Large Infinitary Languages, Model Theory

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 83

Editors

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

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

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

. OXFORD

AMERICAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK

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

1975

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

. OXFORD

AMERJCAN ELSEVIER PUBLISHING COMPANY, 1NC.-NEW YORK

Q

NORTH-HOLLAND PUBLISHING COMPANY-1975

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

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

PREFACE

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

...

Vlll

PREFACE

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

PREFACE

ix

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

..

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

Chapter 3

&----

3 3.D

1 Chapter 2, 5 Chapter 4

1

[Chapter 2,

2.AI

xv

CHART OF INTERDEPENDENCIES

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

la

i Appendix C

Appendix D

Appendix E Chapter 0, Chapter 3,

6 5

4 3.C

I Chapter 0, L

3

I J

CHAPTER 0

PRELIMINARIES

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

2

[Ch. 0,f 1

PRELIMINARIES

Following our friend G. Reyes’ (cf. [l], p. 103) famous motto: “Do not scratch if it doesn’t itch”, we shall be rather brief and sketchy concerning the set-theoretical information used in this book. Any ‘of the numerous good books on set-theory (such as BACHMANN[l], KURATOWSKIMOSTOWSKI [ l ] or SIERPINSKI[l]) should supply the reader with what is missing here. We use the standard definition of ordinals a la Von Neumann, whereby an ordinal coincides with the set of all smaller ordinals. In presence of the axiom of foundation (or regularity) an ordinal can be defined as a transitive set’ on which the membership relation E is connected? Thus the order relation between ordinals coincides with the €-relation, or, what is When there is no the same, with the relation “c”of proper in c lu ~ io n .~ possibility of confusion we shall use the symbols “E’’ or “<” interchangeably to denote this relation. ON shall denote the class of all ordinal numbers. Lower-case Greek letters (with the exception of “cp”, ”)I“ and, possibly, a few others) will be used as variables ranging over ordinals. Given a well-ordered set ( x , <), there is a unique a E ON such that (x, < ) = ( a , E ra)4; such ordinal a is called the order type of the set ( x , <); it is denoted by ( x , <), or simply by X when no confusion is possible. The process that we have just described can be generalized.

DEFINITION 0.1.1. Let K be a class and R a binary relation on K . R is extensional iff for every x,y E K the following holds: (a) (for every u E K , uRx w uRy)

x=y.

(b) R is well-founded iff (i) each non-empty subset of K has an R-minimal element; (ii) for each a E K , the “R-initial segment” { x E K I xRa} is a set.

’ A class S is transitive iff for all x : x E S A binary relation R

JX

cS.

C S X S is connected iff for all x,y E S : xRy V y R x v

x = y . Recall that the domain of a binary relation R C S X S is the set ( x E S I (3y)((x,y) E R ) ) , the range of R is the set { y E S 1 ( 3 x ) ( ( x , y ) E R } , and thefield of R is the union of its domain and its range. We shall always use ‘‘C” for inclusion and “c”for proper inclusion. If R is an n-ary relation on a class X,R c X“,and Y c _ X , then R Y denotes the relation R f Yl “ on Y . If K , , K 2 are classes and R , , R 2 relations on K,,K,, respectively, with the same number of arguments, the symbol ‘‘z”denotes the relation of isomorphism between ( K , , R , ) and

r

(K2>R2).

Ch. 0, 5 I]

SET-THEORETICAL BACKGROUND

3

REMARK. If K is a set, (ii) is superfluous. It is easy to show, using the axiom of choice, that (i) is equivalent to: (i‘) K does not have infinite R-descending sequences, i.e., there is no f : w + K ’ such t h a t f i s one-one a n d f ( n + l)Rf(n) for every n E o . (i’) will be used several times in this book. The fundamental facts about well-founded relations are: (1) (Induction principle) Suppose R is a well-founded relation on K and @ a formula. If for every x E K we have: (for every u E K , uRx =+ @ ( u ) )

-

@(x),

then @(x) holds for every x E K . Corresponding to (1) there is an obvious principle of definition of operations on K by “induction” on R. (2) (The Shepherdson-Mostowski theorem) If R is a well-founded and extensional relation on K , there is a transitive class M such that ( K, R ) = ( M . E M ). Moreover, there is a unique isomorphism defined inductively by the condition:

r

F ( x ) = { F( u ) I u R x }

for x E K .

This map is called the collapsing isomorphism of ( K , R ) onto the transitive class M . A proof of the two last results is given in Appendix A; they will often be used in this book. An initial ordinal is an ordinal a which cannot be put into a one-one correspondence with an ordinal smaller than a . These are the finite ordinals and the “alephs” (or “omegas”). We shall identify cardinals with initial ordinals. It is easy to prove from the axiom of choice that for any set x there is a unique initial ordinal which can be put into one-one correspondence with x. This is denoted by 7 and called the cardinal number of x. CN shall denote the class of all infinite cardinals and CN’ the class of all cardinals. There is a map-usually denoted by o or K-such that (1) w:ON+CN. (2) o is one-one and onto.

’

Throughout this book we shall use the following notation for maps. Dom(f), Range(f) denote the dumuin and the range off, respectively.f ( x ) will always denote the image of x E Dom(f) by f. /[XI,f - ‘I Y ] (square brackets) shall always denote the image set (resp., inoerse image set) of X c Dom(f) (resp. of Y) by f. f:A-+B will always mean that Dom(J)=A, R a n g e 0 B.

4

[Ch. 0, 3 1

PRELIMINARIES

(3) w is strictly increasing: a,P~ONr\aEj3 (4)

waEws.

w is continuous:

-

~ , P E O N A { Y ~ ~ ~
I< P

NOTE.“ U stands for the usual set-theoretical union. If a C ON, U a coincides with the “supremum” (least upper bound) or “limit” of the set a . ”

The number w,, coincides with the order type of the set of all natural numbers with their usual ordering. It is also denoted by w. NOTATION. If { x j l i E I } is a family of sets indexed by the set I , IIiEIxi denotes the set of all maps f : Z + UiE,xjsuch that f ( i ) E x i for every i E Z . If x i = x for all i E I , TIiErxi is just the set of all maps f : Z + x and is If Ti = K;, I I i E I x j is denoted by I l i E , ~ i . If, moreover, the denoted by XI.

-

xj’s are pairwise disjoint, then

=A,

K’

denotes

7 .The

UiE,xiis denoted by 2 i E r ~ i If. K = K and

-

existence of most of these sets requires the

axiom of choice. 9 (x) denotes the power set of x, and 9 ( x ) is denoted by 2 ’ , this notation being consistent with the preceding one. Given an infinite cardinal A, 9 ’ ( x ) shall denote the set of all subsets of x of power < A ; thus, 9,(x) is the set of all finite subsets of x.

DEFINITION 0.1.2. (a) a EON is a successor ordinal iff there is /?EON such that a = p+ 1 (= /3 u { p } ) ; otherwise a is called a limit ordinal. If a = p 1, then j3 is denoted by a - 1. a is a limit ordinal iff a = Ua. otherwise it is (b) a ECN is a successor cardinal iff it is of the form a limit cardinal. If K = ws and a = u p + ,we write a = K + . If a = K + , then we shall write a - = K ; if a is a limit cardinal, a - = a. (c) If a is a limit ordinal >0, cf(a) is defined as the smallest p EON such that there is a sequence { yI 1 E < ,f3} c a such that a = Ut
+

Ch. 0, $ 11

5

SET-THEORETICAL BACKGROUND

SAC denote the classes of accessible and strongly accessible cardinals, respectively. (f) In = CN - AC; WIo = CN - SAC. Elements of In are called inaccessible or strongly inaccessible cardinals; elements of WIn are called weakly inaccessible cardinals. We rephrase the definitions of In and WIn: K K

E WIn E In w

K

K

E

E Reg and K is a limit cardinal; Reg and for all cardinals h < K we have '2

< K.'

(g> Given K E CN we define

30

( ~= ) K,

a,( K ) =

u 3[(

if h is a limit ordinal.

K)

aU denotes SU(u);Z u is called beth-a. (h) Given

K,

h E CN, the cardinal

2

KV

U€CN V < X

is denoted by K<' and this operation called weak exponentiation. We also introduce the operation: K>'=

2

(Kv)+,

v<x

V€CN

which will be called semi-weak exponentiation. With obvious modifications these definitions (and the properties stated below) also apply when K is a finite cardinal > 2. (i) T:ON-+ON is defined by the equality

(By the axiom of choice, T is well-defined.) In what follows we list-without proofs-some properties that will be needed along these notes; their proofs can be found in any of the [ 11, Chapaforementioned books on set theory; cf. especially BACHMANN ters V-VII.

' Cardinals fulfilling the last condition are sometimes called strong limit cordinals.

6

[Ch. 0,8 1

PRELIMINARIES

(1) For every limit ordinal a > O , cf(a) is a regular cardinal, and cf(a) Q a. (2) For every K E CN, cf(K +) = K +,i.e. K E Reg. (3) n is strictly increasing and continuous; hence n ( y + 1) > y or ay+I > wy for all y EON. (4) a EON * cf(o,) < w, < arc,,. ( 5 ) Konig's theorem: 2 i E l< ~IIiE,Ai i whenever K~ w, and cf(2') >A. (7) If o i s cofinal in a then 2'o#Na. In particular 20' is different from Nu, Nu*, Nu+u' etc. (8) ACf(') > A. (9) h"=h*K
3: = (ii) a limit ordinal:

I

if K < > , - ~ ,

2"

if

.

if K > 3 , - , .

K

< cf( aa)(= c f ( a ) ) ,

if cf(a) Q K
3,.

K>

if p < cf(A),

K<'

K'

if cf(A) < p Q A,

KP

if p 2 A.

K<'

if p < cf(A),

K'

if cf(A)
K
ifp>A+.

I

(KA)<~,

K'

K
ifpA+.

(15) K strong limit cardinal e~ 2<" = K = 2>".

Ch. 0,

I]

SET-THEORETICAL BACKGROUND

I

(16) a>O + (w,EWIn w a=w,=cf(a)). (17) w E In C WIn (hence SAC C AC). (18) w,EIn * w , E W I n A r ( a ) = a . (19) K E I R ~ K is a strong limit cardinal and K < ~ = K w for every collection { KiliE I } of cardinals such that < K and K; < K for all i E I ,

niE1Kj
K>’<

(K<’)+.

(21) If X is not a strong limit cardinal (e.g. if X is a successor cardinal), then ~>’>>h. (22) If K>’>>X, then ( K < ’ ) + is the smallest regular cardinal bigger or equal than K > ‘ ; in particular: (23) If K>’ is regular and ~>’>h, then K > ‘ = ( K < ’ ) + ; if K>’ is singular or K>’ < A, then K > ’ = K<’. The generalized continuum hypothesis (GCH) is the following statement: for all K € C N :

2”=

K+

The GCH has many important consequences in the arithmetic of cardinal numbers; we list some refinements of preceding results assuming the GCH: (24) For all a E O N : ..(a) = (Y (i.e., 3, = N,). (25) In = WIn. (26) (1 1 ) holds with Nu instead of 3,. (27) h
A<‘=

h (31) 2<“=

if h regular,

h +

if h singular.

K.

Godel proved in 1938 (cf. [l]) that the GCH is consistent with ZF. Cohen proved in 1963 that its negation is consistent with ZFC. Sometimes it is possible to dispense with the GCH in favor of weaker assumptions. For example, (25) is derivable from the so-called limit cardinal hypothesis (LCH): “every limit cardinal is a strong limit cardinal”

[Ch. 0, 9 1

PRELIMINARIES

8

or, in other words: if A is a limit cardinal and K
DEFINITION 0.1.3. Let ( a , 14 < A) be a well-ordered sequence of ordinals. be a sequence of pairwise disjoint (a) Addition. Let ( ( X c , <[)[<
3b

@

(aEXcr\bEX,A~<~)V(aEXrAbEXrAa<~b),

It is easily seen that this relation well-orders

2

.Ed at=

(E(;Jhxd’

Uc
)

(This is a proper definition, i.e., X,
f

i’g

o

f(to)E g ( t o ) ,where todenotes the largest element of the set { t < A l f ( < ) # g ( t ) } .

Again, it is easily proved that if is a well-ordering; we set:

As usual, the product of two factors is denoted by a.P. (c) As a particular case, if a,=a for all E
Ch. 0, J 1)

9

SET-THEORETICAL BACKGROUND

that will help to make clear the sense in which the ordinal operations are used in this book. Note first that &,xal =C,<xF;in particular, if each a, is a cardinal (i.e. an initial ordinal) and c< a + a,< as, it is readily seen that: P

where K is the (cardinal) sum of the cardinals a,,(<X (cf. p. 4). The ordinal product is an iterated sum: if a,/? EON and a - a for all [
+

+

+

+

The sets R ( a ) are inductively defined as follows: R(0)=0 R ( a ) = U 9 ( R ( / 3 ) ) fora>O. P
+

10

PRELIMINARIES

[Ch. 0,5 1

n=

Let U a E O N R ( aIt) .is easy to prove that the axiom of foundation is equivalent to the statement n = V . (V denotes the class of all sets.) Therefore every set belongs to some R(a). The rank of a set x , denoted by p ( x ) , is the least EON such that x E R ( a + 1). The following properties can be easily proved by the reader:

The “natural” structures are defined as:

It is not difficult to prove that (A) %la+, is a model of VBG H a E I n - { a } , (B) a €In - {a} * Ba is a model of ZF. The converse of (B) is not true as shown by MONTAGUE-VAUGHT [I]. The results that we have just quoted are the basis for showing that the is consistent with the axioms of set theory, ZF say. If statement “In= {a)’’ In - (a}#a, let a. be its first element. By (B), Ba0is a model of ZF; it is not difficult to show that In%,= {a).’ The same idea can be used to show that if the statement “In - { w } #a ” is consistent with ZF (ZFC), then the following statement is also consistent with ZF (ZFC): “there is exactly one a E In, a > a”. Analogous results can be proved replacing “one” by “two”, “three”, etc. Hence each existential assumption about inaccessibles > w requires a particular axiom, Finally, it should be remarked that it is not known whether “In- { w } #0” is consistent with ZF. It is convenient to make at this point a reference to the class L of Godel and a variant of it, L,, which will be mentioned-though not used directly -in this book. For details we refer the reader to GODEL[I].

’

If X is a class and IUI is a model of ZF (ZFC),X* denotes the class corresponding to X in the sense of the model %I?.

Ch. 0, 5 21

‘‘CONSTRUCTIBLY” DEFiNED INACCESSIBLE CARDINALS

11

The definition of the classes L and L, (for a fixed set a ) is done by transfinite induction; L(a 1) consists of all subsets of L(a) definable (in (L(a), E rL(a))) by a formula involving E using parameters from L(a); for limit ordinals a , L(a)= Up<,L( p). L,(a + 1) is analogous, except that the formula involves an additional unary predicate; likewise, L,(a) = UB
+

!j2. “Constructibly” defined inaccessible cardinals

From the information contained in the preceding section it is clear that inaccessible numbers > w are rather large. However, we will go several steps further in the classification of large cardinals. I t should be obvious that all arguments involving inaccessible cardinals are highly speculative, since we do not have at present a consistency proof of their existence. In order to make meaningful our considerations, it is advisable to introduce very strong axioms concerning the existence of inaccessibles. Some examples are:

“In is cofinal in ON’

(Tarski),

or

“Every strictly increasing, continuous function f : ON-+ON has at least one inaccessible number in its range” (Levy), or even stronger ones. Since we want to keep our discussion at an informal level, we leave to the reader to figure out what axioms are necessary in each case.

12

[Ch. 0,5 2

PRELIMINARIES

Mahlo considered, in a series of articles from 1911-13 two different methods in order to arrive at “big ordinals”. The first one is to take fixed points of enumerations of several classes of ordinals. The second method, much more powerful than the first, is based on topological considerations. A more detailed treatment of Mahlo’s method is contained in the recently [ 11. published book DRAKE A. The fixed-point method; --numbers I t is easy to show in ZF that any strictly increasing, continuous function f:ON+ON has arbitrarily large fixed points. These fixed points may be singular, and, consequently, hardly merit the name “big ordinals”. For instance, i f f is the identity map, every ordinal is a fixed point. For this method to produce “big ordinals”, f should have “large enough gaps”. Mahlo fulfills this condition by restricting himself to regular cardinals, which he calls no-numbers; then he takes fixed points of their enumeration in their natural ordering as ordinals, calling them n,numbers; these are the weakly inaccessible numbers >a. In the next step he considers fixed points of the sequence of n,-numbers to obtain the n,-numbers, etc. The process continues in this way into the transfinite, at the limit steps taking intersections of all classes obtained before. Each of the classes obtained in this way is strictly included in the preceding one, unless this one is empty. However, we do not stop here. One may consider numbers p > 0 which are np-numbers (0 is a na-number for every a E ON), and take fixed points of their sequence, etc., repeating the process indefinitely. It is clear that the same result is obtained (one step later) if we replace Reg by WIn. If we start with In the result will not necessarily be the same (except if we assume GCH, for instance). In this case the numbers we hit at the pIh step are called “inaccessibles of type p”. We shall start from a fixed class (or set) X L C N u (0) that we suppose to be indexed in its natural ordering: X = { aX}XEON if X is a proper class, X = ( aX}X<6 for some S EON if X is a set ( a X< ah,,if A
DEFINITION 0.2.1. rI,(X)

=

x;

[ € I l , + , ( X ) w [ € r I , ( X ) , and type

n n<(x)

5
[nI’I,(X) is well-ordered in

if A is a limit ordinal.

Ch. 0, 5 21

13

“CONSTRUCTIBLY” DEFINED INACCESSIBLE CARDINALS

Let EI be a fixed set, 0 9 ON. The enumeratingfunction r ( . , q , X ) of the class II,(X) is defined by:

(. pL,vI,X)=

a

if there is a EIIJX) such that a n II,,(X)

0

if there is no such a.

=p ,

Thus r (O,q,X), r (l,q,X),... are the 1st,2nd ,... elements of n,(X) in their ordering as ordinals; r(p,q,X)=O iff I I , ( X ) has less than p elements. We use rv(5,X) instead of r(E,v,X) and omit X unless confusion arises. The most important properties of r are: (1) r is a strictly increasing function of its first argument:

(2) r is a non-decreasing function of its second argument:

(3) P , v E O N A ~ ~ ( P ) E O* N rV(p)2p. (4) hErI,+l @ A=r,(h) @ r,,(A)€rI,+,. ( 5 ) p > Y and r,(h) EON imply that the following are equivalent: (1) r,,(A) > r”(N (ii) .rr,(h)m,+l (iii) h ~ r I , + ~ .

(6) p > O implies: a = r,,(h) for some h w for every v < p,a = r,(a). (7) ~ > > A A A + ~ ~ X A ~ , , ( A ) E*OrA(p)
(9) A > 0 A h = ~ ~ ( for 5 ) some 5 EON, imply:

A=

{

Oex

rA(o)

if

rA(l)

if OEX

14

PRELIMINARIES

[Ch. 0, 8 2

(10) Let p > X and either p ) # O or rF(X)#O. Then the following conditions are equivalent: ( 0 A(;.) = p); (ii) p = %(A). Either condition implies that h is the least ordinal not in X . NOTE.(1)-(10) above give an almost complete determination of the function r (hence of the r”-numbers). (10) implies (9). In fact, if X > O and h=ax([)for some (EON, by (10) we have that 5 is the least ordinal not in X. If X ~ C N {0}, U we have then: l=O if 0 9 X ;[ = 1 if O E X (because 1 9 X ) . By (4), if X c C N u {0},then the first element > O of II,(X), rt(l,X), is not in I l t + , ( X ) ; in fact, rt(l)# 1 because 1 E X . The same occurs with the first w members of Il,: r t ( n ) # n if n > 1. If [> 1 we also have rC(w)6!!IIt+,; in fact, since [> 1, w E I I , , whereas r&w) En,; hence r , ( o ) # w and (4) gives the result. (9) says, in particular, that if X > 0 is a rA-number(in the case X = Reg) it is the first rx-number >0, i.e., all numbers between 0 and X have been removed at earlier steps. The next results, due to Mahlo, throw some light on the structure of the classes II,. DEFINITION 0.2.2. Let X C O N . L(X), the class of all limit points of X , is the class of all ordinals of the form Ux.,pa?where ( a XI A < p) C X is a strictly increasing sequence and p is a limit ordinal >O. From this definition it follows (1) OSL(X), (2) L is monotone, (3) L( fl ,,,Xi)C fl is,L(Xi). THEOREM 0.2.3 (Mahlo).

x c C N ~ ( 0 )* n , ( X ) - ( O } c X n L ( X ) , X c R e g + H,(X)-{O}=XnL(X).

Ch. 0, $ 2)

“CONSTRUCTIBLY” DEFINED INACCESSIBLE CARDINALS

15

n

The first formula implies WIn = Reg L(Reg), n,(wIn) = WIn f L(Reg), l etc. The second formula is essential to establish the connection between both of Mahlo’s classifications of inaccessibles.

B. Mahlo’s second method; p-numbers; hyperinaccessible numbers The second method considered by Mahlo is much stronger than the first-at least when both are applied to any of the classes Reg, WIn or In-in the sense that one obtains (smaller) classes of cardinals placed very high in the first classification. For instance, the first application of the new operation to WIn-the p,-numbers in Mahlo’s terminology-yields already numbers v which are rv-numbers; moreover, the first of these numbers is not a p,-number. Similarly the operation applied to In yields only inaccessibles of their own type but not the first such inaccessible (cf. Corollary 0.2.8.). In order to explain the principle behind this new classification, consider the first strongly inaccessible > w , say a,,; we have w,o=limE
for [ < a , . In fact, w,l=limt
DEFINITION 0.2.5. A set C CON is relatively closed iff U A E C, for all non-empty subsets A of C that are bounded above in C , i.e., B # A C C A (there is x E C such that for all y E A , y < x) * UA E C .

16

[Ch. 0,5 2

PRELIMINARIES

It is easy to check that C CON is relatively closed iff the set C u { UC } is a closed subset of ON with the order topology. The following sets are easily proved to be relatively closed: (1) Every closed subset of O N in the order topology. (2) Every interval [a,P), a,P EON. (3) Every singleton { a } , aEON. (4) The intersection of any number of relatively closed subsets of ON. ( 5 ) For any set B CON, the rehtive closure of B, g, defined as follows:

B= { a EON I there is A C B , A #B , A bounded

above in B, such that a = U A } .

(6) For any set B CON, L ( B ) is relatively closed. (7) For any set B C ON, B u L ( B ) is relatively closed. As an illustration we prove (6). The rest is left as an exercise.

PROOFOF (6). We prove something stronger, namely, if A r L ( B ) , A #B , then U A E L ( B ) . If A has a last element, then U A coincides with it and therefore U A E L ( B ) . We assume that A does not have a last element. Let u = U A . Since A C Y B ) , for every a € A there are a limit number p ( a )> 0 and a strictly increasing sequence ( y f I X < p ( a ) ) B such that a = UA
c

r o = y o6 0>.

rt+l=first y$+l such that y$+l>max{rC,61}, for ( < p ; T~

=first y$+l such that y$+l> max

i,<, 1

U T ,6,, , for 7 limit


The preceding definition is correct because by induction hypothesis the 7;s are of the form y?, and therefore

Moreover, it is clear that 7(E B for all [ < p and that u = Ut<prtbecause 6, < r,+ I < St+ for all 5 < p. This completes the proof of (6).

Ch. 0,J 21

“CONSTRUCTIBLY” DEFINED INACCESSIBLE CARDINALS

17

DEFINITION 0.2.6. If X CON, we define M ( X ) = { { EON1 there is a set C C X , C relatively closed, such that { = UC }

From property (3) above it follows that X c M ( X ) ; furthermore M is non-decreasing, i.e.,

X

C Y CON * M ( X ) c M ( Y ) .

We shall be mainly concerned with (iterated) applications of the operation M to the classes SAC and AC. We give an illustration of the effect of the operation M. It is clear that ACcM(AC); but many inaccessibles belong to M(AC) as well. For instance, all inaccessibles belong to M(AC) except for (possibly some 04 the fixedpoints in their sequence; i.e.: or, equivalently:

In-II,(In)cM(AC)

{ a EIn 1 there is 5 E ON such that a = ~ ~ ( and 5 ) 5 < ro(5)}C M(AC).

-

(roc)

means here ro(-,In)).

PROOF.Suppose 5 < r0(5)and ~ ~ (E5In; ) then

u

P<5

To(

~ ~ ( is5 regular ) and:

P )
Consider the set C = [( Up
c

The inaccessibles of type 1 which are not limits of inaccessibles of type 1 are also in M(AC):

II,(In) - II,(In) GM(AC). PROOF.Let a E II,- II,; then a = ~ ~ (>55,) for some 5 EON. We want to represent a as UC, where C LAC is relatively closed. From the regularity P ) < a. Consider the set: of a and from a >[, it follows: UB<Er,(

18

PRELIMINARIES

[Ch. 0,9 2

Clearly B c AC; also a = UB because a = no(a)by (4) of the properties of Let C = be the relative closure of B ; since B C C, a < UC ; but B C a implies C C a ([0,a) is relati'vely closed and contains B ) ; hence UC < a ; we have a = UC. Hence it is enough to prove C LAC. Let (no(<,)+I y < p ) C B ( p a limit ordinal) be a strictly increasing sequence bounded ' ; 6 such that no(S)+ is the above by some element of B , say ~ ~ ( 6 ) take least such bound; let p=lim,
since no(S)is inaccessible, the symbol < can be replaced by < ; therefore, p = l i m , ~ ~ n o ( ~ y<)no(S). + If p=.rro(S), p would be regular and we wovld have = no(S); since I y < p ) CS, = no(S)implies no(S)< S ; then we have S = mo(S), i.e., 6 En, (property (4) again). Write 6 as 6 = n,(7) for some 7EON. Since no(&)+E B by assumption, it follows that

p

(5

p

but this is contradictory because the last inequality implies 7 < E, whereas the first shows that p < 7 for all p
This clearly implies that pEIn (otherwise p would be of the form no@)for some A); then p E AC as we wanted to prove. W This process can be repeated all along the finite indices and even more (cf. Theorem 0.2.7). In particular, then

u (n,(w- n,,

or equivalently,

,(In>)GM(AC),

nEO

In- n,(In) CM(AC). This implies CN - M(AC) c II,(In).

Ch. 0, 8 21

“CONSTRUCTIBLY” DEFINED INACCESSIBLE CARDINALS

19

This shows that all members of CN-M(AC) are r,,-numbers for all finite n ; hence they actually must be very rare indeed. However, we shall also consider later transfinite iterations of M. The operation H, dual to M, is defined by: H( Y)=CN-M(CN-

Y).

The members of H(In) (resp. H(WIn)) are called hyperinaccessibles of type 0 (resp. po- numbers). The following theorem of Mahlo establishes the connection between both methods, by showing that the second is vastly stronger than the first in the sense that it produces (in the first step) numbers a which are ra( Y)-numbers, whenever Y LReg. Moreover, the first such number is not hyperinaccessible of type 0 (relative to Y ) .

THEOREM 0.2.7 (Mahlo). (a) Zf Y CReg and r v ( p , Y ) > p , v , then .n,(p, Y ) F H ( Y ) . (b) Zf Y CReg, then the first a > O such that a E l l a ( Y ) does not belong to H( y>* Theorem 0.2.7 can be rephrased to say: COROLLARY 0.2.8. Let Y LReg- (0); then (a) every member a of H( Y ) is a ra( Y)-number; (b) the first a > 0 which is a ra(Y)-number does not belong to H( Y ) . In particular, (c) every p,-number a is a ra-number; (d) thefirst a which is a ra-number is not a p,-number; (e) every hyperinaccessible of type 0 is an inaccessible of its own type; (f) the first cardinal number which is an inaccessible of its own type is not a hyperinaccessible of type 0. The proof of these results can be found in GAIFMAN [I]. We state some elementary properties of the operation M. (1) X LM(X), (2) X C Y * M(X) c M ( Y ) , (3) X CCN * M(X) LCN. More generally, (4) If 2 CON is closed under the limit operation (or closed in ON), i.e., L(Z)C 2, then X Z + M ( X ) c Z . (5) M(AC u (ON - CN)) = M(AC) u (ON - CN). And even more generally, ( 6 ) If Y C O N and Y is closed under the operation of taking the next

20

[Ch. 0,8 2

PRELIMINARIES

bigger cardinal of any of its elements in ON - C N , i.e., (E

Y n (ON-CN)

r\(>o-

T + E Y,

then

M( Y)=M( Y n CN)u ( Y - C N ) u c ( Y), where:

c(Y)=

{ o}

if Y no is finite if Y n o is infinite.

(7) M(AC)=ACuM (Sing).

DEFINITION 0.2.9. (a) If X = ( X , [(YEON) is a decreasing sequence of classes of ordinals indexed by ON, we define the diagonal of X, XD, as follows: XD={aEONIaEXa}. (b) If follows:

X = ( X , I a < 6 ) is indexed by some ordinal 6, XD is defined as XD={aE6jaEXa}U

n x,. a<&

REMARK0.2.10. (1) The meaning of XD can be visualized if we place the elements of X , (in their natural ordering) in the a& row of a matrix. (2) Case (b) is done to make I D coincide with (X*)D where X* = ( X z I a EON) is defined by

(3) If n, = (n,(X)1 a EON), then which are r,(X)-numbers.

(n,)Dis the class of

all ordinals a

As an illustration of our next steps we shall consider different ways of iterating an operation F which associates a subclass F(X) of ON to every subclass X CON. We have in mind here operations like M.

Ch. 0, S 21

"CONSTRUCTIBLY" DEFINED INACCESSIBLE CARDINALS

21

Given one such operation, we can define the iterations of F as follows:

Fo(X) = X , F+'(X)=F(F1)(X)) for -q EON, Fx(X)= U Fp(X) for A limit, h>O. P<X

We also define

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

,.

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

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

of type q if 77
are called the members of CN - lW(§AC). Thus, the members of MD(AC) are the cardinals that are not hyperinaccessibles.of their own type. As an exercise the reader can easily verify that: (8) Under the hypothesis of (6) above the following holds:

M'( Y )= Mq( Y n CN) u ( Y - CN)u c ( Y ) for every 11 EON. It will be convenient for later purposes to introduce the following: DEFINITION 0.2.1 1. For each ordinal v we define F("), the normalized iteration of order v of F, recursively, on subclasses X of ON, as follows:

F'''(X) = X , F'p+')(X)=F(F'p)(X))-( p+ 1) F'"(X)=

[U

P
F@)(X)]-A

for PEON, for h limit, h>0.

22

[Ch. 0,

PRELIMINARIES

We define the operation

F'") by

82

the equation

F'"")(X)= U F'P)(X), PEON

the sequence F ( X ) by

F(X)=(F@)(X) IpEON), and the operation F(D)by D

F'D'(X)= [ f i ( X ) ]

=

{ / 3 ~ 0 N J /E3F ( ~ ) ( X ) } .

The properties of the normalized iterations are collected in the following:

LEMMA0.2.12. (1) For a limit ordinal the following conditions are equivalent: (a) a E up <,F'"(X), (b) a EF(,)(X), (c) a EF(")(X), (d) a E F(D)(X), (2) Let /3 EF(")(X); if pF( /3) is the first p EON such that p EF("(X), then pF( p ) < p; if /3 is a limit number, then pF( /3) < p. ( 3 ) If F is monotone increasing, then F(')LF' - v CF' for all v > 0, F ( ~c)F ~ .

F(") L F",

A final remark about operations. For any F assigning to every X CON, a subclass F(X) of ON, the dual *F is defined by:

(9

*F( Y ) = O N - F ( 0 N -

Y).

