Large Infinitary Languages, Model Theory

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 83

Editors

H. J. KEISLER, Madison A. MOSTOWSKI, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam Advisory Editorial Board

Y. BAR-HILLEL, Jerusalem K. L. DE BOUVERE, Santa Clara H. HERMES, Freiburg i. Br. J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Bristof E. P. SPECKER, Zurich

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

. OXFORD

AMERICAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK

LARGE INFINITARY LANGUAGES MODEL THEORY M. A. DICKMANN C.N . R. S. Universitt de Paris VII Paris, France

1975

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

. OXFORD

AMERJCAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK

Q

NORTH-HOLLAND PUBLISHING COMPANY-1975

AN rights reserved. N o part of this publication may be reproduced, stored in a retrieoal system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Library of Congress Catalog Card Number 73-81530 North-Holland ISBN S 0 7204 2200 0 0 7204 2281 7 American Elsevier ISBN 0 444 10622 7 Published by: North-Holland Publishing Company-Amsterdam North-Holland Publishing Company, Ltd.-Oxford Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017

A mis hermanos latinoamericanos, victimas de la dictadura del capital, que combaten por una nueva sociedad

PREFACE

T6e present book grew out of my lecture notes “Model Theory of Infinitary Languages” published in January 1970 by the Mathematics Institute of Aarhus University, Denmark. Indeed, it is a thoroughly modified and updated version of those notes, which in its present form has only a vague resemblance to the original. This book presents a systematic and largely self-contained development of the model theory of the infinitary languages L,, and L,, (where K,A are infinite cardinals and CQ > K >A). The language L,, admits conjunctions and disjunctions of sets of formulas of power less than K and simultaneous quantifications over sets of variables of power less than A. L,, is, of course, the “union” of the LKA.The basic semantic notions are defined in Chapter 1 as direct generalizations of those for finitary, first order languages. The special case of Lo,, is not treated here; the word “large” in the title is intended precisely to denote this omission. The model theory of this particular language is the subject matter of Keisler’s monograph [3]. The contents of the book can be briefly described as follows. Chapters 0 and 1 contain, for the most part, the necessary set-theoretic prerequisites and a list of the most basic notions, definitions, examples and notations used in the rest of the book; 92 is intended to give the reader something of a grasp on the “size” of large cardinals. I should mention that §5“Partition Calculus”-was originally written jointly with George Rousseau; subsequently I made slight changes. Chapter 2 contains a brief summary of the results from the model theory of L,,, used in this book (§3), a more detailed presentation of some special topics from the model theory of L, (§§1,2), and a list of examples showing the impossibility of extending certain results from L,, and L,,, to other infinitary languages. Thus, a considerable part of the first three chapters is devoted to the presentation of prerequisites and preparatory material. Chapter 3 is concerned with the compactness problem (§§1,2) and a related analysis of various classes of large cardinals (§3), and with Lowenheim-Skolem type theorems (@4,5). It contains several applications of the downward Lowenheim-Skolem theorem and the definition of Hanf and Morley numbers. The central topic of Chapter 4 is the exact evaluation of the Hanf number of infinitary languages with finite quantification. The main result vii

...

Vlll

PREFACE

-the Barwise-Kunen-Morley theorem-is approached through a sequence of steps which include, in. al., a detailed development of the omitting types technique, and the non-definability of well-orderings in the languages LKu. Chapter 5 is devoted to languages with infinite quantification. In $1 we compute upper and lower bounds for the Hanf numbers of these languages in terms of large cardinals with partition properties. In $ 2 we study the preservation of infinitary equivalence under sums and products. $03 and 4 are devoted to a comprehensive exposition of the method of extension of partial isomorphisms (the back and forth method), and to some of its applications. This method is an important research tool, both in logic and in algebra. The presentation here-“algebraic” rather than “gametheoretic”-shows that the infinitary languages L,, are a natural logical framework for the standard forms of the back and forth method. The book contains five appendices. Appendices A, D and E deal with particular topics from set theory which are either repeatedly used in the text or bear a close connection to parts of it. Appendices B and C deal with some constructions and results that require the coding of infinitary formulas; namely, the construction of a formula from with no prenex form, and a survey of axiomatizability, definability and completeness results for infinitary languages, including a proof of Scott’s undefinability theorem. Special attention is devoted to the applications of the methods developed in the text to other topics in logic and mathematics. I believe that methods and ideas from model theory are powerful tools for purely mathematical research and that herein lies one of the most stimulating aspects of the subject; some results obtained in the 1960’s serve to support this belief. This monograph is conceived both as a reference book for researchers and as a textbook for graduate students. It can be used, partially or totally, in one- or two-semester graduate courses. In this case the student should be familiar with some basic results and methods from set theory and from the model theory of first order languages; familiarity with, roughly speaking, Part I of Keisler’s book [3] is also recommended. $1 of Chapter 1 and $3 of Chapters 1 and 2, with their exercises, give a concise idea of the results with which familiarity is required. Some indications concerning the way of reading this book: theorems, lemmas, examples, etc., are enumerated in consecutive order by three figures indicating, respectively, the chapter, the section, and the number of the item. Remarks and observations are numbered only when repeatedly mentioned elsewhere. Exercises are provided at the end of most sections,

