I

T ky J,

for a l l

Proof of claim

-

IF,,, (y) Thus M

1

y.

Let M be a y -model o f T ; t h i s i m p l i e s y < a So i f and T o k >al Hence ] F ( y ) 1 E O n .

since T contains T M -I , hence M

+,

.

1

< IF(C)

pI

F,,, (y)

I

.

I , i t would i m p l y t h a t i n c o n t r a d i c t i o n t o M )= T

< IF(y)

,

t On, h e n c e T b4y J, , and t h e c l a i m i s p r o v e d .

+

7

J,

: O N + O N . MoreI t i m p l i e s t h a t F,,, i s a t o t a l p . r . f u n c t o r o v e r assume y 6 a ; s i n c e w e c a n expand V t o a y-model o f T , i t i m p l i e s V F l i , , h e n c e V k - 1 1 F $ ( y ) \ < I F ( y ) l ; s i n c e F ( y ) and

,

F (y) a r e h e r e b o t h w e l l o r d e r e d , w e c o n c l u d e I F ( y )

I
a. which satisfies PAD(a) < PAD'(cr) < a+.

(d)

-

follows from Th. I V . 4 ,

(1)

cf

( l ) ' , so the proof is completed

Bounding generalized recursive functions of ordinals

211

I V - E X T E N S I O N S OF G I R A R D ' S BOUNDEDNESS THEOREM TO NON p.r. FUNCTORS

Here try to bound it is p.r., we set as those from 5

we consider a generalized recursive function f, and by a functor F ; but instead of requiring that F weaker requirements. The methods of Girard as well I 1 easily yield the results exposed here.

We let 6 denote an ordinal which is admissible or limit of admissible ordinals, and such that L6

every set is countable.

Theorem (Girard) - L e t T be a theory whose conjunction is in ; for each-sentence $ tn L6 , there exists a functor F . O N - t O L such that

V.l

ywlw

gulw n LA

$ .

- i $

F t-(ONbw) $

* F l ( O N p w ) is $

i s 6-finite, and the application 6-recursive

- ii-

F

- 111-

for all ycOn

$

commutes to

3

- hence F

- $ -is Fqy iff 1 ~ ~ UJ

a &-finite functor ( yI t)o n .

Proof - very similar to the proof of Th. 11.1, using the extension of the Henkin lemma. to 3qw V . 2 Theorem -

For every ordinal a

6 the following

are equivalent:

there is a 6-finite sentence 6 ( R ) 6 gu such that a 1 the only ordinal satisfying 3 R e ( R ) ; & every ordinal < a is C-definable from a and fromxarameters < 6 , inside L a + . ______. ___(0)

c

(I)

a+ =

sup I ~ ( a ): F total ~ - f i n i t efunctor I

.

(2) for every a+-recursive function f , there is a total 6-finite functor F such that f(y) < F ( y ) for all y 6 [ a , a + [ .

Proof 11.1. V.3

- Similar to the proof of Th. 1 . 8 , but using V . l instead of Note that Th. 1 . 8 is the particular case 6 = w of this result.

Corollary - The set of ordinals 6 such that for every &+-recursive function there is a total &-finite functor F such that f ( y ) < ~ ( y )for all y t [6,6+[, is cofinal in s q l ( L ) .

218

J.P.RESSAYRE

Proof - For every ordinal a. < , let & ( a o ) be the smallest ordinal 6 >I a. which satisfies our assumptions on 6. It is clear that condition ( 0 ) of Th. V.2 holds when a = 6 = 6 ( a o ) . Hence condition (2) also holds, thus proving the corollary. V.4 Theorem - Let a be an ordinal which is countable inside L a + . Then for every a+- recursive-function f, there is a total a+-finite functor F such that f(y) < F ( y ) , for all y E [a,a+[

.

Proof - One applies Th. V . l in the case 6 = a+, in a way similar to the use of Th. 1 1 . 1 in order to prove 1.6.

Bounding generalized recursive functions of ordinals

219

REFERENCES

-

[A

- R] P. Aczel, W. Richter, Inductive definitions and reflecting properties of admissible ordinals, in Generalized Recursion theory, Fenstad, Hinman, editors, N.H. 1 9 7 4 . B I

[

[

J. Barwise, Admissible sets and structures, Springer, 1 9 7 4 . -

G 1 1

J.Y. Girard, IIi -Logic of Math. Log.

