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. JERO...

Author: H. J. Keisler

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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
3x, . . . xnVy(y

= X I v..

.v

y = x,).

(b) The class of all models isomorphic to the standard model of arithmetic. Add to Peano’s axioms the axiom

Vx(x = o v x = O ‘ V X = 0 ” v . . .). (c) The class of Archimedean ordered fields. (d) The class of Abelian torsion groups. (e) The class of all models (21 = ( A , 0 , - I , U) where ( A , 0 , - I ) is a group and U is a set of generators of 2l. Add to the group axioms the axiom tlX

v

n
3Uo...u,-I

A

...OUn-l).

~ ( U m ) V ~ ( U ~ l ) A X = u g O ( U I O

m
(f) The class of models = (R, F ) where F is a field and R is the ring of all polynomials in one variable with coefficients in F. The language L,,,, is much more powerful than L,,,. For example in L,,,, the following sentence characterizes the class of all well-orderings (21 = ( A , <): The order axioms plus the axiom 1( 3 x O x 1 ~ 2

*) A

n
xn+1

< Xn.

We shall see in Lecture. 10 that the concept of a well ordering cannot be expressed in L,,,.

11

INTRODUCTION

5

Very often it is useful to study well behaved sublanguages L, of L,,, instead of the whole language L,,,. One reason for this is that L,,, has uncountably many formulas. Later on we shall introduce the concept of afragment L, of L,,,, and the stronger notion of a logic Ld over an admissible set d . As this course proceeds we shall use a few important techniques again and again. The reader should watch for these techniques and use them as a way of comparing the model theory of L,,, with the model theory of first order logic. We single out seven basic methods and list them here. A. Constructing models from individual constants (the Henkin construction). B. Skolem functions. C. Indiscernible elements. D. Elementary chains. E. Ultraproducts. F. Adding new symbols to the language. G. Passing from a fragment L, t L,,, to a larger fragment La c L,,,. All of these methods except the last are available in the model theory of first order logic. The presence of the additional method G adds a whole new dimension to the subject, and makes the model theory for L,,, much more than a simple generalization of classical model theory. The most useful result in classical model theory is the Compactness Theorem. The result for L,,, which plays a role most like the Compactness Theorem in classical model theory is the Model Existence Theorem, which appears to be the fundamental result in the subject. Although the Compactness Theorem in its original form is not true for L,,,, it can be generalized in a less straightforward way to the Barwise Compactness Theorem. In a similar manner, many of the other basic results of classical model theory, such as the Lowenheim-Skolem Theorem, generalize in rather unexpected ways to theorems for L,,,. We shall use ordinary set-theoretic notation, and always assume the axiom of choice. Ordinals are defined so that each ordinal a is equal to the set of all smaller ordinals, and cardinals are identified with initial ordinals. The set of all subsets of X i s denoted by S ( X ) . We shall assume throughout these lectures that L is a first order logic

6

11

INTRODUCTION

with countably many relation, function, and constant symbols, the logical symbols and w1 variables

-7,

1'0, u 1 , .

A , v,v,3,

=,

. . u,, . . ., c1 < 0 1 .

The language L,,, has the same symbols as L, but in L,,, the conjunction symbol A and disjunction symbol V may be applied to countable or finite sets of formulas as well as to pairs of formulas. The class of allformulas of L,,, is the least class Xsuch that: (i) Each atomic formula of L is in A'. (ii) If cp E X and u < o1 then (7 cp) E X , Vu,cp E X , and 3u,q E A'. (iii) If @ is a finite or countable, non-empty, subset of X , then A @ E X and V @ E X . All this is done within set theory. For definiteness, we agree that each variable u, is the ordered pair ( u , sc), and each of the other symbols of L is a natural number. Atomic formulas are finite sequences of symbols. 1cp is the ordered pair (1 cp), , A @ is ( A , a), and Vv,q is (V, urn,cp), while V and 3 are treated in the same way. Thus each formulaof L,,, is a set. Informally we shall use parentheses for readability instead of angular brackets and commas. Free variables, models % = ( A , . . .) for L, and satisfaction of formulas of L,,, in %, are defined as usual. We use cp(x, . . . x,) to denote a formula whose free variables are included in the set {xl, ., x,}. A sentence is a formula with no free variables. We shall usually be concerned only with formulas having finitely many free variables. The satisfaction notation

..

..

8 k q[a1 . . . a,]

. ..

means that ( a , , ., a,) satisfies q ( x , x,) in %. The notation % '13 means that 8 and 23 are isomorphic, and % c '13 means % is a submodel of B. We let % = %(L), % = %(Lola), mean that % and 8 satisfy exactly the same formulas in L, or L,,, , respectively. We introduce the abbreviations +, H, in the usual way, for implication and equivalence.

2 Scott's Isomorphism Theorem

THEOREM 1 (SCOTT [1965]). Let U be a countable model for L. Then there is a sentence cp of L,,, such that for all countable models 23 for L, % k cp if and only if 23 2 a.

PROOF. For all a , , . . ., a,, E A and p < wl, define the formula cp,: . . . a m . . . x,,) inductively as follows: 0 'Pat . . . an = A { 8 ( x , . . . x,,): i?l k 8[a, . . . a,,] and 8 is either atomic or the negation of an atomic formula). If p is a limit ordinal, (xl

whenever y < p < w , . Since M is countable, for each a, , . . . a,, E A there exists that for all p 2 a,

a

(Vx1

*

*

xn)d, 7

...a. * V a8t ...a n *

tl

< w , such

8

(2

SCOTT'S ISOMORPHISM THEOREM

It follows that there exists all fi 1 z,

CI

< o,such that for all a , , . . .,a, E A and

'21 ( V X ~* . . X n b : 1 ...a , * ( ~B a ... l a,

*

Let cp be the sentence

A

CP; A

(VX,

* * *

..U,EA

xn)(qzi ... a,

a+ 1 +1.04

... a,).

,
Then 9 ' 1 k cp. Suppose 23 is countable and 93 k cp. We show '21 r 23 by a back and forth argument. It suffices to show that for all a , , . . ., a, E A and b, , . . ., b, E B such that 23 k cp,", . . .a,[bl . . b,],

.

(Van+, E A ) ( g b n + 1 E B ) m 1~:1...a,+j[bl

**.bn+11,

(1)

and (Vbn+,

E B)(Ian+,

€ A ) BC

q:1

...an+l[bl

. bn+,]-

(2)

To show (1): Since 23 k q ! B k cp::.f.,,[b, . . . b,], whence 5 3 k (3x,+,)qz1 ...a,+,[ bl . . . b,], and thus (1) holds. Proof of (2): Again . . . b,], using the fact that B k q,":.f.,,[b, 8 ( V x n + l ) Va,+lEA~zl...a,+l[bl - . b n l whence for Some an+, E A , 23 k yi1...a , + l C b l * . . b n + l I . d The above proof is due to CHANG[1968]. Scott's theorem has the following corollary.

COROLLARY. If 9X and 93 are countable models for L and '21 = B(LmIm), then '3 2 23. CHANG[1969] has shown that the above holds even if L has uncountably many symbols. On the other hand, the exercises show that it fails if we allow 23 to be uncountable. PROBLEMS

-

1. Let '3, 23 be models for L. Suppose that for each n c o there is a relation between A" and B" such that (i) If ( a , . . . a , ) ( b , . . . b,) then ( a , . . a,) satisfies exactly the same atomic formulas in '21 as ( b , . . b,) satisfies in 23.

-

.

.

21

-

,

(ii) If ( a , . . . a,) ( b , . . . b,) then (Van+ . . * an+,) (61 * . . b n + 1 ) (iii) If ( a , . . . a,) ( b , . . . b,) then (Vb,,, (a1

E A)(%,+

,

E B)

N

N

(01

9

SCOTT'S ISOMORPHISM THEOREM

an+,)

N

(b1-

* *

E B)(3a,+l E

A)

bn+l).

Prove that (21 = %(Leo,,).

2. Apply problem 1 to show that 9l = %(L,,,) when: (i) a, '23 are densely ordered structures without endpoints. (ii) 9l = <S(X), S > 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
(%)(POA*

a, 23 are w-homogeneous and . . . x,), . . . of formulas of L,

-

* ~ c ~ + m () 3 X n )

A~ml-

m
for every sequence cpo(xl . . . x,),

'pl(xl

9lk(3xl*..x,)Acp, m<w

iff B k ( 3 x ,... x,)Acp,. mi,

NOTE:3 is w-homogeneous means that for all a,, . . . , a , + , , b , , . . ., b, E A , if (%, a, . . . a,) =- (a, b, . . .b,)(L) then (%,+ 1 E B)(%, a1 . . . a,,,) = (a, b , . . .b,+,)(L). 3. Let the language L have possibly uncountably many symbols. Suppose that %, % are countable models for L and % = %(L,,,). Then

9 z 23.

4. Let L have possibly uncountably many symbols, let

9l be a countable

model for L, and let P c A . Then the following are equivalent: (i) For any Q c A , if (a,P) z (a, Q) then P = Q. (ii) P is definable in L,,,, that is, there is a formula q(x) of L,,, that

(a, P)

vxrqx)

-

cp(X)l.

such

3 Model Existence Theorem

Two basic results for L are the Compactness Theorem (If C is a set of sentences of L and every finite subset of C has a model, then C has a model), and the Upward Lowenheim-Skolem-Tarski Theorem (If a set I: of sentences of L has an infinite model of power a, then it has models of all powers fl > a). The example below shows that both of these results fail for L,,,.

EXAMPLE. Let c o , cl, sentences (VX)

c2,.

. ., c,

be constants for L. Let C be the set of

v (x = cn), c,

# co, c, # c1,.

n
. ..

Then every finite subset of C has a model but C has no model. Moreover, the first sentence in the above list has a model of power o but no uncountable model. The failure of the Compactness Theorem makes model theory for L,,, radically different from model theory for L. Our next theorem can frequently be used in L,,, in situations where the Compactness Theorem was used in L. It gives the essential content of Henkin’s construction which Henkin first used to prove the Godel Completeness Theorem in 1949. The result in the present form is due to Makkai, and it evolved from work of Henkin, Smullyan, and others. We first need some definitions. The set of subformulas of a formula cp of L,,,, denoted by sub(cp), is defined inductively as follows: If cp is an atomic formula, sub(cp) = {cp}. 10

31

MODEL EXlSTENCE THEOREM

I1

s u b ( 7 cp) = sub(cp) u { 7cp}. sub(Vxcp) = sub(cp) u (Vxcp}, sub(A@)= sub(cp) u {A@}, and similarly for 3 and V. It is clear from the definition that for every formula cp of L,,,, sub(cp) is finite or countable. Moreover, if cp has only finitely many free variables, so does every $ E sub(cp). We now introduce a fundamentally important notion, that of a consistency property. Let C be a countable set of new constant symbols, and let M be the language formed by adding each c E C to L. We may then form the infinitary logic M,,, corresponding to M. A consistency property is a set S of countable sets of sentences of M,,, which has certain properties to be listed below. To motivate the definition we mention that the simplest example of a consistency property is the set of all countable sets s of sentences of M,,, such that s has a model with universe A = {uc:c E C } . As we shall see, there are also many other examples.

urp4

Let us first introduce some convenient notation. Given a formula v, of M,,,, the formula cp l, obtained by 'moving a negation inside', is defined inductively as follows: If cp is atomic, cp is q. ( 7 cp) 1is cp. (Arpe@cp) 1is Vrp€@ 7cp. (V,€@cp) 1is Acps@ 140. (Vxcp) is 3x cp. (3xcp) 1is Vx cp. Thus we see that cp is always logically equivalent to 1cp.

DEFINITION. Let S be a set of countable sets of sentences of M,,,. s is said to be a consistency property iff for each s E S, all of the following hold. (Cl) (Consistency rule) Either cp 4 s or ( q ) 4 s. (C2) (1-rule) If (-, cp) E s then s u {cp l} E S. (C3) (A-rule) If (A@) E S then for all cp E @, s u (cp) E S. (C4) (V-rule) If ( V x q ( x ) ) E S, then for all c E C , s u {cp(c)} E S.

12

[3

MODEL EXISTENCE THEOREM

(C5) (V-rule) If (V@) E s, then for some cp E @, s u {cp} E S. (C6) (%rule) If (3xcp(x)) E s, then for some c E C, s u {cp(c)}

E S.

By a basic term we mean either a constant symbol or a term of the form F(c,, . ., c,) where cl, . . ., c, E C and F is a function symbol of L.

.

(C7) (Equality rules). Let t be a basic term and c, d~ C. If(c = d)Es,thensu { d = c}ES. If c = t , cp(t)E s, then s u (cp(c)> E S. For some e E C, s u {e = t } E S. Before proceeding further, some remarks and examples are in order.

REMARK 1. If S is a consistency property, so are each of the following: The set of all subsets of elements of S; The set of all s E S in which only finitely many constants c E C occur; The set of all finite subsets of elements of S. REMARK 2. Suppose S is a consistency property and s c t E S implies sE If S E S and cp, cp -+ $ ES, then s u {$} E S. I f s E S a n d C E C, thens u {c = c } E S. If ~ E Sc,, d, e E C , and ( c = d), ( d = e)Es, then s u { c = e } E S .

s.

Each of the sets below is a consistency property, where s ranges over the set of all countable sets of sentences in M,,,.

EXAMPLE 1. The set of all s such that s has a model %withA = {uc: c E C}. EXAMPLE 2. The set of all s such that s has a model and only finitely many c E C occur in s. EXAMPLE 3. Given any non-empty class K of models, the set of all s such that s has a model in Kand only finitely many c E C occur ins. EXAMPLE 4. The set of all s such that s has models of arbitrarily large cardinality and only finitely many c E C occur in s. The next theorem is the main result which gives the notion of a consistency property importance. It is a very general criterion for the existence of a model. The theorem is due to MAKKAI[1969] and is closely related to earlier work of HENKIN[I9631 and SMULLYAN [1963], [1965].

31

13

MODEL EXISTENCE THEOREM

THEOREM 2. (MODELEXISTENCE THEOREM). If S is a consistency property and so E S, then so has a model.

PROOF. We may assume without loss of generality that every subset of a member of S belongs to S. Let Y be the least set of sentences such that so c

y;

Y is closed under subformulas; If t is a term, c E C, and cp(t) E Y , then p(c) E Y; If (7cp) E Y then (cp 7) E Y ; Jfc,dECthenc=dEY. The set Y is clearly countable. Let X be the set of all sentences in Y, and put X = {cpo, q l , cpz, . . .}. Let T = { t o , ri, t 2 , . . .} be the set of all basic terms of L. We shall construct a sequence so c s1 c s2 c . . . of elements of S as follows. so E S is already given. Suppose we have s,, E S. Then using the definition of consistency property, we may choose s,,+ such that: (1)

sn

c sn+i ES.

(2) If s,, u {cp,,}

E

S then cp,,

(3) If s,, u {cp,,}

E

S a n d cp,, =

(4) If s,, u {cp,}

E

S and cpn

( 5 ) For some c E C, (c

=

E s,,+~.

V@, then for some 8 E @,

= 3xcp(x), then

8 E s,+

1.

for some c E C, cp(c)

E s,,+ 1.

t,,)E s , , + ~ .

We now define a model c1I of so. Let s, = Un<,s,,. For c, d E C, let c

-

d iff (c = ~ ) E s , .

It follows from (C7) that is an equivalence relation on C . For example, suppose c, d, e E C and (c = d ) ES,, (d = e) ES,. Then (c = e ) E X so for some rn, c = e is (P., Tak en 2 m so that (c = d), (d = e)Es,. Then by(C7), s,, u {c = e } ES,whences, u {c = e} E S a n d (c = e)E s,+ I . Let 3 have the universe set A = {c/- : c E C } . By (C7), if ~ ( c , . . c,,) E s, and c1 d , , . . ., c,, d,,, then q(d1 . . . d,,) E s,. Therefore we may interpret the relation, constant, and function symbols of M in in such a way that:

-

.

-

(6) If t is a basic term and c E C, then c1I k c

=

t iff (c = t ) E s,;

14

MODEL EXISTENCE THEOREM

(7) If R is an n-placed relation symbol and cl, k R(c,

. . . c,)

iff R(c, .

[3

. . ., c, E C, then

. . c,) E s,.

The conditions (6) and (7) determine the model a. By an induction on the complexity of terms it can then be shown using (Cl) and (C7) that any atomic sentence or negated atomic sentence in s, holds in a. Finally, using rules (C2)-(C6) and induction on the complexity of sentences, we see that every sentence in s, holds in a. Therefore 9l is a model of s, and hence a model of so .4 REMARK. The model 2l in the above proof is at most countable. In fact, it has the additional property that every U E Ais the interpretation of some c E C, namely each c / - is the interpretation of c. Note that in the proof the rules (Cl) and (C7) were only needed for the case where cp is an atomic or negated atomic sentence. Thus the Model Existence Theorem still holds for a slightly weaker notion of Consistency Property. The following corollary is a useful generalization of the Model Existence Theorem. COROLLARY (EXTENDED MODELEXISTENCE THEOREM). Let S be a consistency property and let r be a countable set of sentences in the language M,,,. Assume that for all s E S and cp E r, s u (9)E S. Then for all s E S, s u r has a model. PROOF.Define

S' =

{SU

r:sEs}.

It is easy to check that S' is a consistency property. Let s E S. Then u r E S', whence by the Model Existence Theorem, s v r has a mode1.i

s

4 Completeness Theorem

For the next few lectures we shall let C be a countable set of constant symbols which are not in the language L, and let M be the language formed by adding each c E C to L. M,,, is the infinitary logic corresponding to M. We shall now take up the task of choosing a list of axioms and rules of inference for La,, and proving a completeness theorem for this language. In this theorem and in many other proofs, we shall use the language M,,, as a tool for obtaining results about our given language L,,, . In what follows, x, y are arbitrary variables, cp, $ are arbitrary formulas of L,,,, and @ is an arbitrary countable set of formulas of L,,,. AXIOMS FOR L,,,: 1. Every instance of a tautology of finitary propositional logic is an axiom.

2.

-

cp)

( 1

3. A @

(cp

1 ) .

cp, where cp E @.

. . ) -+

.

q ( r . .), where cp(x. . .) is a formula, t is a term which is free for x in q ( x . . .), and q ( t . . .) is obtained by replacing each free occurrence of x by t .

4. Vxcp(x.

5.x=x 15

16

7. cp(x. . . ) A t above.

=x

--f

cp(r

COMPLETENESS THEOREM

[4

. . .), where cp(x . . .) and q ( r . . .) are as in

3

RULESOF INFERENCE FOR LaL,:

JI, $ cp, infer cp. 2. From JI cp(x.. .), infer JI -+ Vxcp(x. . .), where x does not occur free in JI. 3. From JI -+ cp, for all cp E @, infer $ + A@. 1. From

4

4

The set of theorems of L,,, is the least set of formulas of L,,, which contains all the axioms and is closed under the rules of inference. kL,, cp means that cp is a theorem of L,,,. Obviously, a necessary and sufficient condition for tLolocp is that there exist a countable sequence cpo, cpl, . . . q a , . ., cppp such that qs = cp and for each CI =< p, cprt is either an axiom of L,,, or is inferred from earlier formulas cpy, y < CI, by a rule of inference. Such a sequence is called a proof of cp. A formula is said to be valid, k cp, iff cp is satisfied in every model for L by every interpretation of the free variables of cp.

.

THEOREM 3. (COMPLETENESS THEOREM FOR La,,). (KAw [1964]). If cp is a sentence of L,,,, then tLmlwcp if and only if k cp. PROOF. Any theorem of L,,, is valid, because the axioms are valid and the rules of inference preserve validity. For any sentence cp of L,,,, we have t-Lolwcp if and only if kMwlwq. Because any proof in L,,, is a proof in M,,,. And given any proof in M,,, , only countably many variables occur in the proof, so we can obtain a proof in L,,, by replacing in a uniform way each occurrence of each c E C by a variable which does not occur in the given proof. Let S be the set of all finite sets s of sentences of M,,, such that only 1A s. finitely many c E C occur in s, and not We claim that S is a consistency property. To illustrate, we check that S satisfies (C5), (C6), and (C7) in the definition of consistency property.

(C5) Let (V@) E s E S but for all cp E @, s u {cp] 4 S. Each s u {cp] has only finitely many c E C, since (V@) E s E S. Therefore k ~ , ~ ,I

41

COMPLETENESS THEOREM

17

A (s u

{cp}) for all cp E @. Since s is finite, we have by RULE1 and AXIOM 1 that FMwlw 1(A S A cp), whence tMol,A s 3 1cp. Then by RULE3, I-Ma,, A s + i cp, SO by AXIOM2, tM,,, A s + 7 V@. Then 1 and RULE1, tMolo1A s. Contradiction. by AXIOM

(C6) Let (3xcp(x)) E s E S but for all c E C , s u {cp(c)} $ S. Take a C E C which does not occur in s. Then tMwlw 1r \ ( s u {cp(c)}), whence tMwlw 1( A s A cp(c)), tMoloA s + 7cp(c). Let y be a variable not occurring in s. Then replacing c by y in the proof we see that tMol,A s + Icp(y). Using AXIOMS 1, 2 , 4 and RULE2, tMwlo A s + 13xrp(x), whence kMwlo 1As. Contradiction. (C7) Let t be a basic term and s E S. Let d E C not occur in s or t, and suppose s u { d = Z} # S. Then tMolo1(s u { d = t } ) , and t M o , , A s + Vy(y # t ) . Then kM,,, A s + t # t , by AXIOMS 4, 1, and RULE1. But by AXIOMS 5 and 7, tM,,,t = 1, and it follows that I- 1A s; contradiction. The other parts of the proof are left to the student.i We now introduce the notion of a fragment of L,,, and generalize the completeness theorem to fragments. One of the difficulties in the logic L,,, is that it has uncountably many formulas. For many purposes it is better to consider ‘nice’ countable subsets of the set of formulas of L,,, . The notion of a fragment is a first, crude approximation to a ‘nice’ set of formulas.

DEFINITION. Let L,,, denote the set of all formulas of L,,,. For any set d , let L, = L,,, n d .We shall call cp a theorem of L,, t,,cp, iff cp has a proof in L,,, such that every formula of the proof belongs to L, . We shall call L, a fragment of L,,, iff (1) sd is a non-empty transitive set. (2) If a, b E d then {a, b } E d ,a u b E at’, a x b E d .

( 3 ) If a E at’ and u is the least ordinal which is not in the transitive closure of a, then u E d . (4) If cp(x . . .) E Ld and t E d is a term of L,,,, then cp(t . . .)EL,.

18

I

COMPLETENESS THEOREM

NOTE:A set d is transitive iff a E b E d implies a E d.The transitiu closure of a is the least transitive set b which includes a, b 3 a. The student should check the following facts about fragments a L,,,. * For any fragment L,, 1. w c d . 2. L, is closed under 1, and under finite A and V. 3. If cp E L, and x E d,then Vxcp E L,, 3xcp E L,. 4. If cp E L,, every subformula of cp is in L, 5. If O(xl . . . x,) is an atomic formula of L and xl, . ., x, E d , the] q x , . . XJ E L ~ . 6 . If Q, is a countable set of formulas and @ ~ then d , A@ E L, VQEL,. Moreover, {-,rp:cp~Q}~d, { 3 x q : c p E Q , } E d f o rx e d Hence if cp E L,, (cp 7) E L, , 7. If cp E L,, there exists a variable x E d which does not occur ii etc. rp. (By (3)). 8. For every countable set Q, c L,,, , there is a least countable fragmen L, such that 0 c Ld.

.

.

.

THEOREM 3&. (COMPLETENESS THEOREM FOR Ld). If cp is a sentence of a countable fragment L, of L,,,, and only if C cp.

then kL,q i

..

PROOF.Let M, be the set of all formulas cp(x, . . . x,cl . c , ~ ) o M,,, obtained from formulas cp(xl . . . x,yl . . . y,) E L, by replacin: each free occurrence of yi by ciE C, 1 i 5 m. Note that i cp(xl . . X,C, . . . c,) E M, and z1 . . z , are variables in d not occurrin: in q(x1 . . . x,cl . . . c,), then ~ ( x ., . . x,zl . . z,) E L,. Using this w see that if 9 is a sentence of Ld and kMdcp, then tL,cp. Let S be th, set of all finite sets s of sentences in M, such that not kMd 1A S. The] as before, one can check that S is a consistency property, and it follow that C rp implies kL,cp. The converse is straightforward, i.e. kL,q im plies i. rp .-I

.

.

.

5 Craig Interpolation Theorem

THEOREM 4 (CRAIG INTERPOLATION THEOREM FOR Lm1,). Let cp, I)be sentences of L,,, such that != cp + I). Then there is a sentence 8 of L,,, such that t cp + 8, t 8 + I),and every relation, function, or constant symbol of L which occurs in 8 occurs both in cp and in Ic/. (The above result is due to LOPEZ-ESCOBAR [1965], who gave a prooftheoretic argument using cut elimination. The result generalizes a theorem of Craig for first order logic. The proof below is due to Makkai, and is an adaptation to L,,, of a proof of Craig’s Theorem for L by Henkin.) PROOF.Let X , be the set of all sentences cp‘ of M,,, such that every relation, constant, or function symbol of L which occurs in cp’ occurs in cp, and also only finitely many c E C occur in cp’. Define X, similarly. Let S be the set of all finite sets s of sentences of M,,, which can be expressed as a union s = s1 u s2 such that: (1)

81

= X,,

s2

= X,.

(2) If 8,, O2 E Xp n X , and k A s1 -+ consistent.

e l , c A s2

-+

02, then 8, A O2 i s

We claim that S is a consistency property. Let us show, for example, that (C5) holds for S. Let s = s1 u s2 E S and V Z E s. Assume first that V Z E S ~Then . V Z E X, and C c X,. We shall show that for some a E Z, s’ = s u {a} E S with s; = s, u {a}, si = s2. Suppose not. Then for all a E Z there exist el,, OZg E X, n X, such that 19

20

and

CRAIG INTERPOLATION THEOREM

el, A 8,,

k

AS, A C

-+

el,,

k AS, -, e,,,

is inconsistent. But then

c A S,

+

v el,,

1 S,

4

aEz

A

e,,.

ad

Thus if we put = VasXOla,8, = /\,EZ8Z,, we see that (1) and (2 are contradicted. This shows that S has the property (C5). The case that V Z E s, is similar. The other parts of the proof that : is a consistency property are left to the reader. We remark that (C6) de pends on the fact that only finitely many c E C occur in each s E S. Fo (C7) we note that we may assume that each relation constant, or functioi symbol of L occurs in cp or $. Recalling that the terms in (C7) are basil terms, we see that all the equations c = t belong to either X , or X , . Since S i s a consistency property, by the Model Existence Theorem eacl s E S has a model. But != cp -, $, so the set {cp, 1$} has no model Therefore {cp, 1$} # S. Also, cp E X,, $ E X , This means that we cai find 8,, 8, EX, n X, such that

*

k cp -, 81, I=1

and

.

-+

02,

8, A 8, is inconsistent. It follows that k 8, .+ -, 8,, k- -, 8,

4

$, whence k

-+

$.

Let us write dl = Bl(cl . . . c,) where cl, . . ., c, are all the c E C occur ring in el. Choose new variables y l , . . .y,,, and let 8 = Vyl. . .ynOl(y . .y,). Then 0 is a sentence of L,,, which satisfies the desired conclusion

.

..

COROLLARY. Let cp(x, . x,), $(xl . . . x,) be formulas of L,,,, sucl that k cp -+ $. Then there is a formula 8 ( x , . . . x,) of L,,, such tha k cp + 8, k 8 4 $, and every relation, function, or constant symbol whicl. occurs in 8 occurs in both cp and $.

PROOF.Replace x l , . . ., x, by new constants e l , . . ., en and apply tht interpolation theorem.4 REMARK: In the Craig Interpolation Theorem we cannot always finc 8 SO that:

51

21

CRAIG INTERPOLATION THEOREM

If the equality symbol For example, consider

occurs in 8, then it occurs in both cp and $.

=

b (Vx, Y ) ( X = Y )

+

(Vx, Y ) ( W

-

NY)).

THEOREM 5 (BETH'STHEOREM FOR L,,,). (LOPEZ-ESCOBAR [1965]). Let P and Q be two new relation symbols with n places. Let q (P ) be a sentence of the language (L u {P}),,,, and let q ( Q ) be formed by replacing each occurrence of P by Q. Suppose that (i)

CP(P)A d Q ) (P(x1 * +

*

-

xn)

-

Q(xl** * xn)),

i.e. cp(P) defines P implicitly. Then there is a formula O(xl L,,, such that (ii) k cp(P) + (P(x,

. . . x,,)

i.e. q(P) defines Pexplicitly.

-

8(x1 .