For operations satisfying the condition:

XCCN

* F(X)CCN

the dual is defined as follows: (i')

*F( Y)=CN-F(CN-

Y).

Thus, *M = H. The dual satisfies the following relations: (ii)

*(*F) = F

(iii)

F(X) c G ( X ) =+ *G(CN- X ) c*F(CN- X ) .

Ch. 0, 5 21

“CONSTRUCTIBLY” DEFINED INACCESSIBLE CARDINALS

23

It is convenient to treat operations by pairs of duals. If in such a pair one of the operations, say F, satisfies the relation X c F ( X ) for every X, the dual will satisfy the reverse inclusion: *F(X) c X, for every X. Similarly, if one is monotone increasing the dual will be monotone decreasing and vice-versa. The iterations of *F are defined by dualizing the corresponding definitions for F; for instance:

(*F)’( X ) = X

( * F ) ~ + ’ ( x=) *F((*F)’(x)) (*F)?x) =

n (*F)”(X)

for every E O N for A limit, A > 0.

P<X

Then it is easy to check that:

(*F)”(x)=oN-F~(oN-x)

for all v EON.

and other similar relations. II,is the “decreasing” operation of the pair (HI,*HI), whereas M is the “increasing” one in the pair (H,M); thus II,should be compared with H and M with *HI, rather than the other way around. This is the content of Theorem 0.2.7(a) which, in this terminology, asserts:

X cReg

* H(X)cl-II‘,m)(X).

In virtue of (3) this theorem admits a dualization relating *H\O0)to M:

X CReg

* *Hclm)(CN-X)cM(CN-X);

for instance, if Y c AC, setting X = CN - Y we will have X c In c Reg, and hence:

We mention finally that *111admits a characterization in topological terms very much like that of M: * I I , ( X ) = { a E O N /there is c c X such that c u { a } is a closed interval of ON and a = U c } (for X CON).

24

[Ch. 0, t 2

PRELIMINARIES

It will be convenient for later considerations to introduce the concept of “generalized iteration” of the operation M. We do so here. This generalization yields classes of cardinals even wider than those considered so far, thus extending the range of validity of the results in Chapter 3. The principle behind this generalization is the following: the usual ordering relation E on ordinals plays a double role when we consider the classes Ma(X) or M(”)(X). On the one hand it is used to define inaccessibles, relatively closed sets, to order the elements of X, etc.; on the other it is used to index degrees of hyperinaccessibility (i.e., transfinite iterations of M). We allow different well-order relations of the ordinals to play those different roles. However, this poses a delicate foundational problem. Among wellorderings of ON occur those that order this class in types “ON O N ’ (the order obtained, for instance, by placing the odd ordinals after all the even ones), “ON’” (ordinal product) or even “ONoN” (ordinal exponentiation). This is to say that there will be relations R which determine initial sections that are proper classes.’ On the other hand we will need ordinal numbers and transfinite induction to enumerate the ordinals in the sequence determined by R, and obviously this cannot be done unless all initial sections determined by R are sets. If ZFC is enlarged by an axiom which ensures the existence of an ! large enough to make our considerations noninaccessible cardinal 2 vacuous, and “ON” is interpreted to mean “the set of all ordinals less than a,” the problem disappears: the well-ordering relations on Q are in one-one correspondence with the ordinals a such that < a < a+; thus the order type of (ON,R) (the unique ordinal a such that Cl < a
+

r

’

Moreover, if R is a well-ordering relation of ON for which every initial section is a set, then (ON,R)=(ON, E ) (cf. GODEL[l]).

Ch. 0, 3 21

between

"CONSTRUCTIBLY" DEFINEDINACCESSIBLE CARDINALS

( T ~ E) ,

25

and (ON,R)). We define Mmv) recursively as follows:

DEFINITION 0.2.13.

M'R,A'(X)= U M'R3c'( X ) if A is a limit ordinal, h > 0. t
Moreover, we define MR as follows:

REMARK 0.2.14. The preceding definition should be thought as a generalization of M"); likewise, MR is a generalization of M'"). The class M('vv) is increase with 5. The set added in the second step to insure that the M(R.E)'s O,, 1 is subtracted, as in the definition of Pq), to insure that the condition a EMR(X) depends only on the relation R restricted to ordinals less than a. The reader can verify by himself that: if a EMR(X)and p ( a ) is the least q EON such that cr EM(R,v)(X),then:

+

(i) ~ ( ais)a successor ordinal, p ( a ) = [ ( a ) +1,

(i) ">8,,,,.

Therefore, whenever a€MR(X), a can be "described" in terms of R restricted to all ordinals smailer than a, the ordinal O,,,, and a relatively closed subset of a, in much the same way singular cardinals are characterized by the number cf(a) and a sequence of type cf(a) converging to a. This analogy will play a significant role in Chapter 3.

In order to illustrate the extent of the operation MR we consider some examples. EXAMPLE 0.2.15. Let R be E; then Ov=q for all q EON. It is a matter of routine to show by transfinite induction on 7 : (*)

M(Esv)(AC)= { a ECN I a < 77 r\a EMa(AC)} U [M"(AC) - (7 + I)].

26

PRELIMINARIES

[Ch. 0,t 2

This implies:

ME(AC)=MD(AC). In fact, if a €Ma(AC), by (*) a EM(E3a)(AC). Conversely, we know by the if a < p(a), by (*) a EMa; if p ( a ) < a , definition of p ( a ) that a EM(E*p(a)); by (*) ~ E M " ( ~ ) ~ Thus, M " . ME(AC) consists exactly of those cardinals which are not hyperinaccessibles of their own type. Our next example is concerned with cardinals that are hyperinaccessibles of their own type.

EXAMPLE 0.2.16. Since for any operation F, we get

(*F)~(x)=cN-FD(cN-x),

HD(In)= CN - MD(AC), i.e., HD(In) consists exactly of all hyperinaccessibles of their own type. Let

po,pI,...,p,, ,... be an enumeration of the elements of HD(In) in their

natural ordering. It is clear that a E C N and a<po implies aEMD(AC), i.e., p o nCN cMD(AC)= ME(AC); moreover, it is obvious that p o nCN is relatively closed; since po is a limit cardinal, po= U (ponCN). Hence po€M(ME(AC)).In a similar way, considering [( lJE<+@+,p,,)n CN instead of ponCN, we can prove that p,,fM(ME(AC)) for all p,, such that p,, > q. This proves that

HD(In) - IIl(HD(In))cM(ME(AC)), in exactly the same way that we proved in p. 17 that In-rI,(In)CM(AC). This suggests starting all over again with HD(In) instead of In, and M €(AC) replacing AC. However, Mahlo's Theorem 0.2.7 has the answer for us; taking Y=HD(In), we get TU(p, H

D ( W EM(ME(AC))

for all those p, v such that nu(p, HD(In))> p, v. This shows that the elements of HD(In) that are not in M(ME(AC)) must be very rare indeed. Of course, if for nothing else than the fun of it, one can continue this classification calling the elements of HD(In) that are not in Mq(ME(AC))for q 2 w (resp. of type q ; by not in M"+'(ME(AC)) for q
Ch. 0, $ 21

"CONSTRUCTIBLY" DEFINED INACCESSIBLE CARDINALS

21

type, and by the argument of Example 0.2.16 it contains also all hyperhyperinaccessibles of their own type [i.e. all elements of HD(HD(In))]that are not fixed points in the sequence of those numbers. We can repeat this process again, and again, and again.. ., ad transfinitum. Thus we have considered the classes M(AC), M'J(AC) (q E ON), ME(AC), M(ME(AC)), M'J(ME(AC)) (q E ON), ME(ME(AC)),. . ., M(ME(ME(.. .(ME(AC)).. .))), .. . . Assuming that none of these classes contain all infinite cardinals, we have seen that each one is larger than the preceding. Moreover, given two well-orderings of ON, R and S, we can define (MS)R by replacing the operation M by MS in the inductive definitions of M(R-q)and MR. Thus we have (ME)€(AC), and it is easily shown that this includes all classes mentioned above. However, certain closure principles can be established to the effect that each of these classes is already of the form MT(AC) for suitable T. THEOREM 0.2.17. Let S,T be well-ordering relations of ON. There are well-ordering relations R,R' of ON such that for any X C CN' the following holds : (i) M ~ ( x )= M ~ ( M ~ ( x ) ) , (ii) MR(x) = ( M ~ ) ~ ( x ) . Moreover, rR= rT+ rs and

T~

= r s -rT.

(i) and (ii) of this theorem correspond to the addition and multiplication of order types, respectively. Similar results corresponding to exponentiation and other recursive arithmetical operations on order types are possible. For instance, the definition corresponding to exponentiation is as follows:

and

28

[Ch. 0, $ 3

PRELIMINARIES

The proofs of Theorem 0.2.17 and of the corresponding result for exponentiation are left as exercises for the reader. A sketch of the first proof can be found in HANF[ 2 ] ,pp. 318-19. All these classes as well as several other, larger ones will be considered in Chapter 3, $ 2 , where we deal with Hanf‘s incompactness results for infinitary languages. $ 3. Filters and ultrafilters; ultraproducts; trees

A. Filters and ultrafilters

DEFINITION 0.3.1. (a) A family % of sets is called a field of sets iff (j1

lJ%€%,

(U)

X,YE~?I*X-Y€%AXLJY€%.

% is afield of subsets o f X iff U % = X . Given an infinite cardinal K, 9 3 is called < K-complete iff G!L

c93 A E < K + U G2L €9.

An element X E 9 3 is an atom iff X # B and for every Y E 3’3 :

Y cX A Y # x + Y = B . (b) A collection 9c $8 is a filter iff it satisfies the following requirements:

(i)

X , Y E 9 + X n Y E ‘3,

(ii)

X E‘3Ax

(iii)

F#S.

Y + YE%’,

’? is proper iff B 9 ‘3 or, equivalently, F # 9. A proper filter Y is principal iff 9~9; otherwise it is non-principal. 5 is prime or an ultrafilter iff it is proper and in addition satisfies:

n

(iv)

X € % *XE%V

u 9- X € T

(c) A filter (prime filter) 5 in a field of sets of the form filter (an ultrafilter) on X . 9 is < tc-complete ( K E CN) iff:

9 ( X ) is called a

Ch. 0, 5 31

FILTERS AND ULTRAFILTERS; ULTRAPRODUCTS; TREES

29

REMARK0.3.2. (a) From (jj) it follows that i3 E % and X n Y E 93 for X , Y E 93 . (b) In the field 3 ' ' ( X ) the atoms are exactly the singletons { x } for x E X . (c) It is clear that any field of sets is < a-complete; 9( X ) is < K-complete for any K ECN; fields with this property are called (absolute&) complete. (d) (iv) is equivalent to the following: (iv')

X,YE% / / X U Y E ~ * X € ~ v Y E ' 3 .

NOTE.Observe that any principal filter is absolutely complete. Notice also that a proper filter is prime and principal iff it contains an atom of $8 ; hence a prime filteris non-principal iff it contains the complement of every atom. In particular, a non-principal ultrafilter on X contains all cofinite (complement of finite) subsets of X . Every filter of % can be extended to an ultrafilter of 3 by use of Zorn's lemma. The following fact is easy to check: Let K be an infinite cardinal, 3 a < K-complete field of sets and 9 C 93 be a family with the following prgerties: (i) F o r e v e r y % , % L % ~ G <=~m % # i 3 (ii) 9 # 91 . Then the set

n% r X ) < K-complete filter containing 9. If n 9 =i3 then n 5 =i3,

T = ( X € % I thereis % C % , E < K s u c h t h a t

is a proper, and 5 is non-principal. In particular, if K =a,3 '3 = C? ( X ) and 3 ' contains all cofinite subsets of X then '3, and therefore 9, can be extended to a non-principal ultrafilter. Given K ECN and a collection { 93;I iE I } of < K-complete fields of iE,%j is a < K-complete field of subsets of X . subsets of X , the family Thus, if 3E. c 9 ( X ) and { 3 ; I i E I } is the collection of all < K-complete fields of subsets of X containing X then iE,93i is the smallest < K complete field of subsets of X containing X . It is called the < K-complete field generated by ?;. , and denoted.by 93%. A remark which we will need later is the following:

n

n

We finally remark that the notion of ideal is obtained by dualizing the by U and C by above definition of filter (i.e., replacing U by 3). All concepts defined and all remarks made above apply, mutatis mutandis, to ideals.

n, n

30

[Ch. 0,

PRELIMINARIES

3

B. Ultraproducts In the present part of this section we introduce the basic definitions that are used in the theory of ultraproducts. We will use this material in Chapter 3, $ 3 . Let ( A , J i E I } be a collection of sets, indexed by the set I ; let 5 be a proper filter on I . We define an equivalence relation between members of H i E I A j as follows: f E F g w {iEIIf(i)=g(i)}ES.

For any f , its equivalence class is denoted by f / % . The reducedproduct of the family ( A j 1 i E I } modulo 4 is the quotient r I j E I A j /= and is simply . 5 is an ultrafilter on I , the reduced product denoted by I I j E I A j / 5 When modulo 4 is called an ultraproduct. If A j = A for all i E I we use the words reducedpower and ultrapower, respectively, and use the notation A'/ 5.In these cases there is a natural function n - : A + A ' / 5 defined by:

where 2 :I + A denotes the function with constant value equal to a. n- is called the canonical map. It is one-one. Suppose now that { R j J i E I }is a collection of relations with a fixed number of places, say n , and R; c (A;)" for all i E I . The reduced product of the relations ( R j I i E I } modulo '3 is the following n-ary relation R among ; f , ,...,f, E r I j E I A j : members of H j E I A j / 9 for

(*I

(fl/

9,. ..,f , / 9)E R

H

{ i E I 1 (f,(i ) , . ..,fn( i > ) E Ri } E 4 .

One has to check that R is well-defined, i.e.,

but this is a simple exercise left to the reader. In the case of a reduced power A I / 5 and a relation S C A " one easily verifies that 7~ is a monomorphism from ( A , S ) into ( A I / 9 , R ) , i.e. that for a , ,.. . , a , EA : or, equivalently:

( a , , ...,a, >Es

,.3 ( n ( a , ) ,

.. .,.(a,)>

ER

Ch. 0, 9 31

FILTERS AND ULTRAFILTERS; ULTRAPRODUCTS; TREES

31

Likewise, when we have a collection { 4.I i E Z} of (say) n-ary operations &.:(A,J"+Ai, we can define their reduced product modulo '3 to be the following operation F : ( I I i E f A i /S)n+.KIiEIAi/ '3: (**)

F(f,/'3,...,f/q=f/ '3

wheref(i)=F,(f,(i), ...,f,( i)) for every ZEZ. Again it is a routine matter to check that F is independent of the particular representatives of equivalence classes used in the preceding equation. It is easily seen that T is, as well, a monomorphism for operations. It is not difficult to prove (by reducing n-ary functions to ( n + 1)-ary relations in the usual way) that (**) is a particular case of (*). In the course of this book we will consider relational systems, this is to say, sets equipped with relations, operations and also distinguished objects (like the neutral element of a group, say). Given a bunch of these systems in which the number and arity of relations, operations and distinguished objects has been specified in a uniform way for all of them, the preceding recipes tell us how to define corresponding relations and operations in the reduced product of the underlying sets; we can also define the reduced product of a collection { c, I i E Z} (c, € A , ) of distinguished objects as the element c / ' 3 of II,,=,A,/%, where c(i)=c,

for all iEZ.

The ultraproduct of relational systems is one of the most important constructions in (first-order) model theory. Its crucial importance stems from the fact that the property (*) not only holds for a given set of specified relations, but also extends to all relations obtained from them by the operations corresponding to the (first-order) logical connectives and fundamental quantifiers. This is the content of EoS'S theorem-the theorem of the theory of ultraproducts-that we treat in Chapter 3, $3, in a version suitable for the purposes of this book. The reader interested in this part of model theory should consult the excellent books BELLSLOMSON [ I ] and CHANG-KEISLER [ 11. C. Trees

DEFINITION 0.3.3. (a) A partially ordered set ( T , <) is called a tree iff for T l y < x } of all <-predecessors of x is every x E T the set S , = { ~ E well-ordered by < . For every X E T the type of the set (Sx, < IS,) is called the length (or height) of x, in symbols h(x). The ordinal sup{h(x)lx E T } is called the length of T, in symbols e(T). For every a < t ( T ) the set L , = { x f T l h ( x ) = a ) is called the ath-levelof T. Any

32

PRELIMINARIES

[Ch. 0, 8 4

subset of T which is linearly ordered (or, equivalently, well-ordered) by < is called a chain of T . Any maximal chain is called a branch or path. Any < -minimal element of T is called a root of T. Let x,y E T; y is called an immediate successor of x iff x
3 4. Measurable cardinals Lebesgue asked in 1904 whether there exists a countably additive extension of his measure to all subsets of the real line which is invariant under translations. The non-existence was proved by Vitali in 1905 using the axiomof choice. In 1914 Hausdorff showed that in space it does not exist even a finite4 additiue measure invariant under rigid motions and giving cubes their correct measure. In 1923 Banach showed that in theplane there exists a finitely additive, invariant measure. In 1929 using the continuum hypothesis, Banach and Kuratowski proved that there was no non-trivial countably additive measure defined on all subsets of the line, even when invariance requirements are dropped. In 1930 Ulam improved the results of Banach and Kuratowski, deriving them from the weaker hypothesis that 2'0 is smaller than the first weakly inaccessible cardinal > w . He proved also several other things: (1) the first cardinal supporting a countably additive, real-valued measure, also supports a completely additiue, realvalued measure (cf. Definition 0.4.7 below); (2) the first cardinal supporting a countably additive, real valued measure is at least as big as the first weakly inaccessible (cardinals supporting completely additive measures are weakly inaccessibles >a);(3) a cardinal supporting a completely additive real-valued measure is either <2'o or it also supports a completely additive, two-valued measure (i.e. one taking only values 0 or l), and these cardinals are strongly inaccessible (this result was obtained independently by Tarski). As Tarski proved later these cardinals are related to several problems in the theory of Boolean algebras, in topology and in algebra. It was also Tarski who posed the problem whether the first strongly inaccessible > a supports one such measure. The work of Hanf in 1960 (cf. [2] and Chapter 2, $2) showed that none of the cardinals described in $ 2 above is measurable. From property (9) p. 13 and Corollary 0.2.8 (e) it follows that any K supporting a two-valued,

Ch. 0, 9 41

MEASURABLE CARDINALS

33

completely additive measure is the K* inaccessible. In 1967 Solovay, Jensen and Fremlin (independently) improved Theorem 0.4.15 below by showing that if K supports a completely additive measure with values in the interval [0, I] it is the K‘“ weakly inaccessible. In this section we develop the “classical,” elementary theory of measurable and real-valued measurable cardinals. We give an orderly (in our opinion) presentation of,it here for the sake of completeness. We omit almost all the proofs, but present the material in a steady progression, so that the reader can reconstruct them without much effort. A. General facts about measures

We shall only consider measures with values in the unit interval [0,1].

DEFINITION 0.4.1. (a) A Boolean a-algebra on a set X is a <w,-complete field of subsets of X . (b) A measure space is a triple ( X , 3 , p ) such that (i) X is a set; (ii) 3 is a Boolean a-algebra on X ; (iii) p : 9+[O, 11 is a countably additive non-trivial measure, i.e., p has the following properties: (1) p ( W = O , (2) A X ) = (3) p ( UnE,A,)= C n E w p ( A , )for any sequence ( A , In ~ w c ’3 ) of pairwise disjoint sets. (4) For all x E X , if { x } E 93 , then p ( { x } ) = 0. The series on the right hand side of (3) may diverge; in this case we consider its sum to be co, with the obvious rules for this symbol. We shall write p ( x ) instead of p ( { x}). The a-additivity of p (condition (3)) implies that p is finitely additive and a-subadditive: ( 5 ) p( U,,,A,) < X n E o p ( A n ) for any sequence (disjoint or not) ( A , I n E o ) C $8. (5) implies: , p(U,,,A,)=O. (6)if ( A f l 1 n E u ) G 9 3 and p ( A , ) = O for all n ~ w then We assume the reader is familiar with the general properties of measures; we only explicitly mention the following: (7) if ( A , I n Eo) c 4’3 is a decreasing sequence of sets, i.e., A , L A , for n < m , n , m E w , then p ( flnEoAn)=limn~mp(An)=inf,,,p(A,).l (8) if ( X , 93 , p ) is a measure space, then every family 9c ’3 of pairwise disjoint sets of measure >O has power < N,. 1 7

I

We only deal with finite measures.

34

[Ch. 0,8 4

PRELIMINARIES

For the rest of this section we will only consider measures defined on all subsets of sets X , i.e., measure spaces ( X , 3 , p ) where 3 = 9’( X ) . Correspondingly, we will omit all references to 3 ,that is, we will write ( X , p ) to denote our measure spaces. Recall that our measures have the property: p(x)=O for all x E X . The next result is related to the concept of measures with degree of additivity higher than countable.

LEMMA0.4.2. Let ( Z ,p ) be a measure space, A E CN be an infinite cardinal. Then the following are equivalent: ( I ) For every f : A + 9 ( Z ) such that f(.$)nf(q)=H for all t,q
u f(0 = x P(f(5)). [
([
(11) For every f:A+$?(Z) such that f(.$‘)nf(q)=H for all t,q
(111) For every f:A+

for all ( < A implies p U f(5) =O. (,
9 ( Z ) the following

p( f (t)) = O for all

holds:

)

t
(IV) For every f : A + 9 ( Z ) such that f([)n f(q)=,0 for all t ,q < A , 5271, the following is true: p ( f ( 5 ) ) I Fa finite subset of A

Notice that by (8) the sum in (I) has

< No non-zero members.

DEFINITION 0.4.3. Let ( Z ,p ) be a measure space, A E CN. The measure p is called A-additive if any of the equivalent conditions (IHIV) is met by p. p is completely additive iff it is A-additive for any infinite cardinal A < 7 The next lemma is the essential step in many of the subsequent results; it allows the formation of new measures out of given ones.

LEMMA0.4.4 (Pivotal lemma). Let ( Z , p ) be a measure space, { X i I i E Z} c 9( Z ) be a family of painvise disjoint subsets of 2 with the following properties: (i) p ( X J = 0 for all i E I ; (ii) p( U i E , X i ) =r>O.

Ch. 0,9 41

MEASURABLE CARDINALS

35

Then there is a measure ,ii defined on 9 ( I ) such that: (iii) { 0,1} C Range ( ,ii)C { s/ r 1 s E Range ( p ) } n [0, I]. (iv) For evety X E CN, < I , if p is A-additive, so is ,ii. (v) If p is completely additive, so is ,ii. REMARK 0.4.5. The point of (iii) is the following: if p is two-valued, then r = I , and Range( p) = { 0,1}, i.e., ji is also a two-valued measure. This will be needed later. Our last observation is that the existence of a measure is invariant under one-one, onto mappings. LEMMA0.4.6. Let ( Z , p ) be a measure space, f:Z+ Y a one-one, onto map. The set function v : 9'( Y)+[O, I ] defined by:

v ( x ) = ~ ( ~ - ' [forx x] )c Y , is a measure on Y . Moreover:

(i) Range( v) = Range( p ) (ii) v has the same properties that p has; for instance A-additivity (for any gioen X E CN), complete additivity, etc.

B. Real-valued measures DEFINITION 0.4.7. (a) (Z, p ) is called a countably real-valued measure spoace iff p is a countably additive non-trivial measure defined in all subsets of Z, with values in [0, I]; in symbols we denote this situation by ( Z , p ) ECRM. We say that an infinite cardinal K is a countably real-valued measura$le cardinal, in symbols KECRM, iff there is a measure p such that ( ~ , pE ) CRM. (b) ( Z , p ) is a real-valued measure space, in symbols ( Z , p ) ERM iff, in addition to (a), p is completely additive. A cardinal K E CN is a real-valued measurable cardinal, in symbols KERM,iff there is a measure p such that (K,~)ERM.

-

The following properties are immediate consequences of the definition: 0.4.8. COROLLARY ( I ) (Z,p)ERM * (Z,p)ECRM; K E R M KECRM. (2) ( Z , ~ ) E C R M A XG Z A G N,*~(x)=o. (3) KECRM * K > N , .

-

X

(4)

( Z , ~ ) E R M A X G Z AaT~<(Tx ) = o .

(5)

K

E RM

K

regular.

36

[Ch. 0,5 4

PRELIMINARIES

The following is simple to prove: LEMMA 0.4.9.

K ~ C R M A A < K A A € C+ N XBCRM, or

K€CRMAK<XAA€CN+ AECRM. The next three lemmata are direct consequences of the pivotal lemma. LEMMA0.4.10. If (Z,p)ECRM, { X i ( i € I } is a partition of Z and = I fZ CRM, then there i s j E I such that p(X,) >0. LEMMA 0.4.1 1. If 7 E CRM and f :I - K N is a function such that f( i) E CRM for all i E I , then 2 f ( i )fZCRM. LEMMA 0.4.12. Let

K~

be the first CRM; then K ~ E R M .

REMARK 0.4.13. Notice that either Lemma 0.4.12 and ( 5 ) of Corollary 0.4.8, or Lemma 0.4.11 show that the first CRM is regular. We ask whether it could ever be a successor cardinal. The next result implies: (1) the first CRM (and therefore any CRM) is bigger than the first weakly inaccessible > w ; (2) any RM is weakly inaccessible. LEMMA0.4.14. For any cardinal of the form (aE K +) of partitions of K + in K sets.

K+

there is a collection

The family % a is constructed roughly as follows. First one defines a doubly indexed collection (A: 1 < K +,5 < K ) of subsets of K with the following properties : (a) A:, n A i 2 = B for all ,$
Ch. 0, 41

31

MEASURABLE CARDINALS

Further details can be found in ULAM[l]. THEOREM 0.4.15 (Ulam). RM c WIn - {a}. PROOF.By (5), (1) and (3) of Corollary 0.4.8, if ( K , ~ ) E R M we , have E Reg and K > No. If K = A + , consider the class { % a la < K } of partitions of K . For each a < K , we have: K = UtGxB:;since p ( ~ ) =1 and p is A-additive, there is .$ < X (depending on a ) such that p(B:) > 0. For every a < K , let f ( a ) be the least such .$ 0 for every a < A + ; in particular, since B: LA: for all .$ < A, we also have y(A,f@))>O.On the other hand, since A + is regular, there is tOERange(f) K

-

-

such that f-'[to]= A + . Otherwise, f - ' [ ~ ]< A for all TERange(f), and we would have:

7

E Range( f)

7E

Range(f)

-

which clearly is a contradiction. Let now 9={ A ? I f ( a ) = t 0 } ; then g=A+ >a. But '3 is a disjoint family of sets of measure > O . This contradicts property (8) of p. 33. Therefore K is not a successor cardinal and the theorem is proved. We have already mentioned the following: COROLLARY 0.4.16 (Ulam). If cardinal > w, then K E CRM.

K

is smaller than the first weakly inaccessible

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

* K+

ECCRM.

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

38

[Ch. 0, 4

PRELIMINARIES

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

K

E2M.

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

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

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

Ch. 0,

41

MEASURABLE CARDINALS

39

REMARK0.4.24. Solovay has proved that the theories:

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

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

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

st={ f ( A )< K

CAA[EA},

40

PRELIMINARIES

[Ch. 0, 5 4

Clearly S,n TE=” and S,u T,G K for all .$
Since p(T,)= 1 for TJ E A - S, by A-subadditivity of p we have:

therefore p ( X ) = 1. On the other hand, if A LA,we have:

and

Thus, (*) and (**) together prove that

Hence X C { f(S)}, and therefore p ( X ) = O , contradicting our previous result. rn COROLLARY 0.4.27 (a) If K E CN is less than the first strong& inaccessible cardinal > a,then K B2CM. (b) If 20’ is less than the first weakly inaccessible cardinal >a, then K CRM * 2“ ECRM; hence, every ‘K E CRM is greater than or equal to the first strongly inaccessible cardinal > a.

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

Ch. 0,5 41

41

MEASURABLE CARDINALS

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

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

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

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

K,

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

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

K

into a measurable

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

1

U f ( t )= 1

L < A

(VI) For any f : A+

* there is t < X

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

9( Z ) :

P

1

u f(D

(*
=sup{

PL(f(t))It
Theorem 0.4.26 of Ulam-Tarski implies, in view of the results mentioned in 52, that it is consistent with the axioms of ZFC to assume that 2M is empty. Corresponding statements for the classes 2CM, RM and CRM can be derived from earlier results in this section.

42

PRELIMTNARIES

[Ch. 0, 3 5

On the other hand, the assumption that there are measurable cardinals is highly risky, as illustrated by the following theorem of Scorr [l]: THEOREM 0.4.29. If V = L , then there are no measurable cardinals. This theorem is proved using a variant of the ultraproduct construction introduced in $ 3 . The concept of normal measure (or its counterpart, that of normal ultrafilter) plays a crucial role in the proof.

DEFINITION 0.4.30. (a) A two-valued measure p on K is normal iff it is completely additive, and for every sequence Gx = ( X , 1 a < K) of subsets of K of measure 1, the diagonal intersection 5XD = { a E K 1 a E f l has measure 1. (b) A non-principal ultrafilter 9 on K is called normal iff it is < K complete, and for every mapf: K + K such that { a E K If ( a )< a} E 9, there exists 7 E K such that f - '(77) E 9. It is not difficult to prove that if p is normal, the ultrafilter defined in Theorem 0.4.28 (B) is normal and conversely. On the other hand, is normal if and only if the element i d / 9 is the K ' ~ element of the ultrapower ( K , € I K ) " / ~ (id is the identity map from K to K ) . We also have:

THEOREM 0.4.31 (Scott). If

K E 2M,

there exists a normal measure on

K.

NOTE.Whenever we refer to measurable cardinals without further specification, we mean members of 2M. f 5. Partition calculus In this section we develop the basic concepts of partition calculus. This branch of set theory has been developed mainly by Hungarian mathema[ 11 and [2], EmOsticians; the classical references are ERDOS-HAJNAL RADO[ I ] and ERDOS-HAJNAL-RADO [ 11. The two central results provable in ZFC-the theorems of Ramsey and Erdos-Rado-have been applied in an essential way to the model theory of first-order languages (these applications are studied in Chapter 2, $ 2 and Chapter 4, $3). Other results depending upon stronger axioms of infinity play a fundamental role in the metamathematics of infinitary [2] and languages (cf. Chapter 3, $ 3 and Chapter 5, $ 1; also MORLEY [I], [2]). SILVER

Ch. 0,5 51

43

PARTITION CALCULUS

For any cardinal r we denote by [X]’ the set of r-element subsets of X

[X]‘={ZCX(T=r} DEFINITION 0.5.1. Let (CJjEf be a partition of [X]’ in not necessarily non-empty) classes:

[XI‘=

7 =K

(disjoint but

u c;.

iEf

A subset Y of X is said to be homogeneous for the partition ( I ) if for some i E Z we have [ Y]l C;. .Notice that to define a partition of [ X ] ’ in the manner described by (1) is the same as to define a functionf: [ X ] ’ + K . The set Y is homogeneous just when f is constant on [ Y]l.