G 2 I

J.Y. Girard, Proof Theoretic investigations of iterated inductive definitions, part I, to appear in the Specker volume in “L’ Enseignement MathEmatique” . Part 2 ,

to appear in Ann.

[ G 3 1 J Y. Girard,

TI:

[ G 4 1 J Y. Girard,

Cours de Th6orie de la Dgmonstration, Universit6 Paris VII.

9 7 9 - 1980, [

M 1

t R 1

-Logic

Part 1 ,

M. Mhsseron, Majoration des fonctions w l C K -recursives par des 6chelles - Thbse de 3 e cyle , Universit6 Paris-Nord, 1 9 8 0 .

I J.P. Ressayre, Logique tous azimuts, Cours de 3 e cycle, Universit6 Paris VII,

[

to appear

1982.

R 2 1 J.P. Ressay e, Models with compactness properties with respect to an admissible language, Ann. of Math. Log., 1 9 7 7 , p. 5

I VdW 1

This volume

PROCEEDLh'GS OF THE HERBRAND SYMPOSIUM LOGIC COLLOQUIUM '81, J. Stem (editor) 0 North-Holland Publishing Company, I982

281

A SUPERSTABLE THEORY WITH THE DIMENSIONAL ORDER PROPERTY HAS MANY MODELS Jfirgen SAFFE U n i v e r s i t a t Hannover

Abstract I n t h i s p a p e r we p r o v e i n a v e r y d i r e c t f a s h i o n by i n t e r p r e t a t i n g graph theory i n a s u p e r s t a b l e theory w i t h t h e dimensional o r d e r property t h a t such a t h e o r y h a s every

x

2

ILI

+

non-isomorphic models of c a r d i n a l i t y

'2

X

for

wl.

0. INTRODUCTION AND NOTATION

A l l t h e n o t a t i o n s i n t h i s paper follow those i n [Sh 11

[Sf 21

or

[Sf 31

which a r e t h e

e x c e p t t h e f o l l o w i n g changes : a t y p e w i l l be usualI l y complete ; we omit t h e d i s t i n c t i o n between t and t* ; two t y p e s p. ES i(A)

same a s

and

[Sh 2 1

a r e c a l l e d weakly o r t h o g o n a l i f f f o r a l l r e a l i z a t i o n s n o t f o r k over

A

B.

1

of

p . t(Fil,AUg2) does

( i f one of them i s s t a t i o n a r y t h i s c o i n c i d e s w i t h S h e l a h ' s d e f i -

n i t i o n ) ; f o r t h e family

z where

2

=

:x

we omit t h e

and u r i t e

(x,h)

for

t y p o g r a p h i c a l r e a s o n s . We assume f a m i l i a r i t y w i t h t h e d e f i n i t i o n s and theorems of s t a b i l i t y t h e o r y a s can b e found i n [Sh 21

[LPI, [ P o l , [Sf 11, [Sf 2 1 ,

a t l e a s t i n c l u d i n g t h e c o n t e n t of c h a p t e r V of [Sh

[Sf 31, [ S h l l ,

11.

I n t h i s p a p e r we g i v e a proof of t h e f o l l o w i n g theorem which i s due t o Shelah :

0.1. Theorem Let

T

be s u p e r s t a b l e . I f

T

has t h e d i m e n s i o n a l o r d e r p r o p e r t y t h e n f o r a l l . h

.

A 2 ( L I + w 1 we have I(X,T) = 2 But c o n t r a r y t o t h e proof i n [Sh 2 1

we a v o i d t h e u s e of c h a p t e r V I I I

of [Sh 11

by i n t e r p r e t a t i n g graph t h e o r y i n a d i r e c t way. 1. THE DEFINITIONS AND THE BASIC LEMMAS

Throughout t h e p a p e r we assume t h e t h e o r y

1.1.

T

bo t e s u p e r s t a b l e .