. . .x,,)

of

. . x,,)),

PROOF.From (i) we have

I= ~P(P)AP(xI * * xn)

+

(q(Q) Q(x1 +

.

-

*

xn)).

By the above corollary there is a formula O(x, . .x,,) of L,,, (1) t cp(P>AP(X,. . x,,) -, q x l . . . x,,), XI . xn) (v(Q) Q(x1 * * * X n ) ) . (2) From the second line (2) it follows that, by replacing Q by P, (3) b e(x, . . . x,,) -+ ( c p ( ~-,) ~ ( x .,. . x,,)). Then by (1) and (3) we conclude that (ii) ho1ds.i

- -

.

+

such that

+

In first-order logic L it is easy to see, using the compactness theorem, that Craig's Interpolation Theorem is equivalent to A. Robinson's Consistency Theorem (which was proved at about the same time, perhaps a bit earlier):

THEOREM. Let L',L" be two expansions of the ROBINSON CONSISTENCY language L such that L' n L" = L (i.e. every symbol common to L' and L" belongs to L). Let T be a complete theory in L and let cp be a sentence of L' and $ a sentence of L". If T u {cp} is consistent and T u {$} is consistent, then T u {cp, $} is consistent.

22

CRAIG INTERPOLATION THEOREM

This theorem fails for the logic L,,,; its proof does not go throu; because the compactness theorem is no longer available. The followi set of exercises explains the situation in more detail. A set T of sentenc of L,,, is said to be a complete theory iff T has a model and for any t\ models %,23 of T we have % = 23(L,,,). PROBLEMS 1. (DOWNWARD LOWENHEIM-SKOLEM THEOREM). If CJ is a countable s of sentences of L,,, and @ has a model, then @ has a finite or countal: model. HINT:Use the Model Existence Theorem. 2. If all countable models of a sentence cp of L,,, are isomorphic ai cp has a countable model, then {cp} is a complete theory. (LO$-VAUG. TEST). 3. (A WEAK ROBINSON CONSISTENCY THEOREM). Let L‘, L” be expansio of L with L‘ n L” = L. Let T be a countable complete theory of L,, Let.cp beasentence of L:,, and $ one of Ud,,. If T u {cp} and T u { each has a model, then T u (cp, $} has a model. HINT:Use Scott’s Theorem.

4. Show that problem 1 fails for some uncountable set CJ. 5. Show that problem 2 fails with ‘model of power o’in place of ‘cou able model’.

6. Show that problem 3 fails for uncountable T. 7. (THEMODELEXISTENCE THEOREM WITHOUT EQUALITY). Assume tl L has no function or constant symbols. Let S be a set of countable s of sentences of M,,, such that each of the rules (Cl)-(C6) of Consistc cy Properties except for the equality rule holds. Assume further tl the equality symbol does not occur in any s E S. Then every s E S ha: model %, and the set of elements of 3 is C.

8. (ALTERNATE DEFINITION OF CONSISTENCY PROPERTY). Let X be a to1 logical space. By a topological interpretation of the language M,,, mean a mapping f on the set of all sentences of M,,, into the set of

51

23

CRAIG INTERPOLATION THEOREM

a) Letf be a topological interpretation of M,,,. Let Scf) be the set of all finite or countable sets s of sentences of Ma,,such that

nfm z 0.

(P-

Prove that S ( f ) is a consistency property. b) Let S be a consistency property. Let X be the topological space where S is the set of all points of X,and a set U c X is open 'iff s E U and s c s1 E S implies s1 E U. Letfs be the function

f&)

= (s E

s:cp E s>.

Prove that fsis a topological interpretation of M,,,

.

c) If S is a consistency property, then S ( f s ) is the consistency property S ( j s ) = {sl: s1 c s j o r some s E S ] .

6 Lyndon Interpolation Theorem

It is easy to show by induction that in a formula cp of L,,,, each o( currence of a symbol is within the scope of a finite number of negatio signs. An occurrence of a symbol is said to be positive iff it is within th scope of an even number of negation signs. Otherwise an occurrence : called negative. Thus a symbol can occur positively, negatively, or both i a formula. THEOREM 6 (LYNDONINTERPOLATION THEOREM FOR Lo,,). (LOPEZ ESCOBAR [1965]). Suppose cp, $ are sentences of L,,, with k cp .+ $. Then there is a ser tence 8 of L,,, such that: (i) 1 cp .+ 8, 1 8 .+ $; (ii) Any relation symbol occurring positively (negatively) in 0 occui positively (negatively) in both cp and $. EXAMPLE.The example k 3x(c = X A R(x)) + 7 R(c) shows thz the theorem is false if we also require that every constant and functio symbol which occurs positively (negatively) in f3 occurs positively ("el atively) in cp and $. PROOF OF THEOREM 6 . For any sentence c of L,,,, let X , be the set c all sentences 8 of M,,, such that only finitely many c E C occur in 8, an each relation symbol which occurs positively (negatively) in 0 also OCCUI positively (negatively) in c. In the argument below we shall consider th sets x,,&,, xTq.

x,,

24

61

25

LYNDON INTERPOLATION THEOREM

Let S be the set of all finite sets s c X, u XIIL which can be written as a union s = s, u s2 such that:

(1) s1

= x,,s.2 = x1,;

( 2 ) I f e , ~ X , n X , , e2EX1, n X l s , a n d k A s , - , 8 , , k A ~ , - , f ? ~ , then 8, A 8, is consistent. We claim that S is a consistency property. The proof is similar to that of the Craig Interpolation Theorem. The only steps which require special explanation are the consistency rule (Cl) and the equality rule (C7). These are the steps where it is essential that all equations of M,,, belong to I , x,, , x,,. and xl$. To prove the consistency rule (Cl), let s = s1 u s2 E S and suppose (Cl) fails, both a and la belong to s. If a E s1 and E s2 , then (2) is contradicted by 8, = a, 8, = la. If, (-,u) E s,, then (2) fails when 8, is -,c = c, O2 is c = c. The other two cases are similar, and we have a contradiction. Therefore (Cl) holds. As an indication of the proof of (C7), assume that s = s1 u s, E S and let t be a basic term, c E C, and (c = t), a(t)E s. We shall show that s u { ~ ( c )E } S. Suppose first that (c = t ) , a(t) E sl. Then k As, -+ O ( C ) and a ( . ) E X,,whence it follows that s u {a(c)> 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

b l , ...,b,

Ea

that

and (a, E) C q [ b , . . . b,].

DEFINITION. A set X t d is said to be X on d iff there is a C-formula q(xy, . . .y.) and elements b , , . . ., b, E d such that

X = { a E d :( d ,E) C q[ab1

.. .b,,]}.

That is, X is defined by a 2-formula with parameters in d.We shall call cp(xbl .. .b,) a 2.-definition of X. The notion of a set X c d being I? on d , and of a lI-dejinition of X,are defined analogously. A set X c d i s said to be A on d i f f it is both2 on d and Ifon d .

36

ADMISSIBLE SETS

[8

NOTE: cp(xb, . . . b,) stands for the pair cp(xy, . . . y,), ( b , . . . b,). Before proceeding further, we give some examples of admissible sets.

EXAMPLE 1. The set d' = R(w) of all sets of finite rank is admissible. Indeed, (d, E) satisfies all the axioms of ZF except the axiom of infinity. The corresponding language L,, is just the classical first-order logic L. For any X c R(w), X E R(o) iff X is finite, X i s A on R(w) iff X is recursive, X is C on R(w) iff X is recursively enumerable. In general, one may think of the property of belonging to d as a generalization of the property of being finite, and A on d , C on d as generalizations of the properties of recursiveness and recursive enumerability. The set R(w) is the smallest admissible set. As we see in the exercises, any other admissible set contains w as an element. EXAMPLE 2. The next smallest admissible set is the set L(a) of all sets constructible before the first non-recursive ordinal u. EXAMPLE 3. If d is a transitive set and (d, E) is a model of all the axioms of ZF except possibly the power set axiom, then d is an admissible set. In particular, let u be any regular cardinal. Then the set L(u) of all sets constructible before a is admissible. The set ff, is defined as the set of all sets x such that TC(x) has power < a. For each regular cardinal a, H, is an admissible set. When a = w,the sets L(o)and H, are the same as the set R(w). The elements of R(w) are sometimes called hereditarily jinite, whence R(w) is denoted by HF. When u = wl,the elements of H,, are called hereditarily countable and H,, is denoted by HC. We remark that KRIPKE[1964] and PLATEK[1966] have introduced the notion of a recursively regular, or admissible ordinal. They showed that an ordinal a is recursively regular if and only if L(u) is an admissible set. Suppose cp(xl . . x,) is a formula of set theory and y is a variable which does not occur in cp. Then the relativization of the formula cp to y , denoted

.

81

ADMISSIBLE SETS

37

.

by cpY(xl, . ., x,), is defined to be the formula obtained from cp by replacing each quantifier (Vz), (32) in cp by the bounded quantifier (Vz E y), (3z E y). Thus for any formula rp of set theory, rpy is a A,-formula, which has all the free variables of rp and the additional free variabley. For relativized formulas we shall sometimes use the notation Notice that if a E d and b,, . . .,b,

E a,

then

For example, the Z-reflection axiom may be expressed in the following form: If 'p(yl . . y,) is a Z-formula, then

.

(d,~ k q () y , ...yn)+(h)[xistransitive

...A ~ , , E X A

A~,EXA

cp"(Y1..

.Y")l.

LEMMA 2. Let d be an admissible set. Then:

(i)Ifa,bEd,thenaxb,anb,avb,(a,b),a-b,and(a,b)Ed. I f f € d andfis a function, then domain(f) and range(f) E d . (ii) (A-SEPARATION PRINCIPLE). If a E d ,X c a, and Xis A on d ,then XE d . (iii) (2-REPLACEMENT PRINCIPLE). If G is a function, G c d x d , G is Z on d ,a E d , and a c domain(G), then Gla E d.

(iv) (DEFINITION BY Z-RECURSION). Let G(,) be a function such that for all a, b E d , G(a, b) E d,and G l d is Z on d .Suppose F is a function with domain d which is defined recursively by

F(x)

= G(x, FITC(x)),

all x

Ed.

Then F is A on d and F maps d into d . Parts (i), (ii), and (iv) are due to Platek, and part (iii) to Barwise.

PROOF.(i) Let a, b E d .By Z-reflection there is a c E d such that a, b E c . By A,-separation, {a, b} ~ dIt .follows that (a, b ) = {{a}, {a, b}} E d . We show that a x b E d . By the above remarks,

38

[8

ADMISSIBLE SETS

( d ,E) =i (Vx E a)(Vy E b)(3z)(z = (x, y)). This is a C-formula, so by Z-reflection there exists c E d such that a, b E c and
Then a x b c c. But a x b = (U E c : (3x E a)@ E b)(u = ( x , J J ) ] ) .

Thus by A,,-separation, a x b E d .The other parts of (i) are similar or easier. (ii) Let cp(xb, . . . b,,) and Il/(xb, . . . b,,) be a C and n-definition of X. For all x E X, 1 < dE,) ~ ( ~ * 6 bn),

--

whence by Z-reff ecfion, x E X implies

( d ,E) I= (3y)(x, b,

. . ., b, E y A y is transitive A cpY(xb, . . . b,)).

Similarly, x $ X implies ( d ,E) C ( l y ) ( x , b , ,

. . ., b, E y A y is transitive A

$Y(xb, . . . b,,)).

Then, since X c a,

( d ,E) I. (Vx E a)[(3y)(x, b , , . , ., b,, E y A y transitive v (3y)(x, b, ,. . .,b,

Ey A

y transitive

A

. . . b,,)) . . . b,))].

cp'(x, bl

A 1+'(xbl

Abbreviate the above as ( d ,E) k (Vx E a)(@v $). Since the formula (Vx E a)(? v $) is Z, there exists d E d such that a, b , ,. . .,b, E d, d is transitive, and (d, E) C (Vx E a)(pv 3).We claim that X E X

iff

< ~ , E ) ~ X E U A @ ' .

Because if ( d ,E) C x E a A pd, then since ? is I;, ( d ,E) k x E a h p, whence ( d ,E) =! cp(xb, . .b,) and x E X. Conversely, if not ( d ,E) C x E a h Fd, then either: x $ a, whence x $ X; or else (d, e ) 1@', in which case (A?,E) != tJd. But 3 is also Z,so ( d ,E) I=@, and hence ( d ,E) C i $(xb, . . b,), SO x $ X . The formula pdis A,. Hence by A,-separation we have X E d . (iii) We first remark that, by (i), every relation R c d x d is also a subset of d . Moreover, it is easy to show using ordered pairs that R

.

.

81

39

ADMISSIBLE SETS

..

.

is Z on d if and only if there is a Z-formula cp(xyzl .z,) and b, , . ., b, E d such that R = {{a, b): ( d ,E) k p[abb, . . bJ]. We may call cp(xyb, . b,) a Z-definition of R. Now let cp(xybl b,) be a Z-definitionof G. Then Gla has the Z-definition X E U A (P(Xyb1 b,),

.

..

...

. ..

so Gln is Z on d .We have, since a c domain(G),

*

bn).

By C-reflection there exists c E such that c is transitive, a, b,, b, E c, and ( c , 4 t= (Vx E 4(3Y)cp(VbI * * b").

. . .,

Therefore Gla c a x c. By (i), a x c E d.Finally, Gla has the l7-definition X E U A (VZ

E C)(p(XZbl

. . . b,) + Z = y).

Thus Gla is A on d .By A-separation, Gla E d. (iv) We first observe that the relation u E TC(x) has the Z-definition: x #

UA

(3u)[x E U A (Vy E u)(y # u + (3z)(z E y n u ) ] .

Hence the relation w = TC(x) has the Z-definition x c w A (w is transitive) A (Vu E w) u E TC(x).

Now let cp(xyzb, .. .b,) be a Z-definition of the binary function G l d . Let F' be the subset of d which has the Z-definition $(xzb, ...b,): (3w)(3f)[w = TC(x)A f i s a function A domain(f) = w A cp(xfzb, A (Vr E w ) ( ~ s ) ( s= TC(r)A q(r fls f(r )b, . . b,)].

.

...b,)

The terms domain(f),fls, f ( r ) can be eliminated by introducing appropriate bounded quantifiers. We show by induction that for all ordinals ct and all a E d n R(a), F'la = Flu and F'la E d. This assertion is trivial for o! = 0 and passes through limit ordinals automatically. Assuming it for ct, we prove it for a+ 1. Let a E R(a+ 1) n d, x E a. Let w = TC(x). Then w E d n R(ct). By hypothesis, P [ w = F'lw E d.Letf = Flw. Then F(x) = G ( x , f ) E d .It then is clear from an examination of the formula $ with this choice of w,fthat the pair (x, F ( x ) ) belongs to the relation

40

ADMISSIBLE SETS

[8

F'. Moreover, w, f are the only sets which satisfy the part of $ within the square brackets, so there is no pair <x, z ) E F' with z # F(x). Therefore F'la = Fla. The set F'la has the Z-definition II/ A X E a. Since F is a total function and F'la = F(a, we have a c dornain(F'1a). Then by the Zreplacement principle proved above, Flu E d.This completes the induction. Hence F maps d into d . We conclude that $ ( x z b , . . . b,) is a C-definition of F. On the other hand, is a Z-definition of d - F , so its negation is a l7-definition of F. Therefore Fis A on d.i

o if x is an atomic formula. Oifx= 7yandF(y)=0. 0 if x = Vv,y, F ( y ) = 0, y E ord n d. F(x) = 0 if x = A@ and F ( y ) = 0 for all y E @. I 1 otherwise. '

81

41

ADMISSIBLE SETS

The set Ld is exactly the set of allx such that F(x) on d.i

=

0, hence Ld is A

LEMMA 4.Let d be an admissible set, d c HC. Then Ld is a fragment of L,,, . Moreover, if L E at’, w E at’, a E d ,and L, is the least fragment of L,,, with a c 99, then L, E d. We leave the proof to the reader.

9 Barwise Compactness Theorem

The next theorem is a generalization of the classical fact that the set of all valid formulas of L is recursively enumerable. THEOREM 10. (BARWISE [1969]). Let sit’ c HC be an admissible set. Then the set of all valid sentences rp E LB is Z, on d .

PROOF.We recall that M is just the language L with the countable set C of extra constant symbols, and thus the preceding lemma holds for MB as well as LB. Let F,cp mean that rp E MB and there exist x E d, u E sit’ such that rp E P(a, x ) , where P is the function on d x d into d defined recursively as follows. Intuitively, rp E P(a, x ) means that cp has a proof in Md of ‘height’ gu such that each step of the proof belongs to x. More formally, P(a, x )

=

0 if a E d - o r d ,

and for a E ord n d, P(a, x ) is the set of all rp E x n MB such that either (1)

cp is an axiom of MI,

(38 E 4 ( 3 e E X ) [ e E P(B, 4 A (0 44 E P(B, .>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(
=x

and in view of Lemma 3,

( 6 ) The relation ‘y E G((u, x ) , f ) ’ is A on d . Then by the A-separation principle, we have (7) G ( ( 4 x>, 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-. ,

It follows that (-01,

E)

I=(ve E o

p E a)(3a E a ) y s~

e) E ~ ( ax,) * .

--f

Since a is transitive and P(a, x) is increasing in x , ( d ,E) b

(ve E o ) ( j X

Ea

y ( s~-+

Let j3 be the least ordinal not in a, i.e. fi tion, fi E d , and hence

( d ,E) k

(ve E O ) ( j a E p)‘(A

=

s -+

e) E P ( X ,

a)’.

ord n a. Then by A , separa-

0) E P(a, a)’.

BY (4),

(d, E) k ‘(A s

-+

A

eEe

e) E ~ ( pa)’,

whence td(A s -, ABEe 0). But V 0 ES, and it follows that I-, A s, contradicting s E S. This proves (C5). i

1

In the above argument, the assumption that d c HC was needed so that L , would be included in Lala)and the Model Existence Theorem

91

45

BARWISE COMPACTNESS THEOREM

would be available. If d is not a subset of HC, then some elements of d are uncountable, and L, will contain uncountable conjunctions. In these lectures we are only interested in the case d c HC. In the next theorem we must assume not only that d c HC but also that d itself is countable, i.e. d E HC. This is the principal theorem in logic on admissible sets. THEOREM 11. (BARWISECOMPACTNESS THEOREM). Let d c HC be a countable admissible set. Suppose X c L, is a set of sentences, X is Z on d ,and every x c Xsuch that x E d has a model. Then Xhas a model.

PROOF. Let S be the set of all s c M, such that s is a set of sentences of M,, s is 1 on d ,every x c s such that x E d has a model, and only finitely many c E C occur in s. We claim that S is a consistency property. Again the only difficult case to verify is: (C5) If V O E S E S then for some 8 E 0, s u { 8 } E S. Suppose V O E s E S but s u { O } q2 S for all 8 E 0.Then for all 8 E O there is an x E d such that x c s and x u { O } has no model. Therefore

(d, E) c (ve E 0 ) ( 3 x ) [ x

c

s

A

( 3 a ) ( 3 y ) ' ( ~x

-+

e) E P ( N ,y)'].

Here x c s is written in terms of a Z-definition of s, and P(a, y ) is the function defined in the proof of Theorem 10. The formula on the right is thus a Z-formula. Thus there is a transitive a E d such that all the constants occurring in the above formula belong to a, and the above formula holds in (a, E). It follows that

(d,~ ~ ( ~)~ E @ ) ( ~ x E u c ) [ S( )x' A ( ~ X ) ( ~ ~ ) ' ( A X - + l e ) ~ ~ ( a , y ) ' ] . ( l ) Let

b=

{UEU:(U,E)~UES}.

Then b c c and b is defined by the A,,-formula U E a A (U E S>"

in ( d ,E). Hence by A,-separation, b E d.Also, since s is 1 on d,we have b c s. Finally, if ( d ,E ) k x E U A ( x c sy, then x c b. It follows from (1) that for all 8 E 0 there is an x c b with t, A x + 78, and thus 1 A x + 8, and k A b + 1 8 . Then C A b + ABEe1 8 , and hence b u { V 0 } has no model. But b u {V 0 }E S(s) n d.This con-

46

BARWISE COMPACTNESS THEOREM

[9

tradiction completes the proof that S satisfies part (C5) of the definition of consistency property. The set X of sentences of Ld mentioned in the hypothesis of the theorem belongs to S. Since S is a consistency property, we may apply the Model Existence Theorem, and X has a model. i

REMARKS. The proofs of Theorems 10 and 11 originally given by Barwise were proof-theoretic, using cut free derivations in L,. Later, Karp gave proofs of Theorems 10 and 11 using Boolean-valued models. The present proofs are more model-theoretic in character than the earlier proofs. Our proofs depend primarily on the Model Existence Theorem rather than on a notion of derivation in L,. The above proof of the Barwise Compactness Theorem is given in MAKKAI [1969b]. Our notation F,cp can be interpreted as saying there is a proof tree for rp in d ,but we use it only as a &definition of I=cp. Using the axiom of choice, one can show directly that P,cp implies kLdcp, that is, if rp has a proof tree in d then it has a well ordered proof using formulas of Ld. The language Ld and the Barwise Compactness Theorem grew out of ideas of KREISEL [1965]. It contains a compactness theorem for o-logic which follows from the Barwise theorem. The Barwise Compactness Theorem has a number of applications to recursion theory and set theory. See, for example, FRIEDMAN and JENSEN [1967] and BARWISE [1970]. We shall givejust one simple illustration here. COROLLARY. Let d be a countable admissible set and let T c L, be a set of sentences such that Tis Z, on d and every cp E Tis true in ( d ,E). Then T has a model (B, F ) such that (4 ( d ,4 = ( B , F), (ii) ( d ,E) # ( B , F ) . (iii) If a E d, b E B, and bFa, then b E d. PROOF.Add to L a constant c, for each a E A? and one additional constant c. Let T' be the set of sentences Tplus the following: c # c,, each a ~ d . vx(x E c,

-,V x = cb), each a E d . bsa

C,

E cb,

each a E b E d .

91

BARWISE COMPA(;TNESS THEOREM

47

Then T' is Z on d. Every Th c T' such that Ti E d has a model; because the set X of all a E d for which c, occurs in TA belongs to d ,and therefore we can obtain a model for Th by interpreting c by any element b E d - X . Then by the Barwise Compactness Theorem, T has a model (B, F). We may interpret each constant c, by a. Then (B, F ) has the properties (i)-(iii). -i PROBLEMS

0. Verify the assertions in Example 3 and Lemma 4, Lecture 8. 1. Show that there exist arbitrarily large cardinals a such that R(a) is an admissible set. HINT: Show that there is a single sentence cp of set theory such that ZF F cp and for all R(a), if (R(a), E) I= cp then (R(a),E) t the A -separation and &reflection axioms, 2. Let d be a countable admissible set, d c HC. Suppose cp, $ are sentences of L, and I. cp + $. Prove the interpolation theorem for L,: There exists a sentence 8 E L, such that t cp + 8, b 8 -,I), and every relation, function, or constant symbol occurring in 8 occurs in both cp and $. 3. Let be any countable set a c HC. Prove that there exists a countable admissible set d with 3?c d c HC. NOTE:Obviously if is countable and admssible, then d E HC. If d is admissible and each a E d is countable, d c HC. 4. Let R c HC. Generalize all the notions and results of Lectures 8 and

9 by replacing formulas of set theory by formulas in set theory with an extra relation symbol P(x) and the model ( d ,E) by ( d ,E, d n R). Show that the resulting improvement of the Barwise Compactness Theorem implies the classical compactness theorem for L. 5. Show that if a2 is an admissible set, then either d = R(o) or o E d.

6. Let d be countable and admissible and let T be a set of sentences of

L , which is X on d . If cp is a sentence in LB and T P cp, then there is some X c T such that X E a2 and X k cp. Moreover, the set of all sentencescpofL,suchthatTCcpisZond.

48

BARWISE COMPACTNESS THEOREM

19

7. Let &’ t HC be an admissible set and define a function Inc(a, x ) on d x d where s ~ I n c ( ax, ) has the intuitive meaning that S E X , A s E L,, and it can be seen in at most tl steps that no consistency property contains s. Give an alternative proof of Theorem 10 which uses Znc(a, x ) in place of P(a, x ) .

BARWISE).If M = ( M , E ) is a model of ZF, let Ord(M) 8. (FRIEDMAN, be the set of all ordinals of ( M , E ) . Let the ordinal standard part of M , osp(M), be the greatest ordinal a such that (ord(M), E ) has an initial segment of order type c1. (i) Prove that for every model M of ZF, osp(M) is an admissible ordinal. (ii) Suppose p < c1 < w I , a is the first admissible ordinal greater than p, and there is a model M of ZF such that c1 osp(M). Then a is the ordinal standard part of some model N of ZF. (Use the Barwise Compactness Theorem to get N . )

10 Undefinability of well order

In this lecture we shall give an example of combining the following two techniques: (a) The Model Existence Theorem. (b) Adding new relation, function, or constant symbols to L. THEOREM 12 (i) (MORLEY[1965a1, LOPEZ-ESCOBAR [19661). Let T be a countable set of sentences of L,,,, and let U, < be a unary and binary relation symbol of L. Suppose that for all u < w l , T has a model 8 = ( A , U, <, . . .) such that < linearly orders U and ( a , <) c ( U , <). Then T has a model % = ( B , V, <, . . .) such that < linearly orders B and ( V , <) contains a copy of the rational numbers. (ii) (MORLEY[1967], BARWISE [1969]). Let &’ c HC be a countable admissible set and let T be a set of sentences of L, which is C on d .Suppose that for all CL E Ord n &‘, T has a model < A , U, <, . . .) such that < linearly orders U and ( u , <) c ( U , <). Then T has a model % = ( B , V, < , . . .) such that < linearly orders B and ( V , < ) contains a copy of the rationals. PROOF.The proof we give differs from the original proofs. We have simplified the argument by using the Model Existence Theorem. It is obvious that (ii) implies (i), but we shall first give the proof of (i) because it is easier. (i) We add to L a set D = {d,.:r rational) of constants. Call the new language L’. Form M‘ be adding a countable set C of new constants to L’, and let M; be a countable fragment of ML,, such that T c M i . 49

50

11'

IJNDEFINABILITY OF WELL ORDER

.>

Let K be the class of all models(A, U, <, . . of L such that the se Ord n U = U'is a countable ordinal and < is the usual ordering on U' Let S be the set of all sets s = so u T u {d, < d,: q < r as rationals: such that so is a finite set of sentences of M;, only finitely many c E C d E D occur in so, say so = so(cl

. . . c,d,, . . . d,,), rl < . . . < r,,

and for all a < w1 there is a model % E K and elements b, , . . ., b, E U such that % I=T,

,(%, b ,

. . . b,)

k (3x,

. . . x,)

A so(xl

. .. x,d,, . . . d,,,),

anda 5 bl,b,+cr 5 b,, . . ., b,-,+a 5 b,. We claim that S is a consistency property. The only parts of the verifica, tion which are not obvious are (C5) and (C7). (C5). Let VO E s E S. Then for each a < w1 , there is a 9, E O suck that

(a, b , . . . b,)

k (3x,

. . . x,) A so A 9,(xl . . .x,

d,,

. . . d,,,),

in the definition of S. Since O is countable, there is a 9 E O such tha 9 = 9, for arbitrarily large a < 0,. It follows that s u {O,} E S. (C7). The interesting case is the following: If d, E D , then for all s E 2 there is c E C such that s u {d, = c} E S. Let c be a constant in C not occurring in so. Let rl < . < rp < I < r P + , < . < r,. Then to show that s u {d, = c} E S we need onlj observe that if p 2 u+u, and /I b,, b,+p S b,, .,b,-,+B 6 b, then setting b = bp+a we have

..

..

a

..

5 b , , . . ., b p + u 5 by b 5 b + u , . . ., b,-,+u 5 b,.

It follows from the hypotheses that

T u { d , < d,: q < r } belongs to S. Then by the Model Existence Theorem the above set has 5 model. It follows that T has a model 2' 3 such that ( V, <) contains a copj of the rationals. (ii) The proof is a refinement of the above argument. Define S exactly as before except that u c wl is replaced by a E Orc:

101

UNDEFINABILITY OF WELL ORDER

51

n d.We wish to show that S is a consistency property. The only case which presents any new difficulties is (C5). Lets E Sand let so = so(cl

. . . ~ , d , ,. . .d,,).

We form the new language L" by adding to L' the new constants

e,,, p < o,p

E ord

nd.