DEFINITION 0.5.2. If for every set X ’ of power rn and every partition of [XI‘ into K classes there exists a homogeneous set of power n, then we write

rn+(n>:. REMARK. It is easy to see that we may replace the word “partition” in the above definition by “decomposition”; by a decomposition we mean a representation of the form (1) in which the C;s need not be disjoint. REMARK. Observe that m-+(n>l, need be considered only for rn 2 n > r . For if 7= m, 7= n and n < r then [ Y]’ has at most one element and every subset Y of X of power n is homogeneous; hence rn+(n); holds. On the other hand, if m < n , clearly rn+(n); is false. Likewise, if m < K and m 2 n > r, the condition rn+(n); is false. LEMMA 0.5.3. Zf rn-+(n)L, rn’2 rn, n‘ < n and

K‘

< K,

then rn’+(n’)L,.

-

PROOF.Let ?’=rn‘ and { C/ I i E Z’}, r” = K ’ , be a partition of [X’]l. Let X G X ’ , 7 = m ; let J be such that Z’nJ=B and Z’uJ = K ; define

I

Or, what is the same, for some set X.

44

PRELIMINARIES

[Ch. 0, 3 5

It is clear that { C , l i ~ l ’ u . T }is a partition of [ X ] ’ into K sets. By assumption there is a Y X , 7= n, homogeneous for { Ci I i E I’ u J}. It is clear that any Y ’ C Y , ?’=n‘, is homogeneous for { C: I i E Z’}.’ Hence rn’+(n’);,. REMARK. Lemma 0.5.3 says that rn+(n)r, is invariant under increasing left hand side and decreasing n, K in the right hand side. The invariance under decreasing r requires a slightly more involved proof. LEMMA0.5.4. Zf m + l+(n+ l);+’, then rn+(n)L. (Hence for infinite m, n, rn-+(n):+ implies rn+(n)‘,.)

’

PROOF.Let 55 =rn and well-order X by < ;’ let x , F X and_ consider - X , = X u { x,} with x, added as first element. Let { Ci I i E I } . 7 = K, be a decomposition of [ X r ; consider the following decomposition of [X,]r+’: fory,, . . . , y r + , E X , and i E Z

(*I

*

( ~ l r - . . , y r + l ) ~ ~ , O

( Y I ~ * . . ~ Y r ECi. }<

+

By assumption, { C: I i E I } has a homogeneous set Yo of power n 1 ; say [ Yoy+lC cj”. Yo has a <-first element, yo; we write Yo={ y o }u Y , where Y =n. Since x, is the first element of X,, we infer that Y CX.It is easy to check from (*) that [ Y]l C C,. Hence Y is homogeneous for { Ci I i E Z} and of power n. 3

The following result shows that we need only consider the relation rn+(n): when r is finite. (Hereafter we do so.)

THEOREM 0.5.5 (Erdos-Rado).

PROOF.By 0.5.3 it is sufficient to show mt.(n),”o. Let M be a set of power m. We well-order [M]’o by a relation < . Elements of [Ml’o will be denoted by A’, Y , .... We place X in C, if X has a <-smaller subset Y , and in C,

’ If 1Y’l’ C Ci(iE J ) then n’ < r, so rn’+(n’);

holds vacuously.

We shall frequently use the device of ordering the base set X. We shall denote by ..., x n ) < any set (x,, ..., x n } for which xI < . . . < xn.

(x,,

Ch. 0,$ 51

45

PARTITION CALCULUS

otherwise. If N is any infinite subset of M then: (i) the <-least denumerable subset X of N is in C, (ii) if X , c XI c ... are denumerable subsets of N and X p is the then X,,, is in C , . Hence N cannot be homogeneous.

< -leiist,

We now prove Ramsey's Theorem (1931) which was historically the first result in the partition calculus.

0.5.6 (Ramsey). THEOREM

PROOF. Assuming No+(No)i = C , + . . . C,. Define

+

we prove No+(N0)i+'.

Mi ({ x , , ...,x, } ) = { x I { x , , ...,x,, x } E Ci }

Suppose [w]'+'

( i = 1,. ..,K).

Note that for every x , , . . .,xr E w , (i) U:= 1Mi( { x 1, .. .,X r }) =

(ii) the M i ( { x l ,...,x,})'s are disjoint for distinct i's. Let 9 be a non-principal ultrafilter on w. Since 9 is prime, (i) and (ii) show that (iii) M , ( { x , , ..., x , } ) 9 ~ holds for exactly o n e j (1 C j C K). , of w as follows; if x,,, We define a sequence of distinct elements x o , x I ... has been defined for all m < n, we may choose x, 'E fI{ Mi( { x,,,,, ...,x,,,}) I 1 C i C K, m,<

- - . < m, < n and

since this intersection belongs to 9 ,it is infinite, and we can also choose < n. Let N ' = {xo,xl,....} and set

x, distinct from all the x,,, for m

c;={ { x,,,,,...,x , , , } I ~ ~ ( { x ,,,..., , , x,,,})E~},

i = I ,..., K.

Then [N']'= C;u * * * u CL and so there exist. an infinite X C N' and a j such that [ X ] r q. It is easy to check that [X]r+'CC,. rn

[Ch. 0,B 5

PRELIMINARIES

46

The following is a finite version of Ramsey’s theorem.

0.5.7. For each finite n, k,r there exists a finite m such that THEOREM

PROOF. 1. Modifying the inductive proof of Ramsey’s theorem. The theorem follows also from the Erdos-Rado theorem given later. 2. The result can be proved also using the compactness theorem for first-order logic. If m+(n); for every mEw, in the language containing k r-ary relation symbols, CI,...,C,, we construct a set Z of sentences “saying” that ( C l , ...,C,) is a partition of a family of r-element sets for which there is no n-element homogeneous set. By the assumption Z has arbitrarily large finite models; by compactness it has an infinite model. This means that p*(n); for some p > No, which by Lemma 0.5.3, contradicts Ramsey’s theorem. In Chapters 3 and 5, the same idea will be used again for different purposes. The details are left to the reader. rn We may ask whether for any n , ~ , (r r finite) there exists m = m ( n , ~ , r ) such that m+(n):. The finite Ramsey theorem shows that the answer is affirmative for n , ~ , rall finite. The theorem of Erdos-Rado shows that the answer is r finite). affirmative for all n , ~ , (r THEOREM 0.5.8 (Erdos-Rado). Zf I+(n)L, m > Za<~K(“).

then m+(n

+ l)L+’, provided

PROOF.Let 7 = m ; [ S ] r + ’ = U,.,KC,,. Fix a cardinal @>m,and write { x o , ..., x , } = { y o , ...,y,} iff both are in the same class C,,. For each x E S we definef,(x)ES (a<@): Suppose that f [ ( x ) has been defined for all . $ < a ; then f , ( x ) = x if f k x ) = x for some .$ < a ; otherwise f , ( x ) is the first y in S - { f&x) I[< a} such that

for all

to<-

* *

< a. (y’s satisfying (*) exist, e.g. x

<.$,-I

CLAIM1. a < P ~ f , ( x ) # x * f , ( x ) < f a ( x ) .

itself).

Ch. 0,8 51

47

PARTITION CALCULUS

PROOFOF CLAIM1. Since (*) holds fory = x andf,(x)#x, we getf,(x)< x ; this proves Claim 1 when f s ( x ) = x . Suppose f a ( x ) # x . Then f s ( x ) E S - - ( f t ( x ) l t < / ? } , and since a < P we obtain f p ( x ) # f a ( x ) . By the second clause of the definition of f a , (*) holds withy=fa(x) (even for all to< < [ r - l < P ) . Thereforefa(x)
.

0

fs( x )
,

Given x E S and to< . . . <&unique v < K such that:

{fs,(.>

,...,f,,-,(.),.}

E C”.

CLAIM2. Let x , z E S; if:

PROOFOF CLAIM2. We shall prove by transfinite induction that ,f,(x) =fa(.) for all a < p(x). This is enough to ensure the conclusion, because j p ( x ) ( x= ) x , f&(z) = z. Let a < p ( x ) and suppose that f & x ) =f&z) for all [ < a . For every t < a < p ( x ) , f i ( x ) < x ; thenf,(x) is:

(**I

{

the firsty E S -

{ fs(z) It< a } such that for all to<.

,

{fzo(z),-.rt , - l ( z ) J } E C”,, ...,

,-I,

x)

* *

<&-I


- C”,, ...,6 ,-I, 2)

But (**) is precisely the definition of f , ( z ) ; therefore f , ( x ) = f , ( z ) , and Claim 2 is proved.

-
CLAIM3. There is x , S~such that p ( x o ) 2 1. PROOF OF

p ( x ) < 1. Given

P < 1, let:

p(x)

E S.

4 P ) = { x E s I d x ) = P 1.

Since I is a cardinal,

48

[Ch. 0, $ 5

PRELIMINARIES

Claim 2 asserts that there is a one-one map from a ( / ? )into the set of all maps Y : / ? I + K ; hence:

/?) < a(

K@')

for each /?
By the assumption S = UB.,ra(/?); therefore:

8<[

a contradiction. This proves Claim 3. Let So= { &(x0)15 < I}. By Claim 3, tion of [Sol' into K classes as follows: A E c"'w A

= I and xofZ So. Define a parti-

u { x 0 )E c,,

for A E [solr.

By hypothesis there is S, So of power. n and some vo< [SlyC C:,. S, = S , u { xo} has power n + 1 and [ SJ+' C,,,.

K

such that

0.5.9 REMARKS AND COROLLARIES. 1. For each n,K,r ( r finite) there exists m = m(n, K , r) such that rn+(n):. (Immediate by induction on r). 2. If n, K , r are all finite, we may take rn to be finite, so we have the finite Ramsey theorem. 3 . We have the following moreprecise estimate of m ( K + , ~ , r ) :

This result is often referred to as the Erdos-Rado theorem. KEISLER[3], lecture 14, pp. 75-77, gives an alternative, somewhat simpler, proof based on a model-theoretic argument. PROOF.Induction on r. For r = 1, (*) is the pigeon-hole principle. It suffices to show Suppose (*) holds for r > 1; we prove ~ , ( K ) + + ( K + ) : + ' . &(K) > 3 r - , (K)+~(z'l. But

x,

Ch. 0, 3 51

PARTITION CALCULUS

49

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

n > r).

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

(m> No,

PROOF.(i) If ~ < m then ,

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

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

LEMMA 0.5.10 (Sierpinski).

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

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

by the above.

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

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

then rn is strongly inaccessible.

50

[Ch.0,8 5

PRELIMINARIES

PROOF.Suppose that m is singular. Then there is a set M of power m which is the union of < m disjoint sets M i( i E I),each of power < m. We place { x , y } in C , if x and y are in the same Mi, and in C2 otherwise. If X is a homogeneous subset of M , then either X c Mifor some j E I or X has at in either case it follows that most one element in common with each Mi; X < m. Thus if m+(m),2, then m is regular. Now suppose that m is not a strong limit cardinal, i.e., 2" > m for some n < m. We consider the least such n; since m is regular, 2<" < m.Let n = Nu and consider the set S of wu-sequencesof 0's and 1's ordered according to first differences. Let < denote this ordering. Now we consider an arbitrary well-ordering iof S and place { x,y } in C , if x and y stand in the same , and in C, otherwise. If m+(m);, then relation regarding < and i 2"+(m):, and so S ha5 a homogeneous. subset of power m; hence S contains a well-ordered or conversely well-ordered chain of power m (with respect to the order <). But S has a <-dense subset of power 2<", vk. the set of all a,-sequences which have a last 1; hence any well-ordered or conversely well-ordered chain has power 4 2<". Thus m < 2<", a contradiction. We have therefore shown that m is a strong limit cardinal.

-

REMARK.Let WC={mIm-+(m)~}.In view of the preceding result the elements of W C larger than w must be rather big, in particular not less than the first strongly inaccessible > w . We shall show in Chapter 3 that the last-mentioned cardinal is not in W C either. On the other hand we shall show that WC contains all measurable cardinals and more. Cardinals in W C have important metamathematical properties. We consider now a stronger partition property. Let each of the maps f , : [o]'-+k

( r = 0, 1, ... ; k finite)

determine a partition of [w]' into k classes. We know by Ramsey's theorem that for each r there exists an infinite set X, homogeneous for f,. Does there exist an infinite set X which is simultaneously homogeneous for allf,? The answer is "no". If A E [ w ] l , let f,(A)=O if min(A) < r, i.e., f,({XI,..

.J,> <>=

0

ifx,
1

otherwise.

An infinite set { a,,a,, . ..}< cannot be homogeneous for any I; with r 2 a, because f , ( { a,, ...,a,})= 0 and f,({ a,, ..,a2,})= 1 since a,, > r.

,,.

,

Ch.0, g 51

PARTITION CALCULUS

51

DEFINITION 0.5.12. If m has the property that given a set M of power m for every system of partitions f, :[M]'+K( r = 0, I,. ..) there exists a set X C M of power n which is homogeneous for all& then we write

Defining [MI<" to be the set of all finite subsets of M we see that m+(n),<w iff for eachj: [M]
f i s constant on [XI' (r=O,l, ...). The result proved above may be stated as follows: LEMMA 0.5.13. N,+(N&,<".

We shall prove that, in fact, any cardinal m satisfying m+(Nd,<" must be very large. On the other hand, m+(m),<" holds for m measurable. We remark that if m+(n),<", then m 1 4 ( n ' ) ~ "for m' > m, n' < n and K' < K. LEMMA 0.5.14 (Rowbottom). Let h be infinite. If ~+(h);",then ~+(h)?.

PROOF. Let f:[K]<"+.p" and let J :w X w+w be a monotone pairing function with "inverses" K : w + w , L : w + w ; i.e., J , K , L are related by the equation : J ( K ( m ) , L ( m ) ) = m , for mEw.'

Define g :[K]<"+.,u

as follows:

By the assumption ~+(h),<".K has a g-homogeneous subset Y of power A. Thus for each m ~ w there , is an a, E p such that g({ y , , ...,y,},) = a,, for every y ..,y, E Y . We may suppose that Y has no last element. It is easy to check that

',.

so that Y isf-homogeneous.

' J(m,n)=2".

* pr, : pw+p

(2n+1) is one such function denotes the projection over the n* coordinate.

52

PRELMNARIES

[Ch. 0, 8 5

For each infinite cardinal h we define ;(A) to be the smallest cardinal rn such that rn+(h)y. It is not clear yet whether, on the basis of the axioms of ZFC, the preceding sentence defines anything at all. The following results show that ;(A) is always a strongly inaccessible cardinal. The proofs are taken from MORLEY [2]. SILVER[l], pp. 83-85, gives model-theoretical proofs of them. LEMMA 0.5.15. For every h E CN,;(A)

is regular.

PROOF.Suppose K = k(h) is singular; then there is a set X of power K and a with 7 < K and < K ( i f l ) . By the definition of -partition X = UiEIXi K there exist partitions

7

f.:[Xi]<"+2

(iEI),

having no homogeneous set of power A. Let Z, = { i € Z l A n Xi#S } for A cX. If I, has only one element, we denote this element by i,. Define a partition of [XI<" in < 4 classes by setting A = B iff or

(i)

- = 9 zB

=

andfA(A)=hB(B)

(ii)

I, , Z, > 1 andf(Z,)=f(I,).

(j)

Y is wholly contained in one X i , and Y is homogeneous forf.,

Obviously, = is an equivalence relation and there can be at most four non-equivalent sets. Let Y be homogeneous for the partition defined by G ; by considering 1- and 2-element subsets of Y we see that either or

(jj) Y intersects each X i in at most one point, and { i E Z I Y n Xi #kJ } is homogeneous for f. In either case we obtain 7
Ch. 0, 8 51

PARTITION CALCULUS

53

property, then we consider the map f’ :[ p]<“+2 defined by

which obviously is constant on [ p]’ and has no homogeneous set of power A; indeed, if Y were homogeneous for f’ and 7 =A, then Y = Y o u Y,, where Yi = ( x E Y I f ( { x } ) =i} ( i = O , I), and either Yo or Y, has power h and is homogeneous for f,a contradiction. Let X = 2 ” be the set of all p-sequences of 0’s and 1’s. For distinct x , y E X , let i ( x , y ) be the first a < p such that x,#y,. Order X by first differences (lexicographically); if x f y : x < y iff x, -0 and y, = 1, where a = i ( x , y ) .

Note that if x < y

< z , then i ( x , y ) # i ( y , z ) . Define g : [X]<”-+2 as follows:

g ( { X,Y,Z 1
0

if i ( x , y )< i ( y , z ) ,

1

if i ( x , y ) > i ( y , z ) ,

g is constant on [ X I ’ and [XI’.

Let Y c X be homogeneous for g. By considering triples of elements of Y we see that either or

(i)

for all x , y , z E Y, x
< z * i ( x , y )< i ( y , z ) ,

(ii) for all x , y , z E Y, x < y < z + i ( x , y ) > i ( y , z ) . We suppose that (i) holds; the case (ii) is completely similar and will be omitted. From (i) we infer at once: (iii) for all x , y , z E Y, if x < y and x < z , then i ( x , y ) = i ( x , z ) , and (iv) for x , x ’ , y , y ’ E Y, if x < x ’ and y < y ‘ , then i ( x , x ’ ) = i ( y , y ’ )w

x=y.

We prove (iv). The implication * is immediate from (iii). Suppose x f y , for instance x < y ; from x < x‘, x < y , by (iii) we get i ( x , x ’ )= i ( x , y ) ; from x < y , y
54

PRELIMINARIES

[Ch. 0,9 5

(iii) and (iv) show that the map x+i(x,x'), where x,x'€ Y and x < x' is well defined and one-one. Thus, the set Z = { i(x,x') I x,x' E Y and x < x'} C p

-

has power 'i;- 1 (Note: if Y is finite, it has a <-last element; otherwise Y-l=T). We show that Z' is homogeneous for f, where Z' = Z if Z has no last element and Z'= Z- {a} if a is the last element of Z.A typical r-element subset of Z' is of the form:

(*I

{ i(vl,Y;),...,i(ur,u:)},

where

obviously we can assume y I <

*

-
Choose x E Y such that x>y: (by the definition of Z' such x's always exist). By definition of g, we have:

(**>

f({ i ( y 1,YJ, * ~(Y,,Y:)})= g ( { Y 1, * * * ,yr,y:,x} <),

whenever r > 2. The right-hand side of (**) is constant. Since f is constant on [ p ] ' , Z' is homogeneous for f. By assumption, 'z'
REMARK. SILVER111, Corollary 4.5.1, p. 84, proves a more general result: if K-.(X):"

and p+(h)c", then K-.(X)>".

Theorem 0.5.16 is an immediate consequence of this result and Exercise 0.5.4. Cf. also Exercise 5.1.4. COROLLARY 0.5.17. For every X ECN, B ( X ) is strongly inaccessible

> o.

PROOF.Immediate from Lemmas 0.5.15 and 0.5.13, and Theorem 0.5.16. H THEOREM 0.5.18. B(h) is a strictly increasing function of A.

Ch. 0,0 51

55

PARTITION CALCULUS

PROOF.Obviously No < A,
: [a]<“+2

with no homogeneous subset of power >A,. Let g:[~]<~+{O,l}be defined by: ({ a,,* * * a n - I > <>* g({ “1, * a n } <>=f% 3

Let Y C K be homogeneous for g. Let p E Y ; then 7
K+(K),<”.

PROOF.(Silver). Let Q be a < K-complete, non-principal ultrafilter on K . If f : [~]<“+2,we define f’ :[~]<“+2as follows: f’(~)== Hi { x l f ( Y u { x } ) = i } E Q ,

,=fL.

and (f, I n Ew ) is defined by induction: fo=f and f,+ be defined by: XES

H

Let the class

5

foreverym,n<wandevery Y ~ [ X ] ~ , f ~ ( Y ) = f ~ + , , ( k 7 ) .

Each element X of S is homogeneous for f: if Y E[ X ] ” ,then f( Y )=fo( Y ) -fn(O’). Moreover, since X E S is a property of finite character, there exists a maximal X E S , Therefore it suffices to show that a maximal element of S has power K : CLAIM.If X

E S’

and

8 < K , then there exists X’ E S

such that X c X’.

PROOFOF CLAIM.,By < K-completeness the complement of X is in Q . For each m,n and for each Y € [ X I ” we have

R (Y )

mf

m + 1( Y

)p f m + n + I(Q‘)*

Hence { x ~ f m ( Y u { x } ) ~ f m + , , + , ( O ‘ ) }for E Qall m,n<w and all Y E [ X ] ” . Form the intersection Z of the last-mentioned classes over m,n<w, and Y € [ X I ” . By the < K-completeness of 9,2 E Q ; therefore, 2 - X E 9 and so we may choose z E Z - X such that

-

f m ( Y u { z 1) f m + n + 1(Q‘)

56

PRELIMINARIES

[Ch. 0,8 5

for all m , n < w and Y € [ X I ” . From this it follows that

X ’ = X u { z }E 5 . COROLLARY 0.5.20. Let 8 be a measurable cardinal; then: (1) (Hanf-Tarski) 8 is the Bth strongly inaccessible. In particular, (2) The first strongly inaccessible > w is non-measurable. PROOF.(1) By 0.5.19, r?(6)= 8. Therefore, by Corollary 0.5.17 and Theorem 0.5.18, {;(A) I A E CN and A < 8 } is a set of power 8 of strongly inaccessible cardinals < 8. (2) is an immediate consequence of (1). W REMARKS. (1) The first conclusion of Corollary 0.5.20 only requires 8 to be a fixed point of the function i.This raises the question whether any fixed [ l ] has proved point of r? is measurable. The answer is “no.” ROWBOTTOM that for each measurable 8 there is a A < 8 such that r?(A)=A (see SILVER [ 11, pp. 22-23, and Exercise 3.3.1 (Chapter 3)). The fixed points of r? are called Ramsey cardinals. (2) In spite of the fact that the partition methods developed here are sufficient to prove the Hanf-Tarski result, the method of Hanf is more refined insofar it proves that 8,, < 8,,, where 8,, denotes the first strongly inaccessible > w and 8,, the first cardinal rn such that rn+(rn);. It is clear that 8,, < 8,, where en denotes the first Ramsey cardinal. However, Silver proved that < holds; furthermore

ewe< r? ( w ) < rz (wl) < en< e,, where 8, is the first measurable. The second inequality is Theorem 0.5.18. K I ( w ~ ) < ~ follows ~ easily from Corollary 0.5.17 and Theorem 0.5.18. The last inequality is the result of Rowbottom mentioned above, and its proof appears in Exercise 3.3.4. (3) It is possible to improve Theorem 0.5.1 by using the existence of normal measures (Theorem 0.4.31). We leave it as an exercise (cf. Exercise 0.5.6). Theorem 0.5.19 was proved before Hanf‘s, but remained unnoticed. The preceding considerations suggest the following: OPENPROBLEMS. (a) Is it possible to give pure partition-theoretical proofs of the following inequalities: 8,, < r?(o),;(a1)< 8,< 8,? (b) Does there exist a partition-theoretical characterization of measurable cardinals?

Ch. 0,f 51

PARTITION CALCULUS

57

EXERCISES 0.5.1. Prove

Conclude that

(2)

if rn+(rn)L, then rn-+(rn): for all finite K.

0.5.2. Prove that every totally ordered set with the Souslin property. has power < 20' . (A totally ordered set has the Souslin property iff every family of disjoint open intervals is countable.) [Hint: use Corollary 0.5.9 (3) and the argument of Lemma 0.5.10.1 0.5.3. Show that 2"+(n+);. [Hint: the ordered set S described in the proof , : or of Theorem 0.5.1 1 has power 2.' but contains no chain of type a

4,I.l

0.5.4. Show that 2"+(3)2. [Hint: with S as in 0.5.1 1 define f:[S]*+a, by settingf({x,y}<)= i ( x , y ) , where i(x,y) is the least 5 such that xr#y,.] 0.5.5. Prove that

0.5.6 (Rowbottom). Let p be a normal measure on K. (a) Assume that fl c S C[K]<" and that (Us1s E S ) is a collection of subsets of K such that p ( U s ) = 1 for all s E S . Prove that the set u= (5 < K I t E max(s)<[ Us } has measure 1. [Hint: for ,$
n

Assuming that p ( U ) # l , use normality of p to show that p ( U J # O for some so€ S.]

58

[Ch.0.8 5

PRELIMINARIES

Note that (a) proves, in particular, that if X = { UaI a < K}is a decreasing sequence of sets of measure 1, the diagonal XD also has measure 1. (b) Let A < K. Prove 'that any f : [ ~ ] < ~ +has h a homogeneous set of measure 1. [Hint: begin by proving the result for all f : [~],+h, r E w ; use induction on r , defining for each a E K: fa

( { a I , .. .,a, } ) =f( { a , a I , .. .,ar} ) for a < a

< . . . < a, < K.

Consider the diagonal of any collection {Ha 1 a E K},where Ha is a set of measure 1 homogeneous for fa.] Note that (b) proves K - + ( K ) , for < " any A < K. For more information on normal measures (ultrafilters), see Exercise 3.3.3. 0.5.7. The partition properties considered in this section can be extended to the case where the right hand side is an ordinal (instead of a cardinal). For aEON, ,+(a); and K - + ( C X )are ; ~ defined by requiring that for every partition f : [ ~ y + p (system of partitions f, : [ ~ l " + p (n = 0, 1,. . .), resp.) there be a set of type a in the natural order of K,homogeneous forf(for all f,, resp.). As in 0.5.15, for each a E O N let ;(a) be the least cardinal v such that ~ + ( a ) ,
a,p EON r\p limit ordinal

Aa

< /3

rZ ( a )< rZ ( p).

Note that (l), (2) imply: if p is a limit ordinal and SUP{ ;( a )la: < P

p < L( p), then

}< (PI

(thus, e.g., there are at least w 1 strongly inaccessibles between t ( w ) and &I)).

(3) Which is the behaviour of the function ;(a) at successor ordinals a? Is there an a EON such that ; ( a ) = ;(a + l)? Is there an a EON such that ;(a) is singular?

0.5.8. Prove that Ramsey's theorem implies the following weak form of the axiom of choice: every infinite family of non-empty finite sets has an infinite subfamily for which there exists a choice function. [Hint: assuming that every infinite subfamily of the given family has empty intersection, argue as in the first half of Theorem 0.5.1 1.1

CHAPTER I

THE BASIC NOTIONS OF INFINITARY LANGUAGES

$ 1. Elementary syntax and semantics of the languages LKA.

A. Description

Among the variety of extensions of first-order languages we shall deal in this book with some of those obtained by admitting expressions of infinite length. where K and h are infinite cardinals, can be described The languages LKA, as follows. Variables: A list of cardinality y = max{ ~ , hof} symbols that we denote by x , y , w,z,uq (17 < y,). As usual, distinct symbols are assumed to denote distinct variables. If p < y we denote by u 1p the sequence (u, 117 < p ) of variables. Often the letters x,y,w, with or without subscripts (but not the u,’s!) will also have the meaning of metalinguistic symbols (variables) denoting variables of the language. Var denotes the set of all variables (we omit here the indices K , A that will often accompany concepts connected with the language LKJ. Logical symbols: Connectives 1,A. Quantifier 3. Equality symbol w . Parentheses of various kinds. Expressions containing symbols other than these will be understood as abbreviations in the usual way, except in a number of situations specified in $ 3 below, where we will also need to consider V and V as primitive symbols. 59

60

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1,

1

Non-logical symbols: These are usually classified in three lists, each of them possibly finite, infinite or even empty: (a) A list Re1 of (finitary) ‘relational symbols. (b) A list Fnt of (finitary) functional symbols. (c) A list Ind of individual constant symbols. These symbols are intended to denote relations, functions and distinguished objects in each interpretation of our languages. Functions (of n arguments) can be considered as particular cases of relations (of n + 1 places) and fixed objects can be considered as 0-place function symbols, by convention. In the case of function symbols this requires special axioms to the effect that the relations in question are functions; sometimes this simplifies considerations to some extent; on other occasions it is convenient to use function symbols. A word of caution is in order: notions and results valid for languages with only relation and constant symbols do not always extend to languages including also function symbols. These extensions, when they can be performed, require special arguments. Theorems 2.3.2 and 2.3.3 (Chapter 2) are examples in which the extension is not possible. The concept of substructure, for instance, is more restrictive when functions are involved (cf. Lemma I . 1.20). The same is true for the converse situation. Likewise, results for logic where the equality symbol is present do not always extend to logic without equality and viceversa. Relations considered in this book are assumed to have only a finite number of places. However some of our considerations would have been valid as well, had we considered relations of infinitely many places. Let us point out that if we wanted to introduce infinitary relations in LKX,these have to have less than K places; otherwise we may be faced with the situation where quantifiers are “too short” to close formulas. For each relation and function symbol we will have to specify a natural number, namely the number of arguments. Thus to specify the language we should give functions aRe,:R e l - w - ( 0 ) and aFnt:Fnt+w; the number aRel(R) is called the arity of the relation symbol R , or its number ofplaces; we will also say that R is n-aty if aRel( R ) = n. Likewise for function symbols. As indicated above individual constants are identified with 0-ary function symbols. The quadruple (Rel, Fnt, aRe,,aFnt)(or simply the pair (Rel, aRe,)if for example Fnt =B ) will be called the similarity type (or simply type) of the p) completely determines the particular language. Thus the notation LKA( language under consideration. The letters p, Y with or without subscripts

Ch. 1, $ 11

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

61

will denote similarity types. Some examples are: (a) The language appropriate to the study of ordered systems is defined by the following stipulations: Rel= { <}, uRel(<) = 2

(i.e.,

< is a binary relation).

(b) The language suitable for the study of group theory (as this is treated in the usual textbooks) is defined by: Re1 =O , Fnt={.,-’,e}, (i.e., e is an individual constant),

uFn,(e ) = 0

(i.e.,

=2

a,,,(.)

uFnt(-’)= 1

- has 2 arguments),

(i.e.,

-I

has 1 argument).

(c) The language suitable for the study of ordered fields is of the following type: Rel= { <}, F n t = { +, .,O,l}, ‘Rel(

<) = 2, aFnt(‘)=2,

uFnt(+)=2,

~F,~(O)=O,

UF,*(

I)=().

It is obvious that a language of a given type may be suitable for the study of many non-related topics; for instance the language suitable for the study of field theory has the same similarity type as the one used for the arithmetic of natural numbers. However, many of our considerations will be independent of the similarity type. Sometimes we will need to consider the addition of new symbols (in particular individual constants) to a given similarity type. Formally the “addition” of two similarity types p, v is defined as follows: we require that uRel, and uRel,coincide on the set Re1,n Rel,, i.e., (*)

URel,

( R ) = uRel,( R ) for every R E Rel, n Rel,

62

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1,

5

1

and likewise for functions. (Notations such as Rel,, etc. have an obvious meaning.) Then p Y (or p u Y) is the type p‘ defined as follows:

+

Rel,, = Rel,

u Rel,,

Fnt,,

u Fnt,,

= Fnt,

aRel,.= aRe],

‘Fnt,.

= ‘Fnt,

u aRel, (which is a function by virtue of (*)),

u ‘Fnt;

However quite often we shall just use notations like: < p , R , S ) or

p u { R , S } or even ( p , ~ ) where ~ ~ ~R ,and S are new relations (i.e. not in

Rel,) of known arity, or C is a set of individual constant symbols such that C n Rel, =B . The meaning will always be clear from the context. The languages L,, differ from each other (and from first-order languages) in the way formulas are defined. We will assume that the notion of transfinite concatenation is understood; strictly speaking one should give an explicit set-theoretical construction of the theory of infinite concatenation, but this is much too lengthy and adds too little to the model theory of infinitary languages, in order to be even mentioned in this book. On the other hand it can be read in Karp’s book [l]. Therefore we go directly to the definition of formulas, which depend upon a fixed (but arbitrary) similarity type p. First we define terms. (1) Any variable is a term. (2) Any individual constant symbol is a term. (3) If F is any n-ary function symbol ( n >0) and t , ? . . , t , are terms then F( t , , .. .,f,)

is a term. (4) All terms are defined by (1)-(3). A term containing no variable is a constant or closed term. The letters t,7 with or without subscripts will be metalinguistic variables denoting terms. Term, will denote the set of all terms or constant terms of the language defined by p. In all cases it will be clear from the context whether terms or constant terms are involved. Note that if there are no function symbols in our particular similarity type the only terms are variables and individual constants.