Definition

T h a s t h e dimensional o r d e r p r o p e r t y (dop i n s h o r t ) i f f t h e r e a r e

(a,w)-saturated

J. SAFFE

282 models

MI,

M,+,

g

and

1-12

and a type

t(M ,M ) does not fork over -1

-2

orthogonal t o

is

1

(g)

such t h a t

(a,w)-prime

<El , -0 M g2 and

M ,+

M1 U

over

-2 ' p is

<M

E2.

and

M -1

,5

.+&

pES

1.2. Remarks and example This d e f i n i t i o n is c l e a r l y equivalent t o t h a t of Shelah i n not have t h e dop then

example of such a theory i s t h e following. Let and

T

E

(which abbreviates

b

guments and we have

T

does

"edge")

P(e, v o , v I )

contain t h r e e p r e d i c a t e s E , V , P

L

has t h e following axioms : Any model of

joint sets

[Sh 2 1 . So i f

s a t i s f i e s some reasonable s t r u c t u r e theorem. The t y p i c a l

T

i s t h e union of the two d i s -

T

and

V

i f f t h e edge

("vertex"), e

P

has t h r e e a r -

joins the verteces

vo

and

between any two v e r t e c e s t h e r e a r e i n f i n i t e l y many edges. The theory i s c l e a r

vl, l y w-categorical but f o r any higher c a r d i n a l i t y i t allows i n t e r p r e t a t i n g graph theory by a c a r d i n a l i t y q u a n t i f i e r and s o t h e r e a r e as many models as graphs f o r any c a r d i n a l i t y

2

w l . The proof of t h e theorem w i l l show t h a t i n some sense it i s

t h e only example. 1.3. The b a s i c lemma

p ES1(N) -

has t h e dop. Then t h e r e a r e (a,w)-satured models M M M and Nand 4' -1 ' -2 according t o t h e d e f i n i t i o n such t h a t f u r t h e r the following hold :

w(M1, M J

=

Suppose

T

1 = w(E2,

q ES1(N) -

F0)

w(Ei,

-I

p

a EM2

2

p1

such t h a t

s t a t i o n a r y and

%

we have

be

MI

t(g'Z,'bll) wlog t(c, w(g,

g, UE2) fll Uy2) is

t(c?;,

M2

M

-1

U

a

and

i = 2 . Choose

We show f o r and

F2

p1 = plMl U

E2. Let

q

is

c

t(M1,

-

%.

Let

N'

-

%Ua2.

Hence

-

= w(a2,

%)

be (a,w)-prime M -1

and

E2)

Uz2 UF)

is

does n o t f o r k over t(c,M1

Ug2 U6)

[Sf 31 f o r a p r o o f ) . Now l e t

(a,w)-isolated. to

bEN

r e a l i z e p . Now choose

(a,w)-isolated.

As

-

is sta-

U

MI Ug2, t(c,M1

% Ua2 (see [SF 2 1 o r M+, Ua2. Clearly w(MV2,flo)

does n o t fork over

is finite.

orthogonal t o p. By r e g u l a r i t y

does not f o r k over

F E N ' because t(K, El Ug2) i s hi1 UF2 U6) t o N' i s orthogonal

M+,)

according t o the d e f i n i t i o n of dop

does not fork over

and t o

(a,w)-prirne over

(5)

i = I,?.. U

i s orthogonal t o

t(g,

1

pfS

So wlog we can assume p t o be r e g u l a r .

is finite for

t(M1 Uc6, kj2)

orthogonal t o

and

does n o t f o r k over

t i o n a r y . Then

-

i s regular.

(even not weakly)

K2.

and

M

Claim 1 : w(Ei, +&) such t h a t

p

is minimal under a l l those examples. There i s a r e g u l a r

which i s n o t

orthogonal t o

and

%, El, E2, N

Proof : Choose such t h a t

%)

is

MI2

i s f i n i t e and

M UM' where -1 - 2 The non-forking extension of

MI2.

over

Hence by minimality

Models of superstable theory

%)

Claim 2 : w(zi, Let

= 1 for i = 1,2.

M+)

n . = w(&,

283

n1 > 1 .

and suppose by way of c o n t r a d i c t i o n

(ci) exemplify t h e d e f i n i t i o n of weight f o r i = 1 , 2 . Now l e t 1 jl , ~is provable in P I ( $ , n t l O ) . P r o o f : For ordinals C I . < E ~let ( c r o ) : = a o and ~~

..., antuantl).

