Model Theory for Infinitary Logic (Studies in Logic and the Foundations of Mathematics, Volume 62)

MODEL THEORY FOR INFINITARY LOGIC LOGIC WITH C O U N T A B L E C O N J U N C T I O N S A N D FINITE QUANTIFIERS

H. JEROME KEISLER University of Wisconsin

1971

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

LONDON

@ North-Holland Publishing Company, 1971

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical,photocopying, record-

ing or otherwise without the prior permission of the copyright owner.

Library of Congress Catalog Card Number 79-140490 lnternational Standard Book Number 0 7204 2258 2

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY

-

AMSTERDAM

-

NORTH-HOLLAND PUBLISHING COMPANY, LTD. LONDON

PRINTED I N THE NETHERLANDS

To Lois

Preface

This book grew out of a Survey Lecture given to the Association for Symbolic Logic meeting in January 1969 and is based on a course at the University of Wisconsin in the spring of 1969. It is planned both as a textbook for an advanced graduate course and as a reference for research work in mathematical logic. We have written for the reader who already has a thorough knowledge of ‘classical’ model theory, that is, model theory for the usual first order predicate logic. The necessary background is given in the book MODEL THEORY by C. C. Chang and H. J. Keisler. In recent years model theory has grown beyond first order logic, and one of the most exciting developments has been model theory for infinitary logic. In this book we have decided to restrict our attention once and for all to the most fruitful part of infinitary logic to date, the logic L,,, .This logic is built up from the first order predicate logic L by allowing countably infinite disjunctions and conjunctions. Our aim is to present the model theory for L,,, in a form which makes the overall structure as clear as possible. We shall give particular emphasis to the methods which are used for constructing models, and on their relation to the methods available in classical model theory. After reading this book the student can go on to the study of the larger infinitary logies L,, in the forthcoming book LARGEINFINITARY LANGUAGES by M. A. Dickmann. The first wave of results in the model theory of L,,, came around 1963 with the work of Karp (Completeness Theorem), Scott (countable modVII

VIII

PREFACE

els), Morley (computation of the Hanf number), and Lopez-Escobar (Interpolation Theorem). A second wave around 1967 began with of Kreisel’s observation that the classification of infinitary languages Lab by the cardinals a, is too crude for many purposes. In particular, the full language L,,, has uncountably many formulas. This wave contains the study of the sublanguages L, of L,,, given by an admissible set d , initiated by Barwise. Another important recent development is the use of Consistency Properties in the model theory of L,,, by Makkai. Many of the early proofs in the subject made heavy use of proof theoretic methods, but the work of Makkai has made a more model-theoretic treatment possible. The earlier portion of this book presents the basic model-theoretic results on L,,, in the light of the work of Barwise and Makkai. Later on, the book continues with topics such as categoricity, models of power wl, and some applications to set theory. It contains some results of the author which have not previously appeared in the literature. The book is divided into four parts. At the beginning of each part a new method of constructing models is introduced: the Henkin construction in Part I, Skolem functions and indiscernible sets in Part ZZ, elementary chains in Part IZI, and ultrapowers in Part IV.The four parts are divided further into lectures. Each of the thirty-four lectures corresponds roughly to one lecture given in the author’s course at Wisconsin in 1969, except that the boundaries between lectures have been adjusted to match the changes of topics. In the earlier portion of the book where the basic material is covered, several problem sets have k e n included. The end of a proof is indicated by the symbol -1. The author wishes to thank the many colleagues who offered helpful suggestions in the preparation of this book, and owes a special debt to Jon Barwise, C. C. Chang, Kenneth Kunen, and Saharon Shelah.

1 Introduction

As we have stated in the preface, we shall assume that the reader already has a knowledge of model theory for first order predicate logic. We shall begin with a brief discussion of various languages which go beyond first order logic. If ct and are two infinite cardinals, then the infinitary logic L,, is like first order logic except that it allows the conjunction and disjunction of a set of fewer than ct formulas, has ct individual variables, and allows universal and existential quantification on a set of fewer than fi variables. In particular, L,, is the classical first order logic. The next simplest case is the language L,,,, which allows countable conjunctions and disjunctions but only finite quantifiers. Many of the results of this book carry over in some form to the languages La,, or sometimes even to the La,. However, we have decided to limit our scope to the language L,,,, with countably many relation and function symbols. The loss in generality is made up for by a much smoother theory. The language L,,, is closely related to several other extensions of first order logic, including the following. 1. w-logic, which is like first order logic except that it has an extra oneplaced relation symbol Nand constants 0, I , 2,3, . .. In the model theory of w-logic we admit only models fl such that the set N of natural numbers is a subset of A , and we interpret N a n d 0, 1,2,3, . . . in the natural way. 2. Weak second order logic, which has variables for finite sequences of elements as well as individual variables, and has an operation of concatenation of finite sequences and a membership relation.

.

3

4

11

INTRODUCTION

3. Second order logic in which the set variables are interpreted as ranging over finite sets. 4. First order logic with the additional quantifier ( Q x ) which means ‘there exist infinitely many’. Of all these languages, L,,, has proven to be the most convenient. The main defect of first order logic is that it is not adequate for expressing most mathematical concepts. There are many important notions from mathematics which can be expressed in L,,, but not in first order logic. Here are some examples of classes of models characterized by a sentence of LUJ,. (a) The class of all finite models,

v

n and 23 = ( S ( Y ) , E) where X , Yare infinite. (iii) % = %(L) and a, 23 are w-saturated models. NOTE: is w-saturated means that for every sequence cpo(x, . . . x,), cpl(xl . . . x,), . . . of formulas of L,

a I=(VX, (iv)

* * * Xn-1)C

A

m E S with s; = s1 u {a(c)> s; = s2. Next suppose that a(t) E s, , c = t E s,. Then k A s1

-+

(c = t

Let s; = s1 u { ~ ( c ) ) ,s; = s,, 8, E X , , n X-,* with b A s;

-+

-+

Cr(C)EX,.

a(.),

and suppose 8, E X , n X , ,

el , c. A s; -,e,.

Then b A s, -, (c = t -,

el),

k A s,

-+ ( C =

r A 8,).

Hence (c = t -,

el)/+

= tA8,)

is consistent, whence 8, A 8, is consistent. Therefore again s' E S. The other cases are similar.

LYNDON INTERPOLATION THEOREM

26

The proof of (C2)-(C6) depends on the fact that if a relation symb occurs positively (negatively)in t~ and if u E Z,then the symbol also occu positively (negatively) in 7CT 1, A Z,Vxt~,VZ, and 3xcr. By the Model Existence Theorem, each s E S has a model, and ther fore {cp, $} 4 S. Hence there are sentences 8,(cl . . . c,,) in X, n 1 and O2(c, . . . c,,) in X-,, n such that cp

Then the sentence of L,,,

-,el,

(el~e2).

$ .+ e2,

e = vx, . . . x,e(x, . . . x,,)

has the required properties (i)-(ii).i

The above theorem also has a version for logic without identity whic we shall need later. THEOREM 6A (LYNDONINTERPOLATION THEOREM WITHOUT EQUALITY (LOPEZ-ESCOBAR [1965]). Assume that L has no function or constant symbols. Suppose cp, $ ai sentences of L,,, in which the equality symbol = does not occur, ar C cp

-,$,

not C

1cp,

not k $.

Then there is a sentence 8 of L,,, in which = does not occur which hi the properties (i) and (ii) of Theorem 6.

PROOF.We follow the notation in the proof of Theorem 6. Define 1 to be the set of all sentences 8 E X , in which the equality symbol does nc occur. Let S be the set of all finite sets s c Y , u Y7# which can be writte in the form s = s1 u s2 such that (1) and (2) hold with Y in place of and s1 and s2 are both consistent (so (Cl) will go through). Then arguir as before we see that S has all the properties (Cl)-(C6) of a consistenc property except the equality rule. Moreover, no member of S contaiI the equality symbol. By the Model Existence Theorem without equalil (Problem 4 in the last lecture), every s E S has a model. Therefore, sin( {cp, 1I)} has no model, {cp, $} 4 S. Furthermore. by hypothesis is consistent and (l$) is consistent. Then as before we see that there a sentence 8 of Lala in which = does not occur with the required prope ties (i) and (ii).i

61

LYNDON INTERPOLATION THEOREM

27

The next result is one of a class of theorems known as 'preservation theorems'. Like Theorem 6 above, it is due to Lyndon for L and to LopezEscobar for L,,,. We say that a sentence cp of L ,,, is positive iff every occurrence of a relation, constant, or function symbol, or the = symbol, in cp is positive. It is easily seen, by 'moving the negations inside', that cp is positive if and only if cp is logically equivalent to a sentence which is built up from atomic formulas using only V, 3, A, and V. A sentence cp of L ,,, is said to be preserved under homomorphisms relative to a iff whenever %, % are models of a, % is a model of cp, and % is a homomorphic image of %, then % is a model of cp.

THEOREM 7 (LOPEZ-ESCOBAR [1965]). Let cp, a be sentences of L,,,. Then cp is preserved under homomorphisms relative to a if and only if there is a positive sentence 6 of L,,, such that b a + (cp c+ 0). PROOF.We see easily by induction that every positive sentence is preserved under homomorphisms; whence if Co + (cp t,e) for some positive sentence 8,then cp is preserved under homomorphisms with respect to u. Assume cp is preserved under homomorphisms with respect to a. We may assume without loss of generality that cp, Q contain no function or constant symbols. For any n-placed function symbol can be replaced by an (n+ 1)-placed relation symbol, and any constant symbol by a l-placed relation symbol, and then a can be replaced by a A 6 where 6 is the appropriate sentence saying that the relations are functions (or constants). Let E be a new binary relation symbol, and let C be the sentence stating that E is an equivalence relation and is a congruence relation with respect to each relation symbol in a, cp. Form cp(E), a(E) by replacing each OCcurence of = by E. Now add a new binary relation symbol E', and for each n-placed relation symbol R of L, a new n-placed relation symbol R'. Form C', cp'(E'), a'(E') by replacing E by E' and each relation symbol R by R'. Finally, let H be the conjunction of all sentences Vxl

. ..x,,(R(x, . . . x,,) -+

R'(x,.

..x,)),

~x,x,(E(x,xz) E'(x1 xz)), for each relation symbol R. Then since cp is preserved under homomorphisms relative to a, -+

28

LYNDON INTERPOLATION THEOREM

1 C A ~ J ( E ) A ~-+~ ( H EA ) C ’ A O ’ ( E ’.+ ) cp’(E’)). The = symbol does not occur in the above sentence. Moreover, the syrr bols E’, R‘ do not occur on the left, and the symbols E, R occur only pc sitively on the right. Therefore, by Theorem 6 ~there , is a positive ser tence B(E), involving only E and relation symbols of L, such that

c

CA

~ J ( E )A

c p ( ~ ) -+

e ( q , c e(E)

.+

( H AC’ A ~ J ’ ( E ’ )

.+

cp’(~’)).