Ch. 1,

5 I]

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

63

Now we can define the expressions of L,, ( p ) , where K,A are infinite cardinals.' DEFINITION 1.1.2. (1) If t l , t 2 are terms, then t l x t 2 is an expression. (2) If R is any k-ary relation symbol and t , , ...,fk are terms, then R(t,,...,t k ) is an expression. (3) If @ is an expression, so is 1@. (4) If p < K and for each 71 < p Qq is an expression, so is

[ A@'o@l * .Q q . .*

3

(5) If 6
REMARK. It should be obvious how to transform this informal definition into a formal definition of set theory, so as to identify expressions with certain well-defined sets. There are several equivalent ways to do this. We let the reader choose this formalization according his taste. A similar remark applies to many definitions found in this book. Expressions such as: * * *

Qq *

* *

3,

[3ua0u,l.. . Uac'

. a], *

will be abbreviated as follows:

Indexing the members of a conjunction or the variables in a quantified expression by means of ordinals often adds no information and produces a cumbersome complication of the notation. For this reason we shall find convenient to use the following notation as well: If '4' is a non-empty set of expressions of LrA( p) of power < K , denotes the conjunction of all the expressions in '4' (defined by (4)). If X is a set of variables of power < A and rp is an expression of LrA( p), ( 3 X ) c p denotes the expression defined by ( 5 ) of Definition 1.1.2. Cf. Definition 1.1.3 for the notion of formula.

64

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, 3 1

We assume that the reader knows the notions of bound and free occurrences of a variable in an expression. An expression without free (occurrences of) variables is called a closed expression. Notice that there might be expressions with “too many” free variables; it is not difficult to construct expressions of L,,,, for instance, with No free variables; since in this case only finitely many variables can be quantified at a time, there are expressions which cannot be closed. This is the reason for the following: DEFINITION 1.1.3. Any expression of LKh(p ) with less than A free variables p). Form (LKA( p ) ) denotes the set of all formulas is called a formulaof LKA( of L,A( p). For every cardinal u A .

A>

Ch. 1, 5 11

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

65

Two languages related to the languages L,. are often considered. The first is called L,, (AECN); the class of its formulas is defined by:

PI)= U Form

Form (,A' (

PI).

U>h

Thus, L,, has arbitrarily long conjunctions and disjunctions, but only quantifications of length < A ; all its formulas have less than h free variables. is defined as The dass of formulas of the second language, L,,, follows: Form(Lww( p ) ) = Hence, L,

U Form(Lm,( P ) ) .

AECN

has unboundedly long quantifications as well.

REMARK.It is easily seen that both the preceding conditions define a class of ZFC. B. Relational structures Once a similarity type has been fixed we should give some general definitions that relate the language under consideration to the mathematical structures that will be studied with its help. These structures will be called relational structures, relational systems, or simply structures of the given similarity type. To each relational symbol of the language it corresponds a relation among the members of our structure with the same number of places; to each individual constant it corresponds an element of our structure; and to each n-ary function symbol it corresponds a map from n-tuples of elements of the structure into elements of it. Thus we make the following:

DEFINITION 1.1.4. A structure of type p is an entity of the form: (U=(A,sR2~),ERe,,~

f E Fnt,

where: (1) A is a set (called the universe of %); (2) for every n-ary R ERel,, n >0, S, c A "; (3) for every 0-ary R ~ F n t ,(i.e., individual constant R ) , S, € A ; (4)for every m-ary f E Fnt,, m >0, F, :A "'-+A.

66

[Ch. 1,

THE BASIC NOTIONS OF INFINITARY LANGUAGES

1

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

%I

'

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

f(z)

if Z E X - Y ,

g(z)

if

LE

Y.

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

C Y , then j;g=

g. In

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

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

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

Ch. 1, 0 11

(2) cp is

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

61

14;then iff

%t-cp[f]

not%k$[f],

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

iff

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

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

iff

5 < K ; then

F<X;then

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

PROPERTIES OF THE NOTION OF SATISFACTION

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

This remark makes meaningful the use of notations such as:

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

where al (or g (x))

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

any aEl%l

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

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

68

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, I 1

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

Z

implies %kcp.

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

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

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

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

-

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

uEZ.

Ch. 1, 8 I]

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

L,,

69

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

K

and A are

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

S't@=S'

for each S E p .

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

I

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

l'ul

c 1BL

S' = SBn I 2l In

for each n-ary relation symbol S E p ;

F'

for each m-ary function symbol F E p ;

=FB

c' =

p

r 1 % Im

for each individual ,constant symbol c E p .

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

70

[Ch. I, $ 1

THE BASIC NOTIONS OF INFINlTARY LANGUAGES

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

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

for each n-ary relation symbol R E p ;

for each individual constant c E p ;

F'rx= F' rX"

for each m-ary function symbol F Ep.

From the preceding definitions it is clear that:

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

''r"

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

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

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

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

Ch. 1,

5

11

ELEMENTARY SYNTAX AND SEMANTICS OF THE LANGUAGES

Lxh

71

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

> * ,*

- 9

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

. . .,yrn)>

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

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

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

12

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, §. 2

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

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

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

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

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

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

'

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

Ch. 1,

21

EXAMPLES

13

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

v

n E o (xxs"_O)]

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

vx[

v

n E w (X%S"_O)]

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

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

v [ v u o . . . u n (o < iv, <

nEo

Then %€Mod(0) w ZEw. See remark at the end of this example.

J < n Ui%Uj)].

14

THE BASK NOTIONS OF INFINITARY LANGUAGES

[Ch.1, 4 2

The members of Fin are sets without structure. 8 can be used, however, to characterize finite systems with structure. In fact, given any finite or denumerable similarity type p and a sentence u of LuI,(p), the class of models 8 ~ equals u the class of all finite models of u. Examples of such u's are, for instance, the L,,-sentence of Example 1.2.1, or the conjunction of all group axioms (an L,,-sentence). This characterization is impossible in ordinary first-order logic if u has arbitrarily large finite models, even using additional predicates, cf. 4 3, Exercise 1.3.2. EXAMPLE 1.2.3. Let A be a class of algebras of similarity type p , ;

< No (i.e.,

p contains only operation symbols and individual constants); furthermore we assume that A is axiomatizable in L,,,, i.e., there is u E Sent(L,J p))

such that A=Mod(u). We also remark that the set Term of all terms of p and the set Termn of all terms of p with < n variables ( n E w ) have both power < N o (we identify terms differing only in the names of their variables). Recall that % € A is finitely generated iff there is a finite set X C 181 such that no proper subalgebra of 8 contains X. We assert that the class Fg(A) of all finitely generated algebras of A is characterizable by a single sentence cp of LuI,(p), It is enough, in fact, to take cp to be the conjunction of u and: T E Term,

nEw

REMARK1.2.4. Notice that even for classes A=Mod(u) where (I is an L,,-sentence, Fg(A) is not characterizable by a set of L,,-sentences. Examples of such A's are the classes of groups, rings, lattices, etc. As remarked in the preceding example Fin(A)-the set of all finite structures in A-can be characterized by a single LWI,-sentence.

EXAMPLE1.2.5. Characterization (in L,,,J of the class W of all non-empty well-orderings, i.e., all relational systems of the form ( A , R ) where A # B and R well-orders A . W is the class of all models of the sentences: VXlR(XJ), VUoUl L)z[ R ( U O ,1)~ A 8 ( 0 1 ,

4 +

Vuou1[R ( 0 0 %0 1) v R ( 0 1, 0 0 ) v (vu

r 0)V ( R nE w

(un,un+ 1)

R (00,02)], 0 11,

v On mum+ 1).

Ch. 1, 8 21

15

EXAMPLES

EXAMPLE 1.2.6. More about well-orderings (in LJ.

We shall here consider systems of the form (A, <, ...) where "<" linearly orders A . Then we can prove (a) For every a E K there is a formula Qa(u0)= q$(uo) of L,,, with exactly one free variable, such that for every structure 9I and every x o EI9Il the following holds:

91b>[x,l

-

(a,EW=(S,~,

<"rSxO),

where Sxo={ z E I9Il I z <"xo} (Le., the set of all <"-predecessors of xo is well-ordered in type a). The formulas q>(uo) are defined by transfinite recursion on a as follows (cf. SCOTT [3], p. 330):

where u is the sentence " < is a linear ordering". The use of uo and equality in the right hand side is a device to avoid substitutions and the overabundance of variables (in fact only uo,ul, and the variables of u occur in q2); a more readable, informal version is:

The preceding statement (a) is proved by transfinite induction on a . Other properties of the q>'s are: (b) If 9I is well-ordered by <', then:

Hence: (c) If a,@EON and one of them is

< K , then:

(a, Era)=rw(p, E

PP) *

a=P.

In fact, let us suppose that a < K ; then ( a , E la) satisfies the sentence in the right hand side of (b); hence so does ( p , E rp), which implies a = P .

76

THE BASIC NOTIONS OF INFINITARY LANGUAGES

(d) If

(*I


[Ch. 1,

2

then

8 + i 3xrp>(x)+~y

V cp,<(

I< a

y ) for all a < K ,

is equivalent to 8 is well ordered.

Notice that there are non-well-ordered systems 8 of power 2 K such that

(*) holds (cf. Corollary 4.4.3 and Exercise 1.2.1).

EXAMPLE1.2.7 A slight generalization of the formulas @a (or p:r in the notation of 1.2.6) is their “relativization” @: to a subset of 181 defined by a formula rp with exactly one free variable; notice that the similarity type of ill, hence of rp, might involve relations and constants other than <. ap is the sentence “< is a linear ordering with field including rp”:

Clearly, (Da can be obtained from @: rp(x)= x m x .

by specializing to the formula

A complete description of the meaning of the @E’s is given by the following:

LEMMA1.2.8. Let K be an infinite cardinal, p a similarity type containing (among other symbols) a binary relation denoted by < . (A) If rp E Form,(L,,( p)) and a < K, then (Dz E Form,(L,,( p)). ( B ) For every structure ? ofIsimilarity Qpe p, every ordinal a and every a E 181, we have: 8 k @:[a] H rp‘ is linearly ordered by is well-ordered by

<‘

<’

and S, n rp’

in type a.

Ch. 1, 5 21

77

EXAMPLES

REMARK1.2.9. A set X L A is an initial segment of ( A , <) iff for all x , y E A ' , EX a n d y < x implyyEX.

The following elementary facts about well-ordered sets are used in the proof of Lemma 1.2.8: (I) Let ( A , <) be a totally ordered structure and p be an ordinal. If for every x E A , S, is well-ordered by < in type < p, then A is well-ordered by < in type < p. (11) If ( A , <) is well-ordered, x,y € A and x < y , then S, has type strictly less than S,. (111) If ( A , <) is well-ordered and of type > a , then for every ordinal /3 E a there is x E A such that S, is well-ordered by < in type p.

PROOFOF LEMMA1.2.8. Clearly, % t-av

H

rp' is linearly ordered by

<'.

The rest is proved by induction on a. a =0: % + @ ~ [ aH] % + 1 3 y ( c p ( y ) ~ y < x ) [ a ]w S,nrp'=H H

S, n rp' has order type 0.

a>O:(*) Assume %+@E[a]. Let bES,nrp', i.e., b<'a and %t-cp[b].By . the induction hypothesis, if &, denotes definition of @.f, %+ V 5 < . @ r [ b ]By the smallest ( < a such that %t- @ r [ b ] ,we have, sbn rp' =&,
b<'a

w %+

v @r[b].

t
If b <'aLa,then b € S, n rp'. By (II), s b n rp' < S, n rp' = a , whence %+ Vc<,@f"b].Conversely, if % + Vc,,@r[b], let tbbe the least (< a such that % + @ r [ b ] ;by induction hypothesis, s b n rp' =tb< a ; then b 4%. Now assume that b = a ; then we have to prove that

By induction hypothesis the right-hand side implies that Sunrp' =[ for

THE BASIC NOTIONS OF INFINITARY LANGUAGES

78

[Ch. 1, Q 2

some ( < a , contradicting the assumption that this set has order type (Y. Hence the right-hand side is false, which clearly implies that the equivalence is true. H The formulas @: and their properties expressed in Lemma 1.2.8 will be used in our incompactness results for L,, of Chapter 3. We include below some examples from group theory. Many classes of groups are known to be axiomatizable in L,, by means of either a single sentence or an infinite number of them; abelian groups and groups of order n (for a fixed number n ) are examples of classes fulfilling the first condition; divisible groups are axiomatizable in L,, by an infinite number of sentences. There are, however, many important classes of groups for which that axiomatization is not possible (e.g., the class of all torsion groups or the class of allp-groups ( p a fixed prime number)). We include in the sequel some examples of classes of groups, which admit a characterization by one sentence of LUl,. This fact will be used in Chapter 3, #4, to give modeltheoretical proofs of some properties of the classes introduced here.

NOTATION. For the sake of simplicity “groups” will be, for us, structures of as the form (G, - ‘ , e ) , “ - ”understood as a binary function symbol, a unary function symbol and “e” as an individual constant; g denotes the corresponding similarity type. Thus our notation is in ‘agreement with the one used by group-theorists. In this way “substructure” coincides with “subgroup”. ‘‘q”denotes the relation of being a normal subgroup. As usual we omit references to the operations, thus H denotes the subgroup ( H , . r H 2 , - ‘ r H , e ) ( H closed under ‘‘.” and “-’”). H , F and G denote groups; A , X and Y denote subsets of G (technically of [GI). If A C G , [ A ] , ( ( A ) G , respectively) denotes the normal subgroup (subgroup, respectively) generated by A in G : [AIG= f l { H I A C H a G }

“-’”

a ,

(respectively, ( A ) G = n { H IA C H A H subgroup of G }). We recall that: LEMMA (1)

( A ) G = { a y l . . .. .a,,“ In € w A m , , ...,m, E Z A a , , ...,a,, E A }.

(2)

LA],=( f i g ,a.r qgi- 1 I n E w A m , ,...,m,EZ i= 1

A a 1, ...,a,, E A A g , , ...,g,, E G

I

.

Ch. 1, !j 21

EXAMPLES

79

(Z= the set of all integers). The proof is left as an exercise; it can be found in any decent textbook on group theory. If a E G , [ale stands for [ { u } ] ~Usually . the subscript will be altogether omitted, unless confusion may arise. y denotes the conjunction of all group axioms (an L,,-sentence). Our first example is almost trivial.

EXAMPLE 1.2.10. Characterization (in LuIJ of the class T of all torsion groups. We recall that a group G is torsion iff for every gE G there exists an n E o such that g " = e . The sentences: vuou 1

4 (00 * (01 'UZN

=((UO'U l>.fJZ>19

where u" is recursively defined by: ul=u

and

U~+~=(U'"W),

characterize T, i.e., T coincides with the class of all models of them.

EXAMPLE 1.2.1 1. Characterization (in LulJ of the class S of all simple groups. 1.2.12. A group G is simple iff its only normal subgroups are DEFINITION {e} and G. S denotes the class of all simple groups. It is easy to check that:

(*I

G is simple -sfor every a E G, a # e implies

[ a ]= G .

The condition in the right hand side of (*> can be expressed by a single sentence cp of LUl,( g):

80

THE BASIC NOTIONS OF INFlNITARY LANGUAGES

[Ch. 1, 3 2

It is clear then that S = Mod(cp A y).

EXAMPLE 1.2.13. Characterization (in LUJ using additional predicates of the class Cs of all characteristically simple groups. For each group G we denote by Aut(G) the set (and the group under the obvious operations) of all automorphisms of G . DEFINITION 1.2.14.A subgroup H of G is called a characteristic subgroup if it is invariant under all automorphisms of G : for eveqfEAut(G),

f[H]=H.

The group G is called characteristically simple iff { e } and G are its only characteristic subgroups.’ Cs denotes the class of all characteristically simple groups.

NOTATION. For each group G and element a E G , set:

It is easy to check that for each a E G , G, is a characteristic subgroup of G containing a and that every characteristic subgroup containing a also contains Go. Hence

(**) G is characteristically simple w for every a E G , a # e implies G, = G . However, the application of (**) to axiomatize the notion of characteristically simple group is not so straightforward as that of (*) to axiomatize the notion of simple group, since our language does not have predicates to “speak” about automorphisms. These are introduced in the following way; we enlarge the language g to g’ introducing the following additional symbols: two unary predicates a ternary relation

G ,M ; A.

( G “describes” the elements of the group; M “describes” a group of automorphisms of G ; the relation A (f,x,y)roughly speaking represents the relation f ( x ) =y). We consider the following LJ g’)-axioms for the new symbols:

’ Since a characteristic subgroup is normal, each simple group is characteristically simple.

Ch. I, 8 21

EXAMPLES

81

(1) y G , the relativization to G of the group axioms. (2) Vy[(,G(y)v M ( Y ) ) A i ( G ( y )A M(y))l (the universe is G u M , and G nM=,0). (3) V f x y [ A ( f , x , y ) - + M ( f )G~( x ) ~ G ( y )(lA relates a member of M with two elements of G ) . (4) V f x y z w [ A ( f , x , y ) ~ A ( f , z , w ) ~ ( x x z ~ y(every x w ) ] element of M is a one-one map). ( 5 ) V&[G(y)+3xA(f,x,y)] (each member of M has range G). (6) Vfxyzwuu[A ( f , x , y ) ~ A ( f , z , w ) ~ u x x~. \zo = y . w - + A (f,u,u)] (each member of M is a homomorphism of G). (7) 3 g V x [ G ( x ) - + A ( g , x , x ) ] (id, belongs to M ) . ( 8 ) Vf3gVxy[A ( f , x , y ) + A ( g , y , x ) ] (the inverse of each map in M is also in M ) . (9) V f g 3 h V x y z [ A ( f , x , y ) ~ A ( g , y , z ) ~ A ( h , x( ,M z ) ]is closed under composition). Every model of (1)-(9) is of the form I1I=(GuM,G,M,A;,-',e), where (G, ., - I , e ) is a group, M is a subgroup of Aut( G ) and A' = { (f,x,y>I f €

M A W E G ~ f ( x =) y } .

Now condition (**) can be formulated by a single sentence 'p of L,,, in the enlarged language g':

-

Let u be the conjunction of y,

(G, . , - ' , e ) E C s

'p

and (1)-(9). Clearly:

there is M c A u t ( G ) such that

(G

u M , G , M , A ,.,-',e)b-a,

(where A is defined from G and M as before).

NOTE.Examples 1.2. I 1 and 1.2.13, as well as those of Exercise 1.2.2 are [ 11. due to KOPPERMAN-MATHIAS

82

[Ch. 1, 5 2

THE BASIC NOTIONS OF INFINITARY LANGUAGES

EXAMPLE1.2.15. Characterization (in L,J of the class AF of all Archimedian fields. The language is that of ordered-field theory. Consider the axioms for ordered fields plus the sentence: vx[

_o < x+vy

v ( y
nEw

where, as usual, the (metalinguistic) notation n.x means x + . * times).

1

+x

(n

EXAMPLE 1.2.16. Characterization (in Lalul)of the class TS of all structures isomorphic to transitive sets (KARP[I], pp. 3-4). Recall that a set S is transitive iff for all x, X E S 3 x C S . TS is the class of all structures of the form ( A , R), R a binary relation, isomorphic to some structure ( S , E IS), where S is a transitive set. TS is not a PC,-class in Law. However, in Lalal(E ) the following characterization is possible:

8, = v x y ~ v z ( z ~ x t t z ~ y ) ~ x x y ~ ,

8 , is the axiom of extensionality and 8, expresses that E is well-founded. TS is the class of all models of 8, ~ 8 , .Indeed, 8, ~ 0 holds , in every transitive structure ( S , E r S ) , and therefore in each member of TS; this is obvious for 8,; for 8, it is proved-using the axiom of extensionality-by the argument mentioned in the remark of p. 3. Conversely, if (A,R ) E Mod(@,A S,), then the Shepherdson-Mostowski theorem shows that ( A , R ) = ( S , E r S ) for some transitive set S . DEFINITION 1.2.17. Let K be an infinite cardinal and x be a set. x is said to be hereditarily of power < K iff X < K and every y fulfilling condition ( + ) below has power < K ; (+)

there are n E w and a chain so,. ..,s, -

such thaty € s o € . . . Es,-

,E x

H(K)stands for the class of all sets hereditarily of power

-

< K.'

'The class of ally satisfying (+) is called the trunritiue closure of x, and denoted by TC(x). Note that xEH(K+)* TC(x)< K + . TC(x) is the least transitive set including x.

Ch. 1, 9 21

63

EXAMPLES

Using the concepts introduced in the preceding example it is possible to give the following: EXAMPLE1.2.18 (Hanf-Scott). Characterization of H(K+)in L,+,+. H(K+)is the class of models of the sentences B,,B, of the preceding example, plus:

The details of the proof can be consulted in KARP[I], pp. 4-5.

r

EXAMPLE 1.2.19. On certain substructures of 23, = ( R ( a ) ,E R ( a ) ) in L,, (where K = Z ’). Recall that the sets R ( a ) are defined by: R (0)=,0,

R ( a ) = U ??(R([)) foralla>O. By induction we define the formulas V0(X)’

1( X W X )

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

i

R(p)

forp
R(a)

f o r p > a.

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

VX

v

P
v,(X).

It is easy to see that ’ZIEMod(u,) Hence

*

Yl=E

for some 6 C B a .

a4

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, 8 2

For more details on the formulas a, and their models, see Exercise 1.2.3.

EXAMPLE 1.2.20. Trees in Lw,w,and LK++((++. It is easy to see that the class Tr of all trees is axiomatizable in Lw,wl( <); it is enough to adjoin the following sentence to the usual axioms of partial ordering for <

Trees in which all branches are of power < K and every element has < K immediate successors can be characterized in LI(++K++ by means of the following sentences; u and the conjunction of:

where S ( x , y ) (‘‘y is an immediate successor of x”) is the formula S ( X,y ) = X
-1

3 Z (X

< Z A Z
Note finally that the condition on branching is expressed by a formula of LK+K+. Another, simpler example concerning trees is the following, proposed by Lopez-Escobar :

EXAMPLE 1.2.21. Trees in Lwlw. The models we have in mind are trees with finitary branching, paths of length < o and only one root. The language has an individual constant 0 (denoting the root) and a unary function symbol P (to denote the preceding “node”). The axioms are: (1) V x [ x w O + + P ( x ) = x ] (0 is the unique x which is its own predecessor); (2) ~ x [ V , ~ , , , ~ , P ” ( X ) ~ P(every ” + ~ element ( ~ ) ] has finitely many pre-

decessors); (3) A n Ew V m E w v x o . * * x m [Ak <m P ( x k ) w P + ‘(xk)+ VO <;<j <m ~e; xi]; (4) v x [ V m Ew v ~ * o. x m [ Ak m P ( x k )x+vO< ~ ;< j < ,x, e xj]] (every element has finitely many successors); where P ” ( n > 1) is defined inductively as follows: P n + ’ ( x ) = P ( P “ ( x ) ) .

-

Ch. 1, 0 21

85

EXAMPLES

EXAMPLE1.2.22 (The expressive power of singular cardinality languages). Let K be a singular cardinal and A < K . Then L,, and L,+, have the same expressive power in the following strong sense: (*)

For every formula cp of LK+,there is a formula @ of L,, (containing the same non-logical symbols) such that Fv(cp) = Fv(@) and for every structure % of the appropriate similarity type and every f:Wcp)+l%I

%t=cp[fl * %+@[fl. Indeed, let { = c ~ ( K ) , and recall that every set of power K can be deconiposed into { non-empty sets of power < K . Define @ by induction on the structure of cp, as follows: if cp is atomic, then @ = cp; if cp is 14,then @= if cp is ( 3 X ) $ , where X CVar, F
14;

@=[

xy'' A A j

4€*

-

if\k=K, if

T
We leave the verification of the last assertion of (*) as an exercise for the reader. It is in general impossible to generalize (*) to L,+,+. For example, there are sentences u of L,+,+ such that for no sentence cp of L,, containing the same symbols, u and cp have the same models (see Exercise 1.2.5).

EXERCISES 1.2.1. Consider the Sentences

(**I

1

< is a linear ordering", V X [ V Y ( Y< X - . 4 ( Y ) ) ~ 4 ( X ) l ~ V X 4 ( X ) , "

where 4 is any L,,-formula with only one free variable. Show that every model of (**) satisfies the sentences (*) of p. 76 for all a < K . Hence every model of (**) of power < K is well-ordered. (However, there are non-well-ordered structures satisfying all the formulas (**).) [Hint: use transfinite induction and property (I) of Remark 1.2.9.1

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THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1,

2

1.2.2. (1) A group G is called transitively normal iff it has the following property: forevery K , H ,

KaHqG

* KqG.

Tn denotes the class of all transitively normal groups. Prove (i) G ETn iff for all finite X , Y c G: if H = [ X ] , , K = [ Y],and K q H q G , then K a G .

(***>

[Hint: prove and use the fact that (***) implies the following condition; for every X , H , K C G :

if K q H a G, K = [ X I , and X is finite, then K a G.] (ii) Using (i), prove that Tn is the class of all models of an L,Jg)sentence. (2) Show that the following class of groups is the class of all models of some sentence of LU,J g): all groups G such that every pair of isomorphic finitely generated subgroups of G are conjugated. [I]. These examples are from KOPPERMAN-MATHIAS 1.2.3. Consider the sentences a, of Example 1.2.19. Prove that (1) 23, EMod(up) for ,f3 > a. (2)%EMod(u,) * % ,C Ba [Hint: use the induction principle for well-founded relations of Chapter 0, § 1.1 (3) The converse of (2) is not true. (4) Every 1-element substructure of some 8, is a model of up for all ,f3 EON, ,f3 >O. ( 5 ) Every extensional and transitive substructure of 23, is a model of ua. [Hint: if 8 c Q % a and ’11 is transitive, then V’, = V,“ n I%\.]

c

1.2.4. (1) Consider the sentences (1)-(4) of Example 1.2.21. (i) Prove that (3)0(4). (ii) Construct models for the sentences (1)-(4) with infinite paths (of length 0). (iii) Prove that in any model of (1)-(3) all paths have length < 0. (2) Construct a sentence of L,+,+ characterizing the K-Souslin trees. 1.2.5. Let K be an infinite cardinal, and let u be the L,+,+-sentence without non-logical symbols asserting “there are exactly K elements”. Show that no L,,(B)-sentence has the same models as u. [Hint: see Lemma 2.4.1, p. 139.1

Ch. 1, S 31

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

81

$3. Classification of formulas, model classes and maps In this section we introduce a number of basic definitions and prove some simple facts concerning formulas, classes of structures and maps between them. These concepts will be needed throughout the rest of the book. In some of the following definitions we shall regard the logical operators A, V, 3,V as truly distinct symbols, rather than considering some of them as abbreviations of expressions containing others. We recall that by definition L,, has the formula property: euey formula of L,, has less than h free variables. We shall often omit reference to the similarity type defining the language. Many of the following concepts defined for the languages L,, apply, mutatis mutantis, to the languages L,, and L,,.

REMARKS. (1) The intersection of arbitrarily many T‘s satisfying (1)-(6) also satisfies these properties; hence the set Sbf(cp) is well defined. (2) The preceding definition gives rise to an’ obvious “induction principle”: if the set of all formulas enjoying a certain property P satisfies (1)-(6), then every subformula of cp has P. Similar remarks apply to the definitions that follow. A set r of formulas is said to be closed under subformulas if whenever cp E r, then Sbf(cp) c I-.

+

DEFINITION 1.3.2. Let +,cp EForm(L,,); < cp iff (9 Q,= i+ or , (ii) cp = (vx)+ for some x c v a r , 7
+

LEMMA1.3.3. Let

(1) The relation

EForm(L,). between formulas is well-founded.

+,Q,

<


such that a

88

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1,

3

(2) $ E Sbf(cp) iff there is a finite sequence r,,. ..,r,,of formulas such that: ( i ) a,=$; ( i i ) r,,=cp;(iii) for every i , O < i, i + 1 < n , r,< r,+,. Moreover, the sequence a,,...,r,,is uniquely determined by 4 and cp. (3) $ESbf(cp) *‘Sbf($)cSbf(cp). (4) The binary relation “$ E Sbf(cp)” on formulas is reflexive, antisymmetric and transitive: hence, by (1) and (2), it is a well-founded partial ordering.

PROOF.(1) is proved by induction on formulas in much the same way that the well-orderedness of w is derived from the principle of mathematical induction. (2) and (3) proceed by induction on formulas. In (4), antisymmetry follows from (1) and transitivity from (3). Details are left as an exercise for the reader. In particular, Lemma 1.3.3 proves that every symbol in a formula occurs within the scope of a finite number of negation signs. More formally, we define: DEFINITION 1.3.4. Let Q be any relation, individual constant or function symbol of p or the equality sign; let rp EForm(LaI,( p)). Q has a positive (negative) occurrence in cp just in case there is a finite sequence r,,. ..,IT,, of formulas such that (i) r,,= cp; r, is an atomic formula in which Q occurs. ri+,for every i, O < i < n - 1. (ii) a,i (iii) { i < n I r,+I = la,}has an even (odd) number of elements. Note that the same symbol may have both positive and negative occurrences in a formula. A formula cp of type p is positive iff no symbol of p or the equality symbol has a negative occurrence in cp. Let Pos,,(p) denote the set of all positive formulas of LKh(p). It is easy to prove that Pas,,( p ) is the smallest set r of formulas with the following properties: (1) r l A t ( p ) , (2) if { c p , l i E Z } G r , T < K , then AiE,cpiEr and

VjE,cp,€r,

(3) if cp E r and X c Var,
Sometimes we shall find it convenient to deal with reduced instead of arbitrary formulas; by Lemma 1.3.6 below there is no loss of generality.

Ch. 1, § 31

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

89

DEFINITION 1.3.5. The set R,, of reduced formulas of L,, is the smallest set such that (1) every atomic or negated atomic formula is in I‘;

r c Form(L,,)

(2) if { r p j l i E I } C r ,7 then VjE,rpj and AjE,rpjare in r; (i.e., I‘ is closed under L,, conjunctions and disjunctions); (3) if rp E r and X GVar, X < A , then (VX)rp, (3X)rp E T (i.e., r is closed under L,, quantifications). The reduced form of a formula is obtained by “pushing” all negation symbols to their innermost places (i.e., right in front of atomic subformulas of rp) and cancelling repeated occurrences by the law -t iri/-ri/. One obtains: LEMMA1.3.6. Evety formula rp is logically equivalent to a reduced formula ri/ having exactly the same non-logical symbols and free variables. Moreover, ri/ can be chosen so that the signs of the occurrences of the relation symbols and equality do not change in passing from rp to ri/.

PROOF.Trivially proved by induction on formulas using the rules i V-A i , i 3 - V i and their duals. In the sequel we introduce several important classes of formulas.

DEFINITION 1.3.7. The class Qf,., of quantifier-free formulas of L,, is the smallest class of formulas of this language containing all atomic formulas, closed under negation and closed under conjunctions and disjunctions of length < K .