Thin by Cantor‘s normal form, for each ordinal exist uniquely determined n and a o , a n such that a=(an, a n , O ) . By an external induction on n and pointwise we prove: induction in P I ( $ , n t l O ) (ao,

...,

O < ~ < E ~ there ,

...,

an+B

=antBhB>OAa>l

+

F ( ao,

a,

. . . ,an+ B ) ( a )2F( a. ,.. .,an ) +1 (a).

Let n=O. B=1 is trivial;&+ B + 1 : FaotBtl(a)=FatE(a) a L >. 5 FaotB(a) 2

-

Fao t 1 (a); h[al+A: FaotX(a)=FaotA[al(a) 2 Faotl(a). Now suppose that the assertion holds for all ordinals ao,...,an,f3o. Now let

al + Fa(a) < Fa(atl) is provable in PI($cntlO). Proof: By pointwise induction on a . a=O: clear; a+a+l: Fatl(a)=

F~tal(at')

L.6 5 '~[atl] (a+l)=FA(atl).

The monotony properties of the Fa's stated in the last three lemmata have already been established by Schwichtenberg in [91.

8. Proposition: For each p.r. function f(a,a) there exists a p.r. function t and a natural number k, such that is provable in PI($,ntlO). b>Fa(t(max(a,a,k))) f(a,a,b)=O an). For a given p.r. Proof: Let ]al,..., an( stand for max(al, function f(a,a) let t be a monotonous p.r. function such that 1, a[f(a,a)l,f(a,a),a,b < t(la,a,bl). For this t, take a natural number kd3 such that t ( I a,a,k I )- l . Then

U.R. SCHMERL

294

i s p r o v a b l e i n P I ( @ , n t l O ) , w h e r e i = O f o r a F a t i ( a ) f o r a > l i s p r o v a b l e i n P I ( Q E Q t l O ) . Here 5 i s t h e o r d i n a l which i s

o b t a i n e d from a i f i n i t s C a n t o r normal form w i s r e p l a c e d by R, i = O f o r acwand i = l o t h e r w i s e .

Proof: T h i s f o l l o w s immediately from t h e p r e c e d i n g theorem. Theorem 1 0 and c o r o l l a r y 11 a l s o h o l d f o r

"O

+

PAI- a>O + Ga(atl)>O. a{atl,b}>O is proved by induction on b.

21. Lemma: The following is provable in PA: (i) For each limit ordinal a of type w , a[klcak+l] and for each limit ordinal a of type Q , Bcy a[5lca[yl holds. +

The Bachmann/Howard ordinal ( i i )a c B A a > l

+

291

Ga(a)

C) = W,

xi1

O(V)

=

B, and

S,

S.

(= xiu)

and

is not. There must is defined but x:+~ 1 Cs [u(xl 1 ,sl>l = C[u(xll,s 1) I and':x 1 be infinitely many such stages s 2 so because the computations involved settle down and for such a stage sl, AS1 = A and Bs Sl+l

=

x i+l

u(xi,sl) and

at any stage

s

>

sl.

crucial property of i.

2.

IF C'

I

= B. Then at stage sl+l, v(xi,sl+l) 1 is defined and case 1 never applies to any marker xj, j i




sl, contradicting the

This contradiction yields the lemma.

0

+I.

T

In this section we describe a construction based on that of 91 which proves the theorem, provided that C' z 9 ' .

Of course, if C' >T

+',

trivial since C' is itself r.e. in C but not of r.e. degree.

then the theorem is The assumption that

R.I. SOARE and M. STOB

308

C is low w i l l allow us t o modify t h e c o n s t r u c t i o n of 5 1 so t h a t t h e outcome i s f i n i t a r y i n nature.

This w i l l a l l o w us t o prove t h e theorem u s i n g a f i n i t e i n j u r y

p r i o r i t y argument.

The technique of r e p l a c i n g an i n f i n i t e i n j u r y p r i o r i t y argu-

ment by a f i n i t e i n j u r y argument u s i n g lowness i s due t o Robinson [ 4 ] .