PREFACE

ix

and have a separate numbering also by three figures. Previous results used in a proof are always explicitly mentioned. The end of a proof is indicated by the symbol The logical interdependencies among the various parts of the book are illustrated in the chart following the table of contents. I wish to express my deepest gratitude to several friends, students, colleagues and institutions whose generous cooperation and help made possible this work and who, in various ways contributed to improve the quality of its contents and style. In the first place to the Aarhus’Mathematics Institute which on several occassions exempted me from other duties to allow me to write this book, and which generously contributed its printing facilities and the time of its typing staff. Within this institution I am specially indebted with Torben Larsen, for his unfailing he‘lp with writing and producing my earlier lecture notes on the subject of infinitary languages; with Brian Mayoh, who encouraged me to write those notes and saw to their publication; and with Lissi Daber and Ursula Engelke for their dedication and effort in producing the typescript both of the lecture notes and of large parts of the present book. Among my other friends and colleagues I shall specially acknowledge the generous help of John Bell, who read the entire manuscript and suggested numerous corfections, improvements of style and additions, and of Richard Gostanian, who read parts of the manuscript and also corrected points of style and suggested additions; the help of Kenneth McAloon was invaluable at the proofcorrection stage. My warmest thanks are due to several colleagues, mentioned in the text, who made available unpublished results and permitted their inclusion in the book. Finally, my thanks to the editorial board of the series “Studies in Logic and the, Foundations of Mathematics,” and to Einar Fredriksson and Thom van den Heuvel of the staff of this series, for their patience, effort and help throughout the production of this book. M. A. Dickmann Paris, April 1974

..

CHART OF INTERDEPENDENCIES The contents of Chapter 0, 5 1 and Chapter I , 9 1 are presupposed and used throughout the book. All the other sections within Chapters 0-2 are conceptually independent from one another, except for the following: In Chapter 0, some concepts and results from $5 3.4 are used in $ 5; the last subsection of Chapter 2 , 9 2.C. uses some definitions from Chapter I , 9 1. The logical interdependencies for the rest of the book are indicated. by chapter and section, in the following chart. The sections under consideration appear in the central columns.

Chapter 3

&----

3 3.D

1 Chapter 2, 5 Chapter 4

1

[Chapter 2,

2.AI

xv

CHART OF INTERDEPENDENCIES

Chapter 5 Chapter 0, 5 5 Chapter 2, 5 2 Chapter 3, 5 5 Amendix D

la

i Appendix C

Appendix D

Appendix E Chapter 0, Chapter 3,

6 5

4 3.C

I Chapter 0, L

3

I J

CHAPTER 0

PRELIMINARIES

0 1. Set-theoretical background Throughout the greater part of the present book, the set-theoretical framework in which our considerations take place may very well be any of the standard systems of set theory, for instance Zermelo-Fraenkel's set theory (ZF), including the axiom of choice (ZFC). In most cases, when we talk about a particular class, we just mean a formula of the language of ZF. Thus, we speak about the class of all ordinals, of all accessible cardinals, of all pairs of natural numbers, etc. All sets defined by a formula of ZF (the set of all natural numbers, for instance) are classes. It should be obvious in each case which is the particular formula involved. The only exception occurs later in these Preliminaries and also in Chapter 3, $2, where we use ordinals to enumerate the class of all ordinal numbers in the sequence determined by an arbitrary well-ordering. For this, the use of a class-type set theory like Von Neumann-Bernays-Godel's (VBG) will do little good, and the same can be said of the corresponding extensions obtained by considering classes of classes, classes of classes of classes, etc., any finite number of times. Some of these arguments can be carried out in the (impredicative) set-class system MK of Morse-Kelley (cf. KELLEY[l], Appendix). The simplest alternative is, however, to extend ZFC by some axiom that guarantees the existence of very large inaccessible cardinals, and assume that the ordinals and cardinals under consideration are all less than some fixed inaccessible number. Such a number should be chosen very large indeed in order to ensure that our statements are not vacuous. Some facts concerning the metamathematical status of such axioms are included below. 1