For each ordinal u E Ord n d ,let rsa be the following set of sentences of L": U is linearly ordered by 5. U(d,J f o r p S n. U(e,,), for eachp < n and p < a. e, < eqy,for p, q < n and p, y < u, where p < q or (p = q and p < y). epa < d,,, for p < n and j? < u. ep0 = d,, forp < n. Observe that if u < p then r,, c rsa. Let K,, be the C.JSS of all models (8,b, . . . b,) such that % E K,b,, . . .,b, E U',and

5 b,, b,+a 6 b 2 , . . .,b,-,+u S b,. Consider any sentence B(c, . . . c,d,, . . . d,,) of M;. Recalling the u

definition of S we see that s u (01 E S if and only if for all u E Ord n sd there is a model (%, b, .. .b,) E K, for which

. .. b,) I=(3x, . . .x,) A s o A B ( x , . . .x,drl.. . drn). We also note that a model (23, bi . . .b:) is isomorphic to a model in the class K,,if and only if it can be expanded to a model of r,. (The con8 k T, (8, b,

stants cog, ep, correspond to the ordinals p, b,+p). It follows that s u ( 0 ) 4 S if and only if there exists a E Ord n d such that

TuI',k

7 ( A p o ~0). (1) Let Q(u, 0) mean that a is an ordinal, O(cl . c,dr, .. .drJ is a sentence of ML, and condition (1) holds. Using Theorem 10 and the Barwise compactness theorem, we see that the relation Q(a, 0) is X on d .Also, u < /3 and Q(a, 0) implies Q(p, 0).

..

52

110

UNDEFINABILITY OF WELL ORDER

We now prove (C5) for s. Assume V 0 E s. Then V 0 E so v T, so it has at most the constants cl, . . ., c,, d,,, . . ., d,,. Therefore for all BEOrdnd, not T u

rSsI= (A so A V 0).

(2)

Suppose that for all 8 E 0, s u (O} $ S. Then

(d, E) F (ve E 0 ) ( 3 u ) ~ ( a el. , By Z-reflection there is a transitive set a E &‘such that 0 E a and (a, E) k (V8

Let fl

=

E @)(~LY)Q(M 8). ,

Ord n a. Then

q,

whence

(ve E WQ(B, 01.

( d ,E) This means that for all 8 E 0,

Tu

rssF

(A so A e),

and hence

T u rsp F

(A

v 0).

This contradicts (2). Hence s u { O } E S for some 8 E 0, and (C5) holds. We conclude that S is a consistency property. It follows from the Model Existence Theorem that, since T u (d,

-= d,: q < r }

E S,

T has a model B where ( V , <) contains a copy of the rationa1s.i

COROLLARY A (LOPEZ-ESCOBAR [1966]). Let cp be a sentence of L,,, such that every model of cp is of the form H = ( A , <, . . .) where < well orders A . Then there is an u c w1 such that every model of cp has order type c u. This shows that the notion of well ordering cannot be expressed by a sentence of L,,,. B. Let T be a countable set of sentences of L,,,. Suppose COROLLARY Thas a countable model H = ( A , U,c ,. . .) such that < linearly orders

101

53

UNDEFINABILITY OF WELL ORDER

U and U is uncountable. Then T has a model 23 = ( B , V, <, . . .) such that ( V , <) contains a copy of the rationals.

PROOF.Any uncountable linear ordering either contains a copy of <), a copy of (or,<), or a copy of the rationals. -I

(wl,

COROLLARY c (MAKKAI[1969a]). Let % = ( A , <, . . .) be a linearly ordered countable model which does not contain a copy of the rationals. Then there is a sentence cp in L,,, such that for all models 23, 23 C cp if and only if 23 2. PROOF.By Scott’s theorem there is a cp such that for all countable 23, cp if and only if 23 z ? IfI rp .has an uncountable model, then by Corollary B cp has a countable model 23 which contains a copy of the rationals and thus 23 %. Hence cp has no uncountable models. -I

23 k

COROLLARY D (FRIEDMAN). Let d c HC be a countable admissible set and let T be a set of sentences of Ld which is Z on d . Suppose that for all a E Ord n a?,T has an increasing chain of models a;, jl a. Then T has a chain of models whose order type is that of the rationals.

-=

PROOF.

For each a < wl, let %a

=

( U &a,

<,R)

If=

where R(B, x ) holds iff x E A ; . Let T* say: For all u E U, the relativization of T to the set ( x : R(u, x ) } ; < linearly orders U, ( v X ) ( v U , 2, E u ) ( U < V A R(u, X ) + R(V, X ) ) ; (VU, 2, E U)(U < V + ( j X ) ( R ( V , X ) A i R(U, X ) ) ; (Vx)(3u E U)R(u,x ) . Then by Theorem 12, T* has a model

where V contains a copy of the rationals. The models BVare the required chain of models of T. -I

11 Omitting Types Theorem

In this lecture we shall give an example of the technique of prov a theorem for a countable fragment L , by applying the Model Existei Theorem to a set of sentences in a larger countable fragment La. This technique of enlarging the set ..dhas no analog in first order 101 and using it we can obtain new results for first order logic L by using methods for Ld. For L the following resuIt is essentially due to Hen and Orey. 13 (OMITTINGTYPESTHEOREM). THEOREM Let L, be a countable fragment of L,,, ,let T c L , be a set of senteni and for each n < o let @,(x, . . .x,,,) be a set of formulas of L , wit1 most the free variables xl, . . .,xp,. Suppose that (i) T has a model;

and (ii) For all n < o and all formulas $(xl . . . x,,) in Ld, if T u {(3x1 xpn)$} has a model, then there exists 'p E @" such that T u {(3x, x,,)$ A 'p} has a model.

PROOF.Let M d be the set of all formulas obtained from form]

in L, by replacing all occurrences of finitely many free variables constants c E C (as before). Let S be the set of all s of the form

...

s = s ~ u T u { V @ ~ ( CcPn):n ~
111

55

OMITTING TYPES TWEOREM

where so is a finite set of sentences of M,, only finitely many c E C occur in so, and S, u T has a model. We claim that S is a consistency property. The only nontrivial step is part (C5) of the definition. Let VO E s E S. CASEI. VO E so u T. Let 9( be a model of so u T. Then for some 0 E 0, 9( k 8, whence s u {e}E S. CASE11. VO = V@,,(c, . . . c,,) for some n < 0 and c l , . . cpnE C. This means that 0 = (Cp(Cl . . . c,,): ql E Q n } .

.

Let dl, .

. ., d,

be all the constants in C - { c , ,

so; thus we may write

. . ., c,,}

which occur in

. . . CPndl. . . d,).

so = S0(CI

Since s E S, the set T u ((3x1

* * .

Xpn)(3Y,

. .Ym) A *

SO(XI

* *

*XpnYl

-

* *

Ym)}

has a model, where the y's are new variables. Then by (ii), there is a ~p E Qi, such that T u ((3x1

xnp)(3Y1 . .

* *

A

~ m )

sO(x1

has a model. It follows that su

because q ( c l

(ql(C1

* *

. . c,,)} *

-

xpnYl

* *

.Yrn)Adx1

*

*

xpJ1

E s,

. . .cpn)E 0 and l u so u

{ql(Cl

. . . c,,)]

has a model. By (i), the set

Tu

{V@,(Cl

. . . c,,):

n < 0 , ci E C }

belongs to S. Then by the Model Existence Theorem and the remark following it, the set (i) has a model % in which every element is an interpretation of some c E C. Therefore % is a model of T, and also of

A

n<m

(VX,

. . . X,")

VQn. -I

56

11 1

OMITTING TYPES THEOREM

The Omitting Types Theorem will be useful later on in these lectures. The name “omitting types” comes from the fact a model of a sentence of the form A (VX, . . . X,) V@,,(X, * . . X,) n
may be thought of as a model which, for each n < w , omits the type of p,,-tuples of elements satisfying the set of formulas { q ( x , . . . x,,): cp E

@,,I.

The following very elementary result points out the importance of sentences of the above special form. It shows that the study of countable fragments of L, can be reduced to the study of models which omit countably many types with respect to L, i.e. to models of sentences of the above special form with each @,, a set of sentences of L.

THEOREM 14. Let L, be a countable fragment of L,,,. Then there is a language L’ formed by adding countably many relation symbols to L and a sentence

CP = A

m
( v ~ *1* * x p , )

V

n
qrnn(X1

* * *

~p,,,)

such that each qmnis a formula of L‘ and: (i) Every model CtT for L has a unique expansion to a model for L‘ with N‘ k cp. (ii) For every formula $(yl . . .yr) E L, with finitely many free variables there is a formula $’(yl . . .y r ) in L’ such that k cp

(iii) For every formula B(yl L, such that

+

. . .y,)

E

k cp -+

($

-

$7.

of L’ there is a formula B1(yl

(e

el).

PROOF.Form L’ by adding, for each formula $(yl placed relation symbol RS(Y, . . yr.). Let us abbreviate

. . .y,) E L,,

,

Y-Yn = ~

1

-

Y *3 - 3

= Y 1 ~ * * * ~ Y r , Yn- 1 , Y n + l ,

* *

‘7

y - y , , = y if n > r.

Yr if n

5 r,

. . .y,: an r.

111

Rvy,e(y)(Y% y . e (y ) (Y

51

OMITTING TYPES THEOREM

Yn)

-Yn)

-

c* ( V y n ) R e ( y ) ( y ) ;

(gYn)&(y)(JT).

The conclusions (i) - (iii) then follow easily. i The Omitting Types Theorem can be used to prove an important theorem in recursion theory. To state this theorem we need some definitions.

DEFINITION. Let U c o. A set X c R ( o ) is said to be recursive in U iff X is A on the structure (R(o),E, U ) ; that is, there are a C-formula ~ ( x and ) a n-formula .(x) in the language of (R(o), E, U) such that X

=

{ a : (R(w), E, U ) k a [ a ] >

X

=

{ a : (R(w), E, U ) 1 .[a]}.

and An ordinal p is said to be recursive in U iff there exists a well ordering X of w of order type j? which is recursive in U. Finally, is defined to be the least ordinal which is not recursive in U. Recall from Lecture 8 that an ordinal a is said to be admissible iff the set L(a) of all sets constructible before stage a is admissible. We shall need two classical lemmas from recursion theory (see PLATEK [19661).

LEMMA A. For any set U c o,the ordinal ordinal. LEMMA B. For any admissible ordinal is C but not A on L(ct).

ci,

is a countable admissible

there exists a set S c

ci

which

58

OMITTING TYPES THEOREM

The proof of Lemma B is just a generalization of Godel's diagonal construction of an r.e. set which is not recursive. We omit the proofs of Lemmas A and B. The following theorem was first proved by SACKS[I9701using a forcing construction. Later FRIEDMAN and JENSEN [1968] found a much simpler proof using the Barwise Compactness Theorem. The proof below is simpler yet and is due to T. Grilliot and S . Simpson. THEOREM 15. A countable ordinal a > 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 a model of T. Moreover, it is obvious that ( 2 ) T is .Z on d. Now let O(x) be the set of formulas: Vy(y E x -+ y is an ordinal) C~ E x, for each p E S -I cs E x, for each /3 E a-S.

111

59

OMITTING TYPES THEOREM

We shall use the Omitting Types Theorem to show that Tu{

3xA0} has a model.

Such a model is said to omit the set S. To do this, consider any formula

$ ( x ) in L, such that T u { 3 x $ ( x ) } has a model. Let I' be the set of all formulas q ( x ) such that T k $ ( x ) + cp(x). By Problem 6 in Lecture 9,

r is C on d .Assume for the moment that 0 c sistent, the sets S

=

a-S=

r. Then, since r is con-

{ p < a: (cg E x ) E r},

{fi
are both Z on d ,whence S is A on d.But S is not A on d ,so there is a formula f3E 0 - r. Then T"

@X)(11/(Xb

1e(x>>>

has a model. It follows from the Omitting Types Theorem that T has a model 8 in which V x VoE8 9, which is equivalent to 13x A@, holds. Then 8 is a model of T which omits S. Let U be the interpretation of cu in 8.Then from the axioms of T we see that U c w and wy 2 a. Assume that my > a. Then there is a Cformula cp(x) which defines a well-ordering Y o n w of order type a in the model (R(o), E, U ) . But (R(w), E, U),and hence Y, are elements of Z. By using the principle of definition by C-recursion in 8, we see that Z must have an at* ordinal, which we may identify with a. Let ~ ( x be ) a C-definition of S in d . Thus Using &separation

s = {fi < u : 8 k a'L'u"(cg)}. in a, there is an s E such that 8 c (Vy)(y E s 4+ o'L(u"(y)).

Therefore

' ~ ? I I = C ~ Efor S BES, 8 k

lcgEs

for P E a - S ,

contradicting the fact that 2 omits S. We therefore conclude that wy = u. -I

60

11 1

OMITTING TYPES THEOREM

The following corollary, due to Friedman, improves Problem 8 in Lecture 9. COROLLARY c. Let a be a countable admissible ordinal and suppose there is a model M of ZF with a 2 osp(M), the ordinal standard part of M . Then a is the ordinal standard part of some model N of ZF.

PROOF.The result is trivial for a = w . Suppose that a > w. Let T‘ be the theory T in the proof of the preceding theorem plus the axioms of ZF. Suppose M C ZF and a osp(M). Then

-=

M k

‘ a is

admissible’.

We may assume that M is countable. Using Cohen’s forcing construction, we may also assume that M k ‘ a is countable’.

Then by Theorem 14 in M ,

kf k (3u)(uC

WAC(

=

u

01).

It follows that if we interpret cu by U and cs by /? for each /? < GI,then M becomes a model of T‘. From the proof of Theorem 15 we see that T‘ has a model N which omits S, and its reduct N must be a model of ZF such that a = osp(N). -1

12 Prime models

In this lecture we give some applications of the Omitting Types Theorem. The results are due to Vaught for L.

DEFINITION. Let L, be a fragment of L,,,. By a complete theory T i n L, we mean a set T c L, of sentences which is maximal consistent. By the completeness theorem, T is a complete theory in L, iff there is a model (ZI for L such that T is the set of all sentences of L, holding in (ZI. Now let T be complete theory in L,. A formula cp(x, . . . x,,) E L, (with finitely many free variables) is consistent with T if and only if T k (3x, . . . x,)cp, i.e. ( 3 x , . . . x,)cp holds in all models of T. q(x1 . . . x,) is said to be compZete (in L,) with respect to Tiff cp E L,, cp is consistent with T,and for every formula $ ( x l . . . x,) E L,, either Tk cp

-+

I) or TI.cp

-+

7

I).

cp(xl . . . x,) is said to be incompletable (in L,) with respect to Tiff cp E LA, cp is consistent with T, and there is no complete O(xl . . . x,) E L,

such that T k 8 -+ cp. A set of formulas @(xl . . . x,) c L, is said to be a type (in L,) with respect to Tiff there is a model 2l of T and elements a , , . . . a, E A such that @ = (q?(x,

The n-tuple (a,

. . .x,) E L,:

(ZI k rp [a,

. ..a,]).

. . .a,,) is said to realize the type @ in (ZI. A model (ZI 61

62

[12

PRIME MODELS

of T is said to realize a type @(xl . . . x,) iff some n-tuple in A realizes @, and otherwise $?I is said to omit Qi. For the remainder of this lecture, we assume L, is a countable fragment of L,,, , T is a complete theory in L,. The above notation will always be used with respect to Ld .

LEMMA A. Let Z be a finite or countable set and for each i E Z, let Qii(x, . . . x,,) be a type which contains no complete formulas, with respect to T. Then T has a countable model which omits each type !Pi, i E I. PROOF. It suffices to show that

...x,,) V q} has a model. Let i E I and let $ ( x l . . . x,,) be consistent with T. If $ # Qi TU { A

(VX,

id

i

r p 4

then (-I $) E and (15, A 11$) is consistent with T. Suppose that $ E dji. Then $ is not complete with respect to T, and it follows that there is a formula 8(x, ...x,,) E Ld such that both $ A 8 and $ A 8 are consistent with T. Moreover, either 8 E ai or (7 8) E Qi. Thus in all cases we have shown that for some rp E Q i i , $ A 1rp is consistent with T. The desired result now follo%sfrom the Omitting Types Theorem. -I

LEMMA B. T has a countable model $?I such that for all n < w , every n-tuple in A satisfies either a complete formula or an incompletable formula with respect to T. PROOF. For each n c w , let @,(xi . . . x,) be the set of all complete or incompletable formulas in the free variables x i , . . ., x,. Let $(xl . . . x:,) E Ld be consistent with T.Then either $ is incompletable, whenceI!,I E @, or there is a complete formula q(x1 . . . x,) E @, such that T k rp -+ $, whence A rp i s consistent with T. By the Omitting Types Theorem, T has a countable model % satisfying the sentence A (VX, . . . x,) VGn. n<m

Clearly % has the required property. -I We shall apply these two lemmas to the study of prime models of T.

121

PRIME MODELS

63

We first introduce the important notion of an elementarysubmodel in L,.

DEFINITION. M is said to be an elementary submodel of 23 in Ld, in symbols M -< Ld23, iff 9I c 23 and for every formula cp(x, . . . x,) E Ld (with finitely many free variables) and all a , , . . ., a,, E A ,

8 C cp[a, . . .a,,] if and only if 23 I=q [ a l . . . a,,].

M is elementarily embeddable in 23 in L, iff M is isomorphic to some elementary submodel of 23 in Ld. We shall say that 8 is a prime model of Tiff M is a model of T and M is elementarily embeddable in every model of T in Ld. These definitions could also be made for arbitrary, as well as countable, fragments of L,,,. The next theorem is due to VAUGHT[1961] for first order logic, and the proof below is a straightforward extension of his argument to L,. THEOREM 16. (i) M is a prime model of Tif and only if M is a model of T of power at most o,and every n-tuple of elements of A satisfies a complete formula in M. (ii) Any two prime models of T are isomorphic. (iii) T has a prime model if and only if there are no incompletable formulas $ ( x , . x,,) with respect to T.

..

PROOF.(i) Let M be a prime model of T.Since T has models of power at most o,M has power at most o.Let a,, . . ., a, E A, and let @(xl . . . x,,) be the type realized by (a, , . . ., a,,) in M. If @ contains no complete formula, then by Lemma A there is a model of T which omits @. But then (11 cannot be elementarily embeddable in 23, because the image of a, ,. . ., a,, under the elementary embedding must also realize @ in '23. Therefore @ contains a complete formula ~ ( x .,. . x,,), and ( a , ,. . .,a,,) satisfies cp in M. Conversely, suppose M is a countable model of T and every n-tuple in M satisfies a complete formula. Let A = {al u z , a 3 , . . .}. For each n < o,let cp,,(x, . . . x,,) be a complete formula satisfied by (a1 . . ., an) in 8. Then T t= 3x,cp,(xl), so we may choose an element b, E B with 23 k p , [ b , ] . In general, if 23 k cp,,[b, . . .b,], then since cpk is complete,

64

PRIME MODELS

112

we have T I=(P,, + ( ~ X , , + ~ ) ( P , , + ~whence , we may choose b , + , E B with % I= cpn+ [bl . . b,,, 3. The mapping a,, -+ b,, on A into B constructed in this way is an elementary embedding of CU into % in L,, because any two n-tuples which satisfy the same complete formula must satisfy exactly the same formulas of L, . Hence (II is prime. (ii) We argue as above, with a back and forth construction between two models having the property characteristic of prime models given in (i). (iii) Suppose there is an incompletable formula q ( x , . . . x,) with respect to T. Then for any model (IIof T, some n-tuple satisfies cp(xl . . . x,,) in a, and thus cannot satisfy any complete formula. Hence by (i) (II is not a prime model. For the converse, suppose there are no incompletable formulas with respect to T. By Lemma B and (i) it follows that T has a prime model. -I

.

COROLLARY c. If for each n < o,T has at most countably many types @(xl . . . x,,), then T has a prime model. PROOF.Suppose ~ ( x ., . . x,,) is an incompletable formula with respect to T. Let Qi(xl. .x,,), i E I, be all the types in T which contain q ( x , . . . x,). Then by hypotheses I is countable. Since cp is incompletable, each ai contains no complete formulas. Then by Lemma A, T has a model (II which omits each Q i , i E I. But T k (3xl , . .x,,)cp, so some a , , . . ., a,, E A must satisfy cp. Then the type realized by (al,. . ., a,,> contains cp. This is a contradiction and shows that T has no incompletable formulas. Hence by Theorem 16 (iii), T has a prime model. i

.

COROLLARY D. T is o-categorical, that is, any two countable models of Tare isomorphic, if and only if

Ti=A (Vx, . . .x,,) V (cp(xl . . . x,,): n<m

PROOF.By Theorem 16 (i) and (ii). i

cp complete w.r. to T}).

13 Skolem functions and indiscernibles

We now turn to an entirely different method of constructing models, namely, from Skolem functions. Let L, be a countable fragment of L,,,. The Skolem language L: for L, is constructed as follows. We first form the language LS). This language has all the symbols of L,, and, for each formula 3xq(y, . . .y,) E L, in which the variables yl, . .,y,, do not occur bound, L ): has a new n-placed function symbol F3xrp.We define LS' to be the least countable fragment which contains all the formulas of L, and all the new symbols &,+,. Proceeding inductively, for each n < w,n 2 1, define L$+') = L:)"). Finally, define

.

:L

=

u L'Z.

nco

Then it is clear that L: is also a countable fragment. The Skolem theory, TSkolem, for L, is the theory in the Ianguage L: whose axioms are all formulas of the form ~ X V ( Y *, *

*

YJ

-+

q(F3xJ~1 *

.

-

*

Y J Y 1 . . * ynh

where 3xq is a formula of L: and y , , . .,y,, do not occur bound in 3xq. Note that the above axioms are formulas of L s , because if 3xq is in Ls) then the above formula is in LS"). Recall that fragments are closed under finite connectives, quantifiers, and substitutions. Let 8 be a model for L. By a Skolem expansion of 8 we mean a model

8* = (8,G3xp)3xgEL.d 61

68

SKOLEM FUNCTIONS AND INDISCERNIBLES

r13

for the language L: which is a model of TSkolem. In order to state the fundamental result about Skolem expansions, we first prove a useful lemma on elementary submodels. LEMMA. Let L, be a fragment of L,,,, and let %, B be models for L. Then 2 ' 3
.

.

The statement for L and proof are due to TARSKIand VAUGHT [1957]. PROOF.We show by induction on the complexity of formulas that for every formula $(yl , , ,y,) E L, and all b,, . . ., b, E B,

'iY k $[bl

. . . b,] if and only if B k $[b, . . . b,].

Since 23 c 2l, this holds when $ is atomic. We note that every subformula of a formula with finitely many free variables has finitely many free variables, so we restrict our attention to such formulas. If $ is of the form 8 or 'r\O and 8, or each element of 0,has the above property, then so does $. Suppose $ = (3x)8(y1 . . .y,). If B t= $[bl . . . b,] there exists a E B such that B k 8[ab, . . . b,], whence % t= B[ab, . . . b,], and % k $ [ b , . . . b,]. Suppose that % t= $[b, . . . b,]. Then by hypothesis there exists a E B such that 91 k 8[ab, . . . b,], whence B t= $ [ b , . . . b,]. -I THEOREM 17. Let L, be a countable fragment of L,,, and let L z be its Skolem language. (i) Every model % for L has a Skolem expansion with respect to L d . (ii) Let a* be a model of TSkolem for L .: Then every submodel B* t %* is an elementary submodel, B*
PROOF. (i) Let = %, L z ) = L,. Suppose n < w and a(") is an expansion of '21 which is a model of TSkolem n L$). Let (yx)cp(y, . . . y,) be a formula in L$) in which the y's do not occur bound. We may then choose the function FjXqsuch that for all b,, . . ., 6, E A , if %(") k 3xcp[b, . . . b,] then a'") k cp[ab, . . . b,] where a = F,,,(b, . . . b,). This makes % 2 " + 1 ) a model of TSkolem n L2"). Then %* = u,<,M(") is a model of TSkolem.

131

69

SKOLEM FUNCTIONS A N D INDISCERNIBLES

(ii) We use the lemma, but for the fragment Llt&instead of L,. Let .y,) be a formula in L.: Choose variables u1 , . . ., u, not occurring in 3xcp and form 3cp(u, . . . u,) by substituting free occurrences of the y’s for the u’s. Suppose b,, . . ., b, E B and %* k 3xcp(y, , . .y , ) [b, . . . b,]. Then from the properties of satisfaction we see that %* k whence U* k cp(F,,,(u, . . . un)ul . . . u,) 3xcp(u, . . . u,)[bl . b,], [b, . . . b,], since %* is a model of TSkolem. Let a = G,,,,,(b, . . b,) in %*. Then (3x)cp(y, .

.

..

.

a* k cp(xu, . . . u,)[ab, . . . b,], hence

a* 1 cp(xy, . . .y,)[ab, . . . b,]. c \u* and b,, . . ., b, E B, we have a E B. The result %* xL.,a*

Since %* follows from the lemma. i

As a first application we have another extension to L, of a result from TARSKIand VAUGHT[1957]. COROLLARY (DOWNWARD LOWENHEIM-SKOLEM-TARSKI THEOREM). Let L, be a countable fragment of L,,, . Let % be a model for L, X c A , and a an infinite cardinal with 1x1 S a 5 IAl. Then % has an elementary submodel % a , a an ordinal, are defined recursively by:

Let a < wl. Then there is a sentence cp E L,,,

which has a model of

70

r13

SKOLEM FUNCTIONS AND INDISCERNIBLES

power za, (and hence models of each infinite power no model of power > 3,.

5

but has

PROOF. Since u is countable we may write o + a = {&: n < w } . Let L have the binary relation E, the one-placed function r (for 'rank'), and constants cp,, n < o.Let cp be the conjunction of the following sentences:

V x

(Vx)(x E cBnc-f

= cBm),

each n < w.

Bm
(W(n
r(csn) = can, (VX, Y > ( X E

Y

+

r(x) E r ( y ) ) .

(Vx, y>[(Vz)(z E x c-f z

Since IR(w+a)l

=

E

y) +x

za, we see that

= y]

('extensionality').

cp has the required property. i

Our next goal is to prove a form of the Upward LGwenheim-SkolemTarski Theorem for L,. If %* is a model of TSkolem and X c A , then the submodel of %* generated by X will be called the Skolem hull of X in %*, and will be denoted by Hull,,(X), (or simply HuN(X) if no confusion will result). Thus from Theorem 17 we see that HUZZ,.(X)
DEFINITION. Let L,, be a fragment of L,,,, and let % be a model for L. By a set of indiscernibles in '$ we mean l a linearly ordered set ( X , <) such that X c A and for any two finite increasing sequences x1 < . . . < x,, and y 1 < . . . < y,, in ( X , <), we have (%XI

i.e. the n-tuples (xl,. L, in 9.I.

* * *

xn) s L L d ( K

Y,

* . *

yn),

. ., x,,) and ( y l , . . .,y,,) realize the same type of

131

71

SKOLEM FUNCTIONS AND INDISCERNIBLES

The order relation < is not necessarily definable in the model a. Consider again a countable fragment L, and its Skolem language L .: Sets of indiscernibles in a model a* of TSkolem are particularly useful. Notice that if ( X , <) is a set of indiscernibles in a*, then it is also a set of indiscernibles in Hull (X), because Hull ( X ) a*. THEOREM 19 (i) (ELEMENTARY EMBEDDING THEOREM) Suppose a*, B* are models of TSkolem for a countable fragment L d , ( X , <) and ( Y , <) are sets of indiscernibles in %*, B*, and finite increasing sequences from Xand Y realize the same types in L; Then every monomorphism f of ( X , <) into ( Y , <) can be extended uniquely to an elementary embedding f of Hull,,(X) into Hull,,( Y). (ii) (STRETCHING THEOREM) Let %* be a model of TSkolem for a countable fragment L,, let ( X , <) be an infinite set of indiscernibles in a*, and let ( Y , <) be an arbitrary infinite linearly ordered set. Then there exists a model B* such that ( Y , < ) is a set of indiscernibles in B*, and the finite increasing sequences from X and Y realize the same types. In particular, a* = L.d'i13*.

.

PROOF.(i) Any element a E HuN(X) can be expressed in the form a = t(x, x,,) where f is a term in L : and x1 < . . . < x,, in ( X , <). If t , and t, are two terms and t , ( x , . x,,) = fz(xl . . . x,,), then any increasing n-tuple y1 < . c y,, in ( Y , <) also satisfies the equation tl(y, . . y,,) = t z ( y l y,,). It follows that there is a function f: Hull ( X ) + Hull ( Y ) such that for all terms t in n variables and all x , < . . < x,, in ( X , <),

...

.. . ..

.

.

..

.

f ( t ( x , . . . x,)) = t ( f x , . . .fxn).

.

(1)

Let 'p(u, . . v,) be a formula of L; and let a, , . ., a, E Hull (X). Suppose ai = t i ( x , . . x,,) where xi < . . < x,, and i = 1,. ., m. Then the following are equivalent, where u l , . . ., u,, are new variables:

.

.

.