To prove t h e theorem, it s u f f i c e s t o c o n s t r u c t s e t s A and B which a r e r . e . C and which s a t i s f y f o r every i Ri

"(wi)

:

iEW

in

or

Oi(Vi)

f; B

or

Zi(B 6 3

c)

P

Vi.

i s an e f f e c t i v e l i s t of a l l s e x t u p l e s of t h e

a p p r o p r i a t e type. Suppose then t h a t we apply t h e c o n s t r u c t i o n of 51 t o Ro. 1.5 t o g e t h e r show t h a t Ro is s a t i s f i e d .

Then Lemmas 1.4 and

Now t o prove Lemma 1.4 (and i n p a r t i c u l a r

(1.7)) we used t h e f a c t t h a t i f s and u a r e s t a g e s of t h e c o n s t r u c t i o n f o r which v(x..s) J

= v(x..u)

J

and V ( X ~ + ~ ,and S ) V ( X ~ + ~ ,aU r e) undefined, then A,[v(x.,s)] J

(This holds i f j i s even; s u b s t i t u t e B i f j i s odd.)

%[v(xj,s)].

=

Thus i f

v ( x j , s ) is d e f i n e d , i t imposes a r e s t r a i n t of t h e same l e n g t h on both A and B.

If

t h i s r e s t r a i n t i s r e s p e c t e d by our a c t i o n f o r o t h e r requirements, then Lemma 1.4 can be proved f o r Ro with t h e same proof a s t h a t of 51. t h e proof of Lemma 1.5.)

(We w i l l d e f e r d i s c u s s i n g

Thus our s t r a t e g y f o r meeting R1 should only a p p o i n t

markers l a r g e r than any v a l u e v ( x . , s ) which is defined. J Lemma 1.4 f o r R, x? s

lo

1

= xso

io

for all s

so and j




x:,

Ro may d e f i n e v ( x i , s )

so l a r g e t h a t C l a t e r changes below v ( x i , s ) c a u s i n g v ( x i , s ) t o a g a i n become ( R e c a l l t h a t v ( x i , s ) is chosen l a r g e enough s o t h a t v ( x i , s )

undefined.

>

xs



a s we s t a t e d i n ( 2 . 4 ) . ) Let so be a s t a g e such t h a t

is t h e f i n a l p o s i t i o n of marker xi and l e t

xfo

u be the " t r u e " use i n e s t a b l i s h i n g a l e n g t h of agreement R(s) Note t h a t i f x

E

A (B) then x

(B,)

E

f o r almost every

S.

l a r g e r than

Now l e t s1

>

xi

.

so be a

t r u e s t a g e f o r C such t h a t

4

Thus, f o r a l l s u f f i c i e n t l y l a r g e [ u ] = A[u], ( B s [ u ] = B[u]) by ( 3 . 9 ) . 1 1 s There i s t h u s no t r u e s t a g e s sl, k(s,) >xi1 v i a t h e c o r r e c t computation. Then

o b s t a c l e towards d e f i n i n g v ( x i , s l + l ) .

Furthermore, v ( x i , s l + l )

i s not defined t o

be too l a r g e s i n c e a l l v a l u e s v ( x < i , j > , s l ) which a r e d e f i n e d s a t i s f y v(x,sl) v(xi,sl+l)

< cs 1 and,

= max {u; v ( x

] = C, [cs 1. Thus i f 1 1 1 d , then C, [ v ( x i , s l + l ) ] = C[v(xi,sl+l)l 1

s i n c e s1 i s t r u e , C[c, ,s),



j

E

Relative recursive enumerability

315

remains defined f o r e v e r , g i v i n g t h e same c o n t r a d i c t i o n t o Lemma

and so v ( x l , s l + l

1.4 a s before. Rather than Eurther d e s c r i b e how t h e requirements f i t t o g e t h e r , we now give t h e construction. each

a

we+'.

E

We f i r s t need some n o t a t i o n .

Requirement Re has markers xa f o r

I n t u i c i v e l y , xa c o n t a i n s guesses a t requirements For i n s t a n c e , xu guesses t h a t t h e marker

RO,...,Re-l.

x

s )undefined i n f i n i t e l y o f t e n f o r Ro and xa guesses V ( X < ~ ( ~ ) > ,i S

marker such t h a t

x < ~ ( ~ ) , ~ (is~ )t h>e l e a s t marker f o r R1 such t h a t

that

is t h e f i r s t

V(X * s ) is

undefined i n f i n i t e l y o f t e n among those markers f o r R1 which guess a ( 0 ) f o r Ro. For s t r i n g s (3e) [(Vi

E

B-(e)




is defined.