2

[Ch. 0,f 1

PRELIMINARIES

Following our friend G. Reyes’ (cf. [l], p. 103) famous motto: “Do not scratch if it doesn’t itch”, we shall be rather brief and sketchy concerning the set-theoretical information used in this book. Any ‘of the numerous good books on set-theory (such as BACHMANN[l], KURATOWSKIMOSTOWSKI [ l ] or SIERPINSKI[l]) should supply the reader with what is missing here. We use the standard definition of ordinals a la Von Neumann, whereby an ordinal coincides with the set of all smaller ordinals. In presence of the axiom of foundation (or regularity) an ordinal can be defined as a transitive set’ on which the membership relation E is connected? Thus the order relation between ordinals coincides with the €-relation, or, what is When there is no the same, with the relation “c”of proper in c lu ~ io n .~ possibility of confusion we shall use the symbols “E’’ or “’>>X, then ( K < ’ ) + is the smallest regular cardinal bigger or equal than K > ‘ ; in particular: (23) If K>’ is regular and ~>’>h, then K > ‘ = ( K < ’ ) + ; if K>’ is singular or K>’ < A, then K > ’ = KO. P<X

We also define

F"O(X)= U F"(X), LIEON

,.

and, finally, the diagonal FD of the operation F is the diagonal of the sequence F: D

FD(X)=[6(X)] = { pEONIPEFP(X)}. For instance, the members of CN - M"+'(AC) are called hyperinaccessibles

of type q if 77 p, v. This shows that the elements of HD(In) that are not in M(ME(AC)) must be very rare indeed. Of course, if for nothing else than the fun of it, one can continue this classification calling the elements of HD(In) that are not in Mq(ME(AC))for q 2 w (resp. of type q ; by not in M"+'(ME(AC)) for q w . He proved also several other things: (1) the first cardinal supporting a countably additive, real-valued measure, also supports a completely additiue, realvalued measure (cf. Definition 0.4.7 below); (2) the first cardinal supporting a countably additive, real valued measure is at least as big as the first weakly inaccessible (cardinals supporting completely additive measures are weakly inaccessibles >a);(3) a cardinal supporting a completely additive real-valued measure is either O . This contradicts property (8) of p. 33. Therefore K is not a successor cardinal and the theorem is proved. We have already mentioned the following: COROLLARY 0.4.16 (Ulam). If cardinal > w, then K E CRM.

K

is smaller than the first weakly inaccessible

An argument similar to that of the preceding theorem shows: THEOREM 0.4.17. K ECRM

* K+

ECCRM.

C. Transition from real-valued measures to two-valued measures DEFINITION 0.4.18. (a) ( Z , p ) is a countably two-valued measure space, in symbols ( Z , p ) E ~ C M ,iff p is a countably additive, non-trivial measure defined in 9 ( Z ) with values in (0, l}. An infinite cardinal K is a countably two-valued measurable cardinal, in symbols K E2CM, iff there is a measure p such that (K , p ) E 2CM. (b) ( Z ,p ) is a two-valued measure space, in symbols ( Z ,p ) E2M, iff, in addition to (a), p is completely additive. A cardinal K ECN is a two-ualued measurable cardinal, in symbols K E ~ Miff, there is a measure p such that ( K , p ) E 2M.

38

[Ch. 0, 4

PRELIMINARIES

DEFINITION 0.4.19. A measure p in Z (such that ( Z ,p) E CRM) is atomic iff there is a subset Y of Z such that: (1) Y ) > 0. (ii) For all X , if X C Y , then p ( X ) = O or p ( X ) = p( Y ) . Any two-valued measure is atomic. The study of two-valued measures is continued in part D below. The next two results are crucial in establishing the connection between real-valued and two-valued measurable cardinals. THEOREM 0.4.20 (Ulam). Let ( Z , p ) ECRM. Then: (I) Zf p is atomic, there is a measure v such that (Z,v)E2CM. Moreover, i f p is A-additive, so is v. (11) Zf p is non-atomic, for every n Eo there is a finite partition an of Z such that: A E an * p ( A ) < l / n . LEMMA 0.4.21. Zf ( ~ , p ECRM ) and p is non-atomic, there is a set Z of power < 2’0 and a partition { A j I i E Z } of K such that p ( A j )= 0 for every i E I . Now we prove our main results: THEOREM 0.4.22 (Ulam). K ERM A K > 2’0*

K

E2M.