. .a,] Hull ( X ) k 'p(t,(u, .. . u,,) . . . tm(ul . . . u,,))[xl . . .x,,] Hull ( Y ) k 'p(t,(u, . . . u,,) . . . t,(u, . . . u,,))Lfxl . . .fx.l Hull ( Y ) k fa, . . .fa,]. Hull ( X ) C 'p[a,.

12

D3

SKOLEM FUNCTIONS A N D INDISCERNIBLES

This shows that f is an elementary embedding. 3 is unique because any extension offto an elementary embedding must satisfy (1) andfis the only extension off satisfying (1). (ii) Let T b e the set of all terms in Lz. If t,(u, . . .on), tz(u, . .u , ) ~T, write t , t2 iff for any x1 < . . < x, from ( X , <),

.

.

N

t,(x,

I . .

x,) = t,(x,

. - x,) f

in %*. Let 2; be the equivalence class o f t , . Let B be the set of all formal expressions t ( y , . . .y,) such that t ( u , . . . u,,) E T has at most the free variables vl, . . ., u,, and y , < . . . < y , E Y. We define an equivalence relation M on B as follows. Let t ( y , . . .y,), t‘(y; . .y:.) E B, where t ( u , . . .v,), t’(u; ..Ar) E T. The yi and yf may be combined into a single increasing sequence z, < . . . < z, in ( Y , <), where y , = z i l , . . . , y n = z,i y ; = z j , , . .,yb. = zj,.. Form terms t o ( u l . . . u,,,) and r;(u, . u,,,) by replacing u,, . ., u, by o i l , . . ., ui, and u ; , . . ., ui,, by ujl, . . ., ujn., repectively. Then we write

.

..

.

..

r(y,

. . .y,)

.

w qy;

. . .y:.)

if and only if to t;. The relation M is obviously reflexive and symmetric. We must show that it is transitive. We need the following fact: N

Suppose t ( y l . . .yo), t’(y; . . .y:.) E B, w1 < . . . < w, in ( Y , <), andy, = wkp,y; = wl,. Form t , ( u , . . u p ) , t i ( u , . .up) by replacing y , by u k , , y: by ulr, respectively. Then t , t i if and only if t ( y l . . .y,) w t’(y; . . .y;.). (1)

.

-

.

The difference between (1) and the definition of M is that z , < . . . < z,,, is the least sequence containing the y , and y f , while w1 < . . . < wp is an arbitrary one. Since X is infinite we can find an increasing sequence x1 < < x, in ( X , <>. Suppose to td, whence to(x, . . .x,) = t;(xl.. . x,). By the indiscernibility of ( X , <), t o ( x q , .. x,,) = t;(x,, . . x,,), where q, is defined so that

...

N

.

.

(21

7

*

2,)

=

<wql

-7

7

.

Wq,).

.

But t , ( x l . . .x,) = f O ( x q l ... x,,), t;(xl . . x,) = t;(xq1 . . x,,). Hence ?,(XI . . . x p ) = t ; ( x l . . x,). By indiscernibility, this holds for all

.

131

73

SKOLEM FUNCTIONS AND INDISCERNIBLES

-

-

increasing p-tuples from (X,<), whence t , t i . By reversing the argument we see that t , t ; implies t, m t ; . This proves (1). The transitivity of x follows easily from (1). Let B be the set of all equivalence classes a", a E B. We treat Y as a subset of B by identifying each y E Y with t(y)" where t ( q ) is the term made up of ul alone. It follows from (1) that: For any a,, . . .,a, E B there exists, for some m < o,terms t , ( u , . . . u,), . . ., t,(u, . . . U,)E T and y , < . . . < y , in ( Y , <) such that a, = t,(y, . . . y,)", . . ., a, = t,(yl . . .y,)?. (2) B can be made into a model B*for L> by defining relations, functions and constants as follows:

R(t,(y, . . . y,)". iff for all x1 <

. . . < x, R(t,(x,

. . t,(yl . . . y,)")

in

B*

in ( X , <),

. . . x,) . . . t,(x, . . . x,))

in a*.

(Here we are committing the abuse of using the same symbol R for a relation of QI* and the corresponding relation of B*.) G(t,(y,

. , . y,)". . . t,(y, . . . y,)")

= (G(t,

. . . u,). . . t , ( ~ , . .. u , ) ) . . .y m ) ' .

( ~ 1

(Here we use G both for a function on B and the corresponding symbol Finally, each constant c of L> 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 , <), the Skolem hull H u N ( X ) has elementary extensions B* >L.d Hull ( X ) of arbitrarily large cardinality. PROOF.Consider a linearly ordered set (Y, < ) of the desired cardinality, with ( X , <) c ( Y , <). Then use Theorem 18. -I PROBLEMS 1. Let d be a countable admissible set, d c HC. Show that the set of formulas L:is A on d ,and so is the set of axioms for TSkolem. Note that in general, if N is the first order language with the symbols of L,: then L: is a proper subset of N d = N,,, n d . 2. Let L, be a countable fragment and %* a model of TSkolem with a set of indiscernibles ( X , <). Show that any automorphism of ( X , <) can be extended to an automorphism of Hull ( X ) . 3. Let a*, ( X , <) be as in the above exercise. Let B* be another model of and (Y, <) a set of indiscernibles in B* such that the increasing n-tuples from X and Y realize the same types. Assume further that X and Yare infinite. Show that HulZ ( X ) and Hull ( Y ) realize exactly the same types in L.: 4. Let %*, ( X , <), B*, (Y, <) be as above. Show that if ( X , <) i , , ,

(Y, <), then Hull ( X )
Hull (Y). (CHANG)(N is as above.)

.

5. Prove that for every formula q ( u l . . u,) E L: there is a universal k cp c)$. formula $(ul . . . o n ) E N,,, with TSkolein 6. (MALITZ).(GCH). Show that for each ordinal a < o1 there is a sentence cp of L,,, such that: (i) cp has a model of power aa+land no models of power < aa+1. (ii) Any two countable models of cp are isomorphic.

14 Erdos-Rado Theorem

In the next few lectures we shall give some applications of indiscernible elements. In order to show that models with infinite sets of indiscernibles exist, we need the following partition theorem of ERDOS and RADO[1956]. The Erdos notation a + where a, p, y are cardinals and 1 5 n < o,is useful here. [XI" is the set of all x c X of power exactly n.

(a)!,

(a):

DEFINITION. c1+ means that whenever 1 x1 = a and the set [X]" is written as a union of y sets, [XJ" = U i E Y C ithere , is a subset Y c X of power I Y [ = such that for some i e y, [Y]"c Ci.Equivalently, given any function f on [X]n into y, there exists a subset Y c X of power IYI = p and an i E y such that for all {yl, . .,y,} E [Y]", f ( { y l , . . .,y.}) = i. Note that the relation a + is preserved under increasing the number on the left and decreasing the numbers on the right. Given a cardinal a, we define a0(a) = OL,and for ordinals 5 > 0,

. (a)!

Zt (a) =

sup (2'<(")).E.g., >,(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 <

. . . < Y , , - ~ < y,,, (by0

,...' b y n - l , c ) ~ C j .

Since by"realizes the same type as c in (H, bs)d
This shows that

[Y]""

c

ci. -1

The above proof is due to S . Simpson, and is simpler than the original purely set-theoretic proof of Erdos and Rado.

15 The Hanf number of Lwl,

In this lecture we shall prove an Upward Lowenheim-Skolem-Tarski Theorem for L,,,. The proof will be an application of the Model Existence Theorem, indiscernible elements, Skolem functions, and the ErdosRado Theorem. It is convenient to use the Extended Model Existence Theorem which was proved in Lecture 3. THEOREM 21. (UPWARD LOWENHEIM-SKOLEM-TARSKI THEOREM FOR Ld). (MORLEY [1965a] in a special case, LOPEZ-ESCOBAR [1966] in general.) Let L, be a countable fragment of L,,, and T a set of sentences of Ld. Suppose that for all a < ol, T has a model of power 3,. Then: (i) T has a model which has an infinite set of indiscernibles in Ld. (ii) T has models of all infinite powers. PROOF. We first prove the implication (i) =, (ii). Assume that (i) holds for all countable fragments Ld and all T. Form the Skolem lanquage L: of L,, and let T* = T u TSkolem. Then L> is a countable fragment and T* is a set of sentences of L>. Moreover, by Theorem 17, every model of T has an expansion which is a model of T*.Hence T*has models of power 3, for all a < ol.Thus by (i), T* has a model 8* which has an infinite set (X,<) of indiscernibles. Let /? be an arbitrary infinite cardinal and let ( Y , <) be a linearly ordered set of power /?. By the Stretin which ching Theorem (Theorem 19 (ii)), T* has a model !8* ( Y , <) is a set of indiscernibles. Then B* has power fl. The reduct % of !B* to L is thus a model of T of power /I. 78

151

THE HANF NUMBER OF

19

La,,

We now prove (i). We remark that (ii) was proved by applying (i) to the fragment L.: We need only prove (i) for an arbitrary L, and T, since the special case of L: and T* is then included. We shall not have to mention Skolem functions in the proof of (i). Let us form the language M by adding to L two countable sets D = {dl, d z , . . .} and C of new constants. Let Md be the least countable fragment of M,,, which includes L,. Let I be the set of all sentences of the forms

8(di, . . . dip)ts 8(dj, . . . d j p ) ,

di = d j ,

1

where 8 ( v , . . . U , ) E L,, 1 5 i, < . . . c i, < w, and 1 s j 1 < . . . < j p < o,and i # j . Then it is easy to see that if 8 is a model for L, and

. . . are elements of A such that (8,a, az . . .) is a model of Z, then 8 has the infinite set of indiscernibles ( X , c )where a,, az,

X = { a 1 , a z ,...I, a i < a j i f f i < j , in Ld. Thus to prove (i) it suffices to show that T u Z has a model. In view of the Extended Model Existence Theorem, it is sufficient to find a consistency property S in M d such that S is non-empty and for all cp E T u I and s E S, sw{cp} E S.

(0)

We shall now define a set S with the desired properties. Let S be the set of all finite sets s of sentences of M, such that: Only finitely many c E C and d E D occur in s, say c,,,dl.. . d,,).

s = s(cl..

.

(1)

For all LY < w , there is a model 8 of T and a linearly ordered set ( X , <) such that X c A , 1x1 = a=, and for all < . . . < a,, in X , (2) (a, a, . . . a,,) C (3, . . . u,,,) A s(u, . . . u,dl . , . d,,).

(I,

.

In the above, it is understood that the constants c1, . ., c,,, are replaced by new variables u1 , . .,u, E -01, and that the elements a, , . . ., a,, are interpretations of the contants d , , . ., d,. We first claim that

.

.

S is non-empty.

(3 1

80

THE HANF NUMBER OF

115

Lmim

For example, consider the logically true sentence (3x)(x = x). Since for each a < wI T has a model % of power a,, and since (3x)(x = x) contains no constants, conditions (1) and (2) hold for s = {(3x)(x = x ) } . Thus {(~x)(x= x)} E S. Next we show that

S is a consistency property.

(4)

All the parts of the definition of consistency property can be verified in a routine way except the condition If V 0 E s E S, then for some 8 E 0, s u { O }

E

S.

(5)

We may write s = s(cl . . . c,dl . . . d,), and for all 8 E 0, 8 = 8(c1 . . . cmdl . . . d,). Let GI < w , . Then cr+n < w , . By (2) we may choose a model Ma of T and a linearly ordered set ( X , , <>of power IX,l = a,+,, X, c A,, such that for all a , < . . . < a, in X,,

Ma b (324,

. . . u , ) A s ( u , . . . umd, . . .d,).

Then since V 0 E s we see that for all a , < . . . < a, in Xu,

yl,

k

v ( 3 U 1 . . . U r n ) ( A S A 8 ) ( U 1 . . . U m d l . . . d,).

ese

We may then choose a function f : [ X , y a , < . . . < a, in X , and f ( a , . . . a , ) = 8,

M, \

+

0 such that whenever

p u , . . . u , ) ( ~s A e ) ( u , . . . urndl. . . d,).

According to the theorem of Erdos and Rado,

an-1(am)+ Then, since a,+,,

=

+

a,,(>,) 2 a,-,(a,)+, aa+n

+

(a:);=. we have

(>ax*

It follows, since 101 S o,that there is a set Y, c X, of power IY,l = a,, and an element 8, E 0,such that for all a, < . . . < a, in Y,,f(al. . . a,) = 8,. Using the countability of 0, we see that there exists 8' E 0 such that for arbitrarily large a < ol,8' = 8,. This means that for arbitrarily large a < o1the set s u {O'} has property (2). It follows at once

151

81

L,,

THE H A N F NUMBER OF

that s u {O'} satisfies (2) for all CL < a,,and therefore s u {O'} E S. This proves ( 5 ) , and (4) follows. It remains only to show that S satisfies (0). Let cp E T u 1, and s E S. If cp E Tit is obvious that (1) and (2) hold fo rs u {cp}, whence s u {p} E S. Similarly, if cp is of the form -- di 3 d j , then it is clear that s u { c p } E S. Suppose now that cp E I, say cp is O(di, . . . d i p )c)B(dj,

...djp).

For this case we must again apply the Erdos-Rado Theorem. Let CL < 0 , . By (2) there is a model 8 of Tan d a linearly ordered set ( X , <) such that 1x1 = &+,, X c A , and for all a, < . . < a, in X ,

.

(3,a,

. . . a,)

t (3u, . . . u,)As(u,

. . . u,dl . . . d,). 2 by putting g(a, . . . a,) = 1 if and only

Define the function g : [XIp if 8 t B[a, . . . a,]. Using the relation 2 p - I(&)+

+

iT3:u

9

whence L + p

+

(&)pz

Y

we can choose a set Y c X of power IYI = Zu such that the restriction < b, of g to [YIPis constant. Then for any a, < . . < a, and b , < in Y, we have

.

.. .

8 t @[a,. . . up]iff 3 t B[b, . . . b,]. Let q

=

. . . < agin Y, (8,a, . . . q q ) k (34,. . . u,) A s(ul . . . u,d, . . . d,) A B(di, . . . d i p ) B(dj, . . . d j p ) .

max (n,i,,j,). It follows that for any a, <

-

Thus the set s u {p} satisfies (2) and belongs to S. This proves (0), and the theorem follows. -I The Upward Lowenheim-Skoiem-Tarski Theorem for Ld can be combined with Example 18 to compute what is known as the Hanf number for L,,, . The Hanf number can be defined in a very general setting.

82

THE HANF NUMBER OF

[I5

DEFINITION. Let Z be a set and for each i E Z let Kibe a class of models for L. By the Hunf number for the family (Ki: i E I } we mean the least cardinal a such that for all i E I, if Ki contains a model of power 2 a then Ki contains arbitrarily large models. The Hunf number of a fragment L, of L,,, is the least cardinal a such that for each sentence rp E L,, if cp has a model of power 2 a then cp has arbitrary large models. That is, iEI} the Hanf number of L, is the Hanf number of the family {Ki: where I is the set of all sentences of L, and Kiis the class of all models of i. For example, the Hanf number of L is a. We thus have the COROLLARY. The Hanf number of the language L,,, is a,,.

PROOF. Example 18 shows that the Hanf number is 1 a,,. Let cp be a sentence of L,,, which has a model of power 2 a,,. There is a countable fragment L, with cp E L,. By the Downward LST Theorem, cp has models of power for each a < al.Then by Theorem 21, cp has arbitrarily large models. Hence the Hanf number is S am,. 4 The definition of the Hanf number is motivated by the following observation of HANF[1964]: REMARK. Every family ( K i :i E Z} has a Hanf number.

PROOF.For each i E I, define ai to be zero if K icontains arbitrarily large models, and to be the supremum of the powers of the models in Kiotherwise. By the axiom of replacement, there is a cardinal p such that for all ~ EaiI < , p. Then /3 has the property that for all i E Z , if Ki has a model of power 2 p then K i has arbitrarily large models. Hence there is a least cardinal a with that property. a is the Hanf number of { K i :i E I } . i

16 The Hanf number of L,

We have seen in the preceding lecture that the Hanf numbers of L and L,,, are, respectively, w and a,,. We shall now generalize the proof to obtain the Hanf number for an arbitrary admissible set a2 c HC. For any transitive set d , let us write o ( d ) = Ord n a2 = least ordinal not in

d.

Thus if &‘ c HC, then w .(at’) 5 wl. We have seen in an exercise that if &’is an admissible set other than R(w),then w E at’ and thus o ( d ) > w. THEOREM 22 (MORLEY[1967], BARWISE[1969]). Let .a? c HC be an admissible set, d # R(o). Then the Hanf number of L, is a0(&). In fact, if cp is a sentence of L,, and for all a < o(at’), cp has a model of power aa,then cp has models of all infinite powers.

PROOF. The proof that the Hanf number of L is 2 is a refinement of Example 18. We need the fact that w E -01, and it must be assumed that L has enough relation, function, or constant symbols to carry the example through. The details are left to the student. We prove the last statement in the theorem, which implies that the Hanf number of Lsr is 5 Suppose cp is a sentence of Lsr and for all a < o ( d ) , rp has a model of power aa. Then there is a countable fragment La E a2 such that cp E La. for Let L: be the Skolem language of La. Then the Skolem theory TSkolem La is a set of sentences of L.: Add two infinite sets of constants, C and 83

84

THE HANF NUMBER OF

116

L&

D = { d , ,d 2 , . . .} to Lg, and let M a be the least countable fragment containing Lz and these new constants. Finally, let I be the set of all sentences of Ma of the forms

l d i ~ d j l, s i < j < w , O(d,,

... dip)

1

i, <

C,

O(dj, . . . d,,),

. . . < i,

where B(ul . . . 0,)

E

Lz,

< a, 1 5 j , < . . . < j , < o.

Then using the A-separation principle we see that each of the sets L,: TSkolem, C,D , Ma,and I are elements of d . We wish to find a consistency property S in the language M,,, such that:

(a)

TSkolem

{cp>

(b) For all s E S and $ E I, s u {$) E S.

Once this is done, the proof can be completed as follows. By the Extended Model Existence Theorem, the set TSkolem

u (cp>

has a model. Let (8*,a 1 u 2 . ..) be such a model. Then the reduct 8 of %* to L is a model of cp. Moreover, the set X = {al, a 2 , .. .}, with the ordering ai < aj iff i < j , is an infinite set of indiscernibles

for the model %* in the language L,: because the set of sentences I holds. It now follows from the Stretching Theorem that cp has models of all infinite powers. We now shall define the consistency property S which is required. Let &(Ma) be the set of all finite sets of sentences of Ma.Then &(Ma) E d. We shall define recursively a function Q on d into d which has the intuitive meaning: Q(p) is the set of all sets s = so u TSkolem u {cp} where so E &,(Ma) and it can be seen in at most p steps that no consistency property satisfying (a), (b) contains s. More formally, Q can be defined recursively by: If x is not an ordinal, Q ( x ) = 0. If fi E o ( d ) , then

Q(p) is the set of all s E sd such that the following for-

161

THE HANF NUMBER OF

85

L&

v (38 E s ) ( ( ~e) E S)

(1)

rc(em < p)(e = II/ A S u w 1 E QW) (4 v (30 E s ) p E~rc(e))(e = A ly A (3$ E Y > ( ~) 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 

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 ))

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’, <) such that X c A , 1x1 2 >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 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<,cpmnn, where cpmn E L. The general case was proved by Barwise. The argument of Barwise, however, was different from the above proof and depended on a forcing construction over the admissible set d.

.

17 Morley's Two Cardinal Theorem

In this lecture we generalize the Upward Lowenheim-Skolem-Tarski Theorem to obtain a theorem on pairs of cardinals. We assume throughout this lecture that U is a one-placed predicate symbol of L, and L, is a countable fragment of L,,,,.

DEFINITION. Let '21 = ( A , U, . . .) be a model for L and let (X,<) be a linearly ordered set with X c A . We say that ( X , <) is a set of indiscernibles over U (for the model '21in the language L,) iff for all uI, . . ., uq E U and every pair of increasing p-tuples x1 c . . . < x p , y1 < . . . < y p in ( X , <), we have

(a, . . . u 4 x , . . . x,) u1

-L,('21,u1

. . . u q y , . . . y,).

DEFINITION. = ( A , Cr, . . .) is said to be a model of type ( K , A) iff IAl = ii and IUI = 1. A set T of sentences of L, is said to admit ( K , A) iff T has a model of type (IC,A). THEOREM 23 (MORLEY[1965a]). Let T be a set of sentences of L,. Suppose that for all c1 < w1 there exists a cardinal K >= 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 , <) of indiscernibles over U , i n L>. 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 , <) is a set of indiscernibles in N,. By the StretchingTheorem, if we let ( Y , <) be a simply ordered set of power 2, then there is a .model (B*, bJuEUsuch that ( Y , <) is a set of indiscernibles for this model in N,, (8*, b),,v = Hull(Y), and increasing p-tuples from X and Y realize the same types in N,. Let 23 = ( B , V, . . .) be the reduct of this model to L. Then 23 is a model of T and its cardinality is IBI = IYI = 1. We claim that V = {b,: u E U } and thus I Vl = w. For let b E V . For some term t ( u , . . . u p ) of N, and some y1 . . . < y , in Y, b = t ( y , . . . y,). Let x1 < . . . < x, in X . Since t ( y , . . . y,) E V, Z(X, . . . x,) E U . Thus for some u E U , t ( x l . . . x,) = u, whence t ( y , . . . y,) = b,. Therefore V = {b":u E U } , and T admits (A, w). Proof of (i). The proof is similar to the proof of (i) in Theorem 21. Add countable sets C, D of constants to L,, forming M,. Let I be the set of all sentences of the form

-=

di (Vu,

=

d j , i < j < o,

. . . u,)[O(u, . . . ziqdi, . . . d i p )

O(u,

t ,

. . . u q d j ,. . . d j J ]

where O(u, . . . uqul . . . u p ) E L, and i, < . . . < i,, j , < . . . < j,. We may assume that T contains a set of sentences which says that U is infinite. It is clear that if (%, a, a2 . . .) is a model of T u I then % is a model of T which has an infinite set of indiscernibles over U. By the Extended Model Existence Theorem, to get a model of T u I it suffices to find a consistency property S such that S # 0, and for all s E S and rp E T u I, s u {cp} E s. We let S be the set of all finite sets s of sentences of M, of the form s = s(cI . . . c,d, . . .d,) such that for all c1 < w1 there is a model = ( A , U, . . .) of T and a linearly ordered set ( X , <) with X c A , 1x1I &(IUl), and for all a , < . . . < a, in X ,

90

MORLEY’S TWO CARDINAL THEOREM

[17

(a, a, . . . a,) t (30, . . . v,) A s ( v l . . . 0,dl . . . d”). The proof that S has the desired properties is like the proof of Theorem 21. The main difference is that in showing that s E S, p E Z implies s u (p) E S, we use a partition relation of the form

au+p(lw ~u(I~I)zpI~l* +

This relation follows from the Erdos-Rado Theorem. -I We remark that any set of indiscernibles over U is a set of indiscernibles. Theorem 21 follows from Theorem 23 by taking U to be a new relation symbol which is not mentioned in the set of axioms T. COROLLARY. Let T be a set of sentences of L,. If there exists K 2 o such that T admits (a,,(~), K ) , then for all A 2 o T admits (A, a). PROOF.By the Downward Lowenheim-Skolem-Tarski Theorem, for all c1 < o,,T admits (aU(ic),K ) . i PROBLEMS 1 . Let L be a possibly uncountable first-order logic and let L, be a fragment of L,,, . Form the Skolem language L: and define TSkolem as before. Show that Theorem 17 still holds for the language L,. Show that the Downward Lowenheim-Skolem-Tarski Theorem still holds if we add the hypothesis c1 2 IL,l.

2. Show that Theorem 19 (Elementary Embedding and Stretching theorems) still hold for an uncountable language L and fragment L, .

3. Let L, be a fragment of L,,, and L: its Skolem language. Let a* = ( A , U,. . .) be a model for L: which has an infinite set of indiscernibles over U. Using the above exercise, show that for all cardinals ;1 2 J A ) there exists a model B* = ( B , V , . . .) of power 1 such that B* E~,%* and I VI = IVl. 4. Let L, be a countable fragment of L,,, and T a set of sentences of L,. Suppose that for all c1 < w1 there exists K 2 au such that T admits ( ~ J K ) ,K ) . Then for all A 2 pL2 w , Tadmits (A, p). HINT:Use Problem 3. Get a set {xl, jx2, . .) of indiscernibles over U, and then a set { u l , u 2 , . .} c U of indiscernibles in the language formed by adding constants for the x i .

.

.

18 Categoricity in power

In the next two lectures we shall give some applications of the TwoCardinal Theorem, Theorem 23. A class & of models for L is said to be K-categorical, where K is a cardinal, iff any two models a, BE& of power K are isomorphic. A set T of sentences is K-categorical iff the class of all models of T is K-categorical. By way of introduction we state a result of MORLEY [1965] for first order logic. MORLEY CATEGORICITY THEOREM. Let T be a set of sentences of L. If T is K-categorical for some cardinal K > o,then T is A-categorical for all ;1 > w. We shall consider the problem of extending Morley's theorem to other kinds of classes of models. We shall first look at the PC, classes of models defined below. DEFINITION. A class& of models of L is said to be a PC,-class iff there is an expansion L' of L formed by adding a finite or countable collection of new relation, function, or constant symbols, and a set T' of sentences of L', such that JI is the class of all reducts to L of models of T'. In other words, a PC,-class is the class of all models of an infinitely long second order sentence of the form W1)(3P,)

* * *

A

n
qn

where each qn is finite first order sentence involving the extra symbols

P,,P,,

. . ..

91

92

CATEGORICITY IN POWER

[I8

The following example shows that Morley's theorem does not hold in general for PC,-classes. EXAMPLE (SILVER).Let J%be the class of all models 8 = ( A , U ) such that IAl 2I"I. T h e n d l is a PC, class. Moreover, dl is ic-categorical if and only if ic = zU for a limit ordinal CL (equivalently, 1 < K implies 2A< ic). Thus if K is of the form zU,CI limit, and A is not of that form, then ic-categoricity does not imply I-categoricity for PC,-classes. Nevertheless we shall now prove that Morley's theorem does generalize to PC,-classes for certain K and A. We shall see later on that it holds when K = ol. For the present we prove: THEOREM 24. Let IC,A be cardinals such that K > o and 1= 3,, for some CI 2 1. Then every PC,-class which is ic-categorical is A-categorical. Theorem 24 and also Theorem 25 below are due independently to G. Cudnovskii, S . Shelah, and the author. To prove the theorem we need some facts about ic-saturated models.

DEFINITION. A model 8 for L is said to be ic-saturated iff for every set X c A of power < K and every set C(u) of formulas ~ ( u ) with , one free variable u, in the language of the model (8,x),,~, if every finite subset of C(u) is satisfiable in (8, x),,~, then C(u) is satisfiable by an element ~ ~is,. the infinite sentence in (a, x ) ~ That

1x1

C

A ( 3 ~A) &du)l

TO€S,(~)

+

(30) A z ( ~ >

holds in (8,x),,~. 8 is said to be saturated iff 8 is ic-saturated where ic is the power of A . LEMMAFOR THEOREM 24. (i) If ic, p > o then every K-categorical PC, class J! has an ol-saturated model of power p. (ii) If 8' is a Ic-saturated model for an expansion L' of L, then the reduct of 8' to L is K-saturated. (iii) If 8, 23 are saturated models of the same power IAI = IBI and if 8 E 123, then 8 z 23. Part (i) is due to MORLEY[1965], and parts (ii) and (iii) are due to MORLEY and VAUGHT[1962]. The proof of part (i) uses elementary chains,

181

93

CATEGORICITY IN POWER

which will be introduced later on. We shall postpone the proof of (i) to Lecture 22. (ii) is simple, and (iii) uses a 'back and forth' construction. We shall not give the details here. be a PC, class which is rc-categorical. PROOFOF THEOREM 24. Let Let T' be a set of sentences in an expansion L' of L such that A% is the class of a11 reducts of models of T'. Let a, 23 E .A have power A. We wish to show that 8 E 23. We first claim that 8 E 23. For if 8 $ 23 then there is a sentence cp of L such that 8 1 cp and 23 k 1cp, whence T' u {cp} and T' u { 7cp} both have infinite models of power A; then they both have models of power K , contradicting the K-categoricity of A. We next claim that 8 and 23 are saturated. Suppose 8 is not saturated. We shall derive a contradiction. Let 8'be a model of T' whose reduct to L is 8.There is a set X c A of power 1x1 A and a set Z(u) of formulas in the language of (8,x ) , , ~ which is finitely satisfiable but not satisfiable in (8,x ) , , ~ . Then I,Z(u)I A. Letf be a one-one function on a subset U c A onto the set Z(u). Let S c A x A be defined by

-=

-=

S(a, b) iff u E U and (8,x),,~ kf(u)[b]. Form the model

a1= (8') u, S ) . ci >= 1 and l U / = Iz'(u)l

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

A n
A

(VU,

< A,

we

have

. . . U, E U)(3U)[s(U, U) A . . . A s ( U n U ) ] A

i (3U)(vU

A(vU E

E

u) v

WL

U)s(UU) A (3x1

.. .X , ) ( V U ) [ s ( U U )

*

. . .X , U ) ] .