Let

then Re r e q u i r e s a t t e n t i o n .

,s)

, s + l ) = max{u(e,xz,s)} u I V ( B , S ) : B

2 ael.

( I t w i l l be e v i d e n t from t h e c o n s t r u c t i o n t h a t f o r a l l 8 , i f v ( 8 . s ) i s d e f i n e d then v ( a e , s + l ) (odd).

Let

2

i n A (B) i f j-1 i s even <j-l> e-1 <j+l> be a number l a r g e r than any p r e v i o u s l y used.

v(B,s).)

s+l

xa

e-1

Enumerate

:X

The proofs t o t h e next two lemmas a r e s t r a i g h t f o r w a r d a s d e s c r i b e d i n our sketch and w e omit them.

Lemma 3.1.

Suppose

Suppose a

(a)

2 B.

d e f i n e d then v ( x a , s ) (b)

xi

2

i s defined. Then

xz

i s defined.

v(xg,s).

Suppose t h a t a < B.

v ( x g , s ) i s defined then

Then

v(xa,s)

xs

defined, where B+ i s such t h a t

Lemma 3.2. a)

< v(xg,s) .

( v i


. Fix e > -1

2 6,.

6,

such that if

S

(3.13)

Be by induction on e.

<j+l>

Then it is evident that

=

is defined for all

Now if s1

s1

s

>

since x

6,

(and hence some a

(\

>

so,

(3.14)

(Vi




s o be any true stage.

at stage s+l.

Thus, it

Then

6, pi Furthermore, case 1 cannot happen at stage t+l since for any a, if

is defined, v(x,,t)




then

For notation, let

is defined for confinitely many

xie

all s

pe+l

and suppose that 6, has been defined; we define

x6e n <j>

often.

= 6

We show that there is a j such that

(3.12)

xs e '

ww

xi is defined for confinitely many s , e S If 6, < a then xa is undefined for infinitely many

a)

Proof. B-l =
, s )

(3.18)

Cs[V(Xg r r < i > ’ t i ) l e

i s defined, =

ct

i

[V(XB n < i > , t i ) l . e

This of course y i e l d s the c o n t r a d i c t i o n t h a t C is recursive. Let t i be the l e a s t s t a g e s a t i s f y i n g (3.16)-(3.18) f i n d ti+l.

W e w i l l suppose i i s even.

f o r i.

W e show how t o

Let ti+l be t h e l e a s t s t a g e such t h a t ti+l

t i and

It i s easy t o argue a s i n Lemma 1 . 4 t h a t t h e r e i s such a s t a g e ; otherwise

s o t h a t a t c o f i n i t e l y many s t a g e s s , hypothesis t h a t every

x

Ben

f o r ti+l (and

v+l such t h a t we

v(xg

,s+l) a t s t a g e s+l ( n e c e s s a r i l y by case 2b). This of course e would serve t o c o n t r a d i c t (3.15). For a t such a s t a g e s+l we must have t h a t (3.20) (3.21)

for a l l k t(e+l,s)




t+l.

xt+' %+

imply t h a t our We claim t h a t , f o r

is defined, and s o

This would be t h e desirgd c o n t r a d i c t i o n .

i f t is true.


>

Iga and positive ones by A - proof is a A"-proof

A.

SECTION 4 : THE PRINCIPAL THEOREM Definition 4.1.

-

The rank function rk 1 / zEx}.

rk(x)

=

Thus

rk(x) = least a

Sup {rk(z)+

Theorem 4 . 2 .

-

Let

from sets to ordinalsis defined by :

F

xEV~+~. be a function from sets to sets, unif-C

over all admis-

sible sets. Then there exists some recursive dilator A such that for any set and any ordinal a, if

Proof let

T

.- Let

rk(x) < a

then

rk(F(x))

< A(a).