PROOF.Let p be such that ( K , ~ ) E R MThen . ( K , ~ ) E C R MSuppose . in addition that p is non-atomic. Since p is 2No-additive, in view of the preceding lemma we arrive at the following contradiction:

Hence p is atomic; by Theorem 0.4.20 (I), there is v such that v) E2CM. Moreover, since p is completely additive, v is also completely additive. Hence (K, v) E 2M. (K,

THEOREM 0.4.23 (Ulam). Zf 2’0 is smaller than the first weakly inaccessible cardinal >a, then: KECRM * K E ~ C M . PROOF. Let ( K , ~ ) E C R M .Assume p is non-atomic and consider the partition of Lemma 0.4.21. Since p( U j , , A j ) =p ( ~ ) =1, apply the pivotal lemma to get a measure p on Z; hence (Z,p)ECRM; by Corollary 0.4.16, I 2 first weakly inaccessible > w ; therefore 2’0 is bigger than the first weakly inaccessible >a, contrary to our hypothesis.

Ch. 0,

41

MEASURABLE CARDINALS

39

REMARK0.4.24. Solovay has proved that the theories:

+ ZFC + “there is a real-valued measurable cardinal < 2’0’’ ZFC “there is a two-valued measurable cardinal”,

are equiconsistent (i.e., if one is consistent, so is the other). Hence the assumption “ K > 2’0” in Theorem 0.4.22 is essential. D. Two-valued measures Most of the measure-theoretical results on 2CM and 2M follow in a fairly easy way from the preceding theorems. We put together in the following theorem all results that follow from previous work: THEOREM 0.4.25. (1) (Z,p)E2M * (Z,p)ERM; K E ~ M * KERM; (Z,p)E2M 3 (Z,p)E2CM; K € ~ M K E ~ C M ; (Z,p)E2CM * (Z,p)ECRM; K E ~ C M 3 KECRM. Hence: (2) K E 2CM * K > first weakly inaccessible > o. (3)2McWIn-{w}. (4) The following results also extend to two-valued measures (replacing CRM by 2CM and RM by 2M): Lemmata 0.4.9-0.4.12. The results below improve (2) and (3) to strongly inaccessibles instead of just weakly inaccessibles.

+ 2“g2CM. THEOREM 0.4.26 (Ulam-Tarski). (a) K E ~ C M (b) 2 M c I n - {a}. PROOF.We will prove (b). By almost copying the proof of (b), one can easily prove (a). ) (3) of the preceding theorem shows that K is regular. (b) Let ( ~ , p E2M; We have to show that A E CN A A < K * 2’< K . Assume, on the contrary, that K < 2’ for some A < K , AECN. Let f : 9 (A)-+2’ be a one-one, onto map. For [ < A let:

st={ f ( A )< K

CAA[EA},

40

PRELIMINARIES

[Ch. 0, 5 4

Clearly S,n TE=” and S,u T,G K for all .$a, then K CRM * 2“ ECRM; hence, every ‘K E CRM is greater than or equal to the first strongly inaccessible cardinal > a.

The following theorem gives a characterization of measurable cardinals in terms of the concepts introduced in 0 3.

Ch. 0,5 41

41

MEASURABLE CARDINALS

THEOREM 0.4.28. (A) The following three conditions are equivalent: (1) K E ~ C M . (2) There is a non-principal, countably complete ultrafilter on K . ( 3 ) There is a non-principal, countably complete prime ideal on K . (B) The following three conditions are equivalent: (4) K € ~ M .

( 5 ) There is a non-principal, (6) There is a non-principal,

< K-complete ultrafilter on K . < K-complete prime ideal on K .