Quantifier-free formulas receive many names: open, Z, no, Boolean combinations of atomic formulas, etc. In fact, it is useful to define in general:

DEFINITION 1.3.8. Let AcForm(L,,). The set Bool,,(A) of Boolean combinations of formulas of A is the smallest set of L,,-formulas containing A, closed under negation and closed under conjunctions and disjunctions of length < K . DEFINITION 1.3.9. (a) The set EX,, of existential formulas of L,, is the smallest set r Form(L,,) such that: (1) every atomic or negated atomic formula is in r; (2) if { r p j l i E I } c T , ? < K , then VjE,rp,,AjE,rpj€r; (3) if q E T , X CVar and
90

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1 ,

53

DEFINITION 1.3.10. (a) The set Unex,, of universal-existential formulas is the smallest set of L,,-formulas containing EX,, and closed under: (i) L,,-disjunctions; (ii) L,,-conjunctions; (iii) universal quantifications over less than h variables. (b) If clause (i) is deleted we obtain a smaller set whose members are called conjunctive-universal-existential formulas and is denoted by Unex,, . (c) The set of existential-uniuersal formulas is defined by the clauses obtained by exchanging “universal” and “existential” in part (a). REMARK.Existential formulas are also called Z,, universal II,, universalexistential &, etc. These names have their root in the case of L,, where formulas are classified in terms of the prefix of their prenex normal form. However in the absence of such a classifying principle in infinitary languages, we prefer to use the names given above; at any rate this will be complete enough, for we shall never consider formulas beyond r12. Prenex normal form theorems are not possible in infinitary languages (cf. Counterexample 2.4.12 and Appendix B); for this reason it is convenient to introduce the class of prenex formulas:

DEFINITION 1.3.1 1. The set Pren,, of prenex formulas of L,, is the smallest set of formulas containing all the quantifier-free ones and closed under both universal and existential quantifications over sets of < h variables. REMARKS. (1) Using (1) of Lemma 1.3.3 it is easy to see that the formulas of Pren,, can be put in the form

where rp EQf,,, X , , . ..,X , c V a r and these sets are of cardinality
Ch. 1, 8 31

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

91

REMARKS. ( I ) I t is easy to check that for all cpEForm(L,,), qr(cp)
X = I in clause (4).

u X 2 ) ) d + 1.

this difficulty is avoided by requiring

(3) Notice that prenex formulas have finite quantifier rank; quantifierfree formulas have quantifier rank 0 and conversely. The converse of the first assertion is false. Owing to the lack of a prenex normal form theorem, the quantifier rank provides the only known (and rather primitive) “measure” of the distribution of quantifiers in an infinitary formula. Next we shall introduce some basic definitions concerning collections of (similar) structures. NOTATION. Let K be a class of structures of similarity type v and let p c v ; by K r p we denote the class of all reducta to p of structures in :

DEFINITION 1.3.13. Let K,A be infinite cardinals such that K > A ; let p be a similarity type, K be a class of structures of type p. (a) K is an LKA(p)-elementaty cluss, in symbols K E EC(L,,( p)), iff there is a sentence 8 of LKA(p) such that K = Mod(8). (b) K is an LKA(p)-elementaty class in the wider sense, in symbols KEEC,(L,,(p)), just in case there is a set of sentences Z c Sent(L,,( p)) such that K = Mod@). (c) K is an LKA(p)-pseudo-e/ementaty class, in symbols K E PC(L,,( p)), iff there is a similarity type p’_>p and a sentence 8 E Sent(L,,( p’)) such that K = Mod,,(8) p. (d) K is an LKh(p)-pseudo-elementaty class in the wider sense, in symbols, K E PC,(L,,( p)), just in case there are a similarity type p’> p and a set Z C Sent(LKA( p’)) such that K = Mod,@) 1p,

r

REMARK. The last two definitions concern axiomatizability using additional predicates. When no confusion is possible we shall use the notations PC, ECA( p), etc., instead of PC(L,,( p)) or EC,(L,A( p)). The preceding

92

THE BASIC NOTlONS OF INFINITARY LANGUAGES

[Ch. 1, 8 3

section $ 2 consisted mostly of examples of EC classes in various infinitary languages; further examples, as well as some of the results known for finitary languages will be found in the exercises at the end of this section. The following facts are easily proved: LEMMA1.3.14. (1) EC c EC, c PC,, (2) EC C PC C PC,, (3) If K = Mod(Z), Z Sent(LKA( p ) ) and v = max{%+,K}, then K E EC(L,,( p)). In other words, if K E EC,(L,,( p)), there is K’> K such that K E EC(L,,,( p)). The same holds for PC instead of EC. (4) If K is a class of structures of type p’, p C p’ and K E PC(L,,( p’)), then K p E PC(L,,( p)). The same holds for PC,. ( 5 ) If K is EC(L,,) or ECA(L,J, then K is closed under -,A, i.e.,

r

% E K A H = , A % * BEK. Corresponding to each class r of sentences we define the concepts of ‘TC-class” and “TC,-class” of structures by requiring that 8 E r and Z c r, respectively, in the last two clauses of Definition 1.1.13. Thus, following the traditional notation, KEUC, KEZC,, KEPUC, etc., mean that K is a “universal class” (i.e., the class of all models of a universal sentence), an “existential class in the wider sense,” a “prenex-universal class”, etc. If necessary, reference to the language will be introduced as before. We will also need an extension of the concepts of PC and PC,-classes [ 11, [2]. due to VAUGHT[ 11 and MAKKAI NOTATION. Let cp be a formula with a single free variable and M a class of similar structures, say of type p ; also Sm(rp) C_ p. Then MT denotes the class (8 rcpa I HEM}. Recall that if 8 is a structure of type p, v c p , A ~1231and 8 r A is defined, then (81A ) 1v = (23 u ) r A . If K is a class of structures, Kfin(Kinf,resp.) denotes the class of all finite (infinite, resp.) structures belonging to K.

r

DEFINITION 1.3.15. (Notation is the same as in Definition 1.3.13.) (a) K is a relative PC(L,,( p))-class, in symbols, K E RPC(L,,( p)) just in case there is M E PC(L,,( p) and a formula cp of the language of M such that K=MTrp. (b) RPCA(L,,( p)) is defined in the same way, replacing PC by PC,. In other words, KERPC,(L,,(p)) iff there is a language p’>p, a formula cp E Form,(L,,( p’)) and a set C C Sent(L,,( p’)) such that for every

Ch. 1, 1 31

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

93

% of type p: %EK

=

there is a % E Mod,.(Z) such that % = (231rp”) r p.

The simplest example of RPC(L,,)-class plicative groups of fields, i.e.,

is the class MG of all multi-

( G , * , e ) EMG w there is a field ( F , + ,.,0,1) such that

(In this case rp(x) is the formula x%O.) It is easy to see that statement (3) of Lemma 1.3.14applies also to RPC. Adding a unary predicate P and the axiom V x P ( x ) , we get (in all L,,):

(6) PC G RPC and PC,

c RPC,.

It is natural to ask whether these inclusions are proper, it turns out that for L,, the answer is “no”:

-

THEOREM 1.3.16 (Makkai). (a) KERPC,(L,,( p)) + KEPC,(L,,( p)) (b) K E RPC(L,,( p)) + KinfE PC(L,,( p)); hence, if K contains no finite structure, K E RPC K E PC. REMARK. The stronger conclusion K E P C in (b) is not valid in general (cf. Exercise 1.3.1(g)). The proof of the preceding theorem is based on the existence of a prenex form for every formula, a property of L,, which does not extend to infinitary languages. It is not known whether the theorem generalizes to the infinitary case. Next we want to classify maps according to the formulas that they preserve; we shall be mainly. concerned with the concepts of L,,-map and L,,-substructure, which are the infinitary analogs of elementary maps and elementary substructures. In the sequel K can be an infinite cardinal or co.

DEFINITION 1.3.17. Let r be a class’ of infinitary formulas of type p and %,% be structures of this type. A map f : I%l+l%l is called a r-map (also r-morphism) iff for every rp E r and every g E %t=rp[gI =+

~b[.fosl.’

r may eventually be a proper class.

* We can make a similar definition for partial maps, i.e., maps f such that Dom(f) c (%I; this case we should demand that g EDom(flFv(V);f may possibly be the empry map.

in

94

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. 1, 5 3

The following facts are easy consequences of the preceding definition: LEMMA1.3.18. ( I ) /f rcA and f is a A-map, then f is a r-map. ( 2 ) I f f is a r-map, then it also is a rA-map and a Tv-map, where these denote, respectiveb, the classes of all conjunctions and of all disjunctions of subsets of r. (3) If x,y E Var, x # y , 1( x = y ) E r and f is a r-map, then f is one-one. (4) I f .f is a r-map and r is closed under negation, then

for all g E IBIFv(rp) and all cp E r EXAMPLES. ( I ) Let %,% be structures of type p such that 1911 @3 and let idx,B denote the identity map from 1911 into 181; then 'u c3 w

is a Qf,,(

p)-map.

(Recall that Qf,,( p) denotes the class of all quantifier-free formula of type p ; by ( 2 ) above Qf,,( p) can be replaced, without loss of generality, by the set of all atomic and negated atomic formulas of type p ; the same remark applies to ( I ) of the next definition and at several other places below.) (2) Let r=Sent(L,,( p)); if 91>1-.~%, then any map f : 19Il+l'FI is a r-map.

DEFINITION 1.3.19. Let %,3be structures of type p.

(1) A Qf,,( p)-map from 'u into 23 is called an embedding or monomorphism. If f is an embedding from 91 into Y we shall write 91 ~ ' 2 3 ; 'u c 3 means that there is an f such that 91 'Y. ( 2 ) An At( p)-map from 91 into 23 is called a homomorphism. (3) If there is a homomorphism from 91 onto 23, 3 is called a homomorphic image of ?I. A monomorphism from 91 onto 23 is called an isomorphism; % = f % means that f is an isomorphism from 91 to 23, and 8 means that there is an f such that 91-"F. (4)A Form(L,,( p))-map is called an L,,-map. (5) If 1911 c "$1 and idu:a is a r-map. 9I will be called a r-substructure of 8 ;in symbols 91
c

use the notation 91

5

LA% with analogous meaning as (I).

Ch. 1,

5 31

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

95

REMARKS. (a) For K = A = w we obtain the concepts of elementary map and elementary substructure. (b) Notice that III % w f[%]<,A%.

sf KA

LEMMA1.3.20 (Tarski's criterion). The following conditions are equivalent for a map f : 1III++Bl: (1) f is an L,,-map; (2) for every cp E Form(L,,( p)) and every g E IIIIIFv(qp),

-

~ ~ c p [ g I Bt-cp[fogl; (3) f is a monomotphism and for every cp E Form(L,,( p ) ) , every partition of Fv(cp) into two sets, XI, X,, and every pair of maps g,EIIIIIX', hEl%lx2, the following holds: Bk-cp[(fog,)*h]

there is ag,EIIII\llx2such that % F c p [ f o ( g l * g 2 ) ] .

COROLLARY 1.3.21. 91 < ,B iff 91 c%and for every formula cp of LKA(p), every partition of Fv(cp) into two sets X , , X , and every pair of maps f , E I % I X I , f 2 E l%lXz, the following holds: if BFcp[fl*f2], then thereisagE1911X2such that %t-cp[f,*g]. COROLLARY 1.3.22. An isomorphism from III onto % is an L,,-map for all K > A (or, equivalently, it is an L,,-map).

A,

K,

Lemma 1.3.20, Corollaries 1.3.21 and 1.3.22, and Lemma 1.3.23 are proved exactly as for first-order logic.

DEFINITION 1.3.24. A set K of structures of type p admits a union iff (i) for every individual constant symbol c of p and every pair %,Bof structures in K, c B = c';

(ii) for every m-ary function symbol in p, every pair Z,% E K and every

x,,. ..,x, E

n I%/, P ( x.,. .,,x,)

.. .

= P ( X , ,x,). ,

THE BASIC NOTIONS OF INFINITARY LANGUAGES

96

[Ch. 1, 3 3

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

c,

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

u ppil= u

1911;

’UEK

S’B=

U

Sgl for every relation or function symbol S ofp;

’u E K

cB = c’

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

Clearly, the condition “K admits a union” insures that U K is well defined. We know, however, that in general U K has different elementary properties-and, a fortiori, infinitary properties-than the elements of K, even in cases where these are pairwise elementary equivalent (cf. Exercise 1.3.1(h)). A well-known result of Tarski states that this is not the case whenever K is directed by the relation < of elementary extension. This result holds for infinitary languages as well, upon strengthening the concept of directed set. DEFINITION 1.3.26. Let ( A , < ) be a partially ordered set and p an infinite cardinal number. ( A , < ) is p-directed (
THEOREM 1.3.27 (Tarski’s union theorem). Let r be a class of infinitary formulas included in some Form(L,,( p)), such that:

Ch. 1, § 31

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CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

(i) r contains all atomic and negated atomic formulas of type p ; (ii) if cp E r, then some reduced form rp* of cp belongs to T; (iii) T is closed under subformulas. Let K be a set of structures of type p, < ql(r)-directed by the relation Then, for euety 8 E K , 3 < U K .

,.

< '.

PROOF.By induction on formulas we shall show that for every V E T and every g E l%lFv(v),the following holds:

If cp has one of the forms V,€,+Lj or true.

the assertion is obviously


COROLLARY 1.3.2. Let K be a set of structures
r = Form(L,,)

u

and apply the preceding theorem.

rn

We introduce next two sets of formulas that play a particularly important role in model theory.

DEFINITION 1.3.29. Let 8 be a structure of type p. Let p' denote the similarity type p u { ii 1 a E I%[} which contains an individual constant symbol i i E p for each a E 181(and a# & for a# 6). If X c V a r and f : X+I%l, let

-

f ( x ) = f ( x ) for each x E X . (1) The diagram of a, q(%), is the following set of sentences of the language LUG(p'): cp(ii,,. ..,a,,)€ 9 (8)iff cp is an atomic or negated atomic formula of type p, Fu(cp) = n and %kcp[a,,...,a,].

98

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch.I , 3 3

(2) The complete L,,-diagram of a, bD;,(Z), is the set of all sentences (of type p') of the form q(f) where qEForm(L,,(p)), f:Fv(rp)+I%l and 2I t- dfl.

ufi:u(%)

verified:

is denoted by 6&?l).

The following equality is immediately

The following facts are easily proved; their proof is left as an exercise.

LEMMA1.3.30. Lei %,!I3 be similar structures and lei f :1911+1!I31 be a map. Then :

EXERCISES In first-order logic there is an extensive theory dealing with the classification of classes of structures; the reader may consult, for example, BELL[ 11, CHANG-KEISLER [ 11, and ROBINSON [ 11. Far less is known in SLOMSON infinitary languages, and most of the known results are for L+u; the reader [3]. Both counterexamples will find an account of these results in KEISLER and known results for larger infinitary logics are treated elsewhere in this book. The first three exercises below tell a part of the story for Lou; they are intended to recall a few basic facts; certainly much more is known. 1.3.1. (a) Give an example of a class K E PC, but not in PC. (b) Show that the class T - W of total orderings that are not wellorderings is an example of a class in PC - EC,. ( c ) Show that the following class is in (PC n UC,) - EC: the class of all structures ( A , S ) such that S A x A and A does not have S-cycles (of any order. ( ( x ,,..., x , ) E A " is an S-cycle (of order n ) iff x , S x , ,..., x, - [ S x , and x n S x , . ) (d) Every EC, class is an intersection of EC classes. Give an example of a PC, class of finite similarity type which is not an intersection of PC classes. (e) Notice that every class of structures is contained in a minimal EC,

c

Ch. 1, 3 31

99

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

class. Determine the minimal EC, class containing each of the following classes: (i) The class of all linearly ordered sets containing a dense subset. (ii) The class of all fields of characteristic different from zero. Show that the first of these classes is a PC. Is the second one a PC,? (f) Give an example of a class of structures closed under isomorphic images which is not a PC,. (g) Give an example of an RPC class which is not PC. (h) Find a set K of structures admitting a union such that the members of K are pairwise elementarily equivalent but no 3 E K is elementarily equivalent to U K. 1.3.2. Let K be a class of similar structures. We denote by Kfin (Kinf, respectively) the class of all finite (infinite, respectively) structures in K. Applying compactness, show that if K contains structures of arbitrarily high finite cardinality, then Kfing PC, (most of the usual algebraic structures-groups, rings, fields, lattices, etc.-are examples of such K's). Show also that if K E PC (PC,, respectively), then KinfE PC (PC,, respectively). Likewise, if K E EC,, then KinfE EC,. But if K E EC, then Kinf is not necessarily an EC class. 1.3.3. Let S(K) denote the class of all isomorphs of substructures of members of K. If p is the similarity type of the structures in K, S,,,(K) denotes the class of all isomorphs of structures of the form '23 v, where BES(K), B isfinite and v is afinite subtype of p. Finally, Thv(K) denotes the set of all universal sentences of p holding in all members of K. Prove: (a) CU E Mod(Thv(K)) w SU,,(CU)c SU,,(K). [S,,,(3) stands for

r

sU,U(P 11.1

(b) If K E PC, and S,,JCU) c SU,,(K), then 3 E S(K). (c) (EoS-Tarski) K E PC, * S(K) E UC,. Note that even if KEPC, the conclusion cannot be improved to UC. However the following weaker statement can be inferred from (a), (b): (d) If KEPC, then S(K)=K H KEUC. From (d) it readily follows: (e) (Tarski) A sentence u is preserved under substructures (i.e., if u holds in %, it holds in all substructures of 3) iff u is equivalent to a universal sentence. This result implies its dual: u is preserved under extensions iff it is equivalent to an existential sentence. [Hint: the proofs of all these results rely heavily on compactness.]

100

THE BASIC NOTIONS OF INFINITARY LANGUAGES

[Ch. I , 5 3

1.3.4. (a) Study in various infinitary languages (e.g., L,, for various values of K wl, LalUl,etc.) the classes considered in Exercises 1.3.1 and 1.3.2. Determine which are EC and which are PC in these languages. Try to generalize to infinitary languages the situations illustrated by these examples. (b) Prove that the class of all scattered ordered systems is a UC(L,I,I) ( ( A , <) is scattered iff no subset of A is dense in the order induced by <). (c) The class of all isomorphs of the structure (a,+, .,O,1) is a uc(L,I,). (d) The class U = { ( A , A x A ) 1 7 4 No} is an EC(LUIwI), satisfies conditions (i) and (ii) of Exercise 1.3.5(b), but is not a PUC(L,I,,). IS it a UC class in the language LUIuI? [Hint: use Exercise 1.3.5(c).] (e) Show that there is a language p in which the class of all metric spaces is characterizable by a single LUl,( p)-sentence. More precisely, define a language p and an La,,( p)-sentence u with the following property: % E Mod( u )

H

there is a map d : 1'11I2+R' such that ( ] % \ , d )is a metric space.

[Hint: introduce denumerably many binary predicates R,, q a rational number.] Note. Since the metric d is not an 'internal' operation, one cannot conclude that the class of all metric spaces is a PC(LUI,)! In this respect, cf. Corollary 3.4.15 and the subsequent remark, in pp. 217-218, Chapter 3, Q 4.c.

(f) If E< K and Sm(%) < K , then the class of all isomorphs of % is a WL,,). (8) Give an example of a denumerable structure 8 of countable similarity type p such that the class of all isomorphs of % is not an EC,(L,J p)). [Hint: cf. Lemma 2.4.1.1 P

Definition. A class K of structures of a given similarity type IJ. is called an EC,,(L,,)-class iff there exists a collection { Ki I i E I } of EC,(L,,( p))classes, indexed by a set I , such that K = U,,,K,.

(h) Show that any EC,,(L,J-class is an EC(L,,)-class. Show that the same holds if K = U ,Kj where each Kj is an EC,-class in some infinitary language of the form LKw. (i) Show that there are EC(Lul,)-classes which are not EC,,(L,,). 0) Prove that if p is finite and does not contain function symbols, then

Ch. 1, § 31

101

CLASSIFICATION OF FORMULAS, MODEL CLASSES AND MAPS

every prenex formula of LKu(p ) is equivalent to a formula in L J,

p).

1.3.5. (a) Prove that for every countable structure 2 of countable similarity type p , there is an existential sentence 9 of Lu1J p ) such that for every 23 23EMod(q)

* 2 _c 23.

(b) Let K E UC,(LxJ; then K satisfies the following conditions: (i) K is closed under isomorphism (i.e., ‘ZI EK and ‘ZI =2‘ 3 * 23 E K). (ii) K is closed under substructures (i.e., 2 E K and ‘ZI C 23 + 23 E K). (c) Let K EPUC(L,,( p ) ) , i.e., K is the class of all models of a set of prenex-universal sentences of LKA(p ) ; then K has the following properties: (iii) For every structure 3 of type p if S,((u) CK, then 2 E K .

(S,(%)denotes the class of all substructures of 2 generated by a set of power
2,< r J lJ 2, . IEI

CHAPTER 2

SOME RESULTS FROM THE MODEL THEORY OF L,, AND L, , I

§ 1. Background on types and saturated structures

In this section we work within L,, that

and p is a fixed similarity type. Notice

Form (L,,( p ) ) =max{

7 , No}.

DEFINITION 2.1.1. A type Z in the variables o0,...,o n - , is a set Z C Form (La,( p)) of formulas containing at most o0,. . .,v,-, free. REMARK. The cardinal of the set of all types (in any finite number of variables) is < 2&,where S=max{ 7 , No}. Arguments involving types are very often perfectly analogous for the one variable case and for the n-variable case; whenever this is so, for simplicity of notation we shall treat only the former case. DEFINITION 2.1.2. A type 2 is realized in '% (B a structure with Sm (a)=p ) if there is a E ['$I such that % t=u[a] for all u € 2 . With obvious meaning we also write 2 , a realize 2 . Z is omitted by or in 8 if Z is not realized in '2[ (i.e., for every a E (%I ( . there is u E Z such that 2l k --p[a]). A set S of types is omitted in 8 if every 2 E S is omitted in 3. A type Z is consistent if the set T,={u(c)laEE} of sentences of LaJ p u { c}) (where c B p ) is consistent. A type Z is compatible with a structure 2l [Sm (8)= p ] if realizes every finite subset of Z. 102

Ch. 2, 3 11

BACKGROUND ON TYPES AND SATURATED STRUCTURES

103

REMARK 2.1.3. The following are immediate consequences of the preceding definition. Their proofs are left as an easy exercise for the reader (cf. Exercise 2.1.1). (1) A type Z is consistent iff there is '% realizing Z. (2) '%=Band Z compatible with '% imply C compatible with %. ( 3 ) 'u, a realize Z and % i 23 imply 23, a realize 2 . The following is an easy consequence of the compactness theorem: THEOREM 2.1.4. (Compactness theorem for types). Z is consistent finite subset of C is consistent.

iff

euety

PROOF.Apply compactness to T,. THEOREM 2.1.5. Euety consistent type is contained in a maximal consistent type.

PROOF. Let %,a realize Z. Take as the required maximal type the following:

REMARK. A maximal type Z is characterized by the fact that for all u with appropriate free variables u E Z or i u E Z. Maximal types are also called complete types. A more refined application of compactness gives

THEOREM 2.1.6. Given a type Z and 'u compatible with Z, there is % > ,U such that 23 realizes Z. Moreouer, by the downward Lowenheim-Skolem theorem, '21 can be taken of power max{ E, F,t?t0]. PROOF. Consider the similarity type p' = p { 5 a E I'ul} u { c}, where cfZ1'%1, c g p . Consider the following set T of sentences of p':

u I

(1) qc('%h (11) u(c) for each u E Z . 'u compatible with Z assures that every finite subset of T is consistent; hence T is consistent. Let '21' E Mod ( T ) and B=B'rp. Since 8' E Mod (6Dc('u)), it follows that '21 > '%.From the fact that B' is a model of all sentences of the form (11) it follows that there is b E 1931, namely b = c ~ ' such , that b realizes Z in %. The last assertion is obtained by a trivial application of the downward Lowenheim-Skolem theorem to 8'. W

104

RESULTS FROM MODEL THEORY OF

L,,

AND

Lo,,

[Ch. 2,

5

1

COROLLARY 2.1.7. Let S be a set of types, % a structure (Sm (a)= p) such that every member of S is compatible with 8.There is B > % realizing all _ _ members of 5. Moreover, B can be taken of power < max { 5 , % , F, No}.

PROOF.Let p = 5 and (Z, 1 CY < p ) be an enumeration (without repetitions) of S. Construct the following tower: Bo=some elementary extension of % realizing Zo (of power < max{ E , F ,N~}),

Ba= some elementary extension of Us
<

max{ U,<,B3,,

F,No}),for 0 < CY < p.

Take % = Ua
C,= {“uo is divisible b y p , ” ( n E X } u {“v0is not divisible byp,,,”(mEX}. Z Clearly all these types are compatible with (o,+, .,O, 1). (If ZCZ,, finite, and no,. . .,nk are all the n’s in X occurring in formulas of Z of the first form, a =p,;p,, * . .. .p,, realizes 2.) By one of the preceding compactness theorems we have: (*) For every X C w there is % > (a,+, .,O, 1) suLh that % realizes C,.

Given %=(o,+;,O,l)

andaEI%I(,define: N ( a ) = { n E W Ip, divides a }

and

% = N o and

Ch. 2, I 11

105

BACKGROUND ON TYPES AND SATURATED STRUCTURES

-

If, in addition, ?i =No, then N ( % ) =No. Condition (*) can be written as follows in this notation:

(***I

Given X X EN(%).

there is % > (a,

+, .,O,

I),

= No

such that

If % = ( w , + , ,O,I), the isomorphism type of % is defined as I,= { B ] B = % } .

From (**) it follows that with every isomorphism type I, we can associate a set N ( Z ) C 9 (w) as follows: N ( I ) = N ( % ) for any % E I . Let 5 be the set of all isomorphism types of denumerable models of Th((w, +, -,O, 1)). Condition (***) tells us

Hence

-

since N ( I ) = N o f=orall I E 5. It follows that T = 2'0, i.e., there are 2u~painvisenon-isomorphic denumerable models of (complete) arithmetic. Corollary 2.1.7 can be further exploited to obtain a very important result of model theory: the existence of saturated structures.

DEFINITION 2.1.9. Let.% be a structure and p be its similarity type; let A be an arbitrary cardinal. Let { c, I a
3

% is A-saturated for all cardinals A. % finite or

>A.

106

RESULTS FROM MODEL THEORY OF

L,,

AND

L,,,

[Ch. 2, $ 1

THEOREM 2.1.10. Let p be an arbitrary similarity type, K an infinite cardinal, > F.Then (1) every infinite structure of type p and power < 2" has a saturated eIementary extension of power < 2"; ( 2 ) i f K is strongly inaccessible > w every infinite structure of iype p and power < K has a saturated elementary extension of power K . K

PROOF.(1) Let '2 be an infinite structure of type p and power < 2". Let a:2"+1'21 be an enumeration of I%[ (i.e., an onto map). Consider the language p2* = p u { c, I a < 2"}, where the constants c, d o not occur in p . Let S be the family of all types Z of formulas in Luu(p2K)such that (a) Z is compatible with (a, a ( a ) ) , < , . (i.e., with the structure in which the constants c, are denoted by the elements a ( a ) ) ; (b) T < K . Clearly S C $7' (Form(L,,( p2"))). Since Form (Lwu(p2"))= 2" and a set of power 2" has (2")" =2" subsets of power K , we infer that 5 < 2". By Corollary 2.1.7, 2X has an elementary extension, 8, of power < 2" such that (!€?,a(a)),

If p< K + , f EI'ZIIP and Z is a type of the language pP compatible with ('2, f (a)),
If g : p+2" is a map such that f ( a )= a ( g ( a ) ) for all a < p and g ( 2 ) is obtained from Z by replacing each c, by cgc,,, then the hypothesis of (*) implies that g ( Z )E S . Hence ( 3 * , a ( a ) ) , < 2 . realizes g ( Z ) ; this clearly implies that (%*, f (a)),
Let 8 = U5 %. A simple induction proves that all the 3;s-hence also 8-are of power < 2". Finally we verify that '13 is K+-saturated. Let p< K + , f €181pand 2 be a type of the language pP compatible with (23,f(a)),,p. Since K + is regular

Ch. 2, f 11

BACKGROUND ON TYPES AND SATURATED STRUCTURES

107

and j?< K + , there is y < K + such that Range( f)c lZ,.l. By (*), Z is realized in (a,.+,,f(a)),
-

Since Form(L,,( Hence

pK))

= K,

for A <

CPA(Form(Lw,(pK))) = K‘=

K

K.

From this point on the proof proceeds as in (I), with the only exception that the sequence of the 3,‘s is of length K.

COROLLARY 2.1 . I 1. Let T be a set of L,,-sentences with infinite models and K an infinite cardinal. Then: ( 1 ) (GCH) T has a saturated model in each cardinal K + ; ( 2 ) if K is strongh inaccessible > w , T has a saturated model of power K . REMARK. The GCH cannot be eliminated in ( I ) of the preceding result. There are several ways of seeing this. Perhaps the simplest is by the following slight modification of Example 2.1.8:

THEOREM 2.1.12. b’vety saturated model of complete arithmetic has power 2 2“. PROOF.First we remark that for each K > W there is an elementary extension of ( w , + .,0,1) of power K, containing K primes. To see this consider the following set r of sentences of type { , .,0,1} u { c, 1 a < K } : (I) Th((w, +. ..O, I)), (11) “c, is a prime” for a < K, (111) ~ ( c , x c p ) for a , p < ~ , a # j ? .

+

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Since there are infinitely many primes in w , every finite subset of r has a model. By compactness, r has a model, which by (1)-(111) has the required properties and which, by the downward Lowenheim-Skolem theorem, can be taken of power K . Let 8 be a structure with the preceding properties, and let { p a 1 a < K} be an enumeration of all primes of 3. For each X K,let Z, be the following type in the language { +, .,0, l } u { c, I a < K } : Z, = { “c,, is a prime” I y E K } u { “c,, divides 0’’ I a E X } u { “cP does not divide u” 1 fl E K - X }.

Clearly, each 8, is compatible with (%,pa)a a. Each Z, is compatible with (93,pa)a
X , Y C K A X ZY

3

b,#b,.

Hence { b, IX C K } is a subset of 1231 of power 2“; therefore,

3 > 2”.

The same argument clearly shows that any model of complete arithmetic which realizes all the types 2, ( X Ca)of Example 2.1.8 has power 2 2No. In particular, this shows that there is no countable, o-saturated model of arithmetic. REMARK.There are examples of theories without saturated models in singular powers, e.g., the theory of linear orderings (Exercise 2.1.4). There is a construction which, for these cardinals, produces a class of structures (called special) with properties in many respects similar to those of saturated models (cf. BELL-SLOMSON [I], Chapter 11, Q 4). The fundamentally important role of saturated structures in model theory stems from the following properties, which we mention without [l], Chapter 11, 0 3). proof (for proofs, cf. BELL-SLOMSON DEFINITION 2.1.13 (1) A structure Iu is &universal iff for every 93, %E%A%
8 is universal iff it is T-universal.

8

5 a.

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109

(2) % is weakly A-universal iff for every 23,

% is weakly universal iff it is weakly 5-universal.

is A-homogeneous iff for every X , X c 181 and < A imply that (3) every elementary map f :X + 8 can be extended to an automorphism of 8. % is homogeneous iff it is z-homogeneous. (4) % is weakly A-homogeneous iff for every X , y , X C 181, 7 < A and y E1 8 1 imply that every elementary map f : X + 8 can be extended to an elementary map g : X u { y } + 8 . % is weakly homogeneous iff it is weakly 8 -homogeneous.’ Clearly, A-universal A-homogeneous

* weakly A-universal + weakly A-homogeneous.