% be the language of set theory with symbol

be the Kripke-Platek theory KP

VY(YEX3 x(Y)). If m = (M,E) is a model of KP

and let

x

$(X,x)

then Dr$,=Wf(h),

for E. By Ville's lemma (truncation lemma in [BAR]),

E

for membership,

be the formula the well founded part of M (Wf(m),

EF Wf(=))

is a

model of KP. F

C -formula A which can be written 32 Ao(x,y,z) where . A 1 A -formula. By hypothesis, M CVx3!yA(x,y) for any admissible set M.

corresponds to a

is a

3zED$ Ao(x,y,Z). B is true in all inLet B be the formula : \/XED$ 3yED$ ductive models because these are isomorph to the transitive models o f KP. Thus we may apply theorem 3 . 4 . with

?B equal to : A,a V X E I $ ~3y€I$A(a) ~ z E I $ ~ ( ~ Ao(x,y,z). )

=:$I Thisprovides the result becauseif M is an admissible set then

{xEM/ rk(x) ON.

Many thanks to Jean-Yves GIRARD, who propounded this question, for his help and constant encouragements during this work.

I- The category GAR and DIL s2

I - 1 - The category GAR I- 1 - 1 DEFINITION that :

-

F

.-

An

61-flower F

is a functor from ON

I - 1 - 2 REMARK (G).2.4.

.- This definition shows that an
1 ; then, since D(w) < R , we conclude that x < R ; clearly D. is an R-dilator.

ii) By definition of SEP, if D = D'+ D" with D" perfect, S E P @ J )=~D' + SEP(D")Y ; then, since by i) D' is an 0-dilator, it is sufficient to show the result when D is a perfect R-dilator # 1 ; in this case, SEP(D)y(~) < D(y+x) for all ordinals x,y : see the definition of SEP in (G) section 3.3 ; then, since D is an R-dilator, we have the result. I- 2 - 4 DEFINITION

.- An

R-bilator

B

is a bilator such that, for all x,y< R,

B(x,y) F-(a)

=

f (x)

H~ = S Y N ( J ; ' ( ~ ) ) , T

T(z,a) = SYN(fa)(z) ; we v e r i f y t h a t

to

D : let

( a ) ) by

- -

Y

X

z < R,

Y

u < F i ( a ) , (remark t h a t t h i s d e f i n i t i o n i s c o r r e c t because and using (B) 1 3 - 8) then

F (a) by I - 1 - 8

sYN(fa)Em(Ha, D ~ i)f we s e t

H

f"EC(Ji,(,),JF X

f a ( u ) = f ( u ) , f o r each f (Fi(a))

= y : define

?(x)

gEI(a,B),

h E I ( z , z ' ) , then

D~ = S Y N ( J ~ (a)) ; l e t , f o r Y

i s a n a t u r a l t r a n s f o r m a t i o n from

T(z',B)oH(h,g)

=

SYN(f')(z')o SYN(Fx(g))(h) = S Y N ( f B ) ( z ' ) o SYN(Fx(g))(z') o SYN(JF (a)) (h) X

= SYN(fBo F x ( g ) ) ( z ' ) o SYN(JF (,))(h)

= SYN(JF ( B ) ) (h) o SYN(fBoFx(g))(z) ;

Y

X

f B o F x (g) = F (8) o f a , then Y

i n a garden,

T(z',B) o H(h,g) =

SYN(JF ( B ) ) (h) o SYN(Fy(g))(z) o SYN(fa) ( z ) = SYN(F ( g ) ) (h) Y Y D ( h , g ) o T ( z , a ) . We extend

T

by d i r e c t l i m i t s . We s e t

0

SYN(fa) ( 2 ) =

SYN(f)

=

UN(T),

then

SYN(f)EIR(SYN(Ji), SYN(J ) ) , and SYN(f) (R) = UN(T) (R) = T(R,R) = Y f a = f : we use t h e f a c t t h a t l-i t m T(R,a) = SYN(f")(R) =

9

=

(n,Ea,) If J

Y

=

9 (a,

EaB)aGB 1 , r E U

Choose i n V a term

7

f o r r.

Cl

since

For each a choose pa,n,

1 i s uncountable so choose p , n and an i n f i n i t e u

_c

h

W.H. WOODIN

370

such t h a t a E G

+

p = pa, n = na.

It n

Thus p

aE0

Using lemma 4 we can now p r o v e lemma 3.