PROOF.We sketch the proof of (B); that of (A) is completely similar. (4) 3 ( 5 ) ~ ( 6 ) .If p is a completely additive, two-valued measure on define:

K,

F = {xc K I J L ( X ) = I}, 9= {xC K (p ( x ) = o } . I t is a routine matter to show that 5 is the ultrafilter required in (5) and 5 the prime ideal required in (6). (5) 3 (4).Given an ultrafilter 5 as in (5), define p: T(X)+{O,l} as follows:

It is easy to check the properties of p that make cardinal. H

K

into a measurable

NOTE.The following result makes it easier to carry out these verifications: Let p be a countably additive two-valued measure on Z , and X ECN. Then the following conditions are equivalent to (IHIV) of Lemma 0.4.2: (V) For any f : A+ 9( 2 )the following is true: p

1

U f ( t )= 1

L < A

(VI) For any f : A+

* there is t < X

such that p( f ( t ) ) = 1.

9( Z ) :

P

1

u f(D

(* 1; we prove ~ , ( K ) + + ( K + ) : + ' . &(K) > 3 r - , (K)+~(z'l. But

x,

Ch. 0, 3 51

PARTITION CALCULUS

49

NOTE. Assuming GCH, (*) may be written Nu+r+(Nu+l)Lm. This was proved independently and at the same time by Kurepa (using GCH). 4. Assuming GCH we have:

n > r).

if rn+(nx, then rn++(n+ I):+'

(m> No,

PROOF.(i) If ~ < m then ,

(for if B = max{K, E} < rn,then K' Q B~ = = S+ < m). (ii) If rn Q K , since rn > n > r, then rn+(nx is false (cf. remark before Lemma 0.5.3). 5.2,- I(K)+isthe best possible estimate of rn(K+,K,r),i.e.,

fai0 in general. Indeed we have the following result of Sierpinski which implies that (**) fails for r = 2 and K = No.

LEMMA 0.5.10 (Sierpinski).

PROOF.Let < be the usual ordering of the real numbers, and let < be some well-ordering of them. Let ( X J } E Co if x and y stand in the same , and let { x,y } E C , otherwise. If there were a relation regarding < and i homogeneous subset of power N, then we should have a subset of R of type w I or w:, but this is contradictory. REMARK. By a similar argument one can show 2m++(rn+)22. NOTE.N,+(N,,):

by Ramsey's theorem; N,*(N,)$

by the above.

This negative result extends much further as we shall see. Indeed we have the following result:

THEOREM 0.5.1 1. Zf rn+(rn):,

then rn is strongly inaccessible.

50

[Ch.0,8 5

PRELIMINARIES

PROOF.Suppose that m is singular. Then there is a set M of power m which is the union of < m disjoint sets M i( i E I),each of power < m. We place { x , y } in C , if x and y are in the same Mi, and in C2 otherwise. If X is a homogeneous subset of M , then either X c Mifor some j E I or X has at in either case it follows that most one element in common with each Mi; X < m. Thus if m+(m),2, then m is regular. Now suppose that m is not a strong limit cardinal, i.e., 2" > m for some n < m. We consider the least such n; since m is regular, 20, F, :A "'-+A.

66

[Ch. 1,

THE BASIC NOTIONS OF INFINITARY LANGUAGES

1

NOTATION. In order to have a less cumbersome notation that relates in a flexible way the underlying language with the relational structure we will use the following: will denote the If % is the symbol denoting the structure in question, I universe of 3 (i.e. the set A in the notation used before); for every non-logical symbol S , S' will denote the corresponding object in % (thus, if S = R, an n-ary relation symbol, R is the n-ary relati,on denoted before by S,; if S=f, an rn-ary function symbol, 'f is the map 5, etc.). S' is called the denotation of the symbol S in 3.

%I

'

In what follows we introduce the concept of satisfaction. We need some notation first. Let % be a fixed (but arbitrary) relational system. Let X , Y be subsets of Var, not necessarily disjoint. Let f : X + \ % l , g : Y - + ( 8 (be arbitrary maps assigning an element of 181 to each variable in X and Y respectively. Then, the map f*g :X u Y-+ 1 % I is defined by:

f(z)

if Z E X - Y ,

g(z)

if

LE

Y.

Notice that if X n Y =,0, then f*g =fu g , and if X the following definitions, Dom(f) =Var.