Then $ holds in a,. Thus the set of sentences T' u {$} has a model of type (A, I U l ) . But T' u {$} is a countable set of sentences in the language M,,, where M is the language of a1.Hence by Theorem 23, T' u {$} has a model Gl = (G',V , R) of type (K, 0 ) . But then the reduct of G' to L is a model in A% of power K , and since $ holds in 6, we see easily that 0. is not w,-saturated. On the other hand, by part (i) of the lemma with p = K , -4 has an

94

CATEGORICIN IN POWER

r18

o,-saturated model of power K . This contradicts the K-categoricity of A. We conclude that '% is saturated. Similarly, % is saturated. Since '% = 8, it follows from part ( i i i ) of the lemma that '% z 23, whence A is 1-categorical. i The proof we just carried out actually gives a somewhat more general result.

DEFINITION. A class .A of models for L is said to be a PC,-class over L,,, iff there is an expansion L' of L formed by adding a finite or countable collection of new symbols and a countable set T' of sentences of I-"," such that A is the class of all reducts to L of models of T'. for some ordinal 01 2 1. THEOREM 24 (General form.) Let 1= Let IC be an uncountable cardinal. Let A be a PC,-class over L,,,. Then: (i) If every model in A of power K is w,-saturated, then every model in A of power 1is saturated. (ii) If& is K-categorical and the model of power K is w,-saturated, then A? is A-categorical. Note that the theorem holds in particular where A is the class of all models of a countable set T of sentences of Lala. It also shows, putting K = A, that if every model in .A of power 1is w,-saturated then every such model is saturated. Theorem 24 is an example of a theorem in classical first order logic which is proved by applying a result in the infinitary logic L,,, . Similarly the general form of Theorem 24 may be regarded as a theorem about a countable fragment Ld which is proved by going to a larger countable fragment La.

19 Homogeneous models

Many familiar sentences in L,,, do not have any o,-saturated models. Thus to enlarge the scope of the results of the last lecture it is desirable to replace the notion of a saturated model by a weaker notion. If L , is a fragment of L,,,, 3, 23 are models for L, X c A andfis a function on X into B, then the notation

(a, X ) X d

=Ld(~'fX)x%X

means that for every formula q ( u , bles and any xl, . . ., x, E X ,

.. .

0,) E

L, with finitely many varia-

2 != q [ x l . . . x,] iff 23 =! q v x ,

. . .fx,,].

DEFINITION. Let L , be a fragment of L,,, and IC a n infinite cardinal. A model % for L is said to be ( K , L,)-homogeneous iff for every set X c A of power 1x1 < K and every functionfon Xinto A , if X),X

=I,JWx),x

then for ally E A there existsfy E A such that

(a?X)X%X"~Y) - L . , d ( ~ ' f X ) x . X " ( y )

*

K-homogeneous means (IC, L)-homogeneous. A model 'i-4 is said to be L,-homogeneous iff it is neous where K = IAl.

(K, L,)-homoge-

We list some elementary lemmas about homogeneous models. 95

96

[19

HOMOGENEOUS MODELS

LEMMA A. If K < rl then every ( I , L,)-homogeneous homogeneous.

model is

(K, Ld)-

LEMMA B. Suppose L,, La are fragments of L,,, with L, c La. Let % be an (o, L,)-homogeneous model. Then for any set X c A and function f on Xinto A , ( 8 9

X)X€X = L , ( ~ r ’ f X ) x , X

iff (8, X),€X

=La(~?fX)x€X.

PROOF.We may assume that L, = L,,,. We prove by induction on the complexity of formulas q ( u l . . . u,) of La,,that for all xl, . . ., x, E A and y1, . .,y, E A , if

.

(% ~

.

1 * -xn) EL,(%,

.

1 * . yn)

~

(1)

and % k q [ x l . . . x,,] then 8 I= q [ y l , . .y,]. The only non-trivial step is where cp is of the form @)I),and I) ha3 the desired property. Assume (1) and % I= q [ x l . . . x,]. Then for some X E A , 8 k $[xl . . . x , x ] . By (0,L,)-homogeneity there exists y E A such that

(87

*

xnx) E L,(%

~1

-

yny).

* *

Then by the inductive hypothesis, % k $[yl . . . y,y], and hence 8 k q [ y l . . . y,]. The conclusion of the lemma follows easily. 4 LEMMA C. Let L,, L, be fragments of L,,, with L, c L,. Then for every cardinal K 2 o,every (K, L,)-homogeneous model is ( K , La)homogeneous. PROOF. By Lemmas A and B . 4 LEMMA D (MORLEY).Suppose 8,23 are both models of power K, are both L,-homogeneous, and both realize the same types in L,. Then

23.

8

PROOF.We shall prove by induction on I that for each cardinal

5

K,

For all X c A of power I there exists a function f on X into B such that (1)

(8, x)x,x and

=L , ( ~ Y f X ) X , X

7

191

91

HOMOGENEOUS MODELS

For all Y c B of power 1 there exists a function g on Y into A such that

(% Y),eY

(2) =LJ%

SY)YPY.

By hypothesis, (1) and (2) hold for all A < w. Assume w 5 A 5 K and (l), (2) hold for all p < A. We prove (1) for 1. Let X c A , 1 x1 = A. We may enumerate X , X = {xu:a < A]. For each a < 1 we shall choose by induction an element y , E B such that Suppose /? < 1 and we have chosen y , E B, a < p, such that

Since (1) holds for the cardinal a 5 p , such that (‘21s

]PI < I , there exists a sequence z, E B,

XU),S@ = L d W 3 ’

Z,)uIkl*

Then

(‘B, Y,),<@

=L,(’B?

Z.)U<@.

By the (ti, L,)-homogeneity of % there exists y s E B such that

(% Y.).g

=L,(B

Z a ) a S @ =Ld(‘21’

Xu)ass*

Thus (3) holds for p+1. It is clear that if y is a limit ordinal y S 1, and if (3) holds for all p y, then (3) holds for y. We conclude that (3) holds for p = L. This proves (1) for A. The proof of (2) for 1 is similar. It follows that (1) and (2) hold for A = K . Using (1) and (2) for all 1 < K , we can construct an isomorphism between and ‘B using the familiar back-and-forth argument. i

-=

LEMMA E. Suppose 4 is a PC, class over L,,, which has a model of has a model ‘21 power 2 &. Let ti be an infinite cardinal. Then of power K such that only countably many types in L,,, are realized in ‘21. PROOF:Let 4 be the class of all reducts of the countable theory T’ in . Let L‘, be a countable fragment containing T‘.We may assume T’ has Skolem functions for L , . T’ has models of power 2 SO by

98

119

HOMOGENEOUS MODELS

Theorem 21 T' has a model W generated by an infinite set (X,<) of indiscerniblesin L;. By the Stretching Theorem we may assume that X has power K and ( X , <) is the ordering of a real closed field of power K. Then any two increasing n-tuples in ( X , <) can be extended to an automorphism of (X,<). Moreover, %' has power K. Let % E A? be the reduct of %' to L. For a, b E A" define a b iff there exist terms t,, . . ., t,, of L' and x1 < . . < x,, y , < . . < y , in ( X , <) such that

-

.

. a, = t i ( x , . . . x,,,), bi = t,(y, . . . y,),

1

5 i < n.

That is, a and b are built from the same Skolem terms using increasing m-tuples from ( X , <). We claim that: For each n < o, is an equivalence relation on A" with countably many classes. (1) N

If a

-

b then

(W, a)

(W, b).

(2)

To see (1) we note that L' has only countably many terms t and every element of A is of the form t ( x , . x,,,). To prove (2), extend xi + y j to an automorphism of ( X , <), then by Problem 2, Lecture 13, extend this to an automorphismf of %'. Since a b, we must have f ( a , ) = b,, . .,f(a,) = b,, whence (2) holds. From (2) we see that if a b then a, b realize the same type in L,,,. Hence by (1) only countably many types in L,,, are realized in the model %. i

..

-

.

N

THEOREM 25. Let K, A be cardinals such that K > 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
=.

3. Let K > 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, <) of indiscernibles of power IYI = K such that B’* and a’* realize the same types in L*. It follows that 93’and 6’realize the same types in L. In particular, since 23’is a model of T, B’ is a model of T. Recalling the way T was defined, this means that B is isomorphic to an end elementary extension of 8. Since B‘* is generated by Y, 6 has power IYl = K. This is the required conclusion. It remains to prove (2). By the Extended Model Existence Theorem, it suffices to find a consistency property S in M,,, such that S # 0 and

104

1.20

END ELEMENTARY EXTENSIONS

for all (PE T u r a n d S E S, s u {cp} E S . We define S to be the set of all finite sets s of sentences of M,,, such that: Each cp E s is either a sentence of M, one of the sentences (I), or a sentence of the form (3) V ( c E e, + c = eb), a E A , C E C. bEa

Only finitely many c E C and d E D occur in

S,

. . . cmd1. . . d,).

s = s(cl

say (4)

%' k (Va)(3x)[a is a cardinal -+ x is a set of ordinals AIXl

(3u,

(5)

< . . .
We claim that S is a consistency property. The only non-trivial step is (C5), which reduces to: c = e,) E s E S, then for some bEa, {c E e, -, c = eb} v s E S.

(6)

If cpv@E S E S, then either s u {cp}

(7)

If vb,,(c E e,

+

E

S or s u {@}E S.

For (6) and (7) we use the fact that the formalization of the Erdos-Rado Theorem holds in the model %. Assume the hypotheses of (6). Let a be a cardinal in %. Let %' k y = 3,(la u a[). For some y E A , ( 5 ) is satisfied with y, y in place of u, x. Then using the partition relation y + (a)~,,in g,we see that for some bEu, the condition (5) holds for a with s replaced by s u {c E e, + c = t?b}- Thus

a' k (Va)(3b)(3x)[a is a cardinal -+ bEc, A X A 1x1 = ClA (vpl . . . p, E X ) ( j 1 < . . . < p, (3u)(A

s ( u B ~- - - P n ) A (ui E e,

with u = ( u l

+

ui =

is a set of ordinals

.,))I,

. . . urn),c = c i .

Using the axiom of replacement in % we see that a fixed element b E u can be chosen independent of a in the above formula. Therefore s u {c E e, + c = eb> belongs to S. The proof of (7) is similar but

20 1

END ELEMENTARY EXTENSIONS

105

involves partitions into two parts instead of la1 parts. Hence S is a consistency property. Any finite set of logically true sentences of L clearly belongs to S, SO S is non-empty. If s E S and cp E T, then s u {cp} E S because 8'k cp. To prove that if s E S a n d cp E Z then s u { c p } E S, one must use once more the Erdos-Rado Theorem in 8.The details are similar to the corresponding step in the proof of Theorem 21. -I The above proof may easily be extended to the case where 8 is a countable model of ZFC by using a lemma which was proved by Solovay using the forcing construction.

LEMMA(SOLOVAY). Let 8 = ( A , E ) be a countable model of ZFC. Then there is a linear ordering 5 of A such that ( A , E, I) is a model of ZFC in which the replacement scheme holds for all formulas involving both E and the new relation symbol 5 , and such that ( A , E, I) k (Vx)(the restriction of

5 to xz is a well-ordering),

that is, every non-empty set is well-ordered by 5 . A proof of the above lemma without forcing was announced by GAIFMAN [1968]. Our proof of the special case of Theorem 26 for models of ZFL also shows the following: Every model ( A , E, 5 ) satisfying the conditions in the above lemma has arbitrarily large end elementary extensions. However, for every end elementary extension ( B , F, 5 ') of ( A , E, 5 >, ( B , F ) is an end elementary extension of ( A , E). Thus it follows that every countable model ( A , E ) of ZFC has arbitrarily large end elementary extensions. The proof in the case that ( A , E ) is only a model of ZF uses similar ideas but is more complicated. Theorem 26 also holds for countable models of ZF with extra relations, provided that the replacement scheme holds for formulas involving the extra relation symbols. Theorem 26 also holds for uncountable models = ( A , E ) of ZF such that the linearly ordered set (ordinals of 91, E ) has cofinality 0. However, it does not hold for all uncountable models of ZF, or even Of ZFL. (See KEISLER and SILVER[1970].)

21 Elementary chains

Beginning with this lecture, we shall add one more method of constructing models in the languages L,. Up to this point we have applied the methods of the Model Existence Theorem, Skolem functions, and indiscernible elements. The new construction we take up next is the elementary chain construction. Let L, be a fragment of L,,,. We recall that 91
.

.

'ill k q [ a , . . . a,,]iff 23 k q [ a l . . . a,,]. This is equivalent to % c 23 and the relation U)oeA =L,(%

a).EA

By an elementary chain in L, of length a, where u is an ordinal, we mean a sequence 21P, /3 < a, of models for L such that whenever y < /3 < u we have 'ill,,
THEOREM 27 (TARSKI AND VAUGHT). Let 91P,/3 < u, be an elementary chain Then for all y < CI, in L,, where L, is a fragment of L,,,.

9l7
u p < . %P.

PROOF.The proof is to show by induction on complexity of formulas 109

110

12 1

ELEMENTARY CHAINS

that for every formula cp(u, a,,*..,an~Ay,

a7c q [ a , . . .a,,]

. . . on)EL,, iff

every y < a, and all

U '21Dt cp[a, , . .a,,].

B
(1)

For example, suppose (1) holds for cp. We prove (1) for (3u,)cp. If 91, t 30, cp[a, . . . a,,] then for some a, E A , , 'iY7 t c p [ q . . . a,,], whence up
-

COROLLARY. Let L, be a fragment of L,,, and let cp = An<,(Vxl . . .xp,) Vm<,cpm,,(xI . . . xPn),where each cpmn E L,. Then: (i) If 8
DEFINITION. Let (LL = ( A , R,. . .) where R is binary. %

=

( B , R', . . .) is

21 1

ELEMENTARY CHAINS

111

a strong end extension of (21 iff (21 c 8,(21 # 23, and for all a E A and b E B - A , aR'b and not bR'a. When R, R' are linear orderings, the notions of end extension and strong end extension are equivalent.

THEOREM 28. Let (21 = ( A , R, . . .) be a countable model for L such that R is a transitive, irrefiexive relation on A . For any formula p of Let,, let (3 arb. large x)cp = (Vz)(3x)(zRxA cp),

where z does not occur in cp. Let Ld be a countable fragment of L,,,. Then the following are equivalent: (i) (21 has a strong end elementary extension. (ii) (21 has a strong end elementary extension of power o (iii) Each of the following are valid in (21:

(4

(VY, z)(3x)(yRxA zRx).

(b) (3 arb. large x)V,<,O, is in L,.

+

V,&

arb. large x)O,, where O V ,,.,

(c) (3 arb. large x)(3y)cp (3y)(3 arb. large x)cp v ( 3 arb. large y ) (3x)cp,where cp E L d . (In (b) and (c), the formulas may have finitely many free variables).

PROOF.We first assume (i) c-, (iii) and prove (i) c)(ii). It is trivial that (ii) -+ (i). Assume (i). Since Ld is a fragment, the formulas (a), (b), and (c) of (iii) all belong to Ld. Thus since (iii) holds for (21, (iii) holds for 223 whenever (21 xLd8.Hence (i) holds for 8 whenever % xLd 23 and 8 is countable. Using (i) o1times, and using Theorem 27, we see that 8 has a strong elementary end extension of power wl. For we may form an elementary chain a@, < w1 in Ld such that %, = (21 and for y < p < ol, is a strong end extension of %?; thus has power at least w1 and is a strong end elementary extension of a. This proves (ii).

uB<,l

We now prove (i) + (iii). Let 23 be a strong end elementary extension of 8. Let b E B - A . (a) For anyy, z E A , we have yR'b A zR'b, whence 8 Is (3x)(yRxA zRx), and hence (a) holds in %.

112

121

ELEMENTARY CHAINS

.

(b) Suppose 'II k (3 arb. !arge x ) V,,, O,[a, . . a,]. Then 23 l= (3 arb. large x) V ,, 8,[al . . . a,]. Hence for some c E B, bR'c and 23 k Vnc,On[al. . . a,,c]. Then for some n c o,23 k O,[a, . . . a,c]. Since bR'c and b E B- A , c E B - A . Therefore for all a E A , aR'c. It follows that 'IIk (3 arb. large x)O,[a, . . . a,], and 'IIk V,,,(3 arb. large x)e,[a,

. . . a,,,].

(c) Suppose 'IIl= (3 arb. large x)(3y)cp[al. . . a,]. Then the same holds in 23. Hence for some C E B, b R c and '23 k (3y)cp[a, . . . a,c]. So for some dE B, '$3 i= cp[al.. . a,,,cd]. Since bR'c, C E B - A . Case I. d E A . Then for all z E A , B ' k ( 3 x ) ( z R x ~cp)[a, . . . a,d], so the same holds in 'II, and 23 i=(3 arb. large x)cp[a, . . .a,d], and % k ( 3 y ) (3 arb. large x)cp[al . . . a,]. Case 11. d E B- A. For all Z E A , zR'd, so '23 I= (3y)(zRyA 3xcp)[al . . . a,], and since the same holds in %, % k (3 arb. large y)(3x)cp[al . . . a,,,].

Finally we prove (iii) (i). Assume (iii). Add to L a constant symbol en for each a E A and a new constant d. Let L[, be the least countable fragment containing L, and these new constants. Let a' = (a, Let T be the set of all sentences O(d) of L, such that --f

(3 arb. large x )

%' k That is, rtr' k (3z)(Vx)(zRx cp =

--f

O(x).

O(x)). Let cp be the sentence

A (Vy) V ( y = eb v enRy).

REA

bsA

Thus T is a set of sentences of LL, and cp is of the form Ancto(Vy)Vmcocpmn, each cpmn E LL. Since R is transitive and irreflexive, aRb implies that not bRa. It follows that for any model '$3" = ('$3, b,, b)aEAof T u (cp}, 2 '3 is isomorphic to a strong end elementary extension of 8. To show that T u {cp} has a model we use the Omitting Types Theorem (Theorem 13). It thus suffices to prove that: T is consistent.

(1)

If (3y)$(y) E L, is consistent with T and a E A , then there is a b E A such that (3y)($(y)A ( y = env e,Ry)) is consistent with T. (2)

21 1

113

ELEMENTARY CHAINS

We first prove (1). We add to L> a countable set C of constants, forming M,. Let S be the set of all finite sets s = s(cl . . . c,d) of sentences of M, such that

rz1' t (3 arb. large x)(3ul . . . urn)A

s(ul

. . . urnx).

Then S is a consistency property. The crucial step, part (C5) of the definition, is taken care of by part (iii)(b). For if O V ,,, E s E S, then

'11' P (3 arb. large x)(3u1 . . . urn)v (As A en), n
'11' P (3 arb. large x) V ( 3 u , . . . urn)(A S AOn), n<w

and by (iii)(b), there is an n < w such that

'11' t (3 arb. large x)(3ul . . . urn)(As

A

On),

whence s u {On} E S. Moreover, S # 0. For by (a) we have

1 ' 1' t (3 arb. large X)(X whence {c

= x),

= c } E S. We claim that

If s E S and $ E T, then s u {$} E S. For we have %' i= (3z)(Vx)(zRx

S'

and by (a),

. . . urn)As),whence (3ul . . . urn)As). From

rz1' C (3 arb. large X ) ( Z R X A(3u, '11' i= (3 arb. large x)($(x)A

+ $(x)),

(3)

(3) it follows that

= { S u T:s E S }

(4)

is a consistency property. Therefore (1) holds. We now prove (2). We claim that

(51

$(d) is consistent with T iff 1 ' 1' i= (3 arb. large x)$(x).

For if 1 ' 1' P (3 arb. large x)$(x) then {$(d)} E S, whence T u {$(d)} E S' and has a model. On the other hand, if $(d) is consistent with T then (3 arb. large x) $(x), whence % C 11(3 arb. large not % i= x) 11$(x), % t (3 arb. large x)$(x). Now suppose (3y)$(d, y ) is consistent with T.Let a E A . Then

1 ' 1 (3 arb. large x ) [ ( 3 y ) ( $ ( x y ) A e,Ry) v (3y)($(XY) A

e,RY)].

1

114

[21

ELEMENTARY CHAINS

Using (b), we have two cases. Case I. 8’C (3 arb. large x)(3y)($(xy)~e,Ry). Then a’ C (3 arb. large X ) ( ~ ~ ) ( $ ( W () yA= eb v e,Ry)), for any b E A . Thus by (51, ( 3 ~ ) ($(dy) A ( y = ebv e,Ry)) is consistent with T and (2) holds. Case 11. a’ C (3 arb. large x)(3y)(ll/(xy)~ e,Ry). Clearly, W C (3 arb. large y)(3x)($(xy)~ e,Ry). Therefore by (ii i)(c)

a’ C (3y)(3 arb. large x)($(xy) A

e,Ry).

Hence for some b E A , 8’C (3 arb. large x)ll/(xeb), whence (3y)($(dy)~ ( y = ebv e , R y ) ) is consistent with T. Again (2) holds. Our proof is complete. -I

22 Another two cardinal theorem

In this lecture we give some applications of Theorems 27 and 28 from the preceding lecture. Our first theorem is a result about end elementary extensions of models of Zermelo set theory. By the rank function we mean unique function r(x) on sets to ordinals such that, for each set x, r(x) is the least ordinal u for which r ( y ) < a for all y E x. In ZF, the rank function is definable and for each ordinal a, the family {x:r(x) < a> is a set, denoted by R(u). THEOREM 29. Let % = ( A , E ) be a countable model of Zermelo set theory in which the rank function is definable and for all ordinals a,the set {x: r(x) < a} exists. The following are equivalent: (i) % has a proper end elementary extension. (ii) % has an end elementary extension of power w l . (iii) % is a model of ZF.

PROOF.Let R be the relation such that aRb means r(a) < r(b). Then R is definable in % and is transitive and irreflexive. Moreover, % satisfies conditions (iii)(a) and (iii)(b) of Theorem 28 when we let Ld be first-order logic L. Rewriting (iii)(c) in the form

(vY)(W(wcp

+

r(4

< a A (3P)(VY)(VX)(cp +

+

r(y) < S)

+

(W(WVY)(cp

+

r(x)

< a),

we see that % satisfies (iii)(c) if and only if % is a model of ZF. The result thus follows from Theorem 28. -I 1 I5

116

ANOTHER TWO CARDINAL THEOREM

122

Theorems 30, 31, and 32 below are all in KEISLER[1966]. The next application is another useful Lowenheim-Skolem-Tarski theorem for two cardinals.

30. Let K , A be cardinals with K > A 2 w. Then: THEOREM (i) Let '$ =I( A , U, .) be a model L type ( K , A) and let L, be acountable fragment of L,,,.Then thereexist models 23,B such that 23
..

PROOF.(i) implies (ii), because there is a countable fragment L, with T c L,, hence if is a model of T then so is 6. We prove (i). By the Downward LST Theorem, there exists a,
(a, R) k

(3 arb. large x ) U ( x ) .

(1)

Moreover, it is easily seen that (a,R)satisfies the condition (iii) of Theorem 28. By the Downward LST Theorem there is a countable model (23, R')
221

ANOTHER TWO CARDINAL THEOREM

117

THEOREM 31. Let A be a PC,class. If A is o,-categorical, then d is categorical in every power I > o. PROOF.A has an o,-saturated model of power w , . The proof is just like the proof of Theorem 24 but uses the Two-Cardinal Theorem 30 instead of Theorem 23. One shows that if dl has a model of power I which is not saturated, then d has a model of power w1 which is not saturated. i The special case of Theorem 31 where A is the class of all models of a theory in L is a consequence of the Morley Categoricity Theorem. The next result is analogous to Theorem 25. For the corollaries it is important that we have defined A to be ic-categorical iff d has at most one model of power K up to isomorphism. Thus if 4 has no models of power K, it is still said to be K-categorical. THEOREM 32. Let A be a PC, class over L,,, . Let L, be a countable fragment of L,,, . Suppose that every model % E 4 of power w1 is L,-homogeneous. Then every uncountable model in 4 is L,-homogeneous. PROOF.Like the proof of the Theorem 25. -i COROLLARY 1. Let A be a PC, class over L,,,. Suppose A is w,-categorical and for some countable fragment L, of L,,,, the model of d of power o,is L,-homogeneous. Then A is I-categorical for every cardinal A > 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
d V l

A

,

. %)EL&

f-,

Call the above sentence 8. By Theorem 30, T' u ( 8 } has a model (B',Y ) of type (wl, w ) . Since 8 holds in this model, every n-tuple in B realizes the same type in L, as some n-tuple in V. But V is countable, so the reduct 93 of 123' realizes only countably many types in L d . Since 123' k T'. B E A. Finally, 8 has power wl. -1 Let 23 be a model for L, X c B, and L, a fragment of L,,, . Form the language L,(X) by adding to L a constant Cb for each b E X and taking all formulas obtained from formulas in L, by substituting finitely many constants for free variables. Let 123, be the model (93, b)b,x and T, the complete theory of $' 3, in the language Ld(X). We recall from Lecture 12 that a type (in one variable) in L,(X) with respect to T, is a set of formulas @ ( x ) c L,(X) such that for some model '21 of T, and a E A , @ = { d x ) E L,(X): ax 1 cP[al>. That is, the set of all formulas of L,(X) satisfied by an element a in some model 8, of T, is a type w.r. T,. When the model 123 is fixed, we may use a briefer notation. By a type in Ld over X we mean a type in L,(X) with respect to T,. In particular, by a type in L, over '$3 we mean a type in L, over B. Since 91B is a model of TBif and only if 8 iLd%, we see that @(x)is a type over '$3 if and only if for some % >L, 123 and a E A , @ =

{cP(x)

L d ( B ) :%B

(P[al}*

221

ANOTHER TWO CARDINAL THEOREM

119

A type in L, over Xis realized in 23 iff it is satisfied in BXby some b E B. In general, there are at most 2''' types in L, over an infinite set X , of which at most (BI are realized in 8. THEOREM 34. Let ,A? be a PC, class over L,,, an let L, be a countable fragment of L,,, . (i) Suppose that for every countable model 23 E A, there are at most countably many types in L, over 8. Then for every infinite model 0.E A, there are at most IC( types in L, over 6. (ii) Suppose that for every model B E ,A? and every countable set X c B, 23 realizes at most countably many types in L, over X . Then for every model Q E ,A? and every infinite set Y c C, Q realizes at most (YI types in L, over Y.

PROOF.(i) Suppose A%is the class of all reducts of models of T' to L. Let B be an infinite model in ,A?, say ( C ( = I. Let 6' be a n expansion of 0: to a model of T'. Assume that there are R+ different types Za(u), u L,B and let ua E C, realize Z,(u) over& We may assume C, n C, = C, IC,l = I,for u # p. Let D = C, v A+. Form the model b =
ua
9 k (Vu, p

EA+)(7 u =

p -+ q(vl

v

... U,Y)€Ld

(3x1 . . . xn E C )

. . x n f ( a ) ) tf ~P"(x, . xnf(B))l)~

1[cpCb(x1 *

A ( v E E 2')

* *

A PEL,

(VX,

. . . X,

E C)[qc(X1

. . . &) * cpCa(X1 -

- xn)].

Call the above sentence $. Also for each 8 E T, 5B k 8'. The model 3 has type ( I + , I). Then by Theorem 30 the set of sentences {&: 8 E T'}u ($1 has a model boof type (a1,w). The interpretation of C in bogives us a countable model 23 E A. The function g makes sure that the inter-

120

ANOTHER TWO CARDINAL THEOREM

[22

pretation of'A in a,, has power wl, and the sentence $ shows that there are at least w1 types in L, over 23. The proof of (ii) is similar. -1 The above theorem was first proved by Morley in a different way in the special case that .A? is the class of all models of a set T of sentences of L. This application of Theorem 30 was suggested by J. Silver. We now have the tools necessary to prove part (i) of the lemma for Theorem 24. Recall that this lemma was stated and used in Lecture 18 but its proof was postponed until now. Let us restate the lemma. FOR THEOREM 24. (i) If K , p > 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 , <) of indiscernibles of order type ( K , <). Suppose there are at least w1 types over M. By the Compactness Theorem for L there is a ! which realizes w1 different types over model B > M of power w1 in A M. 0.has an elementary extension of power K 2 wl, so by the K-categoricity of 4 we may assume that 0. < 23. Then A c B and 23; realizes at least w, types. (2)

-

Let X be a countable set X c Y such that A c Hull ( X ) . For y , y' E Y put y y' iff for all x E X , x < y o x < y' and x = y 9 x = y'. Then the equivalence class of y depends only on the least x E X with y < x if there is one, and on whether y = x. Hence there are only w equivalence classes under -. It follows from the indiscernibility of ( Y , <) in B* that any two m-tuples <~I(Y1

(tl(y;

-

* *

* * *

..