On the right if we replace E’, R‘ by E, R then H becomes valid and w have

c e(E) .+ ( c A ~ J ( E )

.+

cp(~)).

Therefore I. C A

~ J ( E ) -+

( c p ( ~ ) +, e(E)).

Now when we replace E everywhere by =, C becomes valid and B(E) bt comes a positive sentence 8 of L,,, . Then we have the desired result

c tJ .+ (cp c+ e1.i We shall next prove another special interpolation theorem and apply I to obtain another preservation theorem for L,,, . A sentence of L,,, is said to be quantijier-free iff it contains no quar tifiers. A formula cp is universal iff every occurrence of the quantifie V in cp is positive, and every occurrence of the quantifier 3 in cp is negativr A formula cp is existential iff ( cp) is universal. It is easy to show b a simultaneous induction that a formula cp of L,,, is universal (existential if and only if it is logically equivalent to a formula built up from atomi formulas and their negations using only V, A, V(3, A, V), and wit the same free variables.

7 Malitz Interpolation Theorem

THEOREM 8. (MALITZ[1969]). Suppose L has no function symbols. Let cp, $ be sentences of L,,, such that $ is universal and C cp + $. Then there is a universal sentence 8 of L,,, such that k cp -,8, k 8 + $, and every relation or constant symbol occurring in 8 occurs in both cp and $.

PROOF. We may assume without loss of generality that $ is quantifierfree, for the bound variables in $ can be replaced by new constants. Let S be the set of all finite sets s of sentences of M,,, such that only finitely many of the constants c E C occur in s, and s can be written as a union s = s, u s, where s, is quantifier-free, s, and s, have models, and: (*) There is no universal sentence 8 of M,,, with C As, + 8, k As, 7 8, and every relation or constant symbol (including the c E C) which occurs in 8 occurs in both s1 and s,. We claim that S is a inconsistent property. Let us check parts (C4) and (C7), and leave the rest to the student. (C4) Let Vxa(x) E s E S. We must show that for all c E C, { ~ ( c ) } u s E S. Let c E C and writes = s1 u s2 satisfying (*). Since s, is quantifier-free, Vxa(x) E s l . Let -+

s; = s, u {a(c)}, s; =

s2,

s'

=

s u'})C(.{

CASEI. c occurs in both s, and s,. Then s' E S because if I= A s; + 8, 8. k 17 S; -+ 8, then I= S, -+ 8, I= S, -+ CASEII. c does not occur in s, Then s' E S for the same reason as above, but time c does not occur in 8.

.

29

30

MALITZ INTl3RPOLATION THEOREM

CASE111. c occurs in s2 but not in sl. Suppose s’# S. Then there is O(c) with C A si + e(c), C A s; + e(c). Hence C A s1 + (o(c). e(c)). Since c does not occur in sl, C A s1 4 Vx(a(x) + e(x)), when1 C A s1 -+ VxO(x). But C A s2 -+ Vxe(x), contradicting (*). Thi sI E

s.

(C7) Let {a(t),d = t } c s E S, where d E C and t is a constant. P show that s u {a(d)} E S. CASEI. a(t) E sl,d = t E sl. In this case C A s1 -+ a(d), and we si that s u {o(d)} E S by putting si = s1 u { d = t } , s; = s2. CASE11. a ( t )E s1,d = t E s2, d occurs in sl. Then s1 u {d = t} a1 s2 satisfy (*) because if C A (s1 u { d = t}) + 6 and C A s2 + 1 then C A s1 + (d = t + e), C A s2 + (d = t -+ 0). Therefore case applies, and s u {a(d)} E S. CASE111. o(t) E sl,d = t E s2, d does not occur in sl.If I= A (sl u { d e(d), then c A s1 -+ V X ( X = t + e(x)), t } ) .+ e(d), k A s2 -+ C A s2 -+ 1Vx(x = t + e(x)),’ contradicting (*) for s1,s2. Thus (*) satisfied by s1 u { d = t}, s2, and case I applies again. CASEIV. o(t) E s2, d = t E s2. In this case we see that s u { ~ ( d )E} by putting s; = sl,s; = s2 u { ~ ( d ) } . CASEV. o(t) E s2 ,d = t E sl,d occurs in s2. Similar to case 11. CASEVI. a(?) E s2 , d = t E sl,d does not occur in s2. If C A s1 4 e(a C A s2 u { d = t} + e(d), then C A s2 + Vx(x = t -+ 1e(x)), whence C A s2 + e(t). However, C A s1 4 e(t). This contradicts ( for sl,s2 . Then (*) holds for sl,s2 u {d = t } , and case IV applies. Let s E Sand let t be a constant. Let c E C not occur in s. We show th s u {c = t } E CASEI. t occurs in both s1 and s2. Then we see that s u {c = t} E by putting s; = s1 u { c = t } , s; = s2. CASEII. t does not occur in s2. Then again putting s; = s1 u { c = 1 s; = s, we see that s u {c = t } E S. CASE111. t does not occur in sl.Similar to case I1 with 1 and 2 reverse The above paragraph is the place where we would have difficulty L had function symbols, because some of the symbols in t might then o cur in s1 and others might occur in s2. By the Model Existence Theorem, every s E S has a model. Hen1 {p, I +} 4 S. This means that there is a sentence 8 of L ,,, of the requiri kind.4

s.

71

31

MALITZ INTERF'OLATION THEOREM

DEPINITION. Let cr, cp be sentences of Lm1,. cp is said to bepreserved under submodels relative to cr iff whenever %, 23 are models of u, 58 is a submodel of %, and % is a model of cp, then 23 is a model of cp. THEOREM 9 (MALITZ[1969]). Let cp, u be sentences of L,,,. Thencp is preserved under submodels relative to D if and only if there is a universal sentence 8 of L,,, such that =! u -+ (cp ++ 8).

PROOF. We may assume that L has no function or constant symbols. For we may replace functions and constants by relations in the usual way and add to cr a sentence stating that those relations are functions or constants. Let C be a countable list of new constant symbols. By induction we define for each formula $ of L,,, a formula $ of M,,, as follows: If

"

+ is atomic, +" = $.

(A Y)" = A {$": $ E !PI. (1 $1" = 7 ($"). (VX$(X.. .))" = A $"(c.. .).

1

C6C

Thus $" is quantifier-free. Since cp is preserved under submodels relative to u, we have k

~7A

Cp -+ (0" + q").

By Theorem 8 there is a universal sentence 8 of L,,, such that b O A V + 8,

c 8 -,(8

-+

cp").

(u + cp) has no countable Since 8 A (0" 3 cp") has no model, 8 A model, since any countable model can be enumerated by C and thus (u" 3 cp"). From the Model Existence Theorem made into a model of it follows that 8 A (cr -+ cp) has no model at all, whence I= 8 + (u cp). Therefore I= u * (cp t,8).i -+

COROLLARY. Let u, cp be sentences of L,,,. Then cp is preserved under extensions relative to u if and only if there is an existential sentence 8 of L,,, such that I= cr -+ (cp t+ 8). PROOF. cp is preserved under extensions relative to cr if and only if (7 cp) is preserved under submodels relative to u.i

32

MALITZ INTERPOLATION THEOREM

"

Theorem 9 was first proved for first order logic L by LoS and Tarsk using methods quite different than the above proof. The corollary wai proved later by Henkin, and more generally by A. Robinson, for firs order logic. Several other known preservation theorems for L were gen eralized to L,,, by Makkai using the methods of the last two lectures.

PROBLEMS

1. (BARWISE, MALITZ).Suppose L has no function or constant symbols Let cp, be sentences of L,,, such that 4p is universal and k cp 3 $ Prove that there is a universal sentence 8 of L,,, such that k cp + 8 k 8 3 t+b, and every relation symbol which occurs in 8 occurs in both 4 and +. NOTE:To see why we assumed L has no function or constant symbols, consider the counterexample

+

k R(c) + (3x)R(x).

2. Let @ be a finite or countable set of sentences of L,,,, such that @ has no model. Then there are sentences 8,+, E L,,, for each rp E @ such that:

(i) {O,: cp E @} has no model. (ii) k cp -+ 6 , for each cp E @. (iii) For each cp E Qi, each relation, function, or constant symbol which occurs in 8, occurs in cp and in some member of Qi- (9).

3. (MAKKAI). Problem 2 above holds with (iii) replaced by: (iv) For each 4p E @, each relation symbol occurring positively (negatively) in 8, occurs positively (negatively) in cp and negatively (positively) in some member of @ - {cp}. 4. (BARWISE, MAKKAI,WEINSTEIN). Let 8 and 8 be countable models for L. Then can be isomorphically embedded in 8 if and only if every universal sentence of L,,, which holds in 23 holds in 8.

5. (CHANG,MAKKAI).Let 8, '23 be countable models for L. Then '23 is a homomorphic image of 8 if and only if every positive sentence of L,,, which holds in 8 holds in 23.

71

33

MALITZ INTERPOLATION THEOREM

6. (MALITZ).For any two sentences cp, $ of L,,,, valent:

the following are equi-

(i) Any submodel of a model of cp is a model of $. (ii) There is a universal sentence 8 such that t cp

7. (BARWISE). For any two sentences cp, $ of L,,,, valent:

-+ 8

and I= 8 --* $.

the following are equi-

(i) Every countable model of cp can be embedded in some countable model of $. (ii) Every universal consequence of $ in L,,, is a consequence of cp. 8. (LOPEZ-ESCOBAR). Let cp, $ be sentences of L,,,.

Equivalent are:

(i) Any homomorphic image of a model of cp is a model of $. (ii) There is a positive sentence 8 in L,,,

such that cp F. 8, 8 I= $.

8 Admissible sets

In this lecture we shall introduce the notion of an admissible set, which is due to PLATEK [1966].

DEFINITION. By a formula of set theory we shall mean a formula in the first order logic with identity and the binary relation symbol E. If cp is a formula of set theory, we use the abbreviations (Vx E Y)cp for (Vx)(x

EY

+

cp),

(3x E y)cp for (3x)(x E y A q).

A A,-formula is a formula of set theory which is built up from atomic formulas and their negations using only the operations

A,

v, (Vx E Y ) , (3x E Y ) .

Here we mean finite A and V. A Z-formula is a formula of set theory built up from atomic formulas and their negations using the above four operations and (3x). A n-formulu is one built in the same way but with (Vx) instead of (3x). For example, (Vx E y)(3z E w)(x E z A 7 z = u ) is a A,-formula, while (Vx E y)(3z)(x E Z A 1z = u ) is a Z-formula which is not A , .

LEMMA 1. (i) The negation of a A,-formula is logically equivalent to a A,-formula. The negation of a Z-formula is logically equivalent to a l7-formula. The negation of a l7-formula is logically equivalent to a Z-formula.