C Y , then j;g=

g. In

DEFINITION 1.1.5. By induction on the structure of terms we define the value t"[f] in % of a term t for the assignment f : Var+ 1 % 1: (1) if t = x EVar, t ' [ f ] = f ( x ) ; (2) if t = c, c an individual constant of p, t ' [ f ] = c'; (3) if t = F ( t , , . ..,t k ) where k > 0,F E p is a function symbol and t , , . ..,tk are terms,

DEFINITION 1.1.6 (The notion of satisfaction). By induction we will assign to each formula cp of Lrh(p) a set 'p'c 1 % IVar, i.e., a family of maps f : Var+ 1 % 1; instead of f E 'p' we shall use the notation % t-cp,[f] and say that f satisfies cp in %. (1) 'p is atomic. (a) if cp is t , x t 2 , where t , , t 2 are terms,

(b) if cp is R(t,, ...,t k ) , where the ti's are terms,

Ch. 1, 0 11

(2) cp is

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

61

14;then iff

%t-cp[f]

not%k$[f],

(3) cp is A'k, where 'k C Form (LKA( p)), %kcp[f]

iff

for every a ~ \ k , % k a [ f ] ,

(4) cp is ( 3 X ) $ , where X LVar, rUFcp[j]

iff

5 < K ; then

F<X;then

t h e r e i s a m a p g : X + l % l such that%k$[f*g].

PROPERTIES OF THE NOTION OF SATISFACTION

(I) % kcp[f] actually depends only on the variables that occur free in rp: letf,g: Var+IrU/ be such thatfrFv(cp)=grFv(cp); then

This remark makes meaningful the use of notations such as:

where aEElrUl for [ < a and Fv(q)={ua61t<S}, or g:Fv(cp)+l%[l, in place of the more complete

where al (or g (x))

if x = uat for some [< 6 (or x E Fv( rp)),

any aEl%l

if x # u a 6 for all [ < 6 (or xEFv(cp)).

As mentioned before, a formula a is a sentence (aESent(L,,)) iff Fv( a) =a. In particular it follows from (I) that: (11) If a ~ S e n t ( L , J and there isf:Var+l%l such that at-a[f],then for all g : Var+l%l, rU Fa[ g]. This result can also be stated as follows: If a is a sentence, either (i) for all f E I%lVar,% k a[f], or (ii) for nofEI%lVar,% k a [ f ] .

68

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, I 1

DEFINITION 1.1.7. In case (i) we say that u is true in % (or that o holds in %: in symbols 8 ku); in case (ii) u is false in U. DEFINITION 1.1.8. If u is true in a structure % of type p we say that % is a model of (I, in symbols ZEMod,(u) or just %EMod(u). If Z is a set of sentences of LKA(p ) we write %EMod(Z) iff 8 EMod(u) for every u E Z . NOTATION. Sm(%) denotes the similarity type of %. If Z C Form(L,,( p)), Sm(Z) denotes the similarity type defined by the set of all non-logical symbols of p which occur in formulas of 2 . Obviously Sm(Z)Cp. If Z = {cp}, we shall simply write Sm(cp) instead of Sm({q}). Let Z~Form(L,,(p)), 8 be a structure such that Sm(Z)cSm(%). The notation % k Z has the following meaning: for every f E [%Ivar and every cpEX %+cp[fl* DEFINITION 1.1.9. If for every % such that Sm(Z)>Sm(q), 'ZIFcp, we say that cp is valid or logically valid (in symbols kcp). DEFINITION 1.1.10. Let Z u { cp} C Form(L,,( p)); cp is a logical consequence of Z, in symbols Zkcp, iff for every structure % such that Sm(Z u {cp}) c SmW), %t=

Z

implies %kcp.

The reader can easily check that kcp iff fl kcp, for every formula cp. DEFINITION1.1.11. Let Z Sent(L,( p)); Z is (semantical&) consistent iff Mod(Z)#fl . Otherwise 2 is inconsistent. LEMMA1.1.12. Zf Z is inconsistent, then for any sentence u we have Z k u.

DEFINITION 1.1.13. A set Z of sentences of LKA(p)is complete iff for all

u E Sent(L,,( p)), either Z k u or Z k i u .

DEFINITION1.1.14. Let IZI be a structure of type p. By ThKX(8),the L,,-theoly of 8 , we denote the set of all sentences of LKA(p) that hold in %, i.e.,

-

Th"'(8) = { u E Sent(L,,( p)) I % k u } A set Z of sentences is called a theoly iff Z k u

uEZ.

Ch. 1, 8 I]

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

69

DEFINITION 1.1.15. Let %,B be structures of the same similarity type. We write 'uzK,B, 'u is L,, -equivalent to 23 iff Th"'(%) = ThKX(B) (or, equivalently, either one of the two inclusions hold). Remark that the preceding definitions apply when both cardinal numbers, as well as when K or both K and A are 00.