Furthermore, we have the following

THEOREM 2.1.14 (Fundamental theorem).

(1) Homogeneous w weakly homogeneous; . (2) A-saturated w A-universal A weakly A-homogeneous; ( 3 ) saturated w universal A homogeneous; (4) two elementarily equivalent, saturated structures of the same cardinality are isomorphic.2

Using these properties we describe our next, important:

EXAMPLE 2.1.15. (qA-sets). By Theorem 2.1.10, the theory of linear orderings has saturated models of power (2”. Which is the nature of those linear orderings? In partial answer to this question we consider instead the theory of dense linear orderings without endpoints. Let ( A , <) be one such linearly ordered set; consider the types Z,,,={X
Y},

’

The term “A-homogeneous” is sometimes applied to this notion. For non-complete theories T the notions “universal model of T” and “homogeneous model of T” are usually defined by replacing “I, b E Mod( T)” for “Ie8” in (2) and (4), respectively. Some of these statements are considered again in Chapter 5; cf. Theorem 5.3.7 and Exercise 5.3.2.

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where X, Y are (possibly empty!) subsets of A of power < K such that each element of X is smaller than any element of Y (in symbols, X < Y ) . By density, the types 2 x , yare compatible with ( A , <,~,y),..~,~~ and therefore are realized in this-structure. In other words, ( A , <) must have the following property: (A) for each pair X , Y of subsets of A of power < K such that X < Y , there is an a E A such that X < a < Y . Any linear order with this property will be called an q,+-set. For limit cardinals A, qx-sets are defined by properly changing (A). I t is natural to ask whether (A) alone is sufficient to characterize K +-saturated, dense linear orderings without endpoints. The answer is “yes”. One can see this directly, by eliminating quantifiers in the theory of [ l]), thus dense linear orderings without endpoints (cf, KREISEL-KRIVINE concluding that the sets X x , y are, roughly speaking, the only possible maximal types, up to equivalence. The sets H,+ defined below are simple examples of qK+-setsof power 2“ (the construction is due to Sierpinski). For each infinite ordinal a , let U, be the lexicographically ordered set of all sequences of 0’s and 1’s of length a. H , is the sub-linear ordering of U, consisting of all those sequences s : a+2 having a last I ; i.e., there is a y < a such that sy = 1 and sp = 0 whenever y < p < a. (B) Each HK+is an ?,+-set (evidently of power 2“). (C) Each q,+-set contains an isomorphic copy of U,, and even of HK+ (Gillman; cf. Exercise 2.1.6). In particular, each q,+-set has power > 2, showing by a second example that the GCH cannot be dispensed with in Corollary 2.1.1 1 (1). One can construct very simple examples of non-isomorphic, a-saturated linear orderings of the same power > 2No.qA-setsprovide a nicer example: [I]) If 2 # ~ + there , are two non-isomorphic ?,+-sets of (D) (GILLMAN power 2“. All of this shows that the uniqueness assertion (4) of the fundamental theorem can hardly hold for anything short of saturated structures. As a final remark we observe that ?,+-sets of power K+(GCH)are also saturated models of the general theory of linear orderings. Indeed, using a back and forth argument one can easily prove that any q,+-set is weakly K +-homogeneous, K+-universal (in the sense of Definition 2.1.13 (1)) and K+-universal with respect to arbitrary linear orderings (in the sense that any linear ordering of power < K is isomorphically (not elementarily!) embeddable in it; cf. footnote to Definition 2.1.13. This type of argument generalizes Cantor’s well-known method for proving that any denumerable dense linear ordering without endpoints is isomorphic to the ordered set of rationals: it will be studied in detail in Chapter 5, OB3, 4. ,

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111

HISTORICAL REMARKS. The homogeneity and universality properties of the ordered set of all rational numbers have been known since Cantor. Hausdorff invented the q,,-sets as a generalization of them; he derived essentially all of their properties on the basis of the general theory of linear orderings; ?,,-sets should be considered direct ancestors of saturated models. Sierpinski investigated various kinds of universal mathematical systems. The study of first universal, and then homogeneous-universal structures in a general setting was initiated by Jonsson. Saturated structures were first formulated in their present form by Morley and Vaught, who also pointed out their important role in model theory. EXERCISES 2.1.1. Prove the statements (1)-(3) of Remark 2.1.3 and: (4) Give examples of structures 3 and types Z (Sm(Z) = Sm(%)) compatible with 3,such that 3 does not realize 2. (5) For types in a given language p , let Z,-Z2 mean that C,,Z, are realized by the same elements in all structures of similarity type p. If 2 , zZ2, then there are structures 3 (Sm(%)= p ) and a,, a2E I%[ such that a, realizes X i (i= 1,2), but ( ' i ! l , a l ) ~ ( 3 , a 2 ) . ( 6 ) Show examples of (necessarily non-maximal) types Z,, Z, which are realized by some and the same elements in some structure, but Z , r Z 2 . (7) Give an example of a type in a countable language with 20' complete extensions. (8) (Morley-Vaught) Show that a type Z is contained in exactly one maximal type iff for every 'i!l and every pair a , , a, of elements of realizing Z, there is some % > 3 having an automorphism which maps a , onto a,. 2.1.2. Prove the facts stated immediately after Definition 2.1.9. [Hint: the I a E l3l} is never realized in 3.1 type { 1(q,xii) 2.1.3. Give examples showing that

+ A-homogeneous; (ii) weakly A-universal + A-universal. (i) A-saturated

[Hint: consider, e.g., dense and discrete linear orderings, respectively.] 2.1.4. (a) Show that there is no saturated linear ordering of power Nu.

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[Hint: suppose there was and construct by induction a strictly increasing sequence of order type Nu+,; note that if "
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4 2. Models generated by indiscernibles A. Skolem functions

Let p be a given similarity type. We denote by p* the type obtained by adding to p an n-ary function symbolf, for each L,,-formula cp of type p with n + 1 free variables (if n = 0, f, is an individual constant). The sentence (of type p*) vv

,...v, [3vocp(

Ug,

. . .,on)+ cp (f, (0 1, . .

. 7

v,),

019

..

. 3

on)]

is called the defining axiom off,. (Note that the reverse implication is always true by rules of the predicate calculus.) If T G Sent(L,,( p)), T* is the set of sentences of type p* defined by: defining axiom off,

Icp E U Form, + I ( La,( nEo

p))

).

Notice that since Form(L,,( p ) ) =max{F, No}, then we have max{ F, No}.

7=

LEMMA2.2.1. For every cpEForm(L,,( p)) there is a universal formula p*)) such that Fv(cp)= Fv(cp*) and

cp* E Form(L,,(

Tt=cp m T*t-cp*

for every T C Sent(L,,( p)).

PROOF.Every formula is equivalent to one in prenex normal form where all universal quantifiers precede all existential ones and there is at least one [l], 0§42,43). So we can assume cp has universal quantifier (cf. CHURCH that form. Now eliminate all existential quantifiers one by one by introducing Skolem functions in the way prescribed by the defining axioms, into a final formula cp*. The process of going from cp to the prenex normal form of cp and from that to cp* does not change the free variables of cp, so Fv(cp) =Fv(cp*). It is easy to check by induction that

(where u is the “vector” of Fv(9) ordered in some way). However, a stronger proposition is asserted in the statement, and this requires something stronger, namely:

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LEMMA2.2.2. For euery structure % such that Sm(%)=p there is an expansion %* of % of similariq type p* (not necessarily unique) in which all the defining axioms of the Skolem functions hold. PROOF.Let g : 9 (I%[)- {iY)-+181be a choice function for cpEForm,,+,(L,,(p)) and every a l ,...,a,,E/%l,we let 'pN(.,a,,.. .,an)= .{

Now define * :f

: 1%1"+1%1

E

l%l.

For every

1 % b [ x , a , ,.. .,.,I}.

by

..,a,,)E cp"( - , a , ,...,an)if this set is non-empty, from which Hence jt*(al,. it clearly follows that all the defining axioms of fv's hold in a* = (%Yf~*)pEU.,,Form,+,(L,(p)). The rest of the proof of Lemma 2.2.1 is left as an easy exercise. COROLLARY 2.2.3. For euery

'p E Form(L,,(

p ) ) and euey T

Sent(L,,( p)):

TPcp w T * P q

PROOF.Apply (*) above and Lemma 2.2.1. REMARK. The structure %* in Lemma 2.2.2 is not unique in general, but there is essentially one known way to construct all possible such expansions of 8 ; namely, that prescribed by the defining axioms and carried out in the proof of Lemma 2.2.2. It could be interesting to give a more precise form to this intuition. The strengthening of Lemma 2.2.1 when one requires the conclusion to hold for all formulas cp in p* is not true in general. However, it holds if one repeats w times the process of going from p to p*, as follows: p"+'=(p.")* T o =T

T"+'=(T")* (for TCForm(L,,(

and one defines p# = U pn; nEo

T # = U T". n EU

p)))

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Observe that since F*=max{ ,E,N,}, by an easy induction we get p # =max{ &,No}, i.e., the operation # does not increase essentially the cardinality of p . Now Lemma 2.2.1 can be applied to p" instead of p , thus getting LEMMA2.2.4. For every rp E Form(L,,( p # ) ) there is a universal formula rp# of type p # such that Fv(rp)= Fv(rp#) and T # F Vu(rp-rp#)

for any T C Sent(L,,( p # ) ) (or, alternatiuely, 23 t= Vu(rp-rp#) ture 23 of type p # in which all defining axioms hold).

for any struc-

REMARK.The notation T * makes sense for all TCSent(L,,(p#)); it simply means T u the set of all defining axioms of Skolem functions of

P#.

Iterating w times the process carried out in Lemma 2.2.2 we get

LEMMA 2.2.5. For every structure % of type p there is an expansion 2f# of type p # (not necessarily unique) in which all defining axioms hold, i.e., %# EMod(Th(%)*)

or Th(%)* cTh(%#).

So far we have shown how to enlarge any similarity type p to a type p # which is completely endowed with Skolem functions, and how to proceed to enlarge structures of type p to those of type p # . Henceforth we forget about the base type p and we suppose we are given directly a type v with Skolem functions. DEFINITION 2.2.6. (a) A similarity type v is endowed with Skolem functions (or is a Skofem v p e ) iff a one-one map 7 of n,lForm,(L,,(v)) into

u

-

Term, exists such that for every rp in the domain of T , Fv( TJ = Fv( cp) - 1. (b) A set Z c Sent(L,,(v)) is tidy iff v is a Skolem type and for every n 2 1 and rp EForm,(L,,(v)) the following sentence is in Z:

(c) A structure % of type v is ti& iff Th(%) is tidy (i.e., iff all defining axioms hold in %). We say, alternatively, that I is a Skolem structure.

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REMARK. In dealing with Skolem structures we assume, of course, that the corresponding similarity type is endowed with Skolem functions. LEMMA2.2.7. Let %,23 be Skolem structures. Then (1) '11cB =+ % < 23. (2) Every monomorphism f:%+23 is an elementary monomorphism. PROOF.(1) Apply Tarski's criterion. Assume cp E Form,, ,(Lww(v)),ul,. .. , a, E )ZJ,b E 1231, and Bt=cp[a,,. ..,a,, b ] . Since 23 is a Skolem structure there is a term r of v with Fv(r) = n such that Bt=cp[a,,...,a,,? ?e[ al,...,an]].

Since 1 ' 1 is closed under all Skolem functions and I%lclB/,we have TB[a,,.. .,a,] E I%[. Hence 1 ' 1 < 23. (2) follows immediately from (1).

COROLLARY 2.2.8. Let 'ZI be a Skolem structure, B={a€l'11[l lthere is a constant term r such that r'= a } , and 23 = % 1B . Then 8 < %. PROOF.By the proof of Lemma 2.2.7 it is enough to check that B is closed under Skolem functions, but this is obvious. NOTATION. The structure 23 of the preceding corollary (which is uniquely determined by %) is called the Skolem hull of 1' 1, in symbols @('11).

REMARK. a(%)is the smallest substructure of % (% a Skolem structure).

DEFINITION 2.2.9. Let % be a Skolem structure (of type v) and X c,1%2[(. The Skolem hull of X in 1 ' 1 is defined as follows a x , w = @ ( ( ~ 7 X ) x E xr v) .

(Thus @@,

=

@(W)

-

COROLLARY 2.2.10. (1) ( @ ( X , % ) I = { T ~ ,..., [ X , x , ] ( n E w , x ,..., , x , E X , + r term of v, and Fv(r) = n} (2) For every X C I'LL(, @ ( X , % )i %. ( 3 ) @ ( X , % ) is the smallest substructure of 1 ' 1 containing X .

-

PROOF.(1) The constant terms of type v u { X ( xE X } are exactly the expressions of the form 7 ( X I , . . .,Q, where r is a term of v, Fv(r) = n and X I ' . .. , x , E X .

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(3) This is seen in the same way as in the preceding remark. The methods of this section yield a simple proof of the upward Lowenheim-Skolem theorem for Lau(p) in a case not covered by the best known version, viz. when p has cardinality larger than the structure B and

B
THEOREM 2.2.1 1 (MORLEY[2], p. 114). Let B be a strucgre of infinite power A and let K > A". Then there is 8 such that 8 > B and = K.

a

PROOF.Let Sm(B) = p. Consider the Skolem extension B# of 23 (of type p # ) a n d let T = T h ( ( 2 3 # , b ) b , , B , ) = 6Dc(B#); T is of type p' = p # u { 61b ElBl}. Notice that the terms of p' are terms of p # with some of its variables substituted by 6's. Let X c_lB\be of power N,. Define the following equivalence relation on the set of terms of p': ~

ie., iff

~

--

w - F 7v ~ ( ~= ~ )F v ( T ~ ) (=n,say) and for every n-tuple

~

~

holds 7 iff ~7:#, *T :

define the same map X"+)B3); if n=O,

7$* = 7:*.

The number of equivalence classes modulo

~

~

- is

n Eo

Consider the language p" = p'u { ca I a < K } , where the ca's are new individual constants. Let Z be the following set of sentences in p " : (i) all sentences in T for a , p < ~ a, # p (ca=ca) (ii) i ~ n (iii) T,(c,~ ,..., C , ~ ) X T ~ ( C , ~ ,...,can) for all terms ~ ~ in, p' 7 with variables (n Eo) such that T,-T~, and all a,, . ..,a,, < K .

-

7

~

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Every finite subset 2' of Z has a model, namely:

where a,,. ..,gare all the distinct a's such that c, occurs in Z', b , , .. .,bp all the distinct b's such that 6 occurs in Z', the xi's are r distinct elements of X and the denotation of c4 is xi ( i = 1,. ..,r ) . By compactness Z has a model, say 6. It is easily seen that 6 is a Skolem structure. Let 2 be the set of all cF's (a< K ) . Let @(6) be the Skolem hull of 6. Notice that each element of @(6) can be written in the form T % [ Zl,...,z,,]where T is a rerm of p' and z , , . .. ,z, E 2 ; in other words: 1@(59)1=1@(Z, 6 p')I. From (ii) we have Z = K , and since 1@(6)122, @(6) 2 K . From Corollary 2.2.10 (l), the sentences (iii), and the foregoing remark, it follows that:

r

-

-

=

@(6) < Term( p ' ) / -

-

*

U Z" ,

nEw

i.e.,

-

hence @(6) = K . Since @(6) i 6, a=@@) r p is-by the sentences [i)an elementary extension of 8 of power K , as it was to be proved. REMARK. By a theorem of Keisler and Rabin the restriction K 2 A" cannot be improved (at least for reasonably sized cardinals) [Cf. Chapter 3, Q 3.A].

B. Indiscernibles; the Ehrenfeucht-Mostowski theorem For a given theory T it is possible to find 8 E Mod( T ) with an infinite set X /%I such that all the elements of X are alike (i.e., they satisfy in 8 the same formulas of Sm(T) in one free variable). Moreover, 8 can be found containing an X of prescribed cardinality. This is easily seen applying compactness and the pigeon hole principle. We raise a more general question, namely whether it is possible to find an 9l E Mod( T ) with an infinite X 181 such that not only all the elements of X are alike, but also all the n-tuples (for all n E w). In general this is not possible. For example, in a linearly ordered set, half the pairs satisfy q, < u1 and the other half do not. However, linear orderings are about the strongest obstacle.

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DEFINITION 2.2.12. (a) Let i?l be a structure (type p ) , X c 181a set with some ordering, (X, <) (in general < is not definable by formulas of type p!). (X, <) is a set of indiscernibles of 2l (also called a set of <-indiscernibles, of order-indiscernibles, a set homogeneous for a) iff for every n E w , every cp E Form,(L,,( p)) and every two increasing sequences xo< * * . < xny o < * * *
,,

,

akcp[Xo,...,x

n-,l

‘3

-,I

~kcp[Yo,...,Y,

*

(This is the same as saying that (%,xo,.. ., x n - ,)f(‘%,yo,...,y,- ,).) (b) If (X, < ) is a’set of indiscernibles of %, define Form,(X, <) = { cp E Form(L,,(

p ) ) I there is an increasing sequence x from X such that % k cp [ XI}.

Often we will simply write Form,(X). (c) If Z is a set of tidy formulas of type p , 9.l a structure of type p and (X, <) an ordered set, we write X(Z;(X, <);a) to mean (i) (X, <) is a set of indiscernibles of a; (ii) Form,(X) = 1; (iii) 9l = Q(X, a). REMARKS. (1) “There is” in the definition of Form,(X) can be replaced by “for all”. Form,(X) is a consistent and complete set of formulas, in the sense that for every QI E Form(L,,( p)), either cp E Form,(X) or TQIE Form,(X) but not both. (2) If i!l is a Skolem structure and (X, <) a set of indiscernibles of 8, then X (Form,(X); (X, <); @(X,a)). Our first theorem asserts the existence of structures with sets of indiscernibles of prescribed order type. For the reader’s convenience we restate here the combinatorial theorem of Ramsey proved in the Preliminaries (Theorem 0.5.6). THEOREM (Ramsey). Let X be an infinite set and n E o - { O } . Let ( C , ,. ..,C,} be a partition of [X]” in finiteh many classes. There is an infinite subset Y of X and j , 1 < j < k , such that [ Y]”c q. THEOREM 2.2.13 (Ehrenfeucht-Mostowski). Let % be any infinite structure, E be an arbitrary order type. Then there is 8 > containing a set of indiscernibles of type E.

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PROOF.We assume first that E is infinite. Let ( X , <) be an ordered set of type c such that X n 181=B . Let p = Sm(%). Consider the similarity type p’ defined by: p’= p u { ii I a E pll} u { E Ix E X } .

Let 8 be the following set of sentences (of type p’): (I) W ( %()= t h e complete diagram of 8), (11) 1(Ex?) for all x,y E X such that x # y , (111) q ( E o , . . * , X , ) - q ( j j o , . . . , Y T ) for n E o , qEForm,(L,,(p)) and properly ordered sequences xo< . . . < x,- I , yo< . * *
,

(*>

%+q,[x,

1...

,x,,l .s %+qfb,3...7Yn,l.

Let Z c A’ be the set of elements x of X such that X occurs in a sentence of 2’; Z is finite. Since Y is infinite, there is an order preserving injective map f : Z + Y . Let a , , ...,a, be all elements aEI%l such that ii occurs in a sentence of 8’. I assert that %’= (%,a,,..., a , , f ( ~ ) ) is~ a~ model ~ of 2’. Obviously all sentences of type (I) in 8’hold in a’, because all constants occurring there are among a , , . . .,a,. All sentences of type (11) in 2’ hold in 91’ because f IS injective. All sentences of type (111) in Z’ hold in 91’ as a consequence of (*).

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r

By compactness, Z has a model, say 0.I assert that '33 = (5 p is a model as required in the conclusion of the theorem. In fact, '33 > 3, because (5 E Mod( 9 3' (3)).Since the sentences of form (11) hold in '33, 1'331 contains a set isomorphic to ( X , <), namely, { F ' I x E X } ordered in the obvious way. Since (5 is a model for all the sentences of type (III), this is a set of indiscernibles of '33. T o complete the proof we observe that if ( X , <) Y, i ) and (Y, i ) is a set of indiscernibles of '33, then ( X , <) is a set of indiscernibles of '33. Hence, the existence of an infinite set of indiscernibles for 8 implies the existence of sets of indiscernibles for '33 of arbitrary finite order type. 1

c(

IMPORTANT REMARKS. ( 1 ) We can assert the existence of '33's with the properties of Theorem 2.2.13 in any cardinality K such that max{E,F, ? } < K . This follows from Theorem 2.2.13 applying the downward or upward Lowenheim-Skolem theorem-as the case may be-and observing that '33 implies ( X , < ) a set (a) ( X , < ) a set of indiscernibles of 9l and 9l i of indiscernibles of '33. 8 implies (b) ( X , <) a set of indiscernibles of 8, X 1x1 and 9l i ( X , <) a set of indiscernibles of 9l. (2) From the last sentence in the preceding proof and (b) it follows: if ( X , < ) is a set of indiscernibles of 9l (a Skolem structure) and Y X , then ( Y , < Y ) is a set of indiscernibles of @( Y,%). The easy proof of this fact (sometimes known as the subset property of indiscernibles) is left as an exercise (use also (2) of Corollary 2.2.10). As a simple application of the compactness theorem we get the following interesting fact: if ( X , <) is an infinite set of indiscernibles of 9l and (Y, i ) is any ordered set, there is a structure '33, for which ( Y , i ) is a set of indiscernibles and

c

Form,(X, <)=Form,( Y , < ),

THEOREM 2.2.14 (Stretching theorem). Let 9l be a Skolem structure and ( X , <) an infinite set of indiscernibles of 9l such that Form,(X, < ) = E . Then for evety ordered set ( Y , %(2(Y, i

>;w.

<)

there is a structure '33 such that

PROOF.Let p = Sm(9l) and p' = p u { j j Iy E Y }, where no j j belongs to p . Consider the following set T of sentences in p': T = { q(y0,...,y j j ) In E U ,

Q, E Form,(L,,(

p ) ) , Q, €1 and Y O

increasing sequence in Y }

I

*

..

Y,-

I

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The fact that X is infinite allows to map in an order-preserving way any finite subset of ( Y , < ) into ( X , <); this, plus the fact that every ~ € is 2 satisfied by all increasing sequences of X , ensures that every finite subset of T has a model. Then T has a model, say 23’.We may assume thatJ”=y for every y E Y , i.e., that 1 8 ’ 1 2 Y . Let 8”= 8’r p and 8= @( Y ,8”). Since 3 is a Skolem structure and obviously 1 8 1 is closed under functions of we get (Lemma 2.2.7) ’H i ’H”, from which it follows :?c (2;( Y , <); 8).

SO far we have been concerned with the existence of indiscernibles. The next theorem shows, in particular, that if 91 is a Skolem structure and ( X , < ) is a set of indiscernibles of 3 , the hull G ( X , YI) depends only on the order type of ( X , <) and the set of formulas Form,(X, <). THEOREM 2.2.15. Iff: ( Y , <)A( Y ’ , <‘) is a monomorphism, Z a tidy set of formulas, 3 and 91’ structures such that ( Y ’ , <’); Yl‘), YY(2; ( Y , <); 3 ) and 7<(2; then there is a unique elementary monomorphism g:Yl+iY’ extending f; moreocer, g is onto iff f is onto.

PROOF.By (c, iii) of Definition 2.2.12, 3 = *$(Y ,91) and 3’ = @( Y ‘ ,‘3’). Hence the natural extension g o f f : Y+ Y ’ is:

where 7 is a term of type u , Fv(T) = n a n d y , , . . ..yn is a sequence in Y (not necessarily increasing, cf. characterization of @ ( X , 21) given by Corollary 2.2.10). Notice that Dom(g)= 181. It remains to show that g is a function and an elementary monomorphism. Since ( Y , ( Y ’ , <‘) are sets of indiscernibles of 91 and 91’ respectively, the following holds: for any terms T ~ , . . . , T ~ - , .any sequence xo,. . .,x, - I (not necessarily increasing) of sequences of Y of the proper length and any formula 0 with n free variables: <)?

(where f(x,) denotes the sequence of images of members of x,). By incorporating the terms into the formulas and making suitable changes of variables, this becomes a special case of:

Ch. 2, 5 21

MODELS GENERATED BY INDISCERNIBLES

123

By further permutations and changes of variables we can even assume that

x is a strictly increasing sequence of Y . But now we observe that ( # # ) holds for all cp EForm(L,,( p)) and all increasing x in Y ; in fact, this follows from 'x ( 2 ;( Y , < );a) and X (2,( Y ' , < ');%'), since % Pcp[x] and x increasing implies cp E 2, and hence (since formulas of Z are satisfied in

'u' by all increasing sequences of Y ' ) %'I= rp[ f ( x ) ] .The reverse implication is completely analogous. ( # ) guarantees that g is a function, for taking 8 to be the formula o o ~ u gives: ,

-

( # ) also proves that g is an elementary monomorphism of 'u into V. Finally, it is clear that f onto g onto. The converse, which holds but

is not completely trivial will not be used here, and hence not proved either (the proof appears in MORLEY[2], p. 132). rn NOTATION. Since @( Y , % ) depends-when ( Y , <) is a set of indiscetnibles for the Skolem structure %-only on the set Z=Form,( Y , < ) and on the order type E of ( Y , <), we will denote it by Tl(2,~). If Z is a tidy set of formulas of type p, E an order type, then YJt(2,c) is the structure @( Y,'u) for some Skolem structure % of type p and set ( Y , <) of indiscernibles of % of order type E such that Z = Form,( Y , <). Theorems 2.2.13- 15 together give some interesting results. COROLLARY 2.2.16. (Automorphism theorem). Suppose %(Z; ( X , < );3). Then any order preseruing automorphism of ( X , < ) can be extended uniquely to an automorphism of 3.

C. Applications COROLLARY 2.2.17. Every theory with an infinite model has models with large automorphism groups, for example, if the language is countable, a countable model with automorphism group of power 2'0. PROOF.Let T be the theory, p its language -

(cl#

( F Q No). Extend

p to p# as

= No). By Theorem 2.2.13 get a denumerable model 23 of done before T # with a set of indiscernibles of order type q (the type of the rationals). By Corollary 2.2.16, since q has 20' order automorphisms, the group Aut(23) has power > 2'0 (actually = 2'0). Let % = 23 r p . Since clearly Aut(23) cAut(%), we are done. W

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[Ch. 2, 3 2

COROLLARY 2.2.18. Evety theory with an infinite model has a model (actual&, one in each infinite power) which can be mapped elementarily onto a proper submodel. PROOF.Argue as in the proof of Corollary 2.2.17 taking a set of indiscernibles of type w . Details are left to the reader. The next result is mainly of technical interest: COROLLARY 2.2.19. Zj X(Z;(X, <);%), ( X , < ) c ( Y , < ) and Y is such ),8 )given by that Y n 131= X , then the structure 23 such that S ( 2 ;( Y , i Theorem 2.2.14 can be chosen such that 'u i 23. PROOF.% (Z; ( X , <); Tu) implies the hypothesis of Theorem 2.2.14; since Y n J%I= X , one can get a structure 0 such that X ( 2 ;( Y , < );0) and J0ln131= X . By Corollary 2.2.10(2), @(X,E)< 0, and by Theorem 2.2.15 there is an isomorphism f : Q l + @ ( X , E ) such that frX =id,. Let B 2 J3luY , and let g : JQJ+B be a one-one onto map extending both f-' and id, (this can be done because Yn I%/= X ) . Using g to "copy" the structure of 0 in B , we get a structure 23; since g extendsf-' it follows that % i 8 and since g is an isomorphism from 0 onto 23 extending id, we get T ( Z ; ( Y , <); 23).

=s

In the rest of this section we shall make a last application of indiscernibles to construct towers of models realizing only certain sets of types of small cardinality. The construction is based on the following observation: LEMMA 2.2.20. Assume that X c Y c l'ul, X is infinite, ( Y , < ) is a set of indiscernibles o j % (so that @ ( X ,3)i @( Y , a)) and, 3 is a Skolem structure I? is a type realized in @( Y ,Tu). Then r is realized in @( X , 8 ) .

PROOF.Suppose a = ~ ~ [ y. ..,,yn] , realizes r in !Q( Y ,a), Since 'u is tidy we can choose T so that y , < . . .
2)".

. .,on)= @( T( u1,. . .,en)).

r' (in n variables) by r' = {QT I @ E r}.Now observe: @( Y , ' u ) k @ [ a ]* a( Y,'u)k @ [ 7 y Y ,,...,Yn]] * @ ( Y ? ' u ) +WIYl,...?Ynl;

Define the type

and the same holds in %, since @( Y , % )i 'u; hence

r'

Form,( Y , <).

Ch. 2, 3 21

Therefore

MODELS GENERATED BY INDISCERNIBLES

I25

I? is realized in 91 by any increasing sequence of Y . Take

x 1< . . . < xn in X ; since @ ( X , 2 l ) < 2l, the n-tuple ( x l , .. ., x n ) realizes

w,i.e.9

~ ( x > aCp"x,,*..,xnI )b * s(x,%U)+ where b = ~ ' [ x , ., . .,x,] @(X,W.

E Q ( X , 9l)

r'in

Q[~'U[XI,.*.,X~I] * ~ ( x , B ) ~+ [ b ] , and CP E.'I This means that b realizes

r in

THEOREM 2.2.21. Given a Skolem structure 2l of type p , an infinite set ( X , < ) of indiscernibles of iU and a sequence ((X,, <,,) 1 y = ( X , < >, (ii) (X,, < s ) c ( X , , < y > for S < y < A , there is a collecfion S of complete types and a sequence (ay1 y
REMARK.From (iv) it follows that

F } ,and from (vi) that

5. PROOF.Let 910= @ ( X ,3 ) and S the collection of all complete types realized by ao. Given y
6
where Z = Form,o(X,). By (viii), I Us ; % , ) ; moreover, from the proof of Corollary 2.2.19 it is clear that 3 , can be

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La,,

chosen so that in addition:

P,I n U

Xs=xy7

Y

i.e., in such a way that (viii) is also secured in the construction of 91y. Clearly (a) implies (v) and (b) implies (iii), (iv) and (vii). I t remains to check (vi). This follows easily from Lemma 2.2.20, for if T is realized in S y , (iv) and (v) ensure that is realized in any 91, with 6 < y ; by the induction hypothesis, T E S . Conversely, if T E S , then by the induction hypothesis it is realized in any %&(a< y ) , and therefore in PI,,, by (v).

r

We point out several corollaries of Theorem 2.2.21 COROLLARY 2.2.22. Given a set T of sentences (of similarity type p ) with infinite models and given a non-decreasing sequence ( K~ I y < A) of cardinals such that K~ 2 max{ p, No}, there is a set 5 of power < K~ of complete types and a sequence (Xy1 y < A ) of structures with the following properties: ( 1 ) i X 5 , = ~for y all y < A , (2) 71, i YIy for all 6 < y
PROOF.Form the type p # endowed with Skolem functions: -

p*

=max{ ,ENo}: hence K ~ >max{ p # .So}. By the upward or downward Lowenheim-Skolem theorem (according to the case) we get a Skolem structure PI’ of power K~ such that YI’ E Mod( T # ) : clearly we can choose H’ so that K o n 171’1 =,0. Applying Theorem 2.2.13 and the remarks following it, we get PI such that 71 > YI’, YI= K~ and having ( K ~ E . as a set of indiscernibles. Apply Theorem 2.2.2 1 with X , = K <, = E y ( y
r

.