34

81

35

ADMISSIBLE SETS

(ii) Let a, b be transitive sets with a c b, and let c l , . q(xl . . .x,) is a do-formula, then (a, E) t= q[cl

. . .c,]

o ( b , E)

t= q [ c l ...c,].

* ( b , E)

1P [ C ~

If q ( x , . . .x,) is a X-formula, then (a, E)

t= P [ C ~-

.

c,]

If q ( x , . . .x,) is a I?-formula, then ( b , E) t= q [ c ,

. . . c,] 3 ( a , E)

--

C q[cl

*

..,

C,EU.

If

c.1.

. . . c,].

The proof of this lemma is left to the student. DEFINITION. d is said to be an admissible set iff: (i) d is non-empty. (ii) d is transitive. (iii) If x

E

d,then the transitive closure TC(x) of x belongs to d .

..

(iv) ( A o - S ~ ~AXIOM). n 0 ~ If q ( x y l . y,) is a do-formula and b, , . . ., b,, C E d,then { a E c: ( d ,E) t= q[abl . . b,]} E d.

.

(v) (X-REFLECTION AXIOM). If q ( y , and

. . .y,) is a 2-formula, bl ,. . .,b, E d

I, (3) (38~ a)(3e, $, y E x)[rp = ($ -,vu,e) A (u, is not free in $ ) A ($ -, 0) E P(B, 4 1 (2)

-+

or (4)

(30

E P(B, 4

E ~)(31,9 E X)[V

1.

= ($

-,A @ ) A (ve E 0)(3p E a)((@-,e) E 42

91

BARWISE COMPACTNESS THEOREM

43

Notice that (2), (3), and (4) correspond to the rules of modus ponens, generalization, and conjunction, respectively. We may regard P as a totally defined function with domain d by making P(y) = 0 when y is not an ordered pair. We shall first check that the function P maps d into d and is A on d . We observe that P is defined recursively from a function G whose domain includes a?x d .Furthermore, for each u, x and f in d ,we have (5) G(, f)E d . Using (6) we see that

( 8 ) G I d x d is Z on d . We now conclude by the principle of definition by Z-recursion, using (7) and (8), that P maps d into d and is A on d. In what follows we shall write ‘cp E P(a,x)’ to denote the Z-definition of P.We observe that P(a, x ) is increasing in ct and x, that is, if a E /l E d, x c y E d ,and u, B are ordinals, then Let S be the set of all finite sets s c M, of sentences such that not I-, As, and only finitely many c E C occur in s. We claim that

1

(9) S is a consistency property.

Let us first assume (9) and give the rest of the proof of the theorem. Since P is Z on d , kdcp has the X-definition (3x)(3ct)’cp E P(u,xy.

Thus the set of all sentences cp E L, such that F d ’ p is Z on d . Since all axioms are valid and the rules of inference preserve validity, we have If cp E L, and I-,cp,

then I=cp.

44

[9

BARWISE COMPACTNESS THEOREM

On the other hand, suppose cp E L, and cp is valid. Let cp be a sentence. We now apply, once again, the Model Existence Theorem; it says that every s E S has a model, hence { 9)6 S. Then t, A(? cp), whence t d q . Thus if cp E Ld and b cp, then tdcp. This proves that the set of all valid sentences cp E L, is Z, on d . The only step in the proof of (9) which presents any difficulty is the verification of’: (C 5 ) . If V 0 E S E S then for some 0 ~ 0s ,u { O } E S. Assume V 0 E s E S but for all 8 E 0, s u { e } 6 S. We shall get a contradiction. We have ,k A(s u {a)), whence kd(A s -+ e), for all 8 E 0. Therefore ( d ,E) b

-

(ve E @ ) ( 3 ~ ) ( 3 a ) r (sA-, 4)E P(a, x ) ’ ] .

The formula on the right is a C-formula, so by C-reflection there is a transitive a E d such that s E a, 0 E a, and

e) E ~ ( axi-. ,

w is admissible if and only if there is a set U c o such that a = my. PROOF.One direction is given by Lemma A. For the other direction, assume that ci > w and a is a countable admissible ordinal. Let L contain the e-symbol, a constant symbol cu, and a constant symbol cs for each ordinal /? < a. Let d be the admissible set d = L(a). Let S be a subset of a which is Z on d but not A on d. Let T be the following set of sentences of Ld: The 2-replacement scheme. All the axioms of Zermelo set theory. x E cs * V ,,, x = c,, for each /? < a. XECU

+ XEC,.

-=

recursive in cu', for each p 01. For each j?,the last sentence is just the formalization in Zermelo set theory of the definition of '/?is recursive in U'. We observe that (1) T has a model. For example, let U c w be such that 'cB is

-

{(my n ) : 2" 3" E U > is a well ordering of w of order type a. Then for some ordinal y > a, a is interpreted by cz in B*. of L):. One can check that this definition does not depend on the choice of representatives of the equivalence classes under w . By induction on the complexity of formulas it can be shown that an increasing n-tuple from Y satisfies a formula 8(v, . . . u,) of L: in B* if and only if every increasing n-tuple from X satisfies 8 in a*. The Skolem functions are needed for the case where 8 is of the form Vx$ or 3x$. It follows that Y is a set of indiscernibles in B* and increasing ntuples from X and Y realize the same types in L: . -I Note that in the above proof,

B*is also the Skolem hull of Y.

14

SKOLEM FUNCTIONS A N D INDISCERNIBLES

113

COROLLARY. Let Ld be a countable fragment and let T be a theory in the Skolem language L.: If T has a model %* which is a model of TSkolem and has an infinite set of indiscernibles, then T has models of arbitrarily large cardinality. Furthermore, for any model %* of TSkolem which has an infinite set of indiscernibles ( X , L.d Hull ( X ) of arbitrarily large cardinality. PROOF.Consider a linearly ordered set (Y, < ) of the desired cardinality, with ( X , ,(a) = 2". 5't

THEOREM 20 (THEERDOS-RADO THEOREM) Let c1 be an infinite cardinal and n < o.Then Zn(ff)'

PROOF.

+ (a+):+1.

We argue by induction on n. The result for n 75

=

0 is the ob-

76

ERDOS-RADO THEOREM

ti4

vious pigeon hole principle: 'If a set of power x + is divided into a parts, at least one part has power a". (a+):. Let X be a set of power Let n > 0 and assume > n - l ( a ) + >,,(a)+ and let

[ X I " = (J ci i d

where I has power a. We may suppose that I c X and that Ci n Cj if i # j . Let R be the n 2 placed relation on X defined by

+ R(x,... x , , i ) i f f i ~ I a n d { x ,,..., x,,}€Ci.

=

0

Form the model

2l

=

( X , R, i)ier.

Let us say that an elementary submodel B < M is P-saturated relatiue to M iff for every Z c B of power IZI < P, every type C(u) which is realized in (2, b)b& is realized in (23, b),,, . We claim that PI has an elementary submodel 23 of power >,,(a) such that I c B and 2' 3 is >,,-,(a)+-saturated relative to 2. The proof is like the proof of the existence of p+-saturated models of power 28. One need only form an elementary chain

By < % y >,,-I(&)+ such that I c Bo , each Byhas power an(a), and for all Z

c B, of power an-l(a), every type ~ ( u realized ) in (2,b)b,Z is realized in b)b&* This is possible because there are at most >,,(a)'n-'(a)* 2'"-L(u)= >,,(a) such sets Z and types C(u) at stage y.

Since M has power > >,,(a), 2' 3 is a proper submodel of 2. Choose an element c E A - B. We then form a sequence b y 7

Y < an-I(a)+

of elements of B such that for all y, b, realizes the same type as c in (2,b s ) d < f .All the by's are dictinct because c # B. Let U be the set of all by's. Then U has power >n-l(a)f. Partition [U]n into disjoint sets D i , i E I , as follows: for yo < . . . < yn-l < a,,-l(a)+, { by , , . . . . , b y n - , } e D i iff { b , o , . . . , b Y n - l , ~ } E C i .

141

ERDOS-RADO THEOREM

77

By induction hypothesis for n-1, there is a subset Y c U of power IYI > u such that for some i E I,

[Y ] " c Di .

Then for any yo
( ~) v (38 E S)(W

E

v (38 E s)(3$, x E Tc(e))(e=

vX+(X> A

(3c E

(S

v (38 E ~ > (ly3E

(3)

c)py < p>

u W ( c > } E Q ( Y ) ) ) (4)

Tc(e))(e= v Y A (W E Y ) ( ~(S u {$I E PW))

v (38 E s p ~x E,

rc(e))(e = gx+(X)

A

(vc E c ) ( g y

(5)

< p)

(s u {$(c>> E

Q(r)))

v ( S c , d ~ C ) ( c= d ~ s h ( 3 y< p ) ( s u { d = c } ~ Q ( y ) ) )

(6) (7)

v 38 E s(3t, x, $ E TC(e))(3cE C)($(x) is an atomic formula or the negation of an atomic formula, t is a basic term,

0 = $(t), c

= t E S A (37 < P ) ( s u {$(c)1 E Q(r>))

v ( 3 t TC(M,))(t ~ is a basic term A ( V C E C)(3y

(s

u {c

(8)

< p)

t1 E Q(r))>l.

(9)

After looking back at the definition of consistency property, the relation between the above definition of Q ( p ) and consistency property will be clear. (For clarity we used = for the equality symbol of L). Using the principle of C-recursive definition, we see that Q(x) is defined for a11 x E d and that the function Q is A on &’. Let us define S as the set of all sets s = so u TSLolern u {cp} such that so E &(Ma),only ~ ) . claim finitely many c E C and d E D occur in so, and s 4 U a E o ( d ) Q (We that (c) S is a consistency property. As an illustration we verify part (C5) of the definition of consistency property.

86

THE HANF NUMBER OF

Ld

Let V Y E s E Sand suppose that for all $ E Y , s u {$} # S. Let ‘ x ~ Q ( y ) ’ be a L: definition of the relation x E Q(y). Then By Z-reflection there is an a E d such that a is transitive, s E d , Y E a, and ( a , E) 1 (W E ~ Y ) ( ~ Y ) ‘ s u {$I E Q(r)’. It follows that ( d ,E) 1 (W E Y ) ( ~EYo(a))‘s u {$} E Q(r)’. Therefore s E Q(o(a)), contradicting s E 5’. Thus part (C5) of the definition of consistency property holds. To complete our proof it remains only to prove (a) and (b). The proof of (b) is easy, for if s E Sand $ E I and s u {I)}E Q(a), then s E Q(a l), contradicting s E S ; thus s u {$} E S. To prove (a) we shall show by induction that for all p E o ( d ) :

+

.

..

(d) Let s = so u TSkolem u {cp}, let so = so(cl . . c,dl . d,,) contain only finitely many c E C and d E D, and so E S,(M,). Suppose there is a model %* and a linearly ordered set (A’, o.(8+1), and for all a , < . . < a,, in X ,

.