*

tnbl

* * *

~m)>,

* * *

tn(A

* * *

~6))

~ r n )

~6)

22 1

wherejJi

-

121

ANOTHER TWO CARDINAL THEOREM

yI for i

y, <

m, and

. . . < y,,y;

realize the same type in

< . . . < ykin(Y, <),

8;.Therefore

23: realizes only countably many types.

(3)

But since A c X , (3) contradicts (2). This proves (1). We now use (1) to prove the lemma. By (1) and Theorem 34, we have For every model

B E A? of power p, there are at most /A

types over a. (4) Let Bo E A have power p. Using (4) and the Compactness Theorem, form an elementary chain

a,,

a

s

0 1

,

such that each amE A, each B, has power p, and each a,+ realizes every type over a,. In forming the chain we actually form an elementary chain of models of T and take unions at the limit ordinals. Then since every countable set X c A,, is included in some A , , a < ol, the model B,, E A is o,-saturated and of power p. 4

PROBLEMS 1. Prove Theorem 26 for arbitrary models of ZFC whose ordinals have cofinality o.

2. Prove that if CI is the first inaccessible cardinal > o,then ( R ( a ) , E) has no end elementary extensions. 3. Let B be a countable model for L. Suppose that B 2". Thus the converse of Theorem 33 fails. HINT:consider a binary tree of length o 1 .

+

5. Let A be a PC, class over L,,,. If every countable B E A? has at most o automorphisms, then every B E A? of power 1 has at most 1 automorphisms.

122

ANOTHER TWO CARDINAL THEOREM

[22

6. Call two elements a, b of a model % ' in the same automorphism class iff there is an automorphism of l!i mapping a t o b. Let .A be a PC, class over L,,, . If every model of A of power w1 has w1 automorphism classes, every model in A! of power I > w has I automorphism classes.

23 More about categoricity in power

Our next theorem is a generalization of the Morley categoricity theorem to countable fragments L,. It was announced in KEISLER[1969] without proof. THEOREM 35. Let Ld be a countable fragment of L,,,, T c Ld a set of sentences, and K, I > w. Assume that: (I) T is K-categorical. (11) For every countable model % of T, there are models B of arbitrarilylarge power such that $2
REMARK1 . In the special case that L, is the first order logic L, hypothesis (11) holds and (I) => (111). Thus only the hypothesis (I) is needed, and the theorem reduces to the Morley Categoricity Theorem. REMARK 2. In the special cases that either K = w1 or I = w l a , ct 2 1, Theorems 32 and 25 already imply the above theorem. In fact, in these two special cases the result holds for a PC, class A over L,,, and holds without hypothesis (11). REMARK 3. The following questions are open: Does Theorem 35 hold without hypotheses (III)? Does (I) imply (III)? We shall prove a series of Lemmas leading to the proof of Theorem 123

124

MORE ABOUT CATEGORICITY IN POWER

[23

35. In what follows we shall assume all the hypotheses of Theorem 35. Our first lemma reduces the proof to showing something about models of power o1and IC. We need the notation below:

If%!=TaandXcA,then: L,(X) is the least fragment containing Ld and a constant c, for each aEX; 'zlx is the model (aya),sX; Txis the set of all sentences cp E L,(X) true in L> 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
Hu//(Y,)

PROOF.By the Downward LST Theoremythere is a countable model

91’ such that X c A’ and %’ X Ld-.‘$1. By (11) there are arbitrarily large ,models B’ > Ld %’. It follows that T, has arbitrarily large models, since mi, 93; must be models of T,. Then T,* has arbitrarily large models. Then by the Upward LST Theorem and the Stretching Theorem, T; has a model 23; generated by a set ( Y , <) of indiscernibles of type (K, <). Thus (i) holds. Let % = {as:/?< ol}.It follows from (I) and (11) that for each

126

[23

MORE ABOUT CATEGORICITY IN POWER

countable model a’ of T there is an elementary embeddingf’: a’
Then the mapping a, -+ b, will give the embedding f required in (ii). If 01 5 w1 is a limit ordinal and (I) holds for all a’ < a, then (1) holds for a. Thus it suffices to assume (1) holds when ct = y and that b, E Hull (Yl) for all /? < y, and then show that there exists by E Hull ( Y , ) such that (1) holds when a = y+ 1. Here y is an arbitrary countable ordinal. Let ‘W be a countable model such that a’ < L d % and X u {aB:B 5 y } c A’. Then there is a mappingf’: a’ i L d B .Therefore

By (111), B is (a,,L,)-homogeneous.

Hence there exists c E B such that

.

We must show that we can choose c E Hull(Yl). For some term t(vl . . u,,) of L*,(X) and some y1 < . . . < y , in ( Y , <) we have c = t ( y l . . yJ. Since y < w1 there is a y E Y , such that for all j? < y, b, E HuN({y’ E Y Y’ < YI). For i = 1, . ., n let

.

.

y ; = Yi if

Yi < Y the irhsuccessor of y if y

..

yi.

Then y ; , . . .,yi E Y , . Set b, = t ( y ; . yi). Then by E Hull(Yl). It follows from (2) and the indiscernibility of ( Y , <) that

(B, a, b&EX,

p
i1

(3)

ELd(B,f’a,f’aa),€x,8
Since f’:a’ i L d B and a‘ (X) has only countably many different terms and the set f(A) has power wl, there must be a single term t ( v , . v,,) of L*,(X) such that for w1 different n-tuples y1 < . . < y,, in (Yl,<), t ( y , . . .y,) E ~ ( A )Thus . for each /? < wl we may choose a different n-tuple yf < . . . < in (Yl, <) with r(yf . . d )E

..

.

.

23 1

127

MORE ABOUT CATEGORICITY IN POWER

f ( A ) . Consider the greatest integer m 5 n such that there is a set W c w , of power I WI = o, where, for all p, y E W, we have (yf . . . y t ) = = ( y : . . . y i ) . Then 0 6 m < n. (The case m = 0 means that there is no uncountable set of B’s with the same yf). Since (Y , , c) has order type ( 0 ,, <) and m is the greatest such integer it is not hard to see that we can find a strictly increasing sequence & E W, y < a,, such that y < 6 < w1 implies y? < yk+ . Thus the sequences ( y f y. . . &),( y f 6 . . .y.86) have the order relationship

,

y? = y p , . . ., y’,

yp<

B

BS

(4)

= y,,

. . . < y: < ys,’,l < . . . < y.8v < y k + , < . . . <

yfd.

Let Z, = { t ( y p . . .&): y < wl} and order Z, according to the indices y. Since (Y,<) has order type (K, <), we may choose n-tuples y p < . . . < yfly in (Y, <) for all w , y < K such that the order relation (4) holds for all y < 6 < K. Let 2 = { t ( y f ” . . .&):y < K} and order 2 according to the subscript y. Then (Z, <>has type (K, <) and Z , is the set consisting of the first w , elements of 2. Since (Y, <) is indiscernible in 23; and (4) holds, (Z, <) is indiscernible in 8;.By the definition of Z,, 2, c f(A). This verifies (iii). -1

LEMMAc. Let % be a countable model of T. Then there are only countably many types over % in Ld.(Equivalently, the theory T A has only countably many types in L,(A).) PROOF.By hypothesis (II), TA has arbitrarily large models. Hence T,* has arbitrarily large models, whence by Theorem 21 T,* has a model 932 generated by a set ( Y , <) of indiscernibles in L:(A) of power K. Moreover, since BA is a model of T A , % Since L*,(A) has only countably many terms and since the type realized by an m-tuple (fl(y11 yl,,). . t m ( Y m l . .ymnm))of elements of B,, depends only on the terms c,, . ., t,,, and the relative order of the elements y i j in (Y, <), the model BA realizes only countably many types in L,(A). We claim that:

-

. .

.

Every type over % in L, is realized in BA. For let

Q > L d % and let

c1, . . ., c,

E

C realize the type Z(u,

(1)

. . . u,)

128

123

MORE ABOUT CATEGORICITY IN POWER

over % in L,. By the Downward LST Theorem we may assume C is countable. Then by (11), (5 has an extension 8' >L,(5 of power K, and by (I), 23' 8. Thus there is an embeddingf: (5
(3, @ ) a s A = L , m

a)aeA = L , ( w - a ) a E A

=(G ,.'.

a)oeA *

By (111), the (wl, La)-homogeneity of 23, there exist dl, . . ., d, E B such that

(B, a, d1 .

* *

dm)aEA

~L,(@,fa,fc,

* * .

Therefore {dl, . . ., d,> realizes C(v, the result follows. -I

f ~ m ) o eEL,((& ~

. . . v,)

a, ~1

* * *

Cm)aEA*

in BA.Thus (1) holds and

COROLLARY. Let % be a model of T and X c A a set of power 5 o.Then Tx has only countably many types in L,(X). PROOF.We may assume 2l is countable, by the Downward LST Theorem. mapping Z -+ Z n Ld(X) maps the set of all types over % in L, onto the set of all types of Txin L,(X). -1 Recall from Lecture 12 that if T' is a complete theory in L,, then a formula ~ ( v ., . v,,) E L, is complete with respect to T' iff is consistent with T' and for all $(ul . v,) E L,, T' k cp + $ or T' k cp -+ 1$. Moreover, 8 is incompletable w.r. T' iff 8 E L,, 8 is consistent with T', and there is no complete formula cp w.r. T' such that T' k cp + 8. Our next lemma is an application of Theorem 16 from that lecture.

.

. .

LEMMA D. Let % be a model of Tand X c A . Then there are no incompletable formulas 8 E L,(X) with respect to T,.

PROOF.Suppose first that X is countable. By the corollary to Lemma

c, T, has countably many types in L,(X). By Corollary c to Theorem

16, Tx has a prime model, whence by Theorem 16 (iii), there are no incompletable formulas $ E L,(X) w.r. T,. We now consider the general case where X is arbitrary. Suppose 8(vl . v,,) E L,(X) is incompletable w.r. T,. Say 8 = B(vl . . . v,a, . . . a,), where a,, . . u , E X . Let X , = {al . . a p } . For each cp(ul.. . v , ) ~ L,(X,), such that

..

.

.

23 1

129

MORE ABOUT CATEGORICITY IN POWER

cp is consistent with Tx and T, k cp B,(ul . . . v,) E L,(X) such that

both c p ~ 8 , and cph completable). Let X , = {a E

-+

8, there is a formula (1)

8, are consistent with T, (because 0 is in(2)

x:a occurs i n 8,

for some cp] u X,.

XI is countable because L,(X,) is. We repeat the process, considering all formulas q ( u , . . . v,) E L,(X,), andsoon. We thusobtainasequenceX, c XI c X , c . . , ofcountable subsets of X such that for any m < w and formula q ( u , . . . v,) E L,(X,) for which (1) holds, there is a 6 , E L,(X) for which (2) holds. Now let X , = U m < , X m .Then X , is a countable subset of X , L,(X,) = L,(X,), and hence for every cp E L,(X,) satisfying (1) there is a 8, E L,(X,) satisfying (2). This shows that 8 is incompletable w.r. Txo. This contradicts the first paragraph of the proof. i

urn<,

The next lemma should be compared with Theorem 16. LEMMA E. Let 2 be an infinite model of T and X c A . Then T, has a model Bx of power 1B1 = w u 1x1 such that every n-tuple in BX satisfies a complete formula of L,(X).

PROOF.We first show the following: If a, E A for all y < p and each a, satisfies a complete formula with respect to TXvfaa:d satisfies a complete formula with respect to Tx. (1) We prove (1) by induction on y. Suppose y satisfies a complete formula w.r. T,. By hypothesis, ay satisfies a complete formula +(u) w.r. TXv(a6:d
. . . U,V)

=

e(u,

. . . U ~ ) A$(uu, . . . 0,).

[23

MORE ABOUT CATEGORICITY IN POWER

130

Then (asl . . .asmay>satisfies a. We claim that u is a complete formula w.r. Tx.To show this let q ( u , . . .u,u) E L,(X). If

$(uusl . . . us,) + cp(asl . . . usmu),since $ is complete, then Txu{ad:,
ax c

1

. * * usma,17

we have T, t a --t q by the same argument. This shows that a is complete and completes the proof of (1). Now let /3 = 1x1 u w. Then we may enumerate all formulas cp(u) E L,(X) in a list cp,(u), y < j?. Choose a sequence a, , y < j?, of elements of A such that: If T, != (3u)cpy(u) then u, satisfies cp,(u); a,

(2)

satisfies a complete formula w.r. to Tx,(a,:d
Thisis possible because, by Lemma D, the formula q,(u) is not incompletable with respect to T,,,,, :,<,). We now repeat the process with an enumeration cps+,,(u), y < p, of all formulas cp(u) E L,(X u {u,: y < B}), and then repeat the construction again w times. We then obtain a sequence a,, y < /3 o,such that for each n < w , cpg .n+y(u), y < w , is a list of all formulas cp(u) E L,(X u {a, : 6 c /3 * n}), and the analogues of (2) and (3) hold for all y < fi * o.Let B = (a, : y < /3 o}),and 8 the submodel of % determined by B. Then X c B. By (2), '23 % +,, whence 8, is a model of T,. Finally, by (1) and (3), every n-tuple in 8, satisfies a complete formula of L,(X), and B has power 1x1 u a. -I a

-

PROOF OF THEOREM 35. It remains only to show that the hypotheses of Lemma A hold. Let % be a model of T of power w , and let X c A have power S w. Let %,?: (Y, <), and (2, < ) b e as in lemma^; that is'23: is a model of T ' of power K and (i)-(iii) hold. Form the model BxVzand the set Txuzof all sentences of L,(X u Z ) holding in BxUz.By Lemma E applied to Bxuz,Txuzhas a modelQxuzof power (Xu 21 = K in which

231

131

MORE ABOUT CATEGORICITY IN POWER

every n-tuple satisfies a complete formula. Then Ex is a model of T, of power K. We will be finished if we show that Every type in L,(X) which is realized in B, is realized in

a,.

(1)

(By Lemma A). Let c , , . . . c, E C, and let C(u, . . . v,) be the type in L,(X) which ( c , . . . c,) realizes. ( c , . . . c,) realizes a complete formula B(v, . . . 0,) E L,(X u Z ) w.r. to Txuz.Let z, < . . . z,,, be the elements of Z which occur as constants in 8. Thus we may write 0 = O(ul . . . u,zl . . . z,,,). Let z ; < . . . < z; be the first m elements in ( Z , <). Thus z; ,. . .z,!,, E 2,. We write

-=

e’ = q u , . . . U n z ; . . . z : ) . We have Txvz k (3v,

. . . v,,)O,

and for all

CT

E C,

e -,c. Then by the indiscernibility of ( Z , <) in Bx, T~~~c

TxUz, k(3u,

. . . v,)O’

and TxuzlC 8’-+

CT.

By (iii) of Lemma B, Z , c f(A),whence

(ax

9 . f

- lz)z,z‘

k

TXUZ,

*

(3)

ax

But (2) and (3) imply that realizes Z.This proves (1). -I We conclude our discussion of categoricity in power by showing how the Morley Categoricity Theorem folIows from Theorem 35. The Morley theorem may be stated as follows: MORLEYCATEGORICITY THEOREM (Restated). If Ld = L then Theorem 35 holds without hypotheses (11) and (III). We can immediately note the following. REMARK 1. If Ld = L, then (11) holds. For the Upward LowenheimSkolem-Tarski Theorem holds in general for L. REMARK 2. If L , = L and (I) holds, then (111) holds. For by the Lemma for Theorem 24, T has an @,-saturated model of power IC, and every @,-saturated model is a,-homogeneous. From Remarks 1 and 2, Theorem 35 implies the Morley Categoricity Theorem.

24 Extending models of set theory

In the next few lectures we shall apply the Omitting Types Theoremand the elementary chain construction to the study of models of set theory. [1970]. We The results in the next five lectures, 24-28, are in KEISLER first need some notation. Suppose (11 = ( A , F ) , 23 = ( B , F ) are models of Zermelo set theory, and let (11 < B. For each a E A , we let

aE = { b E A : bEa}, and we define a, for a E B similarly. An element a E A is said to be $xed by B iff a, = a E .Otherwise a is said to be enlarged by B.

EXAMPLES. B is an end extension of M if and only if every a E A is fixed by B. (See Lecture 20). If ( a E ,E ) is well-founded and a is fixed by B, then (aF,F ) "= ( a E ,E ) , so (a,, F ) is also well-founded. If B has exactly the same ordinals as M, and 91 is well-founded, then B is also well-founded, because every element of 8 or 8 has a rank. In Zermelo set theory a regular cardinal is defined to be a cardinal with the following property:

m

If 1x1 = m, IyI < m,and x = U { f ( z ) :z E y } for some functionf, then there exists z E y such that lf(z)l = m. We shall find the following generalization of a regular cardinal useful. 132

241

133

EXTENDING MODELS OF SET THEORY

If cp(v) is a formula in the language of the expanded model define cp(P1) = { a E A : MA k cp[a]}.

a,

we shall

p(%)will be called a class in 8 (definable with parameters from A ) . Let cp(%), $(a)be classes in M. We say that cp(8)is regular over $(a)iff the

following holds in MA:

For all x , y , an d f , if ~ ( x and ) $ ( y ) andf is a function and x = u { f ( z ) :z E y } , then there exists z E y such that cp(f(z)). Forexample, theformula 1x1 2 rn is regular over the formula Jyl < m if and only if rn is a regular cardinal in 8. We sometimes speak of the formula q ( v ) being regular over $ ( u ) in 8,when the class cp(8)is regular over $(a). THEOREM 36. Let (21 = ( A , E ) be a countable model of Zermelo set theory. Suppose the class cp(%) is regular over the class +(%)and that the sentences (W(cp(U)

+

I4 2 2), ( V W 4

s 2 $(4) +

hold in a,. Then there is an elementary extension @ = ( B , F ) of % such that: (i) Every a E (~('21)is enlarged. (ii) Every a E $(a)is fixed. (iii) For every b E cp(@), lbFl = ol. (iv) For every b E $(B), IbFJ 5 w.

PROOF. Our first observation is that If

%A

k ( ~ ( au , a,), then a, E (~(9) or a,

This is because 8, k $(2) and cpp(8)is regular over duction we see that If

k ~ ( au ,

E cp(%).

(1)

$(a).By an easy in-

. . . u un),then a, E q(%)for some i 5 n.

(2)

To simplify matters, we observe that: Without loss of generality we may assume that the class cpp(8) is closed under the relation 2 , i.e. 'Ir, k cp(x) r \ y 3 x + cp(y). (3) To prove (3) let cp'(u)

= (3u)(u c u A cp(u)). Then

cp'(8) is closed under

134

[24

EXTENDING MODELS OF SET THEORY

3 , and % t q ( x ) + q ‘ ( x ) , whence q ( N ) c q’(%).The other fact needed to prove (3) is that q’@) is regular over $(91). Suppose r, s,f E A and

k q ’ ( r )$(s) ~

A

f i s a function

A I’ =

u{f ( z ) : z

E s}.

Then there is an element t E A such that 1t

C I.Aq(f).

Let g c A be such that Then

a” t (‘dz)(g(z) = t Cm). a” t t

=

u

{ g ( z ) : z E s},

hence by regularity of q over $ there exists z,Es with %A k q(g(z,)). Then %A t q’(f(z,)), and therefore q(9)is regular over $(a). This proves (3). We assume hereafter that ~ ( 9 1 is ) closed under 3 . Our next step is to prove: Let a E (a). Then there is a countable (21’ and fixes all d E $(a).

>

which enlarges a

(4)

Let L(A) be the first order language of the model aA,and form L(A u {c}) by adding a new constant c. Let Th(aA)be the set of all sentences of L(A) true in a,. Let

T = Th(%A)u { c E ~ u } {C$b : bEA-q(%)}. Consider the infinite sentence x given by

x=

A (Vo)(u$dv V u

d6WJ)

eedr:

= e).

Suppose QAv(e) is a model of T u {x}. Then by 2, 0: fixes all d~ $(a). By Th(21A),Q is an elementary extension of a. We must have c E Q - A , because if c E A then %A t q ( { c } ) and thus T k c # { c } , which is impossible if T u { x } has a model. Therefore by c E a, a is enlarged by (5. Hence to prove (4) it suffices to show that T u {x} has a model. We shall show this by applying the Omitting Types Theorem. We claim that for every sentence 8(c) of L(A u {c}), d ( c ) is consistent with T if and only if

9, k q ( { x E a: d ( x ) } ) . ( 5 )

241

135

EXTENDING MODELS OF SET THEORY

q ( { x E a: O(x)}). Then by the definition of T, Assume that %A k the formula c E b is an axiom of T, where b is the element such that %A k b = {x E a: e(x)}. But then T k (c E a h q c ) ) . Since also TI=c E a, we have TI= -,q c ) . O(c). Then for some b,, . . . 6, E Conversely, suppose that T t A -cp(fl),

T ~ ( ~u J {Jc E a> u {c 4 bl Let 6‘ be the element of A such that (2) we have b’ E A - q(%). Moreover Then

A.

. .A c 4 b,}

%A

k b’ = 6, u

n(aA)u iC E a} u { c + b’

whence and

k

P x E a A x 4 b’

+

1

P x E aAB(x)

+

x E b’,

nAc {x E a : e(x))

I.

e(c).

. . . u b,.

Then by

e(c).

qx),

c b’.

Since b ’ ~ A - r p ( % )and we are assuming q(%) is closed under have the desired conclusion

c

%A

3,

we

V({X E a : e(x)}).

Thus the claim (5) is proved. Now assume that

d E I@), and (3y)B(cy) is consistent with T. BY

(6)

( 5 1 9

%A

k

V({X E

a :( 3 Y ) m ) ) ) .

Then by (1 ) we have either or

%A

q({xEa:(~y)(y4dAe(xy))})

%4

c V({X E a : (3Y)(Y E d A e ( x Y ) ) ) ) .

(7)

(8)

In case (7), we conclude from ( 5 ) that (3u)(u

+ d~ e ( c u) ) is consistent with T.

(9)

136

[24

EXTENDING MODELS OF SET THEORY

In case (8), we use the assumption that q(8)is regular over we see that for some e E d E , k q({x

Ea:

$(a) and

e(x e ) } ) .

Therefore O(ce) is consistent with T,so (3u)(u = e A O(cu)) is consistent with T.

(10)

We have shown that (6) implies (9) or (10). It then follows from the Omitting Types Theorem (Theorem 13) that T u {x} has a countable model. This completes the proof of (4). We note that every countable model = 8 satisfies the hypotheses of the theorem and thus satisfies the conclusion of (4). We next prove

8 has a countable elementary extension 23 which satisfies conclusions (i) and (ii) of the theorem, i.e. every a E q(8)is enlarged and every UE$(%) is fixed. (11) Since 8 is countable, q(%) is at most countable and we may list its elements, q(8)= {a,, a,, . . .}. (If q(%) is empty we may take B = 8 and the theorem holds trivially. We now tacitly assume cp(8) # 0). This list may have repetitions. Using (4) w times we form an elementary chain 8 = $lo < g1< . . . of length w such that each a, is countable, and in passing from anto 8,+ the element a, is enlarged but each b E $(a,) is fixed. By induction on n we see that in passing from 8 to an+,each b E $(a) is fixed. Now let B = u,.=m8,. Then 23 clearly satisfies the conclusion of (11). We remark, as in the case of (4), that since every countable model 8, = 8 satisfies the hypotheses of the theorem it satisfies the conclusion of (11). To complete the proof of the theorem we apply (11) w1 times. We thus form an elementary chain a < o,of countable models such that

8,

=

8;

For limit ordinals a, 8, = uB<,8B; For all a < wl, every a E q(8,) is enlarged in passing to i?l,+,; For all a < wl, every a E $(a,) is fixed in passing to a,+,.

We let 23 =

=

L),<m,%,.By induction on y we see that whenever

241

EXTENDING MODELS OF SET THEORY

137

a E $(2lm)and GI < y 5 wl, a is fixed in passing from to a,. This shows that % satisfies conclusion (ii). Furthermore, if b E $(B)then for some GI < ol, b E $(Bm), whence b, = bEaand so Ib,l 5 o. This shows that 23 satisfies conclusion (iv). Each a E cp(8) is enlarged in passing to ?Il, and thus is enlarged in passing to 23, for 8 < 8,< B.Hence 23 satisfies (i). Finally we verify that 23 satisfies (iii). Let b E cp(B). For some GI < co1, b E cp(?Ia). Then for all y , CI 5 y < wl, b ~cp(8~). Hence for each such y there is an element b, E bE,,+,-bE,.Since the sets bEY,GI 5 y < o1 form an increasing chain of subsets of b,, the set {by:GI 5 y < ol}is an uncountable subset of b,. Thus IbFJ2 ol.But each is countable, so IBI 5 o1 whence lbFl = ol. This verifies (iii) and completes the proof. i

am

COROLLARY A (KEISLER and MORLEY[1968]). Let i?l = ( A , E ) be a countable model of Zermelo set theory and let '2l k (mis an infinite regular cardinal). Then there is an elementary extension B = ( B , F ) of 91 such that: (i) Every element of power 2 m in 8 is enlarged. (ii) Every element of power < m in i?l is fixed. (iii) If b has power 2 m in 23 then l b ~ l= 01. (iv) If b has power < m in 23 then lbFl = w.

25 Short, uncountable models of set theory

A model % of Zermelo set theory is said to be an w-model iff the element a of % such that % t (a = w ) has the property that ( a E ,E ) has order type ( w , <).

COROLLARY B (KEISLER and MORLEY[1968]). Let % be a countable o-model of Zermelo set theory plus ‘wi is regular’. Then there is an omodel 115 > 8 such that if 115 t (b = ol),lbFl = w l .

We shall use the following lemma to obtain extensions having exactly the same ordinals as the original model in many situations. LEMMA C. Let % be a model of ZF, % < 23, and let C = {b E B: For some c1 E A, 23 I. c1 is an ordinal ~b E R(c1)). Then if 6 is the model with universe C,

138

251

SHORT, UNCOUNTABLE MODELS OF SET THEORY

I39

PROOF.We show that for every formula cp(uo . . . u,) and all cl, . . ., c,

E c,

.

If 23 k (3uo)p[cl . . c,] then for some co E C,B k cp[co . . . c,]. (1) It will then follow that 0.i 23. Since also B i 23 and 'Z c Q, we can conclude from (1) that the required formula B i 0.i2' 3 holds. For some a E A , 23 k (a is an ordinal), and

B C c1 E R(Ix)A.. . A c, E R(a). By the reflection principle in 8 there exists fl E A such that 8 k (fl is an ordinal) A a 5 /?,and both the formulas cp, (3uo)cp are absolute for B and the R(P) of B;that is, 8 k u o , . . ., u, E R(B) + (cp(uo . . . u,)

c-)

cpR'B)(~o. . . on)),

(2)

and

B t. u l , . . ., u, E R(B) -+

((3uo)cp(u . . . u,,)c-)(3uo E R(/3))cpR'B'(uo. . . u,)). (3)

Since 'Z i 23 the universal closures of (2) and (3) also hold in 8.Since a

IP,

23 k c1 E R(fl)A . . . A c, E R(P).

(4)

Thus (2) and (3) hold for 23 with u l , . . ., u, replaced by cl,. . ., c,. By assumption, 23 k (3uo)cp[c . c,]. Hence by (3) there exists co E B such that 8 C co E R(P)A rpR(')[c0 . . . c,].

..

Using (2),

23 C c0 E R ( / ~ ) A ~ [.C* cn]. ~.

Therefore co E C and (1) holds for 'p. -I COROLLARY D. Let B be a countable model of ZF plus 'every union of a well-orderable set of well-orderable sets is well-orderable'. Then there is a model 23 > 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<=y s and each ys

E

W ( a ) ,then x E W(94).i

COROLLARY H. Let W be a countable model of ZF. Then there exists an elementary extension 23 > 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<,fa.

%A

b d = {a

 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<,X,, = X . This verifies (1) and completes the proof. i REMARK.If we drop the continuum hypothesis, then our proof gives Theorem 37 with (iii) replaced by the weaker conclusion

Given any sequence ( X u :ct < ol)of closed sets of real numbers of positive Lebesgue measure, B can be chosen SO that (iii*) B n Xu has power w l . for each a < ol,

150

LEBESGUE MEASURE

[26

Solovay has shown that if the original model fl is the particular one he constructed, then one can use a forcing construction to prove the conclusion of Theorem 37 without assuming the continuum hypothesis.