S has l e n g t h wl.

gA

f o r m u l a t h a t d e f i n e s S.

r

y E L(S

To see how suppose n o t .

no r e a l random o v e r L(S,zo).

Thus f o r a l l z t h e r e i s a r e a l random o v e r L(S,z).

F i n a l l y by lemma 2 t h i s must h o l d i n V[c]

be a A.

,(r

$,1 s e t

nmf o r m u l a 1

F i x a p o s i t i v e i n t e g e r m.

Assume f i s

$.

f o r m u l a q ( x x1

f o r a l l r e a l s z1 Let H 1,

t i o n F:

-t H , 1, 1 o f elements o f H ,

s t r u c t u r e (b,E), b).

Ue b u i l d a s t a n d a r d model

91 subsets o f R .

L e t cp*(x,y)

I f A has been chosen i n a reasonable f a s h i o n t h e n g i v e n any

... xk) f ... zk, Ex

.

there

L e t f be a p r o j e c t i v e f u n c t i o n t h a t u n i f o r m i z e s

e f f e c t i v e l y induces a skolem f u n c t i o n f rp(x z1

... zk)

-+

= ( a l a i s h e r e d i t a r i l y countable].

H

i n V[c]

Cm+l-absolute. 1

A F R x R t h a t i s universal f o r

t h a t d e f i n e s A. 1

f o r c Cohen o v e r V i . e .

a contradiction.

o f ZFC, Nm, t h a t i s Zm+l-correct, 1 Choose a

B u t such an r i s random

f o r r random o v e r V.

o v e r L(S,zo) a c o n t r a d i c t i o n .

M.

Choose zo such t h a t t h e r e i s

T h i s i s a p r o j e c t i v e statement i n zo and t h e r e f o r e

by lemma 2 i t must h o l d i n V [ r ]

i s a r e a l random o v e r L(S,c),

-t

From t h i s i t f o l l o w s t h a t f o r a l l z t h e r e i s a

a,z) f o r some a < wl.

We now can c o n s t r u c t

Suppose z i s a r e a l .

i s p r o j e c t i v e s i n c e by r e f l e c t i o n y E L(S,z)

r e a l r random o v e r L(S,z).

We

I t f o l l o w s f r o m lemma 2 t h a t i f x i s Cohen o r

random o v e r V t h e n rp d e f i n e s t h e same sequence i n V[x]. The s e t o f r e a l s i n L(S,z)

-I

a contradiction.

Suppose S i s an uncountable p r o j e c -

t i v e sequence ( o f d i s t i n c t r e a l s ) and cp i s t h e may assume t h a t

'rl# 0,

cp(frp(zl

... zk)

z1

rp

f o r rp i . e .

... Zk).

Me d e f i n s f r o m f a p a r t i a l func-

.

Nm w i l l be L (F) = L ( F ) n H, Choose some c a n o n i c a l coding 1, 1 by r e a l s ( f o r i n s t a n c e code a E H by c o d i n g t h e c o u n t a b l e

1 "1 b t h e t r a n s i t i v e c l o s u r e o f a, and i d e n t i f y i n g a as a subset o f

Done p r o p e r l y t h e s e t o f codes i s

1 3.

Given a code x we denote by ax t h a t

.

element i n H coded by x. Suppose R(a,b) i s a r e l a t i o n on H, We say t h a t R "1 1 . i s p r o j e c t i v e i n t h e codes i f t h e induced r e l a t i o n on t h e r e a l s R (x,y) +* ' x i s a code A y i s a code A R(a ,a ) ' i s o r o j e c t i v e . S i m i l a r l y a p a r t i a l f u n c t i o n X Y * G: H, H is p r o j e c t i v e i n t h e codes i f t h e induced r e l a t i o n on r e a l s , G , 1 "1 i s p r o j e c t i v e where G* (x,y) H G(ax) = a Y' -+

Projective uniformization

371

Suppose Q i s a p a r t i a l o r d e r , Q E E H . Assume t h a t f o r c i n g w i t h Q i s non"1 t r i v i a l f o r i n s t a n c e assume Q i s atomless and s e p a r a t i v e . L e t T,

... Tk

b e terms i n VQ f o r r e a l s such t h a t T1

c i t y one may as w e l l assume t h a t each t e r m

11-

[ ( q , t ) I q E Q,tE w