(a*, a, . . .a,,)k ( 3 u l . . . urn)A s(uI . . . umd, . ..d,,). Then s I Q@). The proof of (d) uses the Erdos-Rado Theorem. It is similar to the argument used in proving Theorem 21. Suppose (d) holds for all y < p, E o ( d ) , but that (d) fails for p. There are ten cases corresponding to the ten disjuncts in the definition of Q(p). Two of these cases require the Erdos-Rado Theorem. One of these cases is where there exists O E I and y < p such that s u {el E Q(y). Say 8 = $(di, . . d i p ) $(d,, . djP).But then using (d) for y and the partition relation

.

(>uI.fl+p)

+ +

(&d;o.p’

whence 1,.(8+1)

+

(%.(v+1))2

..

P

we obtain a contradiction. The other case is when there exists V Y E S such that for all $ E Y there is a y < p with s u {+} E Q(y). This time the

161

THE HANF NUMBER OF

87

L&

partition relation &Ll.(p+l)

+

(&D.pE

is used to get a contradiction. To conclude (a) from (d) we apply (d) in the case that so is the empty set. Then no constants from D occur in s = TSkolem u (40). Since w E d and p E d ,it can be shown that w (a+ 1) E o ( d ) . Then by hypothesis cp has a model % of power &,,.(p+l). Let %* be a Skolem expansion of % in the language L:. Taking X = A and < arbitrary, the hypotheses of (d) are satisfied. Then by (d), s # Q(p). Hence s E S and (a) holds. -I

HISTORICAL REMARK. The theorem was first proved by Morley in the special case that cp is of the form ,A ,, (Vx, . . x,,,) Vrn= w such that T admits (z=(K), K). Then: ( i ) T has a model '21 = ( A , U,. . .> such that U is infinite and there is an infinite set of indiscernibles over U. (ii) For all cardinals A 2 w , T admits (A, a). PROOF.As in Theorem 21, we first prove that (i) (ii). Form the Skolem language L> and let T* = T u TSkolem. Then for all c1 < w1 88

171

MORLEY'S TWO CARDINAL THEOREM

89

there exists K 2 w such that T* admits ( & ( K ) , K ) . Applying (i) to L: and T*, there is a model %* = ( A , U, . . .) of T* with an infinite set ( X , . We may assume that %* is countable, using the Downward Lowenheim-Skolem-Tarski Theorem. Moreover, U is infinite and thus IUI = o.For each U E U add a new constant c, to L>, forming the language N,, and form the model (%*, u ) , . ~ .Each formula of N, contains only finitely many c,. It follows that ( X , = 1 and l U / = Iz'(u)l

Since A = D,,., where A 2 &,,l(lUl). Let $ be the sentence

A n w and A = for some ordinal a 2 1. Let L, be a countable fragment of L,,, and let A be a PC,-class over L,,,. If every model % E A of power K is ( ~ 1 , L,)-homogeneous, then every model 23 E A? of power 2 A is (A, Ld)homogeneous.

COROLLARY 1. Let K, A be cardinals such that K > o and A = a,,. 11, a 3 1. Let A? be a PC,-class over L,,,. Suppose A? is K-categorical , La,,)-homogeneous. Then A? and the model in A? of power K is (o, is A-categorical. We shall first prove the corollary and then the theorem.

191

99

HOMOGENEOUS MODELS

PROOFOF COROLLARY. Suppose A is not I-categorical. Then there are two non-isomorphic models a, B in A? of power I. Since 1 2 a,, we may use Lemma E. Thus the model B E Aof power ic realizes only countably many types in L,,,. Therefore there is a countable fragment L, of L,,, such that for all a, b E C", n < o, ( C , a) E~,(C, b) iff

(c,a) = L , I o ( ~ ,

b).

We simply put into L, a formula to distinguish each pair a, b which do not realize the same type in L,,,. By hypothesis, C7 is (mi, L,,,)homogeneous. It then follows by (1) that 0. is (ol, L,)-homogeneous. By Theorem 25, a and B are L,-homogeneous. Then by Lemma D, and 8 do not realize the same types in Ld. Thus there is a countable set C(u, . . . u,,) of formulas of L, such that, say,

a k ( 3 U l . . . U,,) A z(U1 . . . U,,) and

!8 k

i (3U1

. . . a,,) A z(U1 . . . U,,).

Let T' be a countable set of sentences in a countable expansion L:, of L,,, such that A is the class of all reducts to L of models of T'. Then each of the countable sets of sentences

. . . u,,) A C), (30, . . . V,,)Az)

T' u {(3ul T'

U

{T

has a model of power 1 2 sol. By the Upward and Downward LST Theorem, each of these sets has a model of power K , contradicting the ic-categoricity of A. -I

PROOFOF THEOREM 25. Suppose there is a model B €A? of power

2 12 which is not (A, L,)-homogeneous. Let T' be a countable set of sentences in some expansion LL,, such that A? is the class of all reducts to L of models of T'. There is a set X c B of power 1x1 < I and a funcx),,~ =Ld(B,fx)x,x,but for somey E B tionfon Xinto B such that (8, and all z E B it is not the case that

(% x , Y),d =Ldd(!83'fxY Z)x,x.

100

119

HOMOGENEOUS MODELS

B' be an expansion of B to a model of T' and form the model B" = (B', X , f ) . In 8''the following sentence q9 holds:

Let

A

(VX,

PEL,

A(~Y)(VZ)

1

A

. . . x, E X)(Cp(Xl . . . x,)

(vx1

* * *

xn E x ) ( + ( x I .

*

c*

cp(fx1

* *

Xny) c* +(fxl

.fX"))

. . . jxn z)).

Moreover, since I = z, .= and a 2 1 and 1x1 < I S IBI, we have IBI L 3,1(lXl). Thus by Theorem 23 the theory T' u {$I has a model of type ( K , w ) , say W'= (n', Y , 9). Then the reduct of a' to L is a model in A of power K . Moreover, since q9 holds in W'and IYI = 0 , we see that 'ZT is not (wl, L,)-homogeneous. Our proof is comp1ete.i The corollary can be slightly generalized to: COROLLARY 2. Let K , I satisfy the hypotheses of Corollary 1. Let A- by a PC,-class over L;,, such that A- is the class of all reducts to L- of some class d satisfying the hypotheses of Corollary 1. Then d -is I-categor ical. EXAMPLES. Let A, be the class of all models ( A ; , U ) such that ( A , is a group freely generated by the subset U c A . Then A1is categorical in all uncountable powers and all uncountable models of dlare homogeneous. Let d zbe the class of all free groups. Then d zis categorical in all uncountable powers. Since A2is the class of all reducts of models in d ld ,zsatisfies the hypotheses of Corollary 2. Let d 3be the class of all models of the form ( A , U ) where IUI = 0 . Then d 3is a PC,-class over L,,, and is categorical in every uncountable power. d 3does not satisfy the hypotheses of Corollary 1, because no uncountable model of A3is (wl, L,,,)-homogeneous. However, satisfies the hypotheses of Corollary 2 because any uncountable model ( A , U , c,,, cl, c z , . . .) satisfying the sentence (Vx E U ) v, x C, is homogeneous. 0

)

OPENQUESTIONS

1. Does every PC,-class 4 -over L;,, which is categorical in some uncountable power satisfy the hypotheses of Corollary 2?

191

101

HOMOGENEOUS MODELS

2. Let K > w and let T be a countable set of sentences of L,,, which is K-categorical. Is the model of T of power K (wl, L,,,)-homogeneous? The above questions are most interesting when

IC = w1

or

K =

a,,.

PROBLEMS 1. Suppose 23 is an L,-homogeneous model, IAl 5 IBI, and every type in L, which is realized in iX is realized in 23. Show that there is a model 6 iX such that 6 o and I = a,,. c1 2 1. Let L, be a countable fragment of L,,, and A a PC,-class over L,,, all of whose models of power K are (a1,L,)-homogeneous. Suppose A has exactly ,u S w models of power K. Then A has at most ,u models of power I (up to z ).

20 End elementary extensions

In this lecture we shall apply the technique of indiscernible elements to obtain end extensions of models of set theory. ZF, ZFC, and ZFL denote, respectively, Zermelo-Fraenkel set theory, ZF with the axiom of choice, and ZF with the axiom of constructability. DEFINITION. Consider a model 2i = ( A , E, . . .) where E is a binary relation. A model 23 = ( B , F, . . .) is said to be an end extension of 2i iff 23 is a proper extension of PI and for all b E B and a E A , if bFu then b E A . THEOREM 26 (KEISLERAND MORLEY[1968]). Let % be a countable model of ZF. Then 2 has end elementary extensions of arbitrarily large cardinality. We shall not prove the theorem in its full generality here. Instead we prove a simpler special case. 26. Let (21: be a countable model of ZFL. Then SPECIAL CASEOF THEOREM

2 has end elementary extensions of arbitrarily large cardinality.

PROOF.Add a constant e, to the language for each u E A , and form the model %’ = < A , E , U ) . e A .

.

Let L be the language of (21:’. Add two countable sets C, D = { d , , d 2 , . .} of new constants to L, forming the language M. Let T be the set of all 102

201

103

END ELEMENTARY EXTENSIONS

sentences of L which hold in a’, plus the infinite sentences Vx V (x bEa

E e, + x

= eb),

a

E

A.

(1)

and 8’is a model of T.

Thus T is a countable set of sentences of L,,, Let I be the set of all sentences

‘diis an ordinal’, d i E d j , where 1 5 i < j < w ,

B(di, . . . dip)++ B(d,, . . . d,,),

where B(ul . . . up) E L,

and the sequences of i’s and j ’ s are increasing. Thus I is a set of sentences of M. We shall show that

T v Z has a model.

(2)

First we show that (2) implies the conclusion of the theorem. Suppose

T u Z has a model

%” = (B’, d, d2 . . .) = ( B , F , e,, d, d,

. . .)aEA.

..

Then with respect to the language L, the set { d l , d,, .} with the obvious order is an infinite set of indiscernibles in the model B’. Since ‘iX is a model of ZFL, so is 23 = (B, F). Therefore, since in ZFL there is a definable well ordering of the universe, the model 23 has a Skolem expansion 23* such that for every formula $ of L* there is a formula cp of L with B* k cp t)$. Then the set { d , , d,, . .} is indiscernible in the model B’* = (@*, ea)aEA.

.

Let K 2 w. By the Stretching Theorem there is a model B‘* generated by a set (Y, 0. In the above corollary we cannot conclude that d has any models of power I . However, if we strengthen the hypotheses slightly we obtain a stronger conclusion. 2. Let A be a PC, class over L,,, which has a model of power COROLLARY 2 3 0 , . Suppose A is o,-categorical and the model of A of power o1is L,,,-homogeneous. Then A is I-categorical and has a model of power A for each cardinal I > o.

This corollary is the analogue of the corollary to Theorem 25. We give two more applications of Theorem 30. The following theorem is new.