27 The property of Baire

We shall give analogues to the results of the last lecture with the notion of Baire category in place of Lebesgue measure. A set X c [0,1] is said to be offirst category if€ X is disjoint from some countable intersection of dense open sets. The classical Baire category theorem states that in ZFC any countable union of sets of first category again has first category. Mycielski has shown that the Axiom of Determinateness also has the following consequence: 1'. Every set x c [0, 11 has the property of Baire, that is, there exists an open set y c [0, I ] such that the symmetric difference ( x - y ) u ( y - x ) is of first category. Solovay's model of ZF + I + I1 +I11 also satisfies 1'. Moreover, Kunen has shown that I' and the 'countable choice' property I1 imply

111'. The union of any well-ordered sequence of sets of reals of first category has first category. THEOREM 38. Assume the continuum hypothesis 2'" = wl. Let % = (%, E) be a countable transitive €-model of ZF + I' + I1 +111'. Then there is a transitive €-model 23 = (B, E) such that: (i) ord(%) = ord(%). (ii) There is an elementary embedding$ % < 23 which is the identity on ord(%). (iii') The set [0, 11 n B is not of first category in the real world. 151

152

[27

THE PROPERTY OF BAIRB

Moreover, if 8 is also a model of I and I11 then 23 can be found so that in addition (iii) The set [0, 11 n B has outer Lebesgue measure one in the real world.

PROOF. Since the methods are similar to those we have used before, we shall only sketch the main points of the proof. It is sufficient to show that:

For every countable E-model % = ( A , E) of ZF+l’+Il+TII’ and every set Y c [0, I ] which is a countable intersection of dense open sets, there is a model 8 > 8 which has exactly the same ordinals as and an element c E B such that r

{ n ~ ~ : B k n n c } ~ Y .

=

We may represent Y in the form Y = dense open set of the form

K

=

u

m<w

(1)

nn<,Yn where each Y,, is a

( x m n 9 Ymn), x i j , Y i j

rational,

and Yo 2 Y , 3 Y, 3 . . .. Form the language L(A u {c}) and let T be the following set of sentences:

T = Th(aA)u {c

E

[0,111 u {c d a :

C ( a has first category)}.

We wish to show that

T u {(Vx)(ord(x)

+

V x

asord(%)

= a)} u {

A

V c E (Xmn

ncco m C o

9

Ymn)}

(2)

has a model. (2) will clearly imply (1). The proof of (2) uses the Omitting Types Theorem. The criterion for consistency with T is q ( c ) is consistent with T if and only if

is not of first category).

k ( { x E [0,11: ~ ( x ) } (3)

Thus Tis consistent. To use the Omitting Types Theorem the following two facts must be shown.

If (3y)(y is an ordinal A p(c, y ) ) is consistent with T, then there is an a E ord(8) such that q ( c , a ) is consistent with T. (4) If q ( c ) is consistent with T and n < o,then for some m < W, q ( c ) A c E (x,,,, y,,,) is consistent with T. (5)

271

THE PROPERTY OF BAIRE

153

The proof of (4) uses hypothesis HI’, while the proof of ( 5 ) uses 1’. The first clause of the theorem is proved by using (1) w1 times. The ‘moreover’ clause is proved by alternately using (1) of this proof and (I) of the proof of Theorem 37 w1 times. 4 Without the continuum hypothesis one can prove a weaker form of Theorem 38 like the weaker form of Theorem 37 which we stated above. Theorems 37 and 38 are due to the author, and answer questions [1970]. proposed by K. Kunen. They were stated without proof in KEISLER

28 Second order number theory

We shall now give an application of our methods to models of second order number theory. Second order number theory has two kinds of variables: number variables x , y , z, . . ., set variables X , Y , Z ,

. . ..

In addition it has the symbols = ,E, a constant 0, the successor functions, and + , .. The symbols s, + , *, 0 are applied to number variables to form terms. The = symbol is meaningful between two set variables or two number terms, and E goes between a number term and set variable. The axioms of second order number theory with choice are:

-

Peano's axioms, including induction. Extensionality: X = Y c-, (Vx)(xE X x E Y ) . The comprehension scheme: (3X)(Vy)(y E X c r q ( y , . .)), where X does not occur in q(y, . . .), but other free variables may. The choice scheme: (Vx)(3Y)cp(xY.. .) + (3Y)(Vx)cp(xYx. .), where Y, = { y : 2" 3y E Y } with the obvious abbreviations.

.

.

A model of second order number theory is a model of the form 2 = + , where A c S(N), the number variables range over N, the set variables over A , and all the axioms hold when E has the standard meaning. An w-model is a model A = ( w , A , 0, s, + ,-)where A c S(w)

(N,A , 0, s,

a)

154

28 1

155

SECOND ORDER NUMBER THEORY

+

and 0, s, , . are standard. An o-model is completely determined by the set A c S(o). The notion of an elementary extension is meant now in the sense of second order logic.

THEOREM 39. Let A be a countable w-model of second order number theory with choice. Then: (i) There exists an elementary extension B > 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,<, nm<, V EP

c-,

$(V)

where $ ( V ) is a finite Boolean combination of formulas of the type

158

SECOND ORDER NUMBER THEORY

t28

n E V. We wish to show that the theory has an o-model in which the infinite sentence ( V X ) A V -1X E S , , n<m m < m

holds. We claim that For every sentence (3X)(p(CX) consistent with T and every n < o,there is an m < o such that (3X)(rp(CX)A 1X E Smn) is consistent with T. (6) To prove (6) suppose that for all m < o, Tk

( 3 x ) ( ( p ( c x )A

i

x E s,,).

1

Then for all m,the formula ( ~ X ) ( ( P ( V XA )1X

E smn)

is enumerable. In number theory there is a definable way of associating a natural number with each finite sequence of natural numbers, and hence with each basic closed set. Thus there is a formula + ( n X ) such that for each n, AA C Il/(nX) t* X E the nth basic closed set. Using the axiom of comprehension, the set W = { n E o:( ~ X ) ( ( P ( V X ) A $ ( n X ) ) is enumerable) belongs to A. Moreover, for each m < o,S,,,, is the p Z hbasic closed set for some p E W.Using the choice scheme, the formula or

( 3 4bJE W A (3X)((PW) A ( 3 X ) [ C p ( W A(

W ( U E

WX))I

1

(7)

W A 1WX))I,

can be shown to be enumerable. But since A n S = 0, A n S, = 0, whence A, k (3U)(U E WA i $'(Ux)). It follows that the formula (7) is equivalent in A A to

281

SECOND ORDER NUMBER THEORY

159

which is therefore enumerable. Thus (3X)(p(CX)is inconsistent with T, and (6) holds. Using (3), (4), and ( 6 ) with the Omitting Types Theorem we see that T X E S,, holds. Then using has an o-model in which (VX)A,,, ,,V, an elementary chain of length ol we obtain the desired uncountable o-model B > A such that B n S = 0.i Part (i) of the theorem answers a question of H. Friedman.

29 A three cardinal theorem

We shall prove a three-cardinal theorem for the language L,. Suppose throughout that Ld is a countable fragment of L,,,, and that U, V are one-placed relation symbols. We say that 91 = ( A , U, V , . . .) is a model of type ( K , A, p) iff JAl = K , IUI = A, IVI = p. Let us recall two earlier results:

THEOREM 23. If T c L, is a set of sentences and T admits for some K 2 w , then T admits (A, w ) for all I 5 w. THEOREM 30. If T c Ld is a set of sentences and T admits some K > p 2 w , then T admits (wl, a).

(>,,(K),

(K,

K)

p) for

We shall now prove the natural common extension of these two results: If T c L, is a set of sentences and T admits (&,,l(~),K , p) for some ti > p 2 w , then T admits (A, wl, w ) for all A 1 wl. In the style of Theorem 23, we shall state our theorem in the following more general form.

THEOREM 40. Let T c L, be a set of sentences. Suppose that for all ordinals c( < w1 there are cardinals K > p 2 w such that T admits ( & ( K ) , K , p). Then: (i) T has a model of type (wl, wl, w ) in which there is an infinite set of indiscernibles over U. (ii) For all I 2 wl, T admits ( I , wl, 0 ) . L:

PROOF. We first show that (i) implies (ii). Form the Skolem language and let T* = T u TSkolem. Then T* satisfies the hypotheses. Hence 160

291

161

A THREE CARDINAL THEOREM

T* has a model B* of type (wl, ol, o)with an infinite set of indiscernibles over U. By the problems in Lecture 17 concerning indiscernibles in uncountable languages, T* admits (A, ol, o),whence T does also. Proof of (i): For this proof we shall need a slight generalization of Theorem 28 (Lecture 21). The proof is a simple adjustment of that of Theorem 28. THEOREM 28’. Let B = ( A , R, U ‘ , . . .) be a countable model such that R is a transitive, irreflexive relation on U ‘ . For any formula cp of L,,, let

(3 arb. large x)cp = (Vz)(U’(z) + (3x)(zRxA q)), where z does not occur in q. Let La be a countable fragment of L,,,. Then the following are equivalent: (i’) There exists ‘23 >La % such that

( U & ,R,) is a strong end extension of ( U ’ , R ) . (ii’) There is a ‘23 B such that (U;,R g ) is a strong end extension of ( U ‘ , R ) and U , has power W1.

(iii’) Each of the following are valid in 8: (a’) (t’y, z)( U’(y)A V ( z )+ (3x)(yRxA zRx)). (b’) (3 arb. large x ) V , < , On -, V ,, (3 arb. large .)On,

Vn<,

en E

where

La*

(c’) (3 arb. large x ) ( 3 y ) ( qA U’(y)) + (3y)(3 arb. large x ) ( q A U’(y)) v (3 arb. large y ) ( 3 x ) ( qA U’(y)). We leave the proof of Theorem 28’ to the reader. One simply ‘relativizes’ things to U‘. Now we give the proof of (i). We may assume without loss of generality that T 1 (Vx)( V ( x )+ U(x)). Otherwise replace U ( x ) by U ( x )v V ( X ) . Now add a new one-placed relation U‘ and binary relation R to L, forming LL. Let T‘ be the set of all sentences of L ‘ of the forms (a’), (b’), (c‘) of Part (iii) of Theorem 28’, plus

(VX>((V(X) U’(X>)A ( U ’ W -, W R linearly orders U‘, V is an initial segment of ( U ’ , R ) . +

)

Y

162

A THREE CARDINAL THEOREM

~9

For each a < ol,the theory T' admits ( ~ J K ) , K, p', p); for we may take a model of Tof type ( 3 = ( ~ )K, , p), choose U' such that V c U' c U and U' has power p+, and let R be a well-ordering of U' of order type ,'p thus getting a model of T'. By Theorem 23 part (i), T' has a countable model 58 = ( A , U,U',V , R) which has an infinite set of indiscernibles ( X , <) over U. Add new constant symbols c,, X E X , and let I be the set of sentences of LL({c,: x E X } ) stating that the c,'s are indiscernible over U. Let T" = T' u I. Then in the new language Lz({c,: x E X } ) , all the sentences of the forms (a'), (b'), (c') are consequences of T",because they are all obtained from formulas of type (a')-(c') of LL by replacing free variables by the new constants. Therefore by Theorem 28', there is an elementary extension BXof '$Xx such that (U;,RB) is an end extension of (U',R ) of power wl.Thus the model 93 is of type (a1, wl, ol, o)and in this model ( X , <) is an infinite set of indiscernibles over U.The reduct of 2.3 to the original language L is thus a model satisfying the required conditions for (i). (The set VB is the same as V because I/ is an initial segment of (U',R)). -I The analogue of Theorem 40 for first order logic is due to VAUGHT [19651, the present Three-Cardinal Theorem 40 was announced in KEISLER[19671. Like the two-cardinal theorems, the Three-Cardinal Theorem 40 has a number of applications. We give two examples here. For both of the next two theorems, assume that A is a PC, class over L,,,, Ld is a countable fragment of L,,,, and K, A, p are cardinals with K

> 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,,

n
where each om,E L,. Suppose 'ill has an elementary end extension with , of power ol.Then the following are equivalent: respect to L (i*) 'ill has a strong end elementary extension of power o1such that Q holds in 2 3. (ii*) For every m < w and every sentence in L,(A) of the form where (S) is a string of quantifiers of the types (3u)and (3 arb. large u), if 'illA !=

(S)(% 163

* *

-

xp,w

164

END ELEMENTARY EXTENSIONS WHICH OMIT A TYPE

then there exists n < w such that

I= (S)(3x1 * . - xp,,,)($

A Cmn).

NOTE: Condition (ii) is similar to the condition in the Omitting Types Theorem, except that the string of quantifiers ( S ) is present.

PROOF OF THEOREM 43. We shall rely heavily on results already established during the proof of Theorem 28. The direction (i) => (ii) will not be needed in the sequel, so we only give a bare outline of its proof. First, using the proof of Theorem 28 we see that the model 23 may be chosen so that a class of 23 is unbounded with respect to R if and only if it has power w l . Then (ii) is proved by induction on the length of the string ( S ) . (ii) => (i): Let T be the set of sentences introduced in the proof of Theorem 28. It is in the language Ld(A u { d } ) . Let cp be the ‘end extension’ sentence CP = A (VY> V (Y = e , v e a R y ) beA

a d

of that proof. It suffices to show that T u {cp, o} has an uncountable model. Actually, it is enough to show that T u {cp} has a model (BA,d ) such that condition (ii*) holds for 23. For then we may repeat the process o1times. By Theorem 28, any model of Thas an end elementary extension of power wl. The condition (ii*) obviously passes over limit ordinals, and thus we will have an uncountable model of T u {cp} for which (ii*) holds. It is obvious that any model for which (ii*) holds is a model of o,because in (ii*) we may take $ to be the formula x1 = a, A . . . A x p , = upmfor any p,-tuple u1, . ., up,,,of elements of the model. The condition (ii*) itself can be expressed by an infinite sentence of the form

.

e= A

(vx,

m<W

. . .xqm) v emn, n<w

where Om, E L,. 8 can be described as follows. Fix m c w. Let eO(Y1

*

YqJ

4(Y,

-

*

YqJ’

--

*

be a list of all formulas of the fragment L, of the form 0, = (S)(%

* *

- x,)9,.

301

END ELEMENTARY EXTENSIONS WHICH OMIT A TYPE

165

Let 8" be the infinite formula

Finally, let 8 = A 8". m<m

Then a model 23 satisfies (ii*) if and only if 23 is a model of 8. In Theorem 28 it was shown that T is consistent, and the condition (2) was vtrified so that the Omitting Types Theorem could be used to show that T v {cp} has a model. We now show that in addition

.

If m < o,r < o,(3yl . .y& is a sentence of L,(A u { d } ) which is consistent with T, then either (3Yl

* *

or>

Yq,)bA

1

is consistent with T, or else there is an n < o such that

( 31 ~* * * yq,)[x

A

(S)('X~ *

* *

xp,,,)(+r A ~ m n ) l

is consistent with T.

(2')

We need the fact from the proof of Theorem 28 that x(d) is consistent with Tiff

%A

C (3 arb. large x)n(x).

To prove (2'), suppose (3y1 . . .y,)x(d) is consistent with T, but

Tc

(3yl

. . . yqj(+)

A

er).

Then

T 1 n(d)

+ (S)(3xl

. ..x ~ , ) $ ~ .

Using (5), %A

I= (3 arb. large x)(3yl

. . . yq,)x(x),

whence using (6), %A

k (3 arb. large x ) ( 3 y l

. . . y,J(S)(3x1 . . . xp,)(n(x) A + r ) .

(5)

166

Putting

END ELEMENTARY EXTENSIONS WHICH OMIT A TYPE

( S ‘ ) = (3 arb. large x)(3y1

we have

. . . yqp)(S),

. . . x,,)(+)

% b (s

A

Then using the assumption that (ii*) holds for %, we have some n < o such that (s’)(3x1 * x p m ) ( 4 x >A $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
. . . xp,) V

a,,,,

n
where Q, E Ld. Let u < w1 and let 91fl,p < u be an elementary chain in L, such that go = '$I, each apis countable, and each '$Ip has an end elementary extension in L, of power w1in which u holds. Then gfl has an end elementary extension in L, of power w1 in which u holds.

UB<.

uB
has an PROOF.We have % <=, By Theorem 28, end elementary extension in L, of power wl.Condition (ii*) of Theorem 43 is preserved under unions of elementary chains in L,. Using Theorem 43, (ii*) holds in each afl,hence it holds in afl,and by Theorem 43 again '2Xp has an extension of the required kind. 4

up<.

31 Models of power q

It was shown by EHRENFEUCHT and MOSTOWSKI [1956] that in first order logic, every P c d class A? which has infinite models has a model !XKof each infinite power K such that only countably many types in L are realized in This is an easy consequence of their result that JZ has a model with an infinite set of indiscernibles and ot the Stretching Theorem. For a countable fragment L, we have the following weaker results. For both of them assume is a P c d class over L,,, and L, is a countable fragment:

aK.

1. If A has a model of power 2 a", then for each cardinal K 2 o JZ has a model of power K which realizes only countably many types in L,. (By the Upward LST Theorem).

2. If A has a model of power > 2" then .A has a model of power o1 which realizes only countably many types in L,. (Theorem 33). The above result is not true if we assume only that A has an uncountable model. For example let L, = L and let .A be the PC, class over L,,, composed of all models isomorphic to o-models of second order number theory. Then every uncountable model of A7 realizes uncountably many types, and in fact 4 has no models of power > 2" but does have models of power 2". The next theorem gives a weaker conclusion then Theorem 33 under the assumption that .A has an uncountable model. 168

311

I69

MODELS OF POWER 0 1

THEOREM 44. Let L, be a countable fragment of L,,, and let A be a PC, class over La,, which has at least one uncountable model. Then the set of all types in L, which are realized in all uncountable models in A? is countable. PROOF. Let T’ be a countable set of sentences in a language L,,, such that A is the class of all reducts to L of models of T‘.Then T‘ has an uncountable model, hence a model B’ of power wl.Let R be a well-ordering of B of order type ( w , , < ). Let L,, be a fragment which is countable, contains L,, and contains all sentences in T’ and in the first order language of ( B ’ , R ) . Then there is a countable model (%‘,R’)X L d (B’,R) such that (B’, R) is an end extension of (a’,I?). (To get (W, R’), first take a countable elementary submodel (ao,R,) of (B’, R), then extend it to a countable elementary submodel containing the initial segment of (23,R) generated by a, and repeat o times). Then (a’, R’)and L,, form an end extension pair, and a’ is a model of T‘. Let X be the set of all types in L, which are realized in every uncountable model of A.We show X is countable. Suppose Z = C(u, . up) belongs to X . Then the sentence

..

(VU,

... u p ) v

1 0

aEZ

is not true in any uncountable elementary end extension of (a’, R’) relative to Ld,. This means that condition (ii*) fails. That is, there is a sentence 0, = (S)(3U, * of L,,(A’) such that

(a;,

* *

u,>h

R’) k 0, and for all o E C,

We claim that

If C(u,

. . . up), T(ul . . . up) E X and X

For suppose C,r E X and 0,

=

Or, C #

#

r, then 0,

# 0,.

(1)

r. Then there must be a formula

170

MODELS OF POWER

0 1

and the fact that the quantifiers (3)and (3 arb. large x) commute with disjunction, we see that either (2) or (3) must fail. This contradiction proves (1). Since the language Ldt is countable, and by (1) there is a one-one mapping of X into the set of formulas of L,. ,we conclude that Xis countable. The proof is complete. -I The next theorem shows that if 4 has fewer than the maximum number of non-isomorphic models of power w1,the conclusion of Theorem 44 can be strengthened. THEOREM 45. Let Ld be a countable fragment of L,,, and let A? be a PC, class over L,,,. If A? has fewer than 2"' nonisomorphic models of power o,, then every model in A? realizes only countably many types in Ld * Equivalently, if A! has a model which realizes 2 o,types in L,, then JZ has exactly 2"' models of power w , , up to isomorphism.

PROOF. Let T' be a countable set of sentences of L$,, such that A? is the class of all reducts of models of T'.Assume A? has a model 23 which realizes 2 w , types in L,. Then we may assume 5 8 has power ol.For somep < o,23 realizes o,types in L, withp variables, say C,(Vl

* * *

up), < 0

1 .

Let 23' be an expansion of 23 to a model of T'. Add the following new functions and relations to 23': A well-ordering R of B of order type w , ; A one-one function F of BP onto B;

311

MODELS OF POWER

171

0 1

A set U c B of power o1such that for each u < w1 there isaunique b , , . . ., b, such that B t. &[b1 . . . b,] and F(bl . . . b,) E U ; A function G mapping U one-one onto B. Form the model

B"

(%', R,F, U, G).

=

This model satisfies the infinite sentence (vU1

... U,Ul ...U,)[(U(F(U1 A A b ( U 1 . . .U , ) * b ( U ,

. .. U,))A

. . . Up)) U 1 = U1 A . . A U p = Up].

u(F(U1

. . .Up))

U E L ~

(1)

This sentence states that each type in Ld is satisfied by at most onep-tuple whose F-image is in U.Let L,: be a countable fragment containing L d , T', all sentences of the first order language L" of B", and the sentence (1). By Theorem 28, any countable model W'iLPrd,, Bfthas an elementary end extension in EL..of power o1,hence 8",L'& is an end extension pair. Let 8 6 -&#,

Bf'.

Consider the binary tree of length w1 ,

T=

u "2,

a<mi

with the ordering s 5 t iff s is an initial segment of t for s, t E T. The two successors of s in Tare denoted by SO and sl . We shall construct models 8, and types Z,(ul . . up) in Ld for each is as above): s E T, with the following properties (where

.

If s < t then

is an end elementary extension of '2IS in EL,,.

asis countable. For all s 6 t, at realizes the type Zs. Each

(2) (3) (4)

If s < t and si $ 1, then !I& has an end elementary extension in Ls., of power w1 which omits ZSi (i E (0,11). (5) For example, the first few points of the tree are:

172

13 1

MODELS OF POWER W 1

...

00

.. ... Zol, for instance, realizes the types Z+, Zo, Zolbut has end elementary extensions of power w1 omitting Z1and Zoo. Suppose we have succeeded in constructing the a, and Z,. The three has 2"' branches S E"' 2. For each branch S, let 23s

=

(J as.

s<s

Since Bsis a union of an elementary chain of proper extensions of length a, is countable, Bs is an elementary extension of a, of power 0,. Moreover, the reduct of Bsto L' is a model of T', so the reduct BslL of Bsto L is a member of A. Let S, S' ~ " ' and 2 S # S'. Then there is a least CI such that S(a) # S'(a), say S(a) = 0 and S'(CI)= 1. Let s = Sla = S'la. Then w , and each

so

< s, sl < 9, sl $ s, so g S'.

.

By (4), BSrealizes Zsoand Bs,realizes Z, By ( 5 ) , every for s < S omits Z, , hence BSomits Z, . Similarly Bs,omits Zso.Therefore, since Zsoand C,, are types in L,, '%s\Land B,.IL are not isomorphic. This shows that A has 2"' non-isomorphic models, of power w , . We note that A cannot have more than 2"' non-isomorphic models of power o1 because there are only 2"' isomorphism types of models of power o1 altogether.

as

111

173

MODELS OF POWER W1

as,

Thus the theorem will be proved once we show the existence of Zs satisfying (2>-(5). We show this by induction on the length of s. Suppose we already have constructed C, '$Is satisfying (2)-(5) for all s E T of length less than u (given an c1 < ol). CASE1. c1 is a limit ordinal. For each t of length a, define = a, and let Cf be any type realized by $!If.Then (2), (3), and (4) Gbviously hold. (5) also holds because of Corollary 2 in the preceding lecture. CASE2. c1 = 0. Let C, be any type realized in the given model $!I4. CASE3. c1 = j+ 1. Let s be of length j,s E T. We show how to choose S?llso, aSl, C ,, ,Csl. Let Bo >L"d,,as0 be an end elementary extension of power w1 which omits Cri whenever r c s and ri & s. B0exists by Corollary 1 of the last lecture. Because of the function G, the set U in Bo has power ol, hence by the sentence (I), there are w 1 types in L, realized in B0 Consider the PC, class A" over L,,, of all reducts to L of models isomorphic to end elementary extensions in L;,, of By Theorem 44,only countably many Hence there types in L, are realized in all models in A" of power ol. exists a type Zsoin L, which is realized in Bo but is omitted in some model BIILEAT"of power ol. This model is the reduct of an end elementary of and Bl has power w l . Now let lYso be a countextension B1in Li.. able end elementary extension of %, such that as0 realizes Csoand

at

us<,

.

as.

as,

has end elementary extensions of power w 1 omitting each type Then qs0 C,i where r < s, ri F s. By Corollary 1 we may assume that Bl not only omits Zso but also omits each type Criwhere r c s and ri F s. Arguing as before, B1 realizes w1 types in Ld, hence there exists a type Csl in L, which is realized in Bl but is omitted in some end elementary extension Q of S?lso of power w l . Let aSlbe a countable end elementary extension of lY, which realizes Csland with a s
a s 1
Bl *

Then as0 , Zso, ?Is1, Zslsatisfy all the conditions (2)-(5). complete. i

The proof is

174

MODELS OF POWER W1

REMARK. Notice that the notion of an end extension did not enter into the statements of the last two theorems. However, they were essential in the proofs in order to give a method of taking proper elementary extensions o1times. Theorem 45 gives a new result even for first order logic. In the case of first order logic it can be stated more simply because of the Compactness Theorem.

COROLLARY. Any complete theory T i n L which has uncountably many types has exactly 2"' models of power o1up to isomorphism. It is an open question whether the above corollary holds wlth o1replaced everywhere by 2" (without the continuum hypothesis). It is known that if T has uncountably many types then it has 2" types (equivalently, a countable Boolean algebra with an uncountable Stone space has Stone space of power 2"). The following example shows that the corollary fails for almost all cardinals K > 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
.

182

[3 2

ULTRAPOWERS

(3xo)$(x,,x, . . . x,,). Let F', domain cp(23). Suppose that {x

E

. . ., F" be

definable functions in 23 with

(~(23):8 I: ( 3 x o ) $ [ F ' ( x ) .

. .F"(x)]} E D .

(1)

Let b, . . . b, be all the constants occurring in cp and in the definitions of F', . . ., F". Then there is a formula B(xx,b, . . b,) E L,(B) such that for all x and xo in B,

.

23 I: $ [ x o ~ ' ( x ) .. . ~ " ( x ) iff ] 23 b e [ x x o b , . . . b,]. Since 23 is Skolem over cp(23) there is a definable function F o in 8 with domain (~(23)such that for all x E cp(23),

8 c ( ~ x , ) $ [ F ' ( x ) . . . ~ " ( x ) implies ] 23 I: $ [ F ' ( x ) . It follows that {x

E

..~ " ( x ) ] .

cp(23): 23 I: $ [ F o ( x ) . . . F " ( x ) ] }E D ,

whence by hypothesis IID%

and so flDB

. Fg]

k $[Fi..

I: (3xO)$[Fh.

..Fg].

(2)

The proof that (2) implies (1) is easy. Now assume that ,,A, $,(x, . x,) E L, and the theorem holds for each $., We prove the theorem for ,,A, $., Suppose

..

f l D 8

*.

I: A $,[Fh m
F;].

(3)

Then for each m < w , UDB I: $,[FA

*

- . F;],

whence by hypothesis {x

E

cp(23): 23 I: $,[F'(x)

. . . F " ( x ) ] } ED.

Then by the L,-completeness of D, {x

E

~ ( 2 3 ) :23 I: A

m
$,[F'(x)

The proof that (4) implies (3) is easy. -I

. . . F"(x)]} ED.

(4)

321

183

ULTRAPOWERS

REMARK. The above theorem easily generalizes to the richer infinitary logics Lapwith conjunctions of power < c( and quantifiers of power < B. EXAMPLE. If D is a countably complete ultrafilter then D is L,-complete for every d , whence we always have 23
=

{$(23): $@) c (~(23) and

c E $(&)}.

Then E has the finite intersection property and hence can be extended to an ultrafilter D over ~(23). D is non-principal because for each b E q(B), we have c E .( z b A cp(x))(0.>, whence ( x # b A q ( x ) ) ( % ) = ~(23)- {b} E D . To show D is L,-complete, let

A U Y , . * . Y m ) E L,

n<w

and let F k , . . ., Fg be definable functions in 23 with domain q(8).Suppose that for each n < w , ( x : !BBt O,(F&(x). . . Fg(x))} E D .

Then since the FA are definable in 23, we have

GC t e,,(F&) Hence

. . . F,"(c)), for all n < w.

gCt A e,,p:(c) . . . ~ y c ) ) . nco

184

ULTRAPOWERS

There is a formula

$ 41 ~*

which says that yl

*

ymx)

E LAB)

. .A y,

= Fi(X) A .