-

91,

=

r ~ ~ )

r

7 =max [ p

T.Ky)=Ky

(i.e., ( I ) holds). By Theorem 2.2.2 1 (vi), %, realizes exactly the types in T. Let S be the set of types that are restrictions to p of types in T, i.e., T E S .s there is ZET such that T=ZnForm,(L,,(

p))

Ch. 2. f 21

MODELS GENERATED BY INDISCERNIBLES

I27

Evidently the following holds: if 23 is of type p # , % = ' B ~ pand bEI(P1, when b realizes C in 3,b also realizes C n Form,(L,,( p)) in Z. Hence, all types in S are realized in each 3, ( y < X ) . Conversely, if r (type in p ) is realized by a in aY.let Z be the type realized by a in By:

{

= rp E Form 1 (La,( P"

>> I 3,+ rp [ a ] } .

By construction of aY,C E T . Clearly, r = C n Form,(L,,( p)). Hence by definition of S, TES, as was to be proved. Then (3) holds. COROLLARY 2.2.23. For eoety set T of sentences of type p haoing infinite models, for evety K > max{ p, No} and for eoety cardinal a > K there are models of T of power a realizing at most K complete types. PROOF.Apply Corollary 2.2.22 with A = 2,

K ~ =K , K ,

= a.

COROLLARY 2.2.24. Every denumerable theory with infinite models has in every infinite power models realizing at most countably many complete types. PROOF.This is the particular case of Corollary 2.2.23, where

< No and

K=Ko.

Corollaries 2.2.23 and 2.2.24 can be proved in a more direct manner than as above (see Exercise 2.2.1). Some infinitary properties of (La,-) indiscernibles are studied at the end of Chapter 5, $4. EXERCISES

2.2.1. ( 1 ) Corollary 2.2.24 can be improved to the following: every denumerable theory with infinite models has models 91 in every infinite power with the property that for every countable X Cli)Il, 91 realizes at most countably many complete types over Th((%,a)oEx). [Hint: Skolemize the given similarity type p into p#. Let K be an infinite cardinal and 91 = ' ~ ( C , K )where , C is an appropriate set of sentences. Since X ~131 is countable, there is Y L K such that 7< No and X c4(Y , % ) . How are complete types realized in (3, x ) , ~determined? ~ The result follows easily from the answer to this question.] (2) Let K be an infinite cardinal and T a K-categorical theory in a countable similarity type. Show that for every model 91 of T and every countable X C I%/ there are only countably many complete types realized in any model of Th((i)I,x)xEx).

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3

2.2.2. (1) Prove that for every infinite cardinal K there is a linear ordering of power K with 2“ automorphisms. [Hint: there is a very simple proof using the fact that 71 has 2’0 automorphisms.] (2) Extend Corollary 2.2.17 to get a model with 2“ automorphisms in each infinite power K. (3) Show that the field of complex numbers has 2’”’’ automorphisms. [Hint: use (2) and Steinitz’s theorem (two algebraically closed fields of the same characteristic and the same transcendence degree over the prime field are isomorphic).] (4) The (up to isomorphism) unique model of power K of a K-categorical theory has 2“ automorphisms.

$3. Some results from the model theory of L,,, This section is devoted to give a succinct review of some important results holding in L+,, which will be used or referred to elsewhere in this book. We shall only present the relevant definitions and statements of results; the reader is referred to Keisler’s book [3] for the corresponding proofs and, more generally, for a detailed treatment of the model theory of LUI,and its fragments. At present, LUI, is the best known of all the infinitary languages LKA. Although the compactness and upward Lowenheim-Skolem theorems fail in LUI,-a fact which makes its model theory essentially different from that of L,,-there is a tool that in many cases can successfully be used as an alternative to these basic theorems; namely, the concept of consistency property and the model existence theorem associated with it. These ideas were introduced by MAKKAI [3]: they evolved from earlier work of HENKIN [ I ] and SMULLYAN [I], [2]; they combine the essence of Henkin’s proof of Godel’s completeness theorem with methods akin to Gentzen’s “sequent calculus.” The definition of consistency property and the proof of the model existence theorem can be found in lecture 3, part I of KEISLER [3]. The simplest result proved by this method is a weak adequacy (completeness) theorem for L,,,: (1) It is possible to construct a simple axiom system for LUI, involving one inference rule with denumerably many premises such that its theorems are exactly the logically valid formulas of these languages: moreover: (2) the deduction theorem with one premise (or, equivalently, denumerably many of them) holds for such system; hence:

Ch.2.

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129

(3) the system is adequate for derivations from one premise: iff for arbitrary cp,+ E Form(L,J. However, it is easy to construct non-denumerable (syntactically) consistent-and even (syntactically) complete-sets of sentences without models (cf. Exercise 2.3.1); therefore: (4) unless uncountably long derivations are admitted, no axiom system for LUI, can be built which is adequate for derivations from arbitrary sets of premises. This and other related axiomatic systems for infinitary logic are discussed and described with more detail in Appendix c. A. Interpolation theorems

One central result of first-order logic which does extend to LUI, is Craig’s interpolation theorem (cf. KEISLER[3], part I, lecture 5): THEOREM 2.3.1. Let cp,$ be sentences of La], such that t=cp+; then there is a sentence 8 of LUI, such that: (1) every relation, function or individual constant symbol occuring in 8, occurs both in cp and ( 2 ) l=cp+0 and t=8+$.

+;

The sentences cp,+ are usually called the interpolants; 0 is called the interpolating sentence. The preceding result was proved by L6pez-Escobar. The interpolation theorem admits several refinements; the most important are the following: THEOREM 2.3.2 (Lyndon for L,,; Lopez-Escobar for Lola). Let cp,$ be Lola(y)-sentences such that k=cp++. Then there is a sentence 8 of La,,( y) such that (1) evety non-logical relation symbol occurring positively (negatively) in 0 occurs positively (negatively) in both cp and $; (2) kq+8 and k0+; (3) if y does not have function nor individual constant symbols and if the equality sign occurs in 0, then either k irp, t-+, or it occurs in at least one of cp7

4.

A full proof of this theorem can be found in KEISLER[3], part I, lecture 6 ; the next is proved in lecture 7 of the same book.

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[Ch. 2,

3

THEOREM 2.3.3. Let p be a similarity type without function symbols and let cp,+ESent(LWI,(p))be such that k q + 4 . (A) (Malitz). If 1c/ is universal, then there is a universal sentence 0 of Lwlw( p ) such that: (i) every symbol of p occurring in 0 occurs in both q and 4; (ii) kcp+0 and l=0+. (B) (Malitz-Barwise). If 9, is universal and in addition p does not haoe indioidual constant symbols, then there is a unicersal sentence 0 of LWI, satisfying ( i ) and (ii). The last theorem can be dualized in an obvious way so as to apply to existential sentences as well; one needs only to observe that if q is existential then i q is logically equivalent to a universal sentence and conversely. We shall see in 3 4 of the present chapter that L,, and LUI, are the only languages L,, enjoying the interpolation property of Theorem 2.3.1. Owing to these negative results (which also occur in many other extensions of first-order logic) considerable interest has been attached to the study of “relative” interpolation properties obtained by allowing the -interpolating sentence 0 to belong to a different (usually larger) language than that of the interpolants q,$. In view of those generalizations we introduce the following: CONVENTION. We shall understand by language an entity L consisting of the following two ingredients: a class of objects (“formulas” of L) and a distinguished subclass of the former (“valid formulas” of L). Below we consider pairs of languages L, L’ subjected to the following additional restrictions: (a) All “formulas” of L,L’ are formulas of second order Lmm. (b) The classes of formulas of L, L’ are: (i) closed under the connectives and quantifiers of first order logic; (ii) closed under substitution of non-logical symbols by arbitrary formulas. (c) The classes of valid formulas of L, L’ contain all substitution cases of theorems of first-order logic and are closed under modus ponens and (first-order) generalization. (d) All primitive symbols (logical and non-logical) of L are among those of L‘. (e) All formulas of L are formulas of L’. Some comments are in order here.

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131

REMARKS. (1) “Second order L,,” is the language constructed as L,, (cf. p. 65) but allowing in addition second order variables and quantifications over sequences (of arbitrary length) of these variables. Strictly speaking, condition (a) is not necessary to prove the results below, but it makes clear such notions as “free occurrence of a variable”, “sentence”, “substitution”, “first order connective”, etc., for formulas of L, L’. (2) The “valid formulas” of L,L’ need not be valid formulas of L,, in the usual semantics of this language (though they frequently turn out to be so, for example, in all examples below). (3) Concerning (e). Not all valid formulas of L need be valid formulas of L’. If u is a valid formula of L, we shall write F L u ; u p L rp means P L o - q ~ : this symbolism will always be used in connection with sentences of L. The reason for this notation is that “validity” is frequently defined in terms of some sort of satisfaction relation. (4) We shall restrict ourselves to consider only finitaty relations and function symbols as primitive non-logical symbols. As usual. Sm(a) denotes the set of non-logical symbols occurring in u. (5) The meaning of (b), (c) should be clear from the preceding remarks; for example, (b,ii) means that if rp,u are L-formulas, rp contains R(x l,...,xn) as a subformula (where R is a relation symbol) and Fv(u) = Fv(R(x,, ...,xn)), then any expression obtained from cp by substitution of (possibly only some) occurrences of R(x l,...,xn) in rp by u, is also a formula of L. (6) The restriction (c) can be weakened slightly. (7) Examples. The following are typical examples of pairs of languages that satisfy the preceding convention, to which the definitions and arguments below will apply ( p, p’ are fixed similarity types, p p’): L = LKA( p),

L = La,( p).

h < A’: L’ = any system of second-order (or weak secondorder) logic with p as set of non-logical symbols.

L’ = LKSA,( p’), where

K

Q

K‘,

DEFINITION 2.3.4. Given languages L, L satisfying the preceding requirements, we say that L has the interpolation property relative to L’, in symbols Interp(L, L’), iff for every pair uwul of sentences in L such that F L uo-ful. there is a sentence u in L’ such that: (1) kL‘u0+u and FL,u-+u,: (2) Sm(u) c S m ( u o ) n Sm(u,). Thus, Craig’s theorem is Interp(L,,, L,,) a n d L6pez-Escobar’s Interp(L,,,, La,J The first example of the relative interpolation property

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[Ch. 2, 0 3

occurs in MALITZ[l], [3]: if K is regular, then Interp(L,,,L(?<.,+,,). H. FRIEDMAN [ l ] has recently proved another result of this kind: if cf(K)=w, then Interp(L,+,, L(z
B. Some important consequences of the interpolation theorems

I. Definability theorems. The most important corollary of the interpolation property is Beth’s definability theorem. The original result of Beth dealt with L,, and was proved before the discovery of the interpolation property (cf. BETH[l]); it is an adequacy (completeness) theorem for the notion of definability; it originates in Padoa’s method for proofs of undefinability which dates back to 1900. A short account of the significance of Beth’s result can be found in DICKMANN [l], Vol. 1, pp. 163-165. DEFINITION 2.3.5. Let L, L’ be languages satisfying the requirements set forth in the preceding convention. (a) A relation symbol R of L is explicitly definable in L‘ relative to the L-sentence u iff there is a formula g, of L’ such that: -

=

(1) Fv(g,) = Fv( R ) (= n , say), (ii) Sm(g,) C Sm(o) - { R }, (iii) u F V x , ...x,[ R ( x , ,...,x,)+xp(x,,...,x,)]. (b) We say that an m-ary relation symbol R is implicitly defined by the sentence u iff u A u( $) t= V x , . ..x, [ R ( x , , . . .,x,)*

T ( x , ,. ..,x,)],

where T is any m-ary relation symbol such that TESm(u) and u ( + ) denotes the sentence obtained replacing all occurrences of R in u by T. (c) We say that the language L has the definability property relatiue to L‘, in symbols, Def(L,L‘), iff whenever u is a sentence of L, R is a (finitary) relation symbol of L and u implicitly defines R , then R is explicitly definable in L’ relative to u. The meaning of explicit definability is clear. R is implicitly defined by a sentence u means, roughly speaking, that the relation R is uniquely determined in each model Yl of u (for more details cf. Exercise 2.3.3 at the end of this section). Beth proved Def(L,,, L,J; later Craig realized that the implication Interp(L, L) * Def(L, L) applies to almost any extension of first-order logic. We prove an even more general result which appears explicitly in MALITZ[I]:

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133

THEOREM 2.3.6 Let L be a language with the restriction ( a ) above. Assume that the n-ary relation R of L is implicitly defined by u, a sentence of L. Then there is a valid sentence of L of the form uo+ul such that for all languages L’ with the restrictions (a)-(c) above, the following are equivalent: ( x ) The pair uo,ul has an interpolating sentence in L’, ( x x )R is explicitly definable in L’ relative to u. In particular, Interp( L, L’) Def(L, L‘).

-

PROOF.By assumption R is implicitly defined by a; therefore:

where T is any new n-ary relation symbol. Adding fresh individual constants c,,. ..,c, to L and applying simple tautological transformations we get: t = ~ ( c ~ , , , , , Rc ~ ) ( u-.~,c,,))+[(J(;)+ ( ~ 1 3

T(c1,.. .,c,)].

Let uo be the left hand side and u1 the right hand side of this implication. (x) + (xx) Let O=O(c, ,..., c,,-a formula of L ” = L ( c , ,..., c+be an interpolating sentence for uo, ul. Hence: i.e.,

s m ( e ) c S m ( a u { R ( c l ,. ..,c,)}) n Sm(u($) u { T ( c , ,...,c,,)}), Sm(O)c(sm(a)- { R } ) u{ c I,...,cn}.

we get t= L,, u+ ( R ( c , ,. ..,c,) + O ) . From t=L,,O+ul, replacFrom l=L~~uo-sB ing T by R,’ we getI=LrrO+(u+R(c,,...,c,)). By simple tautological transformations we get: u l=Lt,O(cl,. . .,c,,)-R(c,, .. .,cn). Since c I ,. ..,cn do not occur in u we can quantify over the corresponding variables, thus obtaining:

l=,,vx,...X , , [ R ( ~ , , . .

.,xn)++e(x,,...,X J .

Therefore O explicitly defines R in L’ relative to u. (xx) + (x). It is clear that any explicit definition of R in L’ relative to u is an interpolating sentence in L‘ for the pair uo,ul.

COROLLARY 2.3.7. (1) (Lopez-Escobar) Def(L,,,, La,,). (2) (Malitz) If K is regular, then Def(L,,, L(2<x)+,r). ( 3 ) (Friedman and Chang) If cf(K)= w , then Def(L,+,, L(2
’ Why is this substitution legitimate?

I34

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[Ch. 2, !j 3

In $4 below we shall give examples showing results of the form Def(L,,, Lab) for various ~,X,a,,l?.

11. Preservation theorems. The special versions 2.3.2 and 2.3.3 of the interpolation theorem yield - 3s corollaries an important class of results known under the common name of “preservation theorems”. These results relate certain algebraic properties of the models of sentences of LUI, with the synactic form of the sentences themselves. Very early results in the model theory of L,, include theorems of this kind (e.g., by Birkhoff, Tarski, etc.). In most cases the methods of proof for LUI, differ considerably from those of first-order logic. Many results known in L,, do not extend even to L,,,. The best known preservation theorems in L,,, are the following:

THEOREM 2.3.8 (Malitz). Let QL u ESent(L,,,). The following conditions are equivalent : ( I ) for every pair %,B of structures, i f 8,’HEMod(u), B C % and % E Mod(cp), then ‘f3 E Mod(cp); (2) there is a universal sentence 6’ of LaI, such that FU+++e).

Using the dual form of Theorem 2.3.3 (A), we obtain a preservation theorem dual to the preceding one, where ‘‘2C 23” is replaced by “93 C 8” and “universal” by “existential”. Moreover Theorem 2.3.8 implies: COROLLARY 2.3.9. Let KEEC(L,,,); stronger result is valid in L,,:

then K=S(K) iff KEUC(L,,,).

A

but this theorem does not extend to LWI,(cf. $4 below).

THEOREM 2.3.13 (Lyndon, Lopez-Escobar). Let cp, u ESent(L,,,) be such has a model. Then the following statements are equicalent: that u ~ c p (1) for every pair 8,93 of structures, if %,aE Mod(u), % E Mod(cp) and B is a homomorphic image of 3 , then gEMod(cp); (2) there is a positive sentence 8 of L,,, such that FU+++e).

The proofs of these theorems are given in KEISLER[3], part I, lectures 6 and 7, where the reader will also find a number of corollaries of the theorems stated above, and some related results.

Ch. 2. 8 31

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I35

C. Scott’s isomorphism theorem Another central result in the model theory of LUI,asserts that the countable isomorphism type of any countable structure with countable similarity type is characterized by a single sentence of Lala; more formally:

THEOREM 2.3.1 1. Let ’11 be a countable structure of countable similarity type p. There is a sentence rp, of Lu,,( p) such that for each countable structure 23 type c1:

0s

23kcp,

w %=%.

rpgl is called the (more properly, a ) Scott sentence for ’21.

The situation described by Scott’s theorem resembles that of finite structures 2l of finite similarity type in Lu,. There is one important difference, though: in the finite case there is a sentence qn of L,, such that for euely 23 similar to !B,B+rpn iff BE%.In fact, rp, can-express, among other things, the cardinality of ‘21; however, the fact that Z = H N , cannot be expressed by means of LaI, sentences (cf. Exercise 2.3.5). In further analogy to the finite case we have:

THEOREM 2.3.12 (Chang). If ‘21,B are countable structures of arbitrary similarity type (not necessarily countable!) and’%-ulw23, then % E23. Both Theorems 2.3.11 and 2.3.12 are proved by the method of iterated extension of finite partial isomorphisms. In Chapter 5 , §§3,4 we shall study extensively the method of extension of partial isomorphisms obtaining, in particular, proofs for the abovementioned theorems; we shall be concerned, as well, with various generalizations of the preceding theorems to larger infinitary languages. Scott, Ryll-Nardzewski and more recently Vaught and others applied the isomorphism and interpolation theorems for LUI, to the theory of Bore1 sets. These applications have, in fact, given a new, illuminating focus to some aspects of that theory leading, in particular, to the solution of some of its open problems. We have omitted in this book the study of those applications; the interested reader is referred to DICKMANN [l], vol. I, pp. 210-222, SCOTT[2], [3], VAUGHT[2], and MAKKAI[5]. EXERCISES 2.3.1. (a) Construct a set Z of LUl,-sentences such that every denumerable subset of Z has a model, but Z has no model.

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[Ch. 2, 5 3

(b) Prove that every set with the properties stated in (a) is syntactically consistent (i.e., no contradiction can be derived from it by means of the derivation rules mentioned at the beginning of this section and described in Appendix C (p. 420); see also KEISLER [3], part I, lecture 3). (c) Does there exist a set,with the properties of (a) involving only finitely many symbols? What if denumerably many symbols are allowed? Consider the following set A of sentences in the language (0, +, < ,P } for (abelian) ordered groups plus an additional singulary predicate P ; A consists of the axioms for ordered abelian groups, the sentence of Example 1.2.15 of 2, Chapter I , making the group Archimedian, and the following sentences :

P ( 0 )A v X ( P ( X ) + o 3xcp:(x)

< X)

for each a
Notice that A has the properties stated in (a). (d) Prove that A has no (syntactically) consistent and complete extension. [Hint: prove that any such extension of A defines the 'addition table' of a model for A.] NOTE. Ryll-Nardzewski has constructed a (syntactically) consistent and complete set of La,,-sentences without models (unpublished; cf. SCOTT[3], pp. 3 3 3 4 ) .

2.3.2. Show that the following three statements are equivalent (cf. HANF[2], p. 3 1 1 1: K , X are arbitrary infinite cardinals such that K 2 A): (I) For every r c Sent(L,,) the following holds: if every subset of r of power < K has a model, then r has a model. (11) For every r u {rp} CSent(L,,) the following holds: if rk-cp,.then there is Z cr, E < K such that Zkcp. (111) For every r u {cp} GSent(L,,) the following holds: if r=rp, then there .is Z c r, E < K such that Z = rp. % E Mod(rp)). (Here r= cp means: for every structure 3, 2 E Mod(T) 2.3.3. (a) Prove that for any language satisfying the requirements set forth in p. 130 and any sentence u of L the following'are equivalent: (1) The (n-ary) relation R is implicitly defined by u. (2) For every structure % of type Sm(u) - { R } and every pair P, Q of subsets of I%/'', if (%,P),(%,Q)€Mod(u),

then P= Q .

Ch. 2. $ 31

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I37

(3) For every pair %,B of structures of type Sm(a)-{R}, every 3 P cl%ln,Q1231’’ and every map f :l9lIj.1231, if 8 ~ ~ 2and (%,P),(B,Q)rMod(a), then (%,P)=’(B,Q). (b) Let (a,+, -,O, 1) be the standard model of arithmetic. (1) Show that “.’’ is not defined (implicitly nor explicitly) by the conjunction of all the sentences in Th((w, + ,0,1)). _O, 1) and two (2) Show that there is a structure of the form ( A , distinct operations *,, *2 on A such that

+,

( A , k, * i , _ o , l ) = ( w ,

+, .,o, 1) (i=1,2)

but (A,+,*1,_o,1)~.(A,+,*2,_o,l).

+

(3) Is it possible to choose A = w and _f = in (2)? [Hint: use Presburger’s result that Th((a, +,O, 1)) is decidable.] (c) Let a ( R ) be an implicit second-order definition of the n-ary relation R. Show that 3 9 ( a ( :)A 9 ( x , , ...,x,)) is an explicit definition of R. [9is a (second-order) variable for n-ary relations.] 2.3.4. Let QI E Sent(L,,,( p)), F < No. The following statements are equivalent: (1) { QI}is complete and consistent; (2) rp is No-categorical (i.e., rp has a model and any two denumerable models of rp are isomorphic); (3) QI is a Scott sentence for some denumerable structure of type p. [Hint: use the downward Lowenheim-Skolem theorem for La], (cf. Corollary 3.4.5) and Chang’s Theorem 2.3.12.1 2.3.5 (MAKKAI [4]). The notion of denumerable structure is not EC,(LUI,); Exercise 1.3.4(g) provides a counterexample. Incidentally, note that K E PC(L,I,) implies Kno= { % I % E K and < No} E PC(LUI,). However, the argument below, due to Makkai, shows that for each p there are wide classes of La,,( p)-sentences having only countable models. DEFINITION (a) a will denote the partial order of “sequence extension” in

w<“, the set of all finite sequences of natural numbers:

s a t w Dom(s)rDom(t)

and trDom(s)=s.

E(s) denotes the length of the sequence s. (b) A set S is a well-founded tree (wft) if and only if:

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(i) s E U < W A \ t E S A S Q t* SES; (ii) if sn E S for some n EU, then sm E S for all m EU; (iii) D n S 2 is a well-founded partial order on S . (c) s E S is an endpoint iff for n o n E u , sn E S . (d) A set @ c Form(L,,,) is a well-foundedformula tree (wfft) iff there is a wft S C w < " and an onto map f : S+@ such that (i) f ( R ) = u o x u o (A denotes the empty sequence); (ii) for every s E S , Fv( f ( s ) )= { uo,...,v ! ( ~ ) } . For s E S , let rpS =f(s). (1) Prove the following "induction principle" for wft's: if P is a property of members of and, (i) every endpoint of S has the property P, (ii) if for all n E U , sn has P , then s has P , then every s E S has the property P . [Hint: use the induction principle for well-founded relations of Chapter 0, 0 1.1 Clearly there is a corresponding principle of definition by induction on wft's. If CP is a wfft, by induction on ) : ' ) as follows: the corresponding wft S we define the formulas 4,= I (i) if s is an endpoint of S , $S = 3!u0rpS; (ii) if s is not an endpoint of S ,

-

Clearly Fv(+bS)= l?(s). Let )'()I be the sentence $,. (2) Prove that if @ is a wfft and 8 a structure of the proper type, 8k+(') implies < No. [Hint: use the induction principle proved in (l).]

REMARK. It is instructive to compare the preceding construction with that of a Scott sentence rp, for countable structures rzI (cf. KEISLER[3], part I, lecture 2).

0 4. Counterexamples This section is devoted to give examples showing that many theorems mentioned or proved in previous sections which hold in L,,, or in L,, do not generalize to other infinitary languages. The interpolation and definability theorems for L,,, fall in this category (Counterexamples 2.4.4 and 2.4.5, 2.4.8, respectively), as well as the preservation theorem stated as Corollary 2.3.9 (Counterexample 2.4.1 l), and a stronger theorem related to it which only holds in finitary logic (Counterexample 2.4.10).

Ch. 2, 8 41

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We go on to show that the prenex normal form theorem of L,, fails in most infinitary languages, among others, in all the languages L,,, as well as in many of the form L,A, w < A < K (Counterexample 2.4.12). After studying these elementary examples the reader is encouraged to turn to Appendix B, which contains a more involved example showing that the prenex normal form theorem does not extend either to L,,,, or to L,, for uncountable regular K . We close the section with some examples of WEINSTEIN [ I ] and results of [I], [2], related to the problem of extending the Chang-EoS-Suszko NEBRES theorem of L,, to infinitary languages. In connection with these counterexamples we mention some of the problems which still remain open. We will need the following important: LEMMA 2.4.1. Let a ESent(L,,(0)), i.e., a involves equality but no non-logical symbols. Then u is true in all structures of power > A or in no structure of power > A .

PROOF. Let cp be any formula of L,,(D)),A, B be sets such that A C B and A > A, and let f EA Fv('+'); then:

Obviously the assertion of the lemma follows from (*). This is proved by induction on the structure of cp; the only non-trivial case of this induction is when c p = ( 3 X ) + , where X GVar, ?
-

> A, there is a set K L A such that Range( g) u Range( f)
z,

A

(Cy

'3

k)cERange(B;kE M

and

(A

3

h(k))cERange(B;kEM .

Therefore it follows that (A)k+[f*(h*og)]; since L o g € A X , we have (A)kcp[f ] as was to be proved. rn Lemma 2.4.1 has an interesting application, namely, a characterization of EC, classes of typeD in LKA.

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DEFINITION 2.4.2. Let K be a class of similar structures. The spectrum of K is defined as follows:

With this notation, Lemma 2.4.1 can be expressed as follows:

2.4.1a.Let u E Sent(L,,(0)); then: LEMMA CN - Sp(Mod( u)) C h or

Sp(Mod( u)) C A.

(Recall that CN denotes the class of all infinite cardinals.) THEOREM 2.4.3. Let K be a class of structures of type 0 . Then the following are equiualent : (1) K = Mod@) for some X Sent(L,,(0)). (2) K is closed under isomorphism and either (a) CN - Sp(K) C A, or (b) h is singular and Sp(K) G A, or (c) A is regular, Sp(K)GA and Sp(K) < K .

-

PROOF.(2) * (i). For each cardinal p
-

(*I

%hpp e % = p .

For instance take

We remark

(**I

IUEK,-,Z=Z + BEK,

(***I

Z€Sp(K) w IUEK.

B are !I of ,type 0 ; since K is From = it follows that B = % because % closed under isomorphism, BEK. To prove (***): if 8 F K , by (**) there is no BEK such that z=E; hence EeSp(K).

Z

Ch. 2,

8 41

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COUNTEREXAMPLES

There are three cases to be considered: (a) CN - Sp(K) c A. We claim that K = Mod(Z),

where

Z = { -ITp 1 I-1 < A A P ESP(K)}.

If % E K, then E E Sp(K) which implies E# p for all p g Sp(K); by (*) it follows that % t- irp,, for all p
(AllS
and

v

P
p€Sp(K)nA,

qp)).

Proof of claim. Let % E K . Then by (***) and (b), z < A , and since A is singular, %= p
then 4 t- qp for some [< p and some p E Sp(K) nA,, which by (*) implies that E=p€Sp(K); hence, by (***), %EK. Observe that the formula of the (cj In this case K = Mod({ V,,Espw~qp}). right hand side is in L,, because Sp(K) < K and Sp(K)cA. The remaining details are left to the reader. (2). Suppose that K = Mod@). Clearly, (1 j

-

Conversely, if p E n,,,Sp(Mod(u)),

then for each U E Z there exists a set

A, of cardinality p such that (A,)Pu. By Lemma 2.4.1 we conclude that (p> F u, whence ( p ) E Mod@), and this implies that p E Sp(Mod(Z)).

By Lemma 2.4.1awe get

CN-Sp(K)ZA

* Sp(K)CA.

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[Ch. 2, 8 4

Therefore, if A is singular the alternative (b) holds trivially, showing that it suffices to prove (c); this, in turn, follows from:

A regulai A Sp(K) c A

*

Sp(K) <

K.

-

If A = p + , from Sp(K)cA it follows that Sp(K)Gpu { p } , because all members of the spectrum are cardinals; therefore Sp(K) Q p < A Q K . Let A be regular and limit, i.e., weakly inaccessible. If K > A there is nothing to prove; hence assume K = A . In this case it is easy to prove by induction on formulas:

+. there is p < A such that 9, E Form(L,).

rp E Form(L,,)

-

For each rpEForm(L,,) let pq denote the smallest p such that rpEForm (LJ; then pq A = K , then each u € 2 holds in some structure of power > pa; whence, by Lemma 2.4.1, it holds in all structures of power > pa; in particular,

(a 1 @ > A]

hence

therefore, Sp(K) 2 CN Sp(K) H

-

cMod(u),

A which is a contradiction. We have proved that


COUNTEREXAMPLE 2.4.4. The interpolation theorem fails in all languages L,, with K > w , ; moreover if K , V are infinite cardinals such that K > v + , then

PROOF. Let the language p have one individual constant ca for each ordinal

a < v + ;let

u, = v u

v

a
(UXC,),

Ch. 2, 5 41

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The sentence u,+u2 is obgously valid. Indeed, if % is a structure of type p such that 8 k u,, we have % < v, and letting xa={

PI v < p < u

+

A c; = c," } , -

xa

we also have UaEuXu = [v, v+), whence there is (Y E u such that > u ; this clearly implies that % t- u2. If u is an interpolating sentence of L,, for the pair u,,u2, then u should obviously be of type 0 . From t=ul+u it follows that u holds in some structure of cardinality v. By Lemma 2.4.1, u holds in all structures of power 2 v. Since t-u+u2, then u2 has the same property; but this is a contradiction because u2 fails in some structure of power v f (e.g., it fails in ([vP+)?~),<,<,+). rn The next counterexample is also due to Malitz. It depends on two theorems proved in later chapters. For the reader's convenience we repeat them here: (I) (Corollary 5.2.6 and Theorem 5.2.2) For every K E C N and uESent(L,,) there is a v 2 K such that for all structures %;,B;(i=1,2) with 1811 n 1911,J= lBl1n 1B21=0,the following holds: % i - y y 8 i ( i = 1,2)

* (%l@%2ku w

B1638,t-u)

(11) (Corollary 3.4.2; the downward Lowenheim-Skolem theorem) Let % b e 2 structure of type p ; let K , A be cardinals suckthat max{ P , K } < A = A " < g . Then there is a structure 8 i% such that B=h. ( is enough for Counterexample 2.4.5.) KK

=KK

COUNTEREXAMPLE 2.4.5. Beth's definability theorem fails in all languages L,, such that K 2 A 2 wl; moreover there is a valid sentence uI+u2 of L,,,ul that implicitly defines a relation which is not explicitly definable in any language L,A: 1Def(LUIuI, LrA)

for all K 2 A 2 w ,

or, alternatively,

By Theorem 2.3.6, Craig's interpolation theorem also fails in all these languages.