118

[22

ANOTHER TWO C A R D I N A L THEOREM

THEOREM 33. Let A? be a PC, class over L,,, and let L, be a countable fragment of L,,,. Suppose A contains a model % of power > 2". Then A . ! contains a model '$3 of power w1 such that only countably many types in L, are realized in '$3. PROOF. There are at most 2" sets of formulas of Ld, hence altogether at most 2" types in L,. Let T' be a set of sentences in an expansion L& of L,,, such that A is the class of all reducts of models of T'. Then 2l is the reduct of some model 8' of T'. Since 9 realizes at most 2, < IAl types in L,, there is a set U c A of power I UJ = 2" such that

(a', U)k A (VX, . . . x,)(3u, . . . u, E U) cp(X, . . . x,) q ( u , . . . 24,). n w then every K-categorical PC, LEMMA class 4 has an w,-saturated model of power p.

PROOF.Let .A? be the class of all reducts of models of a theory T* in an expanded language L". We may assume that T* has Skolem functions. We claim that:

' 1E 4,there are only countably For every countable model 1 (1) many types over M. By Theorem 21 (or by the classical result of Ehrenfeucht and Mostowski for L), the theory T* has a model generated by an infinite set of indiscernibles, whence by the Stretching Theorem T* has a model B* generated by a set ( Y , is the Skolem language for L,; T* = T u TSkolem, T; = TX u Tkotem.

ax;

LEMMAA. Suppose that for every model 'zl of T of power w1 and every finite or countable X c A , there is a model B of T of power K such that X c B, Bx = L,(xl ax,and every type in L,(X) which is omitted in !2lx is omitted in BX.Then the conclusions of Theorem 35 hold.

PROOF.By Theorem 32 it suffices to prove Every model of T of power w, is L,-homogeneous.

(1)

T is @,-categorical. (2) Suppose the model of T of power o1 is not L,-homogeneous. Then there is a countable Y c A and a functionfon Y into A and an element b E A such that

(a>U).EY =L J ~ > f 4 . € Y

(3)

but there is no f b E A with

Let X

=

Y u ( f a : a E Y >u ( b } . Let

Bx be a model of power K

such that

L,Bx and every type in L,(X) omitted in ax is omitted in Bx. By (III), B is (a1, L,)-homogeneous. Since (3) holds for aYit also holds for 23. But this means that there existsfb E B such that (4)holds for 23. 'zlx

However, since (4) does not hold for any fb E A , the type in L,(X) which is realized by fb in 'Bxmust be omitted in BlX.This contradiction shows that % is L,-homogeneous.

231

MORE ABOUT CATEGORICITY IN POWER

125

To prove (2) it suffices to show that any two models of T of power o1 realize the same types in L,, in view of Lemma D of Lecture 19. Suppose %, %’ are models of T of power o,,% realizes a type Z(v, . . . 0), in L,, and %’ omits C ( u , . . . v,). Let X = {xl . . . x,} where ( x , . . . x,) realizes C in U. By (1) there is a model 23, of power K such that (x, . . . x,) realizes C in 23. Also, there is a model B’ of T of power K such that every type in L, omitted in 8‘is omitted in 23’. In particular 23‘ omits Z. Thus 23 and 23‘ are not isomorphic, contradicting (I). -I

LEMMAB. Let U be a model of T, IAl = w,, X c A , (XI 5 o.Then there is a model $3‘; of T; of power K such that: (i) 23; is generated by a set (Y, of indiscernibles of order type (4

.

(ii) There is an elementary embedding f: gX B such that (i) B has exactly the same ordinals as B. (ii) For every b E B such that 23 C (b is not well-orderable), we have lbFl

=

01.

Moreover, if

B is well-founded so is B.

140

SHORT, UNCOUNTABLE MODELS OF SET THEORY

REMARK. The conclusion (i) implies that every well-orderable u E A is fixed, and also if b E B and 8 k ( b is well-orderable), then lbFl 5 w. Conclusion (ii) implies that every non well-orderable a E A is enlarged. Examples of well-founded models M satisfying the hypotheses of the corollary and in which not every set is well-orderable (i.e. choice fails) have been given, e.g., by MATHIAS [1970]. PROOFOF COROLLARY D. Let q ( u ) be the formula ‘ u is not well-orderable’ and $ ( u ) be ‘ u is well-orderable’. It then follows from the hypotheses that q(M) is regular over $(a). Hence by Theorem 36 there exists 8’> CU such that every ordinal of M is fixed and for all non-well-orderable b E B’, IbPl = ol.By Lemma c the submodel B of all b E 23’ such that 93’.It follows that for some CI E A , B’ k b E R(a), is such that 9l < B i 8 has exactly the same ordinals as a, i.e. (i) holds. Since bF = bF,for all b E B, (ii) also holds. The ‘moreover’ clause follows at once from (i). -I Our next aim is to generalize the last corollary to arbitrary models of ZF. LEMMA E. There is a formula W(u) of ZF such that in every model % of ZF, W(M) is the least class in such that: (i) Every singleton of % belongs to W(M). (ii) W(M) is closed under unions of well-orderable subsets, that is, if uE c W(M) and % C (a is well-orderable), then % k W ( u a). PROOF.For each ordinal a, define W, to be the least subset of R(a+ 1) which contains all singletons in R(a+ 1) and is closed under unions of well-orderable subsets. This definition can be expressed in ZF. If a < j, then W, n R ( a + 1) is obviously closed under unions of well-orderable subsets and contains all singletons in R(a + l), whence W, c W,. Define W ( v )c, (3a)(a is an ordinal A v E W,). Then W(u)clearly satisfies (i) and (ii). Let M be a model of ZF and suppose a class q(M) satisfies (i) and (ii). Since each R(a+ 1) is closed under unions of subsets, each set q(%) n R(a+ 1) is closed under unions of well-orderable subsets. Therefore each W, c cp(%), whence W(%) c q(%). This shows that W(M) is the least class satisfying (i) and (ii). i

251

SHORT, UNCOUNTABLE MODELS OF SET THEORY

141

LEMMA F. There is a formula W"(u)of ZF with variables c(, u, such that in ZF: (i) Wo((u)t)u is either empty or a singleton. (ii) If a is a limit ordinal, W"(u)c-f (3p cc)Ws(v),intuitively W" = U B < a Ws(iii) For any ordinal a, W"+'(u)c-f u = x for some well-orderable x such that (Vy E x)W"(y), intuitively W a + l = x: x c W" and x is well-orderable}. (iv) W ( u )c,For some ordinal a, W"(u).

-=

u

{u

The proof of (i)-(iii) is by restricting to R(P+ 1) as in Lemma E, and then (iv) follows at once. G . For any model W of ZF, the class A - W(W) is regular over the LEMMA class of ordinals of W.

PROOF. If x

=

us W such that 23 has exactly the same ordinals as '21 and for all b E B - W(23),lbFl = wl.

PROOF. By Theorem 36 there exists 23' > W such that each ordinal of &L is fixed and for all b E B'- W ( B ' ) ,IbF.l = w1 (using Lemma G ) . Let 8 be the submodel of 23' such that B

=

(b E B': For some ordinal a of

2,23' b b E R(a)}.

By Lemma c, W < 23 < 23'. It is easy to see that 23 has exactly the same ordinals as W and that for all b E B, bF = bF'. The result follows. i COROLLARY I. Let W be a well-founded model of ZF and let 23 > W have exactly the same ordinals as W. Then every U E W(W) is fixed. Thus if 9 k ZFC or even W k (Vu)W(u), then we must have 23 = W.

PROOF. By induction on the ordinals a of W, show that every a E W"(W) is fixed. i A set d is said to be Dedekind iff neither Id] < w nor w 5 (dl. That is, d is neither finite nor infinite. In ZFC it can be proved that there are no

142

SHORT, UNCOUNTABLE MODELS OF SET THEORY

125

Dedekind sets. Halpern and Levy, and later Mathias, have given examples of countable €-models of ZF in which every set can be linearly ordered but the set of real numbers has a Dedekind subset. These models satisfy the hypotheses of the next corollary. COROLLARY J. Let % be a countable model of ZF in which the following holds: If 0 < Ix,I < o for all n < o,then the sequence (xn: n < o) has a choice function. Then there is a model 23 > % which has exactly the same ordinals as 9.l and such that for all b E B, if 23 C b is Dedekind then lbFl = ol.Thus Dedekind sets are ‘much larger’ than the class of all ordinals when % is viewed from the outside. PROOF.In view of our previous arguments it suffices to show that the class cp(%) of all Dedekind sets in 2 is regular over the class I)(%) of all ordinals of a. Let x E cp(%), y E I)(%), and

fl C (fis a function and x

=

u

{ f ( z ) :z E y } ) .

Suppose that for all zEy, % k If(z)l < o.Since 2 C ( y is an ordinal), we may assume without loss of generality that

% C ( z E z‘

Ey

+ f ( z ) , f ( z ’ )are disjoint and non-empty).

For we may replace f ( z ’ ) by f ( z ’ ) - u z s z , f ( z )and ‘delete’ z’s with f ( z ) empty. If k y < o,then % C 1x1 < o,contradicting % C x is Dedekind. Thus 2 C y 2 w . Using our choice assumption,

8 C (3g)(g is a choice function for ( f ( n ) :n < 0)). Let % C u = { g ( n ) :n < w}. Since the f ( n ) ’ s are disjoint, % k IuI = to A u c x . But this contradicts 2 I= (x is Dedekind). We cannot have Yl k If(z)l 2 w. Hence for some zEy, % C ( f ( z ) is Dedekind), i.e. ‘!I k cp(f(z)).-I Corollaries H and I can be explained in terms of transitive E-models. If \u = ( A , E) is a transitive €-model of ZF, then by ord(%) we mean the

251

SHORT, UNCOUNTABLE MODELS OF SET THEORY

143

least ordinal not in A , which coincides with the set of all ordinals u E A . Using Corollaries H and I we can draw the following conclusions: Let T be a complete extension of ZF. Then T has a transitive ernode123 of power u1such that ord(23) is countable if and only if T has a countable transitive €-model and T k (3u) W(u). T has a transitive €-model 23 of power o1with ord(23) = a, (a a given countable ordinal), if and only if T has a countable transitive model 91 with ord(9l) = u and T b (30) 1W(u). Theorem 36 above appeared in KEISLER [1970], Corollaries A and B appeared earlier in KEISLER and MORLEY [1968]. The models mentioned in connection with Corollary J are constructed in HALPERN and LEVY [1970] and MATHIAS [1970]. The first examples of transitive €-models 23 of ZF such that ord(23) is countable but 93 has power w1 were constructed by Cohen and were not published. His proof used the forcing construction. His method was refined by Easton, Solovay, and Sacks to obtain transitive €-models 23 of ZF such that ord(23)is countable and 23 has power 2" (also unpublished). They evidently did not settle exactly which complete extensions of ZF have such models, as in Corollaries H and r above. The precise connection between the models of this kind constructed via forcing and via the Omitting Types Theorem has not been worked out. The question of whether Corollary H holds with 2" in place of o1is open.