K

and A are

DEFINITION 1.1.16. (1) The reduct of a structure % of type-p' to a type pep' is denoted by % l p and defined as follows: I w p i = PI?

S't@=S'

for each S E p .

(2) 'u is an expansion of 2' 3 iff Sm(B) C Sm(%) and % r Sm(23) = 23. (3) In the particular case where Sm(%)=Sm(B)u C , C is a set of individual constants, C n Sm(B) =If and there is a map f : C- 123 such that %=@, f ( ~ ) ) , , ~(i.e., c"=f(c)), % will be called an inessential expansion of B.

I

DEFINITION 1.1.17. (1) Let %,Bbe structures of type p . We say that % is a substructure of 23, in symbols U LB, just in case the following conditions are fulfilled:

l'ul

c 1BL

S' = SBn I 2l In

for each n-ary relation symbol S E p ;

F'

for each m-ary function symbol F E p ;

=FB

c' =

p

r 1 % Im

for each individual ,constant symbol c E p .

(2) Given a class K of similar structures, S(K) (resp. S,(K),A an infinite cardinal) is the class of all (isomorphs of) substructures of members of K (resp. of cardinality 0) and every x , , ...,xm in X , F"(x,, .. .,xn)EX.

70

[Ch. I, $ 1

THE BASIC NOTIONS OF INFINlTARY LANGUAGES

The structure %rX,the restriction of % to X (or, briefly, % restricted to X ) is defined as follows:

iwxi = x R' r x= R'n x "' cNrx = c'

for each n-ary relation symbol R E p ;

for each individual constant c E p ;

F'rx= F' rX"

for each m-ary function symbol F Ep.

From the preceding definitions it is clear that:

LEMMA1.1.19. For every set X C I % I veribing (a) and (b), % r X C%. REMARK. If v ~ and p X L 131 satisfies (a) and (b) above, ( % r X ) r u = (% r v ) r X . No confusion will arise from the use of the symbol in both the sense of Definitions 1.1.16 and 1.1.18.

''r"

Next we introduce the definition of certain operations which will be used in various parts of this book, mainly in Chapter 5, $2.

DEFINITION 1.1.20. Let p be a similarity type containing only relation symbols and let %, 23 be structures of type p such that I % 1 n 123 I = D . The cardinal sum of % and 8,in symbols %@%, is the structure B defined as follows:

Cardinal sums will also be called simple cardinal sums, to avoid confusion with the concept defined in Definition 1.1.22. (In several branches of mathematics it is also known as "disjoint sum").

DEFINITION 1.1.21. Let p be a similarity type without restriction (i.e., eventually containing also function or individual constant symbols); the direct product of the structures % and 23 of type p , in symbols % x 23, is the structure 0 defined as follows:

Ch. 1,

5

11

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

Lxh

71

where R is an n-ary relation symbol. The denotation in Q of function and individual constant symbols of y is defined in the obvious way, e.g. ~ ‘ ( ( x~ ,l

> * ,*

- 9

<xrn,yrn>)= ( ~ ‘ ( x 1 *.. , ,xrn>,F’(Y,,

. . .,yrn)>

for arbitrary xl,. . .,xm E I % I, yl,. . .,ymE IB I. REMARK. This is a familiar concept throughout mathematics; in algebra is sometimes called “direct sum”. DEFINITION1.1.22 (Full cardinal sum). (a) Given a similarity type y containing no function symbols, we shall denote by y‘ the similarity type obtained, roughly speaking, by “duplicating” y; more precisely p’ consists of: (i) for each symbol S of y, two symbols Sl,Szof the same category and arity as S (we write p i for { Sj I S E y}, i = 1,2);

(ii) two extra unary predicate symbols P I ,Pz. (b) Given two structures %, B of type y such that 1% I n 123 I =a,the full cardinal sum of % and 23, in symbols %+B, is the structure Q of type p’ defined as follows:

REMARK. (1) The restriction that the similarity types p1,y2 be disjoint copies of the same similarity type y is not an essential one. It should be obvious how to extend the preceding definition to the case where y I , yz are arbitrary disjoint similarity types containing no function symbols. The corresponding operation will be called extended full cardinal sum, and denoted by % 8. (2) The full cardinal sum % + B contains full information about each of its summands; not only does it “glue” 9l and B together but-unlike the simple cardinal sum-it also “keeps track” of what goes on in the factors (in particular, it “knows” which are these summands). The simple cardinal sum and direct product are, in an obvious way, definable within the full cardinal sum. These notions of sum and product of structures are very particular cases of the generalized products considered by FEFERMANVAUGHT[I].