=

Fz(X).

Then

Q, t ( 3 ~ 1 . . Yrn)[$(yl* *

* *

y r n c ) ~A On(y1 n<m

-

* *

~rn)l-

The above sentence belongs to Ld(B u { c ) ) , hence by the definition of D, {X:

BEt ( 3 Y 1 * .

Therefore {x:

- Ym,[$'(Yl

*

B~ c A

ncw

--

YmX)"

A

n<m

8,(Y,

*

-

Yrn)]) ED.

en(F;(x) . . . ~g(x))}ED.

This shows that D is L,-complete.

-I

The next corollary describes the relation between the models % and

nDBof the last theorem.

COROLLARY. Suppose B is Skolem over cp(B), 0.>L, 23, c E cp(Q), and D is an ultrafilter over q ( B ) such that for every class $(8)c cp(B), c E I)(&)if and only if

$(B) E D.

L e t 8 n D B+ Q be the mapping defined by f(F%iD)

=

for each definable function F, in B with domain cp(B). Then f is an elementary embedding,

f :flD%

PROOF. Using the proof of the preceding theorem we see that D is L,complete and hence for all 8(x, . x,,) E L, and all definable FA, ., F:, the following are equivalent:

..

f l D B t 8[FkD

..

--

9

FkD];

{x E ~ ( 823 )t:8 ( F i ( x ) . . . Fg(x))} E D ;

Q c e[F:(c). . . F;(c)].

-1

33 Ultrapowers of models of set theory

In this lecture we give an application of the ultrapower to models of set theory. For simplicity we shall consider models of first order logic, but we shall see in the next lecture that the results can be used to give information about models of L,. Suppose L has among its relations the binary relation E. We shall denote by Z the Zermelo set theory formulated in terms of E. A model 23 = ( B yE, . .) for L is said to be a model of Z iff
.

BE k (3f) Gf is a function with domain b A ((3y)cp( x y ) + cp(x,f(x)))]. Recalling an earlier definition, if a, b E B we say that b is regular over a iff the formula 1x1 2 161 is regular over the formula lyl 5 lal, i.e.,

8 k If 1x1 2 lbl, Iyl 6 x =

u

M Z )

la], g is a function with domain y, and : z EY}, then (32 Ey)lg(Z)I 2 PI.

We recall that in Theorem 36 we proved the following (but with b and ~ ( 2 3 )regular over t+b(%)): Let 23 be a countable model of Z. Suppose that a, b E Byb is regular over a, and k lbl 2 2 A IQI 2 2. a replaced by classes

Then there is an elementary extension Q = (C, F, . . .) of 23 such that a is fixed and lbFl = ol. 185

186

ULTRAPOWERS OF MODELS OF SET THEORY

[33

Theorem 36 was stated only for the case where 23 = (B, E ) with no additional relations, but the proof also holds in case 23 = (B, E, . . .) is a model of Z in which the axiom of subsets holds for all formulas of L. Using ultrapowers we now prove a similar result in which the hypothesis that 23 is a model of Z for L is replaced by assuming 23 satisfies choice over b, and the hypothesis that 23 is countable is weakened to the assumption that a certain part of % is countable. The main interest in the theorem is that in certain cases it allows us to handle uncountable models. 48. Let 23 = (B, E, . . .) be a model of Z and let a, b E B. THEOREM Assume: 23 satisfies choice over b; 23 k Ib x bl = Ibl A Ibl 2 2 A la1 2 2; b is regular over a; lcEl = w where 23 1 c = ba. Then there is an elementary extension 0.> 8,0. = ( C , F, . . .), such that

and

a is fixed, i.e. aF = a E , IbFl

=

wl*

PROOF. We first observe that since 93 satisfies choice over b, 8 i s Skolem over the class b,. Moreover, the definable functions F i n 23 with domain b, correspond to the elements f E B where

23 1f is a function with domain b. We shall therefore speak about thef’s instead of the F’s. Let L, be the smallest countable fragment of L,,,(B) which contains the formula AXxZd. dEa

Since cEis countable and 23 k la1 5 IcI, we see that a, is countable, hence the above conjunction is countable. We claim that There exists a non-principal L,-complete ultrafilter D over 6,. (1)

To prove (1) we begin by making an enumeration cE

=

i f 0 , f l ,f2

9

* * *>.

33 I

ULTRAPOWERS OF MODELS OF SET THEORY

187

Using the regularity of b over a, we may choose a sequence e, , e, , e,

,...E B

such that for each n < w ,

B 'F (en+l

c en c

b A le,J = ( b (r\f,le,, is constant).

We take for D any ultrafilter over b, which contains all the sets en,, n < w . D is non-principal because for each d E b, there is an f n which is the characteristic function of {d)in 8,hence en must not contain d. We must now show that D is L,-complete. That is, we must show that for everyfe B such that SB'F (fis a function with domain b), then

If for all dEu, { x E b : 'BE k f ( x ) # d ) E D , ( x E b : %B 'F Af(x) # d } ED. dEa

This is equivalent to showing: If { x E b : BEk f ( x ) E a } E D then for some dEa, {XEb: 'BB ' F f ( x ) = d ) ED. Suppose {XEb: B B ' F f ( x )E U } E D . Let g be such that in

B,

g ( x ) = f ( x ) whenf(x)

E

a, and g E

Then g = f n for some n. Hence in 23,glen is constant, say with the value dEa. Then in '23, f ( x ) = d whenever x E en and f ( x ) = g(x), It follows that { x E b : f ( x ) = d i n %} E D , and (2) holds. Thus (1) is verified. Applying (1) we form the ultrapower n,S.We have by the fundamental theorem that d: B < Ld IZ, B,and the elementary embedding enlarges b. It also leaves a fixed because

BB 'F (Vx)(x E a

--f

A

1

dEa

x # d),

188

ULTRAPOWERS OF MODELS OF SET THEORY

[33

hence IID8also satisfies the above sentence of LJB). Thus IIDB r 0; for some elementary extension Q > B which leaves a fixed and enlarges b. Finally, we note that lCFl = 0. (3) This is because each element of cF corresponds to an element of IIDB which is an equivalence class of an elementfE B such that that is, But

BBkfis

a function on b into c,

BB k f E

’((”a).

BB k Ib((”a)I= Ibxbal= Iba1

= Ic(

because by hypothesis B B

k Ibx bl = Ib(.

Therefore, since c, is countable, there are only countably many such f. This proves (3). By (3) we see that the model 0. satisfies all the hypotheses of the theorem. It follows that we may repeat the construction w1 times and form an elementary chain Ba, a < w 1 such that Bo = B and at each step a is fixed and b is enlarged. The hypotheses of the theorem obviously are preserved by union of countable elementary chains, so we define Ba = UBXu BBwhenaisalimit ordinal < wI.TheunionQ = Ba is then the desired elementary extension of B . -I

uu
REMARK.The theorem can be easily generalized by replacing a by a countable collection a,, n < 0,of elements of B. Theorems 46-48 above were announced in a somewhat different form in KEISLER [1965].

34 The Seven Cardinal Theorem

The two-cardinal result in Theorem 30 (If T admits (K, p) then T admits (wl ,w)), and the three-cardinal result in Theorem 40 were both first proved using ultrapowers. Essentially, their original proofs were based on Theorem 48. The proofs we have given above based on the Model Existence Theorem were discovered later and were preferred here because the Model ExistenceTheorem is a more widely applicable method than the ultrapower construction for L,,, . The original proof of Theorem 40 (ii), however, is of some independent interest and is given below. Let Ld be a countable fragment of L,,,. THEOREM 40 (ii). Let T c Ld be a set of sentences such that for all u < w1 there are cardinals K: p >= w such that T admits (&(K:),K, p). Then for all A 2 al,T admits (A, wl, 0). PROOFUSING ULTRAPOWERS. It is understood that the language L has two distinguished one-placed relation symbols U and V to which the notion of a model of type (a, B, 7) refers. We may assume that

=-

T =! (Vx) V ( x ) -+ U(x).

As in Theorem 40 before, we add a new relation U' and the axiom vX(

V(X)

Then T with (4) admits

-b

u ' ( X ) ) A u'(X)-+

(&(J4 K, p+, P)

for we may take U' of power p+. 189

(X(u)).

(4 )

190

133

THE SEVEN CARDINAL THEOREM

For each a < w1 we may 'impose' a model of Z o n the model of T u (4) of type ( L ( K > , K, P+7 PI in the following way. We may form a model

Bu= < R ( K + ) ,E, &(ti),

K,

P + , P,

..

-)

such that the reduct

Bi

= <&(K), ti, P+, P,

L,, sat, ~ P ) r p ~ ~ ,

- . .>

is a model of T u (4). The set L, is thus represented by a relation of the model and there is a constant for each element of L,. The relation sat denotes the relation between L, and R ( K + )such that for each q ( u l . . . u n ) E L, and x E R ( K + ) , sat (cpx)

iff x E >,(K)" and 58; t= cp[x].

Each Buis a model of the set of sentences T' which is described as follows: correspond to the indicated relation symbols: Let the relations of

~LK), K , P+, p, E w UU' v

< R ( K + ) , E,

*

-,L,, L,

sat, cp> 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.

References

BARWISE, J. [19681 The syntax and semantics ofinfinitary languages, edited by J. Barwise, Lecture Nofes in Mathemafics, 72, Springer, Berlin. [ I968al Implicit delkability and compactness in infinitary languages, in BARWISE [1968], 1-35. [1968b] Applications of strict II: predicates to infinitary logic, J. Symb. Logic, 34, 409-423. [I9691 Infinitary logic and admissible sets. J. Symb. Logic, 34, 226-252. Duke Math. Journal, 36,631-637. [1969a] Remarks on universal sentences of L,,,. [I9711 Infinitary methods in the model theory of set theory. Logic Colloquium '69, edited by R. 0. Gandy and C. E. M. Yates, North-Holland, Amsterdam. BARWISE, J., and EKLOF P. [I9691 Lefschetz' principle. J. Algebra, 13, 554-570. [19701 Infinitary properties of abelian groups. Ann. Math. Logic, 2, 25-68. BARWISE,J. and FISHER, S. [I9701 The ShoenfieId absoluteness lemma. Israel J. Math., to appear.

BENDA,M. 119691 Reduced products and nonstandard logics. J. Symb. Logic, 34,424-436. CHANG, C. C. 119661 Ultraproducts and other methods of constructing models. Sers, Models and Recursion Theory, edited by J. N. Crossley, North-Holland, Amsterdam, 85-121. [I9681 Some remarks on the model theory of infinitary languages, in BARWISE [1968], 36-63. 193

194

BIBLIOGRAPHY

[ 1968al Infinitary properties of models generated from indiscernibles. Logic, Mefhodology and Philosophy of Science, 111, edited by B. van Rootselaar and J. F. Staal, North-Holland, Amsterdam, 9-22. [I9701 Sets constructible using LKK. Proceedings of the set theory institute, UCLA,

1967. To Appear. [1970a] Two interpolation theorems. T o appear, Proceedings of the Conference in Model Theory held in Rome, Italy, 1969.

CHANG,C. C. and KEISLER H. J. [I9701 Model Theory. To appear, Appleton Century Crofts, New York. CRAIG,W. [I9571 Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J. Symb. Logic, 22, 269-285.

CUDNOVSKII,G.

[I9681 Some results in the theory of infinitely long expressions. Math. Dokl., 9, 556-559. [I9701 Algebra and Logic, 9, 80-120 (in Russuian).

CUTLAND,N. [I9711 The theory of hyperarithmetic and Z12 models. Thesis, Univ. of Bristol, England. DICKMANN, M. A. [I971 ] Large Infnitary Languages, Model Theory. North-Holland, Publ. Co., Amsterdam. To appear. EHRENFEUCHT, A. [I9611 An application of games to the completeness problem for formalized theories. Fund. Math., 49, 129-141. EHRENFEUCHT, A., and A. MOSTOWSKI [I 9561 Models of axiomatic theories admitting automorphisms. Fund. Math., 43, 50-68. ENGELER, E. [I9611 Unendliche Formeln in der Modelltheorie. Z. Math. Logik Grundlagen Math., 7, 154-160. [I9631 A reduction principle for infinite formulas. Math. Annalen, 151, 296-303. [I9671 Algorithmic properties of structures. Math. Systems Theory, 1, 185-195. [I9681 Theory of geometric constructions, in BARWISE[1968], 64-76.

BIBLIOGRAPHY

195

ERDOS,P., and RADOR. [1956] A partition calculus in set theory. Bull. Amer. Math. Soc., 62, 427489. FEFERMA , s. [1968] Persistent and invariant formulas for outer extensions. Comp. Math., 20, 29-52. [1968a Lectures on proof theory. Proceedings of the Summer School in Logic,Leeds, 1967, edited by M. H. Lob, Springer, Berlin, 1-108. FEFERMAI S., and KREISEL, G. [I9661 Persistent and invariant formulas relative to theories of higher order. Bull. Amer. Math. Soc., 72, 480485. FRA'I'ss~,R. 119541 Sur quelques classifications des systkmes des relations. Publ. Sci. Unir. Alger. Ser. A , 1, 35-182. FRIEDMAN, H. [1970] On the ordinals in models of set theories. To appear, J. Symb. Logic. [I9711 The Beth and Craig theorems in infinitary languages. Israel J. Math., to appear. FRIEDMAN, H., and R. JENSEN [I9681 Note o n admissible ordinals, in BARWISE[1968], 77-79. GAIFMAN, H. [I9681 Two results concerning extensions of models of set theory. Notices Amer. Math. SOC.15, 947, (Abstract). GREGORY, J. [1970] Incompleteness of a formal system for infinitary finite-quantifier formulas. Technical Report, Univ. of Maryland. [ 1970al Elementary extensions and uncountable models for infinitary finite quantifier language fragments. Preliminary report. Notices Amer. Math. SOC.17, 967-8. GRZEGORCZYK, A., MOSTOWSKI A. and RYLL-NARDZEWSKI C. [1961] Definability of sets of models of axiomatic theories. Bull. deI'Acad. Polon. des Sci., Sir. Math., IX, 163-167. HALPERN, J. D. and LEVY,A. [1968] The Boolean prime ideal theorem does not imply the axiom of choice. To appear.

196

BIBLIOGRAPHY

HANF,W. [I 9641 Incompactness in languages with infinitely long expressions. Fund. Math., LIII, 309-324. HENKIN,L. [I9491 The completeness of the first-order functional calculus. J. Symb. Logic, 14, 159-166. [I9541 A generalization of the concept of w-consistency. J. Symb. Logic, 19, 183196. [I9571 A generalization of the concept of w-completeness. J. Symb.Logic, 22, 1-14. [1961] Some remarks on infinitely long formulas. Infinitistic Methods, Warsaw, 1961, 167-183. [I9631 An extension of the Craig-Lyndon interpolation theorem. J. Symb. Logic, 28, 201-216. KARP,C. [I 9621 Independence proofs in predicate logic with infinitely long expressions. J . Symb. Logic, 27, 171-188. [1964] Languages with Expressions of Infinite Length. North-Holland, Amsterdam. [I9651 Finite quantifier equivalence. The Theory of Models, edited by J. Addison, L. Henkin, and A. Tarski, North-Holland, Amsterdam, 407-412. [ 19671 Nonaxiomatizabilityresults for infinitary system. J. Symb.Logic, 32,367-384. [1968], [1968] An algebraic proof of the Barwise compactness theorem, in BARWISE 80-95. JENSEN, R. and KARP,C. [I9701 Primitive recursive set functions. Proceedings of the set theory insritute, UCLA, 1967. To appear

KEISLER, H. J. (19621 Some applications of the theory of models to set theory. Logic, Methodology and Philosophy of Science, edited by E. Nagel, P. Suppes and A. Tarski, Stanford Univ. Press, 80-86. [I9631 A complete first-order logic with infinitary predicates. Fund. Math., LII, 177-203. (19651 Extending models of set theory. J. Symb. Logic, 30,269 (Abstract). [1965a] Finite approximations of infinitely long formulas. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, 158-169. [1965b] Some applications of infinitely long formulas. J. Symb. Logic, 30, 339-349. [I9661 Some model-theoretic results for w-logic. Israel J. Math., 4,249-261. 119671 A three-cardinal theorem for w-logic. Notices Amer. Math. SOC., 14, 256 (Abstract).

BIBLIOGRAPHY

197

[1967a] A survey of ultraproducts. Logic, Meihodology and Philosophy of Science, 111, edited by B. van Rootseiaar and J. F. Staal, North-Holland, Amsterdam, 112-126. 119681 Formulas with linearly ordered quantifiers, in BARWISE [1968], 96-130. [I9691 Categorical theories in infinitary logic. Notices Amer. Math. SOC.,16, 437 (Abstract). [I 969al Infinite quantifiers and continuous games. Applications of Model Theory to Algebra, Analysis and Probability, edited by W. A. J. Luxemburg, New York, 228-264. [I9701 Logic with the quantifier “there exist uncountably many”. Ann. Math. Logic, 1, 1-93. [I9711 A survey of model theory. Proc. of the Int. Cong. of’Math.heldat Nice, 1970, to appear. [1971a] L+(Q). To appear.

KEISLER,H. J. and MORLEY, M. [1967] On the number of homogeneous models of a given power. Israel J. Math. 5 , 73-78. [1968] Elementary extensions of models of set theory. Isruel J. Math., 6,4945. H. J. and SILVER, J. KEISLER, [1970] End extensions of models o f set theory. Proceedings of the set theory institute, UCLA, 1967. To appear. KINO,A. [1966] Definability of ordinals in infinite logic. J. Symb. Logic, 31, 365-375. Correction, J. Symb. Logic, 32, 343-344. G. KINO,A. and TAKEUTI, I19631 On predicates with constructive infinitely long expressions. J. Math. SOC. of Japan, 15, 176-190. KOPPERMAN, R. D. [I9671 On the axiomatizability of uniform spaces. J. Symb. Logic, 32, 289-294. [1967a] The L,,,,m, theory of Hilbert spaces. J . Symb. Logic, 32, 295-304.

R. D. and MATHIAS, A. R. D. KOPPERMAN, [I9681 Some problems in group theory, in BARWISE 119681, 131-138. G. KREISEL, [1965] Model-theoretic invariants. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, 190-205. [I 9681 Choice of infinitary languages by means of definability criteria; generalized

198

BIBLIOGRAPHY

recursion theory, in BARWISE[1968], 139-151. [1968a] A survey of proof theory. J . Symb. Logic, 33, 321-288. KREISEL,G. and KRIVINE,J. L. [I9671 Elements of mathematical logic, North-Holland, Amsterdam. KRIPKE,S. [I9641 Transfinite recursion on admissible ordinals, I, 11 J. Symb. Logic, 29, 161162 (Abstracts). KUEKER,D. [I9681 Definability, automorphisms, and infinitary languages, in BARWISE[1968], 152-165. [I 9701 Generalized interpolation and definability. Ann. Math. Logic, 1, 423468. KUNEN,K. 119681 Implicit definability and infinitary languages. J. Symb. Logic, 33, 446-451.

E. LOPEZ-ESCOBAR, 119651 An interpolation theorem for denumerably long sentences. Fund. Math., LVII, 253-272. [1965a] Universal formulas in the infinitary languages Lap. Bull. Acad. Polon. Sci. Ser. Sci. Math., 13, 383-388. [I9661 On definable well-orderings. Fund. Math., LVIX, 13-21, and Fund. Math., LVIX, 299-300. [ 19671 A complete, infinitary axiomatization of weak second-order logic. Fund. Math., 61, 93-103. I1967al Remarks o n a n infinitary language with constructive formulas. 1. Symb. Logic., 32, 305-318. 119681 Well-orderings and finite quantifiers. J. Math. SOC.Japan, 20, 477-489. Lo$ J.

[I9541 On the categoricity in power of elementary deductive systems and some related problems. Colloq. Math., 3, 58-62. [I9551 On the extending of models (I). Fund. Math., 42, 38-54. [1955a] Quelques remarques, t h k o r h e s et problemes sur les classes dkfinissables d’algtbres. Mathematical Interpretations of Formal Systems, North-Holland, Amsterdam, 98-1 13.

LYNDON, R. [1959] An interpolation theorem in the predicate calculus. Pacific f. Marh.,9, 129-142. [ 1959al Properties preserved under homomorphism. Pacific J. Math.. 9 , 143-154.

BIBLIOGRAPHY

199

MAEHARA, S. and TAKEUTI, G.

[I9611 A formal system of first order predicate calculus with infinitely long expressions. J. Math. Soc. Japan., 13, 357-370.

MAKKAI, M.

[I9681 A generalized interpolation theorem. Amer. Math. SOC.Norices, 15, 804805 (Abstract). [I9691 Preservation theorems for logic with denumerable conjunctions and disjunctions. J. Symb. Logic, 34, 437-459. [1969a] Structures elementarily equivalent to models of higher power relative to infinitary languages. Notices Amer. Math. SOC.,16, 322 (Abstract). [1969b] An application of a method of Smullyan to logics o n admissible sets. BUN. de I'Acadimie Polon. des Sci., Ser. Math., 17, 341-346. [ 1969~1Preservation theorems for pseudo-elementary classes. Notices Amer. Math. SOC.,16, 425 (Abstract). [1969d] Regular relations and Svenonius formulas. Notices Amer. Math. Soc., 16, 981 (Abstract).

MALITZ,J.

[1966] Problems in the model theory of infinitary languages. Ph.D. Thesis, Berkeley. [1968] On the Hanf number of complete Lml, sentences, in BARWISE [1968], 166181. [I9691 Universal sentences and infinitary logic. Duke Math. J., 36, 621-630.

MANSFIELD, R.

[I9701 A completeness theorem for infinitary logic, To appear.

MATHIAS, A. R. D.

[1970] The order extension principle. Proceedings of the set theory institute, CJCLA, 1967. To appear. [I9701 Surrealist landscape with figures. A survey of recent results in set theory. Proceedings of the set theory institute, CJCLA, 1961. To appear.

MORLEY,M.

[I9651 Categoricity in power. Trans. Amer. Math. Soc., 114, 514-538. [1965a] Omitting classes of elements. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, 265-273. [1967] The Hanf number for 0-logic. J. Symb. Logic, 32.437 (Abstract). [I9681 Partitions and models. Proceedings of the Summer School in Logic, Leeds, 1967,edited by M. H. Lab, Springer, Berlin, 109-158. [I9701 The number of countable models. J. Symb. Logic, 35, 14-18.

200

BIBLIOGRAPHY

MORLEY, M. and MORLEY, V. [I9671 The Hanf number for x-logic. Notices Amer. Math. SOC.,14, 556 (Abstract). MORLEY, M. and VAUGHT, R. [I9621 Homogeneous universal models. Math. Scand., 11, 37-57.

J. MYCIELSKI, [1964] On the axiom of determinateness. Fund. Math., LIII, 205-224. [1966] On the axiom of determinateness (11). Fund. Math., LIX, 203-212. J. and SWIERCZKOWSKI, S. MYCIELSKI, [1964] On the Lebesgue measurability and the axiom of determinateness. Fund. Math., LIV, 61-71. A. MOSTOWSKI, [I9611 Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic. Essays on the Foundations of Mathematics. edited by Y. Bar-Hillel, E. I. J. Poznanski, M. 0. Rabin and A. Robinson, North-Holland, Amsterdam, 269-286. [I9661 Thirty Years of Foundational Studies. Helsinki and New York.

NEB=, B. [ 19691 A syntactic characterization of infinitary sentences preserved under unions of models. Amer. Math. SOC.Notices, 16, 423 (Abstract). OREY,S. [I9561 On o-consistency and related properties. J. Symb. Logic, 21, 246-252. PLATEK, R. [ 19661 Foundations of recursion theory. Ph.D. thesis, Stanford.

A. PRELLER, [I9681 Quantified algebras, in BARWISE [1968], 182-203.

RESSAYRE, J. P.

[I9691 Sur les thiories du premier ordre catkgoriques en un cardinal. Trans. Amer. Math. SOC. 142, 481-505

ROBINSON, A. [I 951 ] On the Metamathematics of Algebra. North-Holland, Amsterdam. [I9561 Note on a problem of L. Henkin. J. Symb. Logic, 21, 33-35. [1956a] A result on consistency and its application to the theory of definition. Zndug. Math., 18. 47-58.

BIBLIOGRAPHY

20 1

119631 Introduction to Model Theoryandto the Meramathematics of Algebra, NorthHolland, Amsterdam.

F. ROWBOTTOM, 119641 The E d conjecture for uncountable theories. Amer. Math. Soc. Notices, 11, 248 (Abstract). SACKS,G . [I9701 Countable admissible ordinals and hyperdegrees. To appear. SCOTT,D. [I9651 Logic with denumerably long formulas and finite strings of quantifiers. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, 329-341.

SCOTT,D. and TARSKI,A.

[1958] The sentential calculus with infinitely long expressions. Colloq. Math., 6 , 165-170.

S. SHELAH, 119701 On stable theories. Israel J. of Math., To appear. [1970a] On languageswith non-homogeneous strings ofquantifiers. IsraelJ. of Marh., To appear. [1970b] On the number of non-almost isomorphic models of T in a power. To appear. [197Oc] O n finite diagrams stable in power. Ann. Math. Logic, 2, 69-118. [1970d] A note on Hanf numbers. To appear. [1970e] Solution of Lo6 conjecture for uncountable languages. Notices Amer. Math. SOC.,18, 968 L(abstract).

SIMPSON, S. (19701 Model-theoretic proof of a partition theorem. Notices Amer. Math. Soc., 18, 964 (abstract). J. SILVER, [1966] Some applications of model theory in set theory. Ph.D. thesis, Berkeley.

R. SMULLYAN, [I9631 A unifying principle in quantification theory. Proc. Nut. Acad. Sci., 49, 828-832. [1965] A unifying principle in quantification theory. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski. North-Holland, Amsterdam, 443434.

202

BIBLIOGRAPHY

SOLOVAY, R. [1970] A model of set theory in which every set of reals is Lebesgue measurable. To appear. SUZUKI, Y. [1968] No-standard models for set theory. Bull. de I’Acad. Polon. des Sci., Ser. des Sci. Math., XVI, 265-267. SVENONIUS, L. [1965] On the denumerable models of theories with extra predicates. The Theory of Models, edited by J. Addison, L. Henkin and A. Tarski, NorthHolland, Amsterdam, 376-389. TAIT,W. W. 119681 Normal derivability in classical logic, in BARWISE[1968], 204-236. G. TAKEUTI, [1968] A deterministic logic, in BARWISE [1968], 237-264. TARSKI, A. [I9521 Some notions and methods on the borderline of algebra and metamathematics. Proc. of the Int. Cong. of Mathematicians, Cambridge Mass., 1950, Vol. I , Providence, 705-720. [1954] Contributions to the theory of models I, 11. Indag. Marh., 16, 572-588. [1958a] Models of universal sentences in predicate logic with infinitely long formulas. Amer. Math. SOC.Notices, 5 , 67 (Abstract). [1958b] Remarks on predicate logic with infinitely long expressions. Colloq. Math., 6, 171-176. [1958c] Some model theoretic results concerning weak second-order logic. Amer. Math. SOC.Notices, 5 , 673 (Abstract). [1961] A model-theoretic result concerning infinitary logics. Amer. Math. SOC. Notices, 8, 260 (Abstract). [1961a] Representable Boolean algebras and infinitary logics. Amer. Math. SOC. Notices, 8, 154-155 (Abstract). [I9621 Some problems and results relevant to the foundations of set theory. Logic, Methodology and the Philosophy of Science, edited by E. Nagel, P. Suppes and A. Tarski, Stanford Univ. Press, 125-135. TARSKI,A. and VAUGHT, R. [1957] Arithmetical extensions of relational systems. Comp. Math., 13, 81-102. VAUGHT, R. [1961] Denumerable models of complete theories. Infinitistic Merhods, Pergamon

BIBLIOGRAPHY

203

Press, New York and P.W.N. Warsaw, 303-321. [1965] A Lowenheim-Skolemtheorem for cardinals far apart. The Theory ofModels, edited by J. Addison, L. Henkin and A .Tarski, North-Holland, Amsterdam, 390-401. [1965a] The Lowenheim-Skolem theorem. Logic, Merhodology ond Philosophy of Science, III, edited by B. van Rootselaar and J. F. Staal, North-Holland, Amsterdam, 81-92. WEINSTEIN, J. [I9681 (a1, o)properties of unions o f models, in BARWISE [1968], 265-268.

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

L:

8* =a

Hull (X) K-categorical PC, class K-saturated model PC, class over L,,, K-homogeneous ( K , L,)-homogeneous L,-homogeneous ZF ZFC ZFL 3 arb. large x r (XI a, Th(8'4) w-model

w,w,

48, 142 48 63 67 67 67 67 70 91 91 92 94 95 95 95 102 102 102 111 115 132 134 138, 155 146

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