26 Lebesgue measure

In this lecture we shall consider models of ZF in which every set is Lebesgue measurable. Our first result is another corollary of Theorem 36.

COROLLARY. Let '21 be a countable model of ZF such that

'21 =! (any well-ordered union of sets of reals of Lebesgue measure0 has Lebesgue measure 0). Then there is a model 8 > '21 which has exactly the same ordinals as % such that if 23 i=b = S(o) then lbFl = ol. PROOF, Let q ( u ) say that u is a set of real numbers and u does not have Lebesgue measure 0. Then it follows from our hypotheses that q(%) is regular over the class of all ordinals of '21. Since when we identify S ( o ) with the real unit interval it has Lebesgue measure 1, the desired result follows from Theorem 36 and Lemma c. i

From now on it is convenient to identify S ( o ) with the real unit interval [0, 11 in the canonical way (binary expansions). We shall use p ( x ) to denote the Lebesgue measure of x , p* and p* denote inner and outer Lebesgue measure, respectively. If '21 = ( A , E) is a transitive €-model of ZF, then [0, 11 n A E A and [0, I ] n A satisfies the formula u = [O, 11 in 21. For this reason it is better to state our results in terms of transitive E-models instead of well-founded models. We recall that every wellfounded model of ZF is isomorphic to a transitive €-model of ZF. In what follows we shall show that it is possible to have transitive E-models 144

261

LESBESGUE MEASURE

145

\zI of Z F in which every set of reals is Lebesgue measurable and [O, 11 n A is quite large. It was shown by Mycielski and Swierczkowski that the Axiom of Determinateness has the following consequences: I. Every set x c [0, 11 is Lebesgue measurable. 11. Every sequence (x,,: n < w ) of non-empty sets x, c [0, 11 has a choice function.

It is not hard to show that I and I1 imply 111. The union of any well-ordered sequence of sets of reals of Lebesgue

measure 0 has Lebesgue measure 0.

Moreover, it was shown by Solovay that if Z F has a transitive €-model with an inaccessible cardinal then ZF has a transitive E-model in which 1-111 above hold (and also the axiom of dependent choice). THEOREM 37. Assume the continuum hypothesis 2" = wl. Let % = ( A , E) be a countable transitive €-model of ZF+I+II+III. Then there is a transitive E-mod61 B = ( B , E) such that: (i) ord(%) = o r d ( 8 ) . (ii) There is an elementary embeddingf: % < B which is the identity on ord(8). (iii) The set [0, 11 n B has outer Lebesgue measure 1 in the real world, p*([O, 11 n B ) = 1. REMARK. The condition p*([O, 11 n B ) = 1 is best possible in the sense that we must have p*([O, 13 n B ) = 0. This was pointed out by Kunen. [0, 11 n B must be a proper subset of [0, 11 since B contains only countably many ordinals and thus there are elements of [0, 11 which code ordinals not in B (i.e. they are of the form {2"3": ( n , m) E y } where y is a well ordering of w of order type not in B). Moreover, [0, 11 n B is a subgroup of [0, 11 under addition modulo 1, and it is a classical result that any proper subgroup of [0,1] has inner measure 0. PROOFOF THEOREM 37. We use the fact that every subset of [0, 11 which meets every closed set of positive measure has outer measure 1. We shall prove:

146

[26

LESBEGUE MEASURE

+

Let ( A , , E , ) be a countable well-founded model of Z F I +I1 +III, and let X c [0, 11 be a closed set with p ( X ) > 0. Then there exists r E X and a model ( A , , El) > ( A , , E , ) with the same ordinals as ( A , , E , ) such that for some b E A , , r = {n < w: (A,,,!?,) k n e b } . (1) Suppose we have proved (1). There are 2" = w1closed subsets of [0, 11. We apply (1) w1 times, once for each closed set X c [0, 11 of positive measure, and thus form an elementary chain u < wl,starting with fl and taking unions at the limit ordinals. The union 23' of this elementary chain is an elementary extension of fl, , has exactly the same ordinals as a,, and for each X c [0, 13 of positive measure there is a b E B' with

am,

{n < w : 2 3 ' C n E b } E X . Thus the transitive realization 23 of 23' has the same ordinals as fl and p * ( [ O , I ] n B ) = 1. Thus it suffices to prove (1) for the theorem. We now prove (1). We may assume that ( A , , E,) is a transitive Emodel, since it is isomorphic to one. Thus we may assume ( A , , E,) = ( A , E). Let X c [0, 11 be a closed set with p ( X ) = 1 > 0. X may be X,, where X , 3 X , 3 X , 3 . . . and each written in the form X = X,, is a finite union of closed intervals with rational endpoints, say

n,

Xn =

Cx.13

u*

~ n , l

* *

u CXnpn 9

~np,l*

The x i j and y i j , being rational, belong to A . For each n let a,, = X,, n A . Then a,, E A , and indeed

fl k an = [ X n I

3

~ n lu l *

.

*

u [xnp,

3

~ n p ~ l .

For a E A let psl(a)be the Lebesgue measure of a in the sense of fl. Then for each n, psl(an)= p ( X n ) and limpsl(an)= 1 > 0. n+ OD

Note that the above limit is computed outside fl and I does not necessarily belong to A . Let L' = L ( A u { c } ) where c is a new constant symbol, and let T be the following set of sentences of L':

261

147

LEBESGUE MEASURE

Th(%,) u {c

E

# a : a E A and lirn pa(a n a,)

[0,1]) u {c

=

n-r oc

0).

We claim that

A sentence cp(c) of L' is consistent with T if and only if h ( b n a,) > 0, where b is the element of A such that %A

!= b =

{XE

[0, 11: cp(~)}.

(2)

PROOFOF (2). Let b be as defined above. Suppose first that ~ ( c )is such that lim h ( b n a,) = 0. n-rm

Then TI=c # b . Hence T ! =c # [0, l ] v l cp(c). But T ! =C E [0, 11, so TI= cp(c), and cp(c) is not consistent with T. Next suppose that T != 1q(c). Then for some b,, . . ., b, E A , we have lim h ( b i n a,) = 0 for i = 1 , . . ., m, n-tm

and

~ h ( % , ) ,c E [O, 11, c C b, . . .) c C b, != 3

Let 6'

=

6, u

. . . u b,.

Therefore

n-r m

1 lirn k ( b i n a,)

k

XE[O,

= 0,

i s m n-rm

Th(%J, c E [0, 11, c # b' != %A

Hence

(c).

Then

lim k ( b ' n a,) 5 and

1~

1]AX$! b'

--f

c~(c).

i

i Cp(X).

Z A =! x E [0,1] A q ( x ) + x E b',

whence b c 6'. It follows that lirn pa(b n a,)

n-l m

This proves (2). When we apply (2) with consistent with T, whence

CE

5 lirn h ( b ' n a,)

= 0.

n-rm

[0, 11 for q ( c ) , we see that

T is consistent.

CE

[0, 11 is (3)

148

LEBESGUE MEASURE

I26

Our next claim is:

If (3y)(y is an ordinal A q(c, y ) ) is consistent with T, then there is an ordinal a E A such that q(c, a ) is consistent with T. (4) Suppose

( 3 y ) ( y is an ordinal A q(c, y ) )

is consistent with T. Let $(c, y ) be the formula which says ‘ y is the least ordinal such that q(c, y)’. Then Tb

~VP(C,

v)

c*

(jy)ll/(c, Y ) ,

whence (3y)$(c, y ) is consistent with T. Let b E A be such that %A

cb

BY (2),

= {x

E

[O, 11: (3y)$(x, Y ) } .

lirn p%(b n a,) > 0. n-r w

(5)

For each a E ord(%) let f a E A be the element where %A ! = f a

= {x

E [O,

11: $(x, a)}.

If a < p then, from the definition of $(c, y ) , we have f a n f , = 0. Using the axiom of replacement in %, there is an ordinal p E ord(%) such that f, = 0 whenever p 6 y. Then b = The set (h:a < p) belongs to A and is well-orderable in %. Let d E A be such that

Ua O}.

Then by the countable additivity of Lebesgue measure in % (a consequence of 11), we have %A b Id1 6 o. Moreover, by 111, %A P@- U f J = 0 , aEd

because b - Uasd f a = UyE,--d f,is a well-ordered union of sets of measure zero in a. We claim that For some c1 E d, lim k ( f a n a,) > 0. n+ m

(6)

261

149

LEBESGUE MEASURE

For suppose (6) fails. Let S > 0 be a real number. By countable additivity there exist al,. . ., a,, ~d such that p*(b-tfa,

".

*

+

Ufa,,))

< 8.

Taking n sufficiently large we can also get

6 m

for i

p % ( f a , n a,) < -

=

1,.

. ., m

because (6) fails. But then m

~ d nba,> 5 6 +

C pa(fai n an) 2 26

i= 1

and this contradicts ( 5 ) . Thus ( 6 ) is proved. By (2) and (6) there exists a E ord('u) such that $(c, a ) is consistent with T. But T C $(c, y ) + q ( c , y ) , so q ( c , a ) is also consistent with T. This verifies (4). Using (3) and (4), we apply the Omitting Types Theorem to obtain a model (BA,c ) of T i n which the infinite sentence ( V y ) ( y is an ordinal +

V y

= ct)

asord(!X)

holds. But this means that B is an elementary extension of 'u which has exactly the same ordinals as 3. Let

r = {n<w:(BA,c)knEc}. For eachn we have T k c # [07I]-a,, and T k C E [O, 11, whence T k c E a,,. But then r Xn = C x n 1 7 Y n I l u * . u [ X n p , ,~ n p - 1 7

because the order relation between real numbers of 2' 3 is the same as that between the corresponding actual real numbers. Thus we have r E fin A which is an w-model of power ol. (ii) Let S ( o ) have the ‘product topology’ under the identification of S( o) with 2”. Suppose S is a countable union of closed sets in S(w) such that S n A = 0. Then in (i), B may be chosen so that S n B = 0. In particular, if S is countable and S n A = 0, we can find B with S n B = 0. I

l

PROOF. We first prove part (i). Add a new constant to the language for each element of A , and form the model A , of this expanded language L(A). We shall call a formula cp( V ) of L(A) with one free set variable ‘enumerable’ (in A ) if and only if A, b (3 V(VV)(cp(V>

+

(W(V=

Q).

Intuitively, this means that

{ VE A : A, b cp( V ) } c { vx:x E o}, so that U ‘enumerates’ cp( V ) . It is easy to see that If cpl( V ) and cpz( V ) are enumerable, then cpl v cpz( V )is enumer(1) able. Add a new constant symbol C to the language L(A) and let T be the

156

SECOND ORDER NUMBER THEORY

set of sentences:

T = Th(A,) u

{l O(C): O(V) is enumerable).