12

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, §. 2

All operations defined above can be extended in an obvious way to an arbitrary number of factors. The reader should be able to state the corresponding definitions without difficulty. In Chapter 3 we will refer, without proofs, to (finitary) formulas involving higher order variables and quantifiers. In addition to the customary logical symbols an nthorder language (n > 1) contains, for each 1 \< k < n, a list of variables:

xgk,x;,...,x,,k . ..

The lists of non-logical symbols can be extended accordingly. The variables with upper index k are interpreted as ranging over kth order sets; thus, Iut- (VXk)cp means that

A formula involving higher order quantifiers is in prenex form if all the quantifiers appear at the beginning of the formula in a string, those of highest order first and so on. A prenex formula cp is called a I l L (Z;, respectively) iff: (i) the highest order variables occurring in cp are of order n + 1; (ii) the first quantifier is universal (existential, respectively); (iii) there are m alternating ( n + l)* order quantifiers (i.e., m ( n + order quantifiers, all but the last of which is followed by a quantifier of the opposite kind). For further details the reader is referred to CHURCH [ 11. $2. Examples

In this section we include some examples showing the expressive power 'of the languages LEA.Most of them will be used later. The missing proofs can easily be supplied by the reader.

EXAMPLE1.2.1. Categorical characterization in L,,, of the arithmetic of natural numbers, i.e., the class N of all structures Iu such that Iu zz ( w , +, .,',O). The similarity type p includes the operations "+" and the successor operation '"" which is a singulary function, and the individual constant _O. "*",

'

This definition corresponds to the hierarchy of simple types. The correspondingpresentation for the ramified hierarchy (i.e. when, in addition, the arity of higher order relational variables is brought into the language) can be reduced to the case mentioned here.

Ch. 1,

21

EXAMPLES

13

(a) N coincides with the set of all models of the Lola- sentence a, which is the conjunction of all first-order instances of Peano axioms.' and the sentence vx[

v

n E o (xxs"_O)]

where the terms s"x are inductively defined as follows: sox=x and s lx = (s"x)'. (b) Another, simpler alternative is to consider the axioms: +

vx[

v

n E w (X%S"_O)]

REMARK. A word of caution: the axiom of induction is an axiom-schema; the corresponding LoI,-sentence to be considered is:

EXAMPLE 1.2.2. Characterization (in Lw,J of the class Fin of all finite sets. Consider the following sentence 0 (in the language p =0,i.e., without non-logical symbols):

v [ v u o . . . u n (o < iv,
O. By induction we define the formulas V0(X)’

1( X W X )

v, (X) = syaVY [ Y Ex-+ V&Y )I. It is clear that Vp””=

i

R(p)

forp a.

Let uq be the following sentence: 0, = V x y [ V z ( z ~ x + - + z E y ) - + X x yA]

VX

v

P 1 such that p contains an m-ary function symbol, and for every xI,.. . , x mE UzEKIi!ll, there exists i!l E K such that x l , ...,xm E 131. REMARKS (1) If p does not have individual constant or functions symbols, then any set K of structures of type p admits a union. (2) Recall that a partially ordered set ( A , < ) is said to be directed (or upper directed) iff for every pair x,y E A there is z E A such that x < z and y < z . Then, if K is directed by C, K admits a union. In particular, if K is it admits a union. totally ordered by

c,

DEFINITION 1.3.25. If K#B and K admits a union, the union of K, in K or UgIEKIU, is the structure 3 of type p defined as follows: symbols

u ppil= u

1911;

’UEK

S’B=

U

Sgl for every relation or function symbol S ofp;

’u E K

cB = c’

for any % E K and any individual constant symbol c of p.

Clearly, the condition “K admits a union” insures that U K is well defined. We know, however, that in general U K has different elementary properties-and, a fortiori, infinitary properties-than the elements of K, even in cases where these are pairwise elementary equivalent (cf. Exercise 1.3.1(h)). A well-known result of Tarski states that this is not the case whenever K is directed by the relation < of elementary extension. This result holds for infinitary languages as well, upon strengthening the concept of directed set. DEFINITION 1.3.26. Let ( A , < ) be a partially ordered set and p an infinite cardinal number. ( A , < ) is p-directed (