We claim that

cp(C)is consistent with Tif and only if cp(V) is not enumerable. (2) Obviously, if q ( V ) is enumerable then q ( C ) is inconsistent with T. Conversely suppose cp(C) is inconsistent with T, i.e. T!=7cp(C). For some enumerable formulas el( V ) , . . ., e,(V), we have Then

T ~ ( A , ) , ~ B ~ ..., ( C )le,(c), , A,

whence

c:

e,(v)A

... A

vw

e,(v)

-+

v(V>,

A , 1 c p ( ~-,) e,(v)v . . . v e,(v).

.

By (l), Q , ( V )v . . v On( V ) is enumerable, whence cp( V ) is enumerable. This proves (2). It follows that T is consistent. (3 1 For if Tis inconsistent then by (2) the formula V = Vis enumerable, and

A, I= (3 U)(VV)(3X)(v = UJ. But using comprehension,

A, k (W)(Vx)(x E v-

x E U,).

1

The above two formulas contradict each other. We conclude that (3) holds. (This is essentially Cantor's proof that 2" is uncountable). Next we prove If (3y)cp(Cy)is consistent with T then for some n < W , cp(cn) is consistent with T. (4) To prove (4), assume that for all n < w , T k each n, cp( Vn) is enumerable, thus

cp(Cn). By (2), for

A, k ( ~ u ) ( v v ) (-+ ~ (3x)V ( v ~ =) U x ) .

281

157

SECOND ORDER NUMBER THEORY

Since A is an o-model, A, I= ( V Y > ( W ( V ~ ) ( C p ( ~ Y )(3x)V = +

Q.

By the choice scheme,

c ( w v Y ) ( ~ v ) ( C p ( V Y )(3X)V

A,

+

=

(k),).

But using comprehension,

A, b (3W)(VX, Y)(U,), =

W2’3Y.

Thence (3W)(VY)(VV)(Cp(VY)-+ (3X)V = %w),

A,

and whence

A,

c (3W)(VY)(VV)(Cp(VY)

4 4

+

(3z)V = WZ),

b ( T ( V V ) ( ( ~ Y ) C p ( V Y ) (3z)V = WZ)). +

This shows that (3y)~p(l/Y) is enumerable, and by (2), T b 1(3y)q(Cy). Therefore (4) holds. Using the Omitting Types Theorem (modified to second order logic), we see that Thas a model (B,, C ) satisfying the infinite formula (Vx) ,V ,, x = n. Thus 23 is an o-model, and A < B. For each D E A , the formula V = D is enumerable, so TI= C = D. It follows that C E B - A . We have shown that: Every countable w-model of second order number theory with choice has a proper elementary extension which is an o-model. (5) Using (5) o1times and taking unions of the elementary chain at limit ordinals, we obtain an uncountable o-model B > A . The proof of (i) is complete. PROOF OF (ii): Let S = S, where each S, is a closed set. Then each S,, can be written in the form S, = S,, where each S,, is a basic closed set. A basic closed set P is determined by a condition

u, w p h o , A h 3,,(P>.

THEOREM 41. Let L have a unary relation symbol U.Suppose that every L,)-homogeneous. Then every model '$X €.A? of type (K, ol)is (ol, model 2.3 E Aof type (A, p) is (p, L,)-homogeneous. PROOF.Like the proof of Theorem 25 or 32. THEOREM 42. Suppose that for every model 58 E A of power K and every countable set X c A , (II realizes at most countably many types of Ld over X. Then for every model 2.3 €.A? of power A and every set Y c B of power p, 2.3 realizes at most p types over Y. PROOF.Like Theorem 34.

30 End elementary extensions which omit a type

In the next few lectures we shall obtain further results on models of power w t for the language L,,,. The method of elementary chains makes it possible to say much more about models of power w1 than we could say about models of larger powers. One of our important tools so far has been Theorem 28 on end elementary extensions. We shall now prove a stronger version of Theorem 28 which concerns end elementary extensions in which countably many types are omitted. The next three results, Theorems 43-45, are special cases of theorems in KEISLER [1970]. THEOREM 43. Let i!l= (A, R,. . .) be a countable model for L such that R is a transitive, irreflexive relation on A. Let Ld be a countable fragment of L,,,. (These are exactly the hypotheses of Theorem 28). Let Q be an infinite sentence of the form d

=

A (VX, . . . xp,) 2

m<w

v

Q,,

nA $r A urnn).

-

Using (5) again we conclude that

-

( 3 ~ 1* * Yq)(S)(gx, is consistent with T, whence (33’1

-

* *

yqJ[44

- - - xpm)(n(4A

A (S)(3x1

*

-.

$r A cmn)

Xp,)($r

A

0m)l

is consistent with T. This proves (2‘).

Now using (2) and (2’) and the consistency of T, we see by the Omitting Types Theorem that T u {cp, d} has a model. By our previous remarks, this completes the proof. -I In the above theorem, the string of quantifiers ( S ) was needed in order to have a condition (ii*) which would continue to hold in some countable end extension of a, so that a chain of length o1could be formed. Notice that in the proof of Theorem 43, the condition (ii*) was used in the given model 8 with a longer string ( S ’ ) in order to show that (ii*) holds in a new model 8 with the string (S). We shall give some applications of Theorem 43. In what follows we shall say that a model 8 = { A , R,.. .> and language L, are an end extension pair iff they satisfy the hypotheses of Theorem 43, that is, 8 is countable; R is a transitive irreflexive relation over A ; L, is a countable fragment of L,,,; 8 has an end elementary extension in L, of power ol.

COROLLARY 1. Suppose 8, L, is an end extension pair. For each m < o let c, = (Vx, . . xpm) V a ,

.

nca,

301

167

END ELEMENTARY EXTENSIONS WHICH OMIT A TYPE

where each om,E L,. Suppose that for each m < o,% has an end elementary extension Bmin L, of power w 1such that Bm I= am. Then '$I has an end elementary extension B i n L, of power w 1such that B 1 A .u,, PROOF.The condition (ii*) for each omis equivalent to (ii*) for A m < o om*

COROLLARY 2. Let %, L, be an end extension pair and let a = A (Vxl m 2".

EXAMPLE. Let L have one-placed relations R o yR , ,R2,. . . only. Let T be the theory of all models l!l such that each consistent finite Boolean combination of the R,'s is infinite in l!l. Then T has 2" types. For each infinite cardinal A 5 2", T has 2A non-isomorphic models of power A. However for cardinals N, > 2", T has lct12w non-isomorphic models of

311

MODELS OF POWER Wt

175

power 8,. For a model is determined by the number of elements in each of the 2" types. E.g. T has just 2'- models of power 22m,etc.

EXAMPLE. The theory of dense linear order has just countably many types but still has 2"' non-isomorphic models of power ol.(However if countably many individual constants are added then the resulting theory has 2" types). Thus the converse of the above corollary fails. The problem of finding a necessary and sufficient condition for a first order complete theory to have 2"' non-isomorphic models of power o1 is open.

32 Ultrapowers

Up to this point in the course we have made use of three main methods of constructing models in the study of infinitary logic: The Henkin construction and Model Existence Theorem; Skolem functions and indiscernibles; Elementary chains. All of these constructions are also of basic importance i n first order model theory. There is one more method of constructing models which is basic to first order model theory but has not yet been used at all in this course, namely the ultraproduct construction. As a matter of fact, there is a way to use the ultraproduct construction in L,,,, and several of the results we have proved were first discovered using ultraproducts. In first order model theory ultraproducts can always be used as a substitute for the Compactness Theorem and there are many results whose only reasonable proofs use ultraproducts. (For a survey and historical account see KEISLER [1967a].) However, in the model theory of L,,, ultraproducts can only occasionally be used as a substitute for the Model Existence Theorem, and there are very few results to date whose only good proof uses ultraproducts. For this reason we have chosen a treatment of the subject which emphasizes the Model Existence Theorem in preference to ultraproducts. We shall now describe an ultraproduct construction appropriate L,,, . Its theory is not yet in a satisfactory state of development, but we will be able to prove some results which indicate that there ought to 179

180

ULTRAPOWERS

be things that can be done in L,,, using ultraproducts which cannot be done in other ways. We presuppose a knowledge of ultraproducts in first-order logic. In the following let L, be a countable fragment of L,,,. If q ( x ) ~ L,(B) and '$3 is a model, recall that the set q(%) = { b E B : BBC q [ b ] }

is called a class of 23. Similarly, if q ( x , y ) E L,(B) and BEk q ( x , y ) A q ( x , z ) + y = z, then the partial function FQBdefined by

FqB(b) = c iff

BEk q [ b , c]

is called a definablefunction of '$3. The domain of the function FQBis the class ((3Y)rp)W.

If necessary for clarity we may add the phrase 'with respect to L,' to the above two definitions, and say 'class in 23 w.r. L,' or 'definable function in '$3 w.r. Ld'. DEFINITION. Let ~ ( ' $ 3 ) be a class in 23 (w.r. Ld). '$3 is said to be SkoIem over q(w.r. L,) iff for every formula $(x, y ) E L,(B) there is a definable function F$Bin 23 (w.r. L,) whose domain is the class q(%) such that for all x E q(23),

BE

(3y)$(x, y ) imp1iesBE

$(xF$B(x))*

DEFINITION. Suppose B ' is Skolem over q(%)and D is an ultrafilter over q ( B ) .The Skolem ultrapower n, '$3 is defined as follows. Let C be the set of all definable functions in '$3 with domain ~('$3). Given F#B and FOB E C , we say that F,,

-DFOBiff{X E (~('$3):

FSB(X) =

FsB(x)) ED.

For F E C, let FD be the equivalence class of F under w D .The universe set n D Bof nD '$3 is the set nDB

= {FD:

F

E

c}.

The relations of IlDBare defined as follows: For all F',

...,F" E C and

321

181

ULTRAPOWERS

n-ary relation R, R,(F;.

. . F;) iff {X E ~(113):

R ( F ' ( x ) . . . ~"(x))}ED.

As in the case of ordinary ultraproducts one can check that x Dis an equivalence relation and the relations RD are well defined. Functions and constants of l7, 8 are treated similarly. This defines the model nDB. The natural embedding of 113 into UD113is the function d: 113 + nD23 which maps each b E B to the equivalence class of the constant function at b. All the constant functions are obviously definable in 113. The above definition is very similar to Skolem's original construction used to get a non-standard model of arithmetic. DEFINITION. Let ~(113) be a class in 113. An ultrafilter D over ~(113) is said to be L,-complete iff for every formula A,, 8,(yl . .y.) E L,(B), if F', ., F" are definable functions in 8 with domain (~(8) and for all m < O, { x : BB C. O,(F'(X). . . F"(x))} E D , then

.

..

{x: BBk A O,(F'(x).

. . F"(x))} ED.

We now prove two theorems which hold for uncountable as well as countable models 113. THEOREM 46 (THEFUNDAMENTAL THEOREM). Let ~ ( 2 3 )be a class in 113, let 113 be Skolem over ~(113) (both w.r. L,), and let D be an L,-complete ultrafilter over ~(23). (i) Then for all formulas +(xl . . . X,)E L, and all definable functions F', . . .,F" in 113 (w.r. Ld), with domain cp(B), we have IIDB

c + [ F A . . . F;S]

iff{x E ~ ( 8113 ) k:+ [ F ' ( x ) . . . F"(x)]}ED.

(ii) The natural embedding d is elementary, d : 113 sat cp

.

Then T' has axioms which state: Zermelo set theory sat(cp, x) for all sentences cp E T, U is a set, i.e. (3x)(Vy)(y E x f--r U(y)), V is a set and U' is a set and V c U' c U, Choice over U', U' is regular over V , (Qx)(x E Ld4-P v *EL, x = c p x IU'x U'I = IU'I and IU'I 2 2 and IVI 2 2, There is a one-one function of L, into V, The rules giving the recursive definition of sat for cp

(*)

E

Ld hold.

All the above axioms of T' are finite except for (*). Let T" be the theory with all the axioms of T' plus a new function F and an axiom stating that

331

191

THE SEVEN CARDINAL THEOREM

F maps W onto the universe. Using the Downward LST Theorem we see that T" has a model of type (>=(K), K, p', p) for each CI < w l . Then using Morley's Two-Cardinal Theorem (Theorem 23) we see that T" has a model of type (A, w, w , w ) for each 1 2 wl. Call this model 8". Moreover, 23'' may also be chosen so that the set "U of all functions in 23'' from U' into U is countable. The reduct B' of this model to the language of T' is then a model of T'. Furthermore, the interpretation of Win 23' has power A because of the function F of T". By Theorem 48, there is an elementary extension 6' > B' such that the interpretation of Win 6 has power A, the interpretation of U' in 6' has power wl, but Vis fixed. Furthermore, 6' is obtained as the union Gf an elementary chain of length w1 of models in which the interpretation of "U is countable (as in the proof of Theorem 48). Therefore the interpretation of U in 6' has power wl. Let 6 be the reduct to L of the restriction of B' to W. Then B is a model of type (A, wl, w). Since V is fixed and there is a one-one function of Ld into V in B', the relation Ld is also fixed by&'. Therefore the infinite sentence (*) holds in 0.'. It follows by 6' > '23' that 6' is a model of T'. Then we conclude that sat has the intended meaning in B', and therefore 0.is a model of T. -I

Notice that this proof of Theorem 40 proceeded by first getting a model of type (A, w, w ) and then extending it to one of type (A, wl, 0).The other proof we gave first constructed a model of type (a1 ,wl, w ) and then extended it to one of type (A, wl,0). Thus there seems to be an essential difference between the two proofs. We conclude with one rather special example of a theorem in L,,, whose only known proof uses ultrapowers. It is given not for its own sake but as an indication that it is possible to do things in L,,, with ultrapowers which cannot be done as readily in other ways. If d' is admissible let (K, ' d (1,v ) mean that every sentence rp E Ld which admits

(K, p )

admits ( I , v).

THEOREM 49 (THE SEVENCARDINAL THEOREM).Let &' be a countable admissible set. Suppose that (K, p ) -)& (A, w ) , and A 2 w1 ,p < K. Then

192

THE SEVEN CARDINAL THEOREM

whenever w S a < p, / ? ( a + ) (K,

[33

p, we have 89.)

+s4

(AY

0 1>

w).

PROOF.Assume o E d . Let cp E Ls4 admit ( K , p, a), say 93 is a model of cp of type ( K , p, a). Let Ld0 be a countable fragment of L,,, such that cp E Ld0 and Ldo E d . Imposing a model of set theory on 93 we form the model

The three dots stand for the relations of 93, and K is identified with the universe of 8. The theory T' of the preceding proof can now be expressed by a single sentence cp' of Lb. By hypothesis, the set " + pof all functions in 93 on a+ into has power p("+) 5 p, and therefore cp' has a model 8; of power 1 in which the interpretation of " + p is countable. Then as before there is a model B' > 93; of power 1in which the interpretation of p has power o1 and the interpretation of a is fixed. Then the reduct Q of B' restricted to the interpretation of IC is a model of cp of type (A, ol, 0). The case when o # d is similar. In this case we have d = R(o)so Ld is just first order logic. Then in forming the model 8'we do not need Ld0 and sat. To make sure that the sentence cp' is a single sentence of L, we need only make the universe of %' be R ( K ++ 1). -I The same argument gives the following result in first order logic.

-=

COROLLARY. Suppose that 1 2 wl, p K, and every theory T i n L which admits ( I C , ~admits ) (A,w). Then every theory T in L which admits ( K , p, a ) admits (A, w1 o), whenever

o Ia c

p,

/I("+'

5 p.

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Author index

Barwise, J. viii, 5, 32, 33, 37, 42, 45, 46,48,49, 83,87 Beth, E. 21 Cantor, G. 156 Chang, C. C. vii, viii, 8, 32, 74 Cudnovskii, G. 92 Cohen, P. 60 Craig, W. 19 Dickmann, M. vii Ehrenfeucht, A. 70, 168 Erdos, P. 75, 77 Fraenkel, A. 102 Friedman, H. 46,48, 53, 57, 60, 159 Gaifman, H. 105 Godel, K. 10, 58 Gregory, J. 110 Grilliot, T. 57 Hanf, W. vii, 81, 82 Halpern, J. D. 141, 143 Henkin, L. vii, 5, 10, 12, 19, 32, 54, 179 Jensen, R. 46, 57 Karp, C. vii, 16, 46 Keister, H. J. vii, 92, 102, 105, 110, 116, 123, 132, 137, 138, 143, 153, 162, 179, 188 Kreisel, G. vii, 46 Kripke, S. 36, 37 Kunen, K. viii, 145, 151, 153 Levy, A. 141, 143

Lopez-Escobar, E. vii, 19, 21, 24, 26, 27, 33,49, 52, 78 LoS, J. 22, 31 Lawenheim, L. 5,22,69,78 Lyndon, R. 24,26,27 Makkai, M. viii, 10,12, 19,32,46, 53 Malitz, J. 29, 33, 74 Mathias, A. 140, 141, 143 Morley, M. vii, 49, 69, 78, 83, 87, 88, 91, 92, 96, 102, 120, 137, 138, 143 Mostowski, A. 70, 168 Mycielski, J. 145, 151 Orey, S. 54 Platek, R. 34, 36, 57 Rado, R. 75,77 Robinson, A. 21, 22, 32 Sacks, G. 57 Scott, D. vii, 7 Shelah, S. viii, 92 Silver, J. 92, 105, 120 Simpson, S. 57,77 Skolem, T. viii, 5, 22, 67,69, 70, 78 Smullyan, R. 10, 12 Solovay, R. 105,145,150 Swierczkowski, S. 145 Tarski, A. 10, 31, 68,69,78, 109 Vaught, R. 22,61, 63, 68, 69,92, 109, 116, 162 Weinstein, J. 32 Zermelo, E. 58, 102 205

Index of definitions

admissible ordinal 36 admissible set 35 admit ( K , 1) 86 automorphism class 122 axiom of determinateness 144 axioms for L,,, 15 Baire Category Theorem 151 Baire, property of 15 1 Barwise Compactness Theorem 4 5 basic term I 1 Beth’s Theorem 21 cardinal 5 categorical, K-categorical 91 categorical, w 64 choice over b 185 choice scheme 155 class in 9.I 133 Compactness Theorem 10 complete formula 61 complete theory 22 complete theory in L, 61 completeness theorem for L, 18 completeness theorem for L,,, 16 comprehension scheme 155 consistent with T 61 consistency property 10 Craig Interpolation Theorem 19 Dedekind set 141

definable function of b 180 definition by Z-recursion 37 determinateness, axiom of 1 4 4 Downward Lowenheim-Skolem-Tarski Theorem 2 2 , 6 9 elementary chain 109 elementarily embeddable 63 Elementary Embedding Theorem 71 elementary extension I55 elementary submodel 63 end extension 102 end extension pair 166 enlarged 132 Erdijs notation 75 Erdijs-Rado Theorem 75 existential formula 28 Extended Model Existence Theorem 14 first category 151 fixed 132 formula 6 formula of set theory 34 fragment 17 Fundamental Theorem of Ultrapowers 181 Hanf number 82 hereditarily countable 36 hereditarily finite 36 homogeneous model 95 206

INDEX OF DEFINITIONS

incompletable formula 61 indiscernibles 70 indiscernibles over U 86 Interpolation Theorem for L, isomorphic 6

47

L,-complete ultrafilter 181 Lebesgue measure 144 logic, o- 3 Lo6-Vaught Test 22 Lyndon Interpolation Theorem 24,26 Malitz Interpolation Theorem 13 model, o- 138, 155 Model Existence Theorem 13 model of type ( K , A, p ) 160 model of type (K, A) 86 Morley Categoricity Theorem 91, 13 1 Morley Two Cardinal Theorem 86 negative occurrence 24 omit a type 62 Omitting Types Theorem 54 ordinal 5 ordinal standard part 48 Peano's axioms 155 positive formula 27 positive occurrence 24 preserved under homomorphisms 27 preserved under submodels 3 1 prime model 63 proof 16 quantifier-free 28 rank 115 realize a type 61 realize a type over X 119 recursive 57 recursively regular ordinal 36 reflection, C 35 regular cardinal 132 regular over y(%) I33 regular over a 185 relativization 36

replacement, Z 37 Robinson Consistency Theorem 21 16 rules o f inference for L,,, satisfaction 6 saturated model 92 saturated relative to ?1 76 Scott Isomorphism Theorem 7 second order number theory 155 sentence 6 separation, A, 35 separation, A 37 set of indiscernibles 70 set of indiscernibles over U 85 Seven Cardinal Theorem 191 Skolem expansion 67 Skolem hull 70 Skolem language 67 Skolem over y 180 Skolem theory 67 Skolem ultrapower 180 Stretching Theorem 71 strong end extension 1 I 1 subformula 10 submodel 6 theorem of L, 17 ,, 16 theorem of L transitive closure 18 transitive set 18 type in L, 61 type over X 1 1 8 universal formula 28 Upward Lowenheim-Skolem-Tarski Theorem 10 Upward Lowenheim-Skolem-Tarski Theorem for L, 78 valid formula 16 weak second-order logic 3 Zermelo set theory 58 Zermelo-Fraenkel set theory 102

207

Index of symbols

w-1ogic

S(X) k

- 9

?

-

Mu,,

A,-formula Z-formula n-formula A ,-separation Z-definition n-definition TCW A on .d Z . on J&' H U

HF HC (Py

A-separation C-recursion n L l

L,-complete ultrafilter

3 396 3 5 6 6 10 34 34 34 35 35 35 35 35 35 36 36 36 37 37 37 180 181

ord(8) OSP

(-4