PROOF.Let K be the class of all structures of the form (%@B,j),where:

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[Ch. 2, 9 4

(1) Sm(a) = Sm(B) = { U , < }( U a unary, < a binary relation symbol), (2) I%/ n 181=a, (3) P = p q , P = B , (4) <’ well-orders 1x1,
r

K = Mod(u). It is a well-known theorem of set theory that there is at most one isomorphism between two well-ordered sets. By the result of Exercise 2.3.3(a), $3, it follows that u implicitly defines F. Suppose that the formula q~ defines F explicitly in some language LKA. Let (I* be u( g) (= the substitution of F(x,y) by cp(x,y) in every occurrence of F in a). Obviously (I* ESent(L,,,,( U , <)), where K ’ = max{ K , ~ , w , } and: ,

(*I

Yf@B€Mod(u*)

H

(3), (4) hold and (131, <’), (1231, <‘)have the same order type.

Let I Z K‘ be as in (I) above. Now define: %,=B,=(22’,22’,

E),

a2be a disjoint isomorphic copy of (22v,@,E). From (*) it obviously follows that

Taking into account that 2’=(2”)’, by (11) there -is B2
-

B,@%2)1u*,

and from (*) it follows that (22’, E ) is isomorphic to (2”, E), a contradiction. This proves that F is not explicitly definable in LKA.

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145

Counterexample 2.4.5 leaves open the question whether Beth's definability theorem fails in the finite-quantifier languages L,, with K > w2, and Lm,. These questions were partially answered by FRIEDMAN [ I ] using methods from metarecursion theory, and later completely solved by GREGORY [ 11 using model-theoretical methods. Gregory's construction shows that definability fails for these and other related languages. Counterexample 2.4.8 and Theorem 2.4.9 below give a complete picture of the situation after Gregory's, results. We shall need several theorems proved in later sections as well as some preliminary work. These are the facts needed: (111) (Counterexample 5.3.44) For every uncountable regular cardinal K there are rigid structures 8,6 of power K and similarity type of power < K containing only relation symbols, such that 23 0 0 but 8 =m K Q . (Recall that a structure is rigid iff its only automorphism is the identity.) (IV) (Theorem 5.2.7) For any infinite cardinal K , L,,-equivalence is preserved under extended full cardinal sums. (For the definition of extended full cardinal sum, see pp. 71 and 301.) (V) (Corollary 5.3.33) For every structure % of similarity type p there is a ( such that for any structure % of type p, sentence cpx in L m Kp)

Let L be a language of the form L,, or L,A, and let 2l be a structure of similarity type p. Recall that an element aEl%l is L-definable iff there is a formula E Form,(L( p ) ) such that 2l t= +[a]and % F 3!t' +(t').

+

LEMMA2.4.6. Let K be an infinite cardinal. A n element a € [ % [ is LmKundefinable iff there is bEliU1, b # a , such that (iU,U)EmK(%,b).

PROOF. Let E Sent(L,,(

+=

cp(a,a> be the sentence given by ( V ) ; p u { c})), where c is a new individual constant.

+= +(c)

Assume that a is LmK-undefinable. Since ( % , a ) P +(c), we get is obtained by replacing u for c in +. Since +' cannot be a definition of a, there is b E 131, b# a, such that 2l k +'[b],i.e., (%,b)F +; hence (%,b)=mK(8,U). (e) If (%,b)EmK(%,a), then for any formula cp€Form,(L,,( p)), (*)

%I= +'[a],where +'=+(u)

Since b # a, a is L m Kp)-undefinable. (

rn

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[Ch. 2, 8 4

LEMMA2.4.7. Let B,6 be rigid structures of infinite cardinality K and and similarity v p e p,,E < K , containing only relation symbols,such that 2 3 ~ 6 B=,,,B. Then B has an .LmJ p)-undefinable element. PROOF.Otherwise, for every b E 1231 there would be a formula such that for all b’EIB1:

@b EForml(LmK(

p))

B k ~ ~ [ be’ ]b = b ’ .

Let 8 be the conjunction of the following sentences: A R ( u , ,. . . ,u,)] for any n-ary relation symbol (i) 3 u , . . .u,,{Al=l@bi(uj) R E p such that B k = R [ b l..., , b,,]; ~ ) r \ for any n-ary relation symbol (ii) 3 t . , . . . ~ ~ [ A ~ ~ , @ i~R~((uu,,...,u,,)] R E p such that Bk=i R [ b , ,..., bn]; (iii) vuVbq81@b(u).

It is clear that 8 E Sent(L,,( p)), and the reader can easily verify that if the map of B into B given by b+ “the unique element of @?”is an isomorphism. On the other hand %==,,6 implies that El-8, and . is a contradiction. Hence B has LmK( p)-undefinable therefore 2 3 ~ 6This elements. w ZE) k 8, then

COUNTEREXAMPLE 2.4.8. If

K

is an uncountable regular cardinal, then i Def(L,+,,

LmK).

PROOF.Using (V) and Lemma 2.4.7, choose a rigid structure a, of power K having at least one L,,-undefinable element a,. Moreover, Z, can be < K . Let chosen so that Sm(g0)= po contains only relation symbols and p, be a disjoint copy of p,, and

where Po, P, are two new distinct unary predicates. p’ is the similarity type appropriate for extended full cardinal sums of structures of types p o u { a 1 a E la,l} and p,, respectively. If So is a symbol of p,, S , will denote the corresponding symbol in p,. Finally, consider two distinct binary relation symbols F,G, not in p’. Let + ( F ) be ,the conjunction of the following L,+,-sentences of type p‘ u { F } : (1) v u i (P,(u) A pl(u)>A v u (P,(u) v pdu)); (2) v u (p,(u)-v, E I P , p 3));

Ch.2, 41

147

COUNTEREXAMPLES *

-

( Z w a ' ) for a , a ' ~ I % , I , a # a ' ; (3) i (4) R o ( q ,..., a,,) whenever R,Epo is an n-ary relation symbol, a l , . . .,a,, E I%,l and ZOt-R,[a,, . . .,a,,]; ( 5 ) i R , ( T , . . ., a,) whenever R o E p o is an n-ary relation symbol, a l ,...,a,,EI8,J and %,+ i R o [ a ,..., , a,]; (6) v u w ( F ( 0 ,W)+(Po(u) A Pl(W))); (7) v u (P,(u)+(3! w ) F(u, w ) ) ; (8) V w ( P l ( w ) + ( 3 ! c )F(v,w)); ( 9 ) V u , ...U , W , ... W , , [ A \ = ~ F ( UW~ ~, ) + ( R , ( ~ , ,. . , ~ , , ) t ) R ~ ( w ..,~w,,))] ,. for all n E w and all n-ary relation symbols R , in p,. Let +(G) denote the sentence obtained by replacing G instead of F in + ( F ) . We have: (a)

+ ( F ) A +(G) t= v t . w ( F ( u , w ) H G ( u , w ) ) .

PROOFOF (a). Let E E M ~ ~ ( + ( F ) A + ( G )By ) . conjuncts (2), (3), (4) and ( 5 ) it is clear that the map a+Z" of %, into (ErPf)rpois an isomorphism. Conjuncts (6)-(9) tell us that f= F" and g = G" are isomorphisms of ( E r P f ) r p l onto (ErPF)rpl. Therefore, g-'of is an automorphism of ((5 r P f ) r p o . Since this structure is isomorphic to %, it is rigid. Therefore g-'of=id6,pg, whence F " = f = g = G " , i.e., 0 I= v u w ( F ( u , W ) H C ( O , w ) ) . (a) asserts that F is implicitly defined by + ( F ) . We now show: p') relative to + ( F ) ; that is, there (b) F is not explicitly definable in LmK( is no formula B E Form,(L,,( p')) such that

+ ( F ) I= v u w( F ( u , w)*O(u, w ) ) . PROOFOF (b). Assume there were such a 8. Let be a disjoint copy of a, and let f be the copying isomorphism of %, onto a1. Let 23= ( S o + % ~ , f , ~ ) T~h u~s ~Sm(23)= ~ ~ , .p ' u { F } , a'= a , and F ' = f . Clearly, 23 I= @ ( F ) . Since a, is L,,-undefinable, there is ~ , E I % , I such that (TU,,a,) =mK(%o,bo)and a,#b,. Since %,=fi?ll it follows that ( % , , f ( a o ) ) =,,(%,, f(b,)) and f(ao)# f(b,). Since (%o,u)u~luol is L,,-equivalent to itself, and L,,-equivalence is preserved under extended full cardinal sums, we get (%074uElNo,

*

*

~ ~ l ~ f ~ ~ o ~ ~ f m ~ ~ ~(%19f(bO)) [ o , ~ > , , ~ , ~ ,

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[Ch. 2, $ 4

or, equivalently,

where p" = p'- { q}. Since 19 defines F explicitly, then 231.p'k8[a,,f(ao)]. The last equivalence implies that 23 1p't= 0 [ao,f(bo)].Since f = FB is a function and f ( a o ) # f ( b o ) , we have reached a contradiction. This contradiction proves (b). Since + ( F ) E Sent(L,+,( p'u { F } ) ) , we have proved that Beth's theorem fails in the following form: 1Def(L,+,, LmK).rn Counterexample 2.4.8 implies that Beth's definability theorem fails for several other languages as well. Indeed,. we have: THEOREM 2.4.9. (1) 1Def(L,,, LmK)for any infinite cardinal K ; (2) i Def(L,,, LmJ ( 3 ) If K is a singular cardinal and o < h < K , then i Def(L,,,

(4) If

K

LmA) and

i Def(L,+,,

LmJ;

i Def(L,+,,

L,+,);

is a singular cardinal, then i Def(L,,,

L,,)

( 5 ) If

K

is regular, K > w , then

( 6 ) If

K

is infinite, then

and

PROOF.Observe that the statement 7Def(L, L') is invariant under increasing first coordinates and decreasing second coordinates. By this we mean that if Li, Lj ( i = 1,2) are languages such that every ,sentence of L, (resp. L;) there is a sentence of L; (resp. L2) having the same structures as models, then 1Def(L,, L2)

*

1Def(L',, L;).

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This is obvious owing to the form of the statement 1Def(L,L’). Thus, Counterexample 2.4.8 immediately implies (5) and 1Def(L,++,, L,,+)

-

for any infinite cardinal

K,

and this statement implies (1) and (6). Likewise (3) (4). (2) is a particular case of (1). To prove (3), notice that if h < K , then Counterexample 2.4.8 implies 1Def(LA++,,LWA+), and this implies 1Def(L,,,L,,), since A + + < K < K + and h < h + .

-

The following are known positive definability results for finite-quantifier languages. They are consequences of corresponding interpolation theorems (see pp. 131-132 and Corollary 2.3.7). (A) (Malitz) K regular Def(L,,, L(,<.)+,); in particular: (B) K infinite * Def(L,+,, L(Z.)+r+). and (C) K strongly inaccessible > w =+. Def(L,,, L,,). (D) (Friedman and Chang) cf(K)=w * Def(L,+,, L,,<.,+,); in particular: (E) Cf(K)=W 3 Def(L,+,,L,,), and (F) K strong limit cardinal and cf(K)=w * Def(L,+,,L,+,). Thus, Theorem 2.4.9(5) and (when cf(K)=w) Theorem 2.4.9(6), (3), show the impossibility of improving these results. The following problems still remain open: (I) Does Def(L,+,, L,+,) or Def(L,+,, L,+,+) hold when K > w and cf(K) (without using LCH). (11) Does Def(L,+,, LmK) hold for cardinals K such that w < cf(K) < K? Problem (11) would automatically get a negative answer if Counterexample 5.3.41 could be extended to cardinals K such that w < cf(K) < K . = w?

Our next counterexample shows that the result: K E E C =+ S ( K ) E U C (cf. Exercise 1.3.3), which holds in L,, result clearly implies Corollary 2.3.9.

does not even extend to Lwlw.This

COUNTEREXAMPLE 2.4.10. There is u E Sent(LUI,) such that S(Mod(u)) gEC,(L,,) for all infinite cardinals K .

I50

RESULTS FROM MODEL THEORY OF

L,,

AND

L,,,

[Ch. 2,

4

PROOF.Let K be the class of all isomorphs of ( w , S ) , where S is the set ((x, x + l ) J x € w } .

Let

IJ

be the conjunction of the following LOI,-sentences: Vx3y R(x,y); VXyZ

[ R (X,y)A R (X, Z)+Y

X 21;

Obviously, K = Mod(a). Let B = ( E , 0 ) where E denotes the set of all even integers. Clearly S n EZ=$3, so that B?ES(K). For every formula 'p of L,,(R), let @ be the formula obtained from 'p by replacing every atomic subformula of the form R ( x , y ) by x x y A 1 ( x x y ) . Evidently @ is of type $3 and for every set C and c l , . . .,c, ?E C we have: (*)

....cnl

-

(C>t=@t c17 .. . ? C , l .

Let g, be any sentence of L,,(R) such that Bt=g,. By (*), ( E ) i = @ ;by Lemma 2.4.1, for every infinite set C we have (C)t=@,whence, by (*):

(Cfi ) t=

cp.

Hence, if S(K)= Mod@) for some 2 cSent(L,,(R)) and some K, it will follow that (C,B ) E Mod(2) = S(K) for every infinite set C. But this is clearly absurd because S(K) contains only countable structures. W In Corollary 2.3.9 it was mentioned that for every LOI,-sentence 'p, Mod('p) is closed under substructures iff g, is equivalent to a universal sentence. The next example, due to Tarski, shows that this is not true for L,I,I-sentences.

Ch. 2, $ 41

151

COUNTEREXAMPLES

COUNTEREXAMPLE 2.4.1 1. Let u be the L,I,I-sentence saying “there are at most denumerably many elements”:

( 3 u Yw)(Vx)

v xzu,

nEw

Clearly Mod(u) is closed under substructures. We show that u is not equivalent to any universal L,I,I-sentence. An easy induction on- the ~) then for every structure of cp shows that if Q I E S ~ ~ ~ ( L ,is~ ,universal, structure 8: if

(*)

forallB,Bca/\ii?
then

* ~t-cp,

9It-q.

The proof of (*) is left as an exercise for the reader (cf. Definition 1.3.9 (b) and Exercise 1.3.5 (b) and (c)). Condition (*) implies:

(**I

if

QJ

is a universal L,,,,(B )-sentence and t- u+cp,

then Mod(cp) has members of all cardinalities.

Infact,ifA i s a n y s e t a n d B c A , E < K , , w e h a v e ( B ) k u ; hence(B)I-cp; by (A)+cp. Since all members of Mod(u) are countable, it follows from (**) that there is no universal sentence cp such that t-u-+cp, and a fortiori such that (*)5

t-uecp.

The question whether the preservation theorem Corollary 2.3.9 extends to the languages L,, with K > w , is still open: OPENPROBLEM. Is it true that for cpESent(L,,), K > w ~ , Mod(cp) is closed under substructures iff cp is equivalent to a universal sentence? The next counterexample shows that the prenex normal form theorem fails for many infinitaiy languages. COUNTEREXAMPLE 2.4.12. An L,+,(B )-formula, for K a limit cardinal, which is not equivalent to a conjunction of prenex L,,(fl)-formulas, for any T 2 K + (hence it does not have a prenex form in LmK). Let a, be the sentence that says “there are < K elements” ( K any infinite cardinal):

152

RESULTS FROM MODEL THEORY OF

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AND

L,,,

[Ch. 2,

4

Clearly a, is, in general, an L,+,(B)-formula; but if { h l h < ~ , h E C N ’ } (or, in other words, if K = Na with 0 < (Y < Na), then uK is in L,,(B). Notice that, in particular, for K = w we obtain the L,,,(B)-sentence which says “there are finitely many elements”. Suppose that K is a limit cardinal and that u, is equivalent to a conjunction, say of prenex L,,(B)-sentences, for some T > , K + . Then each qi has the form


q ; = ( Q ~ c ; ~ A ;(Qi,o;,IAi,)+;, ).-where (a) t+bi is quantifier-free; (b) n j E w ; (c) each Q] ( j = I,...,n;) is t/ or 3; (d) (by eventually renaming variables) each A; ( j = l . , . .,n,) is a cardinal


Let X;=max{X~,...,hi,}; then hi< K and, since K is a limit cardinal, hi+< K . Therefore, by its own definition, u,-hence also rp-has a model of

power hi+;since q ; ~ S e n t ( L , ~ + ( B )by ) , Lemma 2.4.1 ‘pi holds in all structures of power 2 hi+: then it is clear that u, has a model of power > K , a contradiction. For K = W we can give an alternative proof using compactness. In fact, in this case each 4; is an infinitary conjunction or disjunction of atomic or . . negated atomic formulas of type B , i.e., of formulas of the form u i x o ; of . . i (uixu;); since the h;’s are finite (i.e., there are only finitely many variables u;), a simple induction on the complexity of shows that ;)I is equivalent to an L,,(!iY)-formula. Therefore we can consider. without loss of generality, that the ‘pi’s are finitary sentences; then clearly we have:

+;

Mod(u,)

I

= Mod( { ‘pi i E I

}).

By the definition of u,, T = { q;I i E I } is a set of first-order sentences with models of all finite cardinalities; a simple application of compactness shows that T has an infinite model, which is also a model for u,, a contradiction. Notice that when K is a successor cardinal, K = X + , U , is a formula of L,,(B) =LA+,,+@), and i f does hace a prenex form; namely:

(3uTh)Vy

v (c,xy).

a<X

In fact, this is a particular case of a weak prenex normal form theorem for L,,, K a regular infinite cardinal, which states-roughly speaking-that

Ch. 2, $ 41

COUNTEREXAMPLES

153

any finite quantifier-rank formula of this language is equivalent to a prenex formula (cf. Exercise 2.4.3). Note that the preceding Counterexample 2.4.12 still leaves open the question whether there is a general prenex normal form theorem for the languages L,, when K is a regular uncountable cardinal. The answer to this question is “no”; a counterexample has been constructed by M. Jones; this counterexample is presented in Appendix B. The counterexamples given below show the impossibility of a simple, direct extension of the Chang-KoS-Suszko theorem to infinitary languages. DEFINITION 2.443. (a) K is an w-fami& iff K admits a union and whenever I KI and 3 < No, there is 3 E K such that B c I%[. In a similar way one can define a K-family for any infinite cardinal K . (b) K is called an a-chain iff it is totally ordered by the relation “substructure of”, and has cofinality w, i.e., there is a denumerable family { i ? l , , I n E w } C Ksuch that: (i) anC%, whenever n < m , (ii) for every 23 E K there is an n ~w such that 23 CS,,. (c) A sentence rp is preserved under unions of chains [unions of w-families, respectively] iff for every chain [a-family, respectively] K, the following holds:

B

u

for all 9 l E K 3 F r p

-

U Kt-rp.

THEOREM (Chang-EoS-Suszko). Let rp be an La,-sentence. Then the following are equivalent: (1) rp is preserved under unions of chains; (2) rp is preserved under unions of w-families; ( 3 ) rp is equivalent to a universal-existential sentence (cf. Definition 1.3.10(~)).

COUNTEREXAMPLE 2.4.14. A universal-existential sentence of L,,, which is not preserved under unions of chains. The sentence u, asserting “there are finitely many elements” is obviously equivalent to the disjunction ‘p of the sentences:

(i.e., rp,, says “there are


elements”).

154

RESULTS FROM MODEL THEORY OF

L,,

AND

LWI,

[Ch. 2, 9 4

Clearly cpEUnex,,,. Let K = { ( A , ) l n ~ o , n > O } , where A , is the set ( 0,..., n - 2 ) ; clearly (A,,)kcp,,, whence (A,)kcp for all n E w , n>O. However, (w>= UnEo,,,>O(An) and ( w > ) - ~ p .H

COUNTEREXAMPLE 2.4.15 (Kueker). An LoI,-sentence preserved under unions of chains but not under unions of a-families. We need the following:

LEMMA2.4.16. There is a collection { U,, 1 n E w ) of subsets of w with the following properties : ( I ) Every finite subset of w is included in some U,,. (2) U,,- Um#O and Urn- U,, #0 for all m , n E o , m # n . PROOF.Let P be the set of all primes and let { F, Ip E P } be an enumeration without repetitions of all finite non-empty subsets of w . For given p , q E P let: h ( p ,q ) = first number of the formp" not in F,.

Since the Fq's are finite, h is defined for all pairs ( p , q ) E P z . Let

F, = Fpu { h ( p , q ) I q E P A q + p ) . II

,.

L e t p , q E P , p # q . By definition h ( p , q ) E F , an$ h ( p , q ) € F,; hence F, - Fq #,0. If h(p,q)EF,, we would have h ( p , q ) E F , - Fq; hence h ( p , q ) would be a power of q ; but by Gefinjtion h ( p , q ) is a power of p ; hence p = q, a contradiction. Therefore F, - F, f 0 , whenever p # q. Reindexing the collection { F, Ip E P } (setting for instance U,, = Ffn, where p n is the nth prime) we obtain the desired conclusion. Let the type p be { P, ( m E w } , where all the Pm's are unary predicates; let q,,be the following sentence:

where { U,, I n ~ w is} a fixed collectiop with the properties of the lemma. Let q = V n E w V n * (a) rp is not preserved under unions of w-families. Let 23 defined by:

Ch. 2, 3 41

155

COUNTEREXAMPLES

For each n E w, let

a,,

be:

The set {a,, In E w } is an a-family, and 23 = UnE,an. Moreover, if m E U,, P $ # 0 ; if mBU,,, P,"n=0; therefore, %,,/='p,, for all n E w ; hence %,,l-'p for all n E w . But not B Fcp, because P:#0 for all m Eo. (b) 'p is preserved under unions of chains. Let K be a chain such that Zt-q for every % E K . We shall show that:

-

i ? l t - ' p , , ~ % € K ~ r # n Z F 1qr.

For if r # n, by (2) of the preceding lemma there is an m E U, - U,,. Since % F q,,and m B U,, we get P," =0; therefore not % t= ArnE ,,3xPrn(x), and %I= l q r . If no E w

is such that there is % E K, % k p,,,, we shall show that:

Moreover, the following holds:

Q E K A O I = ' ~*~ s = n o . Suppose that the antecedent is true and sfn,. Case (1) Q C Z ; then P," c P," for all m Eo. Using (2) of Lemma 2.4.16 let m E Us- &;,.Since m ?j U,,, and % t= p' ,, P," =0; therefore P," =O. Since m E Usand 0 t-q,, P," #0 and we have reached a contradiction. Case (2) % c Q; apply a similar argument, choosing m E U,,,, Us. Using (*) it is not difficult to see that U KF'p,,,; hen7e U K k q , as we wanted to prove. Notice that the sentence constructed in Counterexample 2.4.15 is also universal-existential. We thus see that in La], fail some of the relations holding among ( 1)-(3) in the Chang-KoS-Suszko theorem; namely: (1)

+ (21,

(3)

+ (1)

and

(3)

+ (2).

Then, it is natural to ask which of the remaining implications hold in Lwlw. One should also consider the following deeper and more interesting questions: (a) finding a syntactic characterization of LwI,-sentences preserved under unions of chains and unions of a-families; and, (b) finding a model-theoretic characterization of La,,-universal-existential sentences. Some of these problems have already been solved.

156

-

RESULTS FROM MODEL THEORY OF

L,,

AND

L,,,

[Ch. 2,

54

The implication (2) ( I ) holds in Lwiw;this is an immediate consequence of the following theorem, proved by means of a downward Lowenheim-Skolem type argument:

THEOREM (WEINSTEIN [I]). Euety LWi,-sentence preserced under unions of a-chains is also preserved under unions of (arbirraty) chains.

The (very elegant) proof of this theorem is given in detail in DICKMANN [I], vol. I, pp. 198-200. Question (a) above is settled by the following preservation theorem, which also establishes the implication (2) + (3).

THEOREM (NEBRES[I], [2]). Let rp,# E Sent(LWI,(p)). The following are equivalent: ( I ) There is a conjunctioe-universal-existentialsentence u of Lwiw( p ) such that krp-+u and i=u+$;’ (2) the union of any a-famib of models of rp is a model of #. The proof is a direct generalization of Makkai’s direct proof of Theorem 2.3.8 (cf. MAKKAI [3]). However, Nebres’ theorem does not settle all of the questions posed above. The following still remain: OPEN PROBLEMS. ( I ) Find a syntactic characterization of L,!,-sentences preserved under unions of chains. (2) Find a model-theoretic characterization of universal-existential sentences of Lalw. EXERCISES 2.4. I . Formulate Counterexample 2.4.4 in a language containing finitely many symbols. [Hint: use two ordering relations.] 2.4.2. (a) Prove that all universal-existential sentences of L,,, are preserved under unions of a,-families. Does this generalize to other languages L,,? (b) Prove that all conjunctive-universal-existential sentences of LWI,(cf. Definition 1.3.10(b)) are preserved under unions of a-families. (c) (WEINSTEIN [I]) Prove that all sentences preserved under unions of a,-chains are also preserved under unions of ,a,-families. [Hint: if K Mod(rp) is an a,-family, using the downward Lowenheim-Skolem Theorem 3.4.1, represent U K as the respective unions of two “sand-

c

’ Cf. Definition 1.3.10 (b). Caution, u need not be an interpolating sentence for 9.4.

Ch. 2, J 41

COUNTEREXAMPLES

157

wiched” increasing chains of length wl, (aaI a < wl), (BaI a < wl), such that for all a
2.4.3. Prove the following weak prenex normal form theorem: if K is a regular uncountable cardinal, then any formula QI of L,, such that qr(rp) < w is equivalent to a prenex formula of L,,. [Hint: induction on the structure of QI.]

CHAPTER 3

GENERAL RESULTS

0 1. Incompactness of accessible cardinals In this chapter we will study a fundamental problem of the metamathematics of the languages L,,, that of compactness. The set-theoretical background for this is given in Chapter 0. I t is well known that the languages L,, have the compactness property: (1)

For any similarity type p and any set r of sentences of La,( p): if every finite subset of r has a model, then r has a model.

This property is a simple corollary of (and actually equivalent to) the completeness theorem for L,,: (2)

Any consistent set of sentences of La,( p ) has a model.

However, purely model-theoretical proofs of (1) can be given using the axiom of choice (cf. FRAYNE-MOREL-SCOTT [l], Theorem 2.10, p. 216, or BELL-SLOMSON [l], Chapter 5, $4,pp. 101-103); the advantage of this method is, that a model for r is obtained by a uniform algebraic construction from given models for each finite subset of r. The proof is well-known and will not be included here; a generalization of it serving different purposes is included later in this chapter. The first stumbling block in the study of the model theory of L,, ( K > o) is the fact that the natural generalization of (1) fails, except for very special values of K , if any at all. In fact, the question whether there is a K > W for which L,, has the compactness property depends essentially on strong set-theoretical assumptions and relates to very important problems in the foundations of set theory (this question will receive due attention in $52.3 below). 158

Ch. 3, 5 11

INCOMPACTNESS OF ACCESSIBLE CARDINALS

159

From the results proved below and well-known results in the metamathematics of set theory it follows that it is consistent with the axioms of set theory to assume that such cardinals do not exist; on the other hand it is not known whether it is consistent to assume that they exist. In spite of so many uncertainties we shall plunge into the artificial paradise of large cardinals. DEFINITION 3.1.1. Let K , A, y be infinite cardinals, K > A. The language L,, is said to be y-incompact iff there are a similarity type p and a set of sentences r of L,, ( p ) with the following properties: (i) r = y. (ii) r has no model. (iii) Every subset of r of power < K has a model. If this is not the case, we say that L,, is y-compact. REMARK. If L,, is y-incompact and A < A’ < K then L,,, is y-incompact.

DEFINITION 3.1.2 (a). A cardinal

K is y-incompact iff L,, is y-incompact. (b) K (and L,,) is incompact iff it is y-incompact for some y 2 K . (c) K (and La,) is strongly incompact iff it is K-incompact. (d) I (SI) denotes the class of all incompact (strongly incompact) cardinals. Incompact cardinals are sometimes called weakly incompact. Set C = C N - 1 and WC=CN-.SI. The elements of C (WC) are called compact (weakly compact) cardinals. Compact cardinals are also called strongly compact.

REMARKS. (a) From the definition it is obvious that SIC1 and hence C WC. From (1) it is clear that w E C. Whether this is the only member of C (or even of WC) is an open problem; from the set-theoretic results mentioned in Chapter 0, 0 1, it follows that it is consistent to assume that this is the case. It is,interesting to mention that C#{w} implies C Z W C (cf. 33.C below). (b) Definition 3.1.2 is a natural generalization of (1):

c

(3)

(4)

K E Ciff for every similarity type p and every F c Sent (L,, (p)) the following holds: if every subset of r of power < K has a model, then r has a model.

iff for every similarity type p and every r cSent(L,,( p)) such that < K , the following holds: if every subset of r of power < K has a model, then r has a model. K E WC

160

GENERAL RESULTS

[Ch. 3, 5 1

Certain incompactness results have been known to Tarski for some time. He established many theorems about Boolean algebras and related topics that applied to accessible cardinals, and later on he realized that the properties studied by him were very closely related to the incompactness of the cardinals involved (some of them actually equivalent). In this way the incompactness of all accessible cardinals was established; the results are implicitly stated in SCOTT-TARSKI[I]. An extensive account on the mathematical properties considered by Tarski-among many others-is given in Keisler-Tarski’s inaccessible paper [ 11. We give in this section new and slightly improved metamathematical versions of Tarski’s results; they are due to HANF[2]. Minor changes in Hanf‘s original proofs enable us to extend his results from L,, to L,, whenever K is an accessible cardinal.

THEOREM 3.1.3. If A, y <

K

and

K

< h Y then

L,, is XY-incompact.

REMARK. This result is a purely propositional matter. In order to avoid working with propositional symbols and their models. we give a slightly different proof. PROOF.The similarity type used to construct the set of sentences that makes L,, AY-incompact consists of one unary predicate R and one individual constant ctl, for every pair (&q) of ordinals 6 < y and 71
121 where 121 is meant to include one sentence for each function f:y-+A. Obviously, r has power A Y . (a) r has no model. In fact the conditions 11 I and 121 clash. Suppose r has a model 91; for each t
contradicting the particular sentence of type 121 obtained for f=fo. (b) Given a function f : y - 4 , let

Ch. 3, I 11

161

INCOMPACTNESS OF ACCESSIBLE CARDINALS

r, has a model. Let

a,

be the following structure:

pq={oJ},

R E / ={ 1 ) . It is easy to check that 111 and the sentences of type 121 which are in in illp i.e., %,E Mod(r,). (c) The set .'I of all sentences of type 12 1 has a model. Let 'H be the following structure:

r, hold

~'H~={O,l}, cg=O

R'=

for

t< y and q
{I}.

Clearly, 23 E Mod(r'.). (b) and (c) together prove that every proper subset of completes the proof of the theorem.

r has a model; this

All structures to be considered from now on are of type (2, .. .), i.e., R, is a binary relation; we shall write x
131

"

< is a total ordering of the universe"

In the original proof of incompactness of the logics L,,, Hanf considered well-orderings instead of linear orderings. He included the sentence:

(*I

r ( ,/?w

1(30 a)

0,

+ I < %)

as well as 131, and in his proof the formulas:

for q < K (which characterize the vth element of the corresponding wellordered structures) played a essential role.

162

[Ch. 3, 5 1

GENERAL RESULTS

We dispense with (*), and instead of (**) we consider the formulas

@F(x) introduced in Chapter 1, $2, which are their reasonable analogs in

L,,. Using them we improve Hanf's results to assert the strong incompactness of L,,, instead of L,,, when K is accessible. 3.1.4. If THEOREM

K

is a singular cardinal, then L,, is K-incompact.

PROOF.By hypothesis there are p < K and a sequence (yg I [< p ) C K such that K = Ug
151

r has power K . In virtue of Lemma 1.2.8, sentence 141 "says" that in any , such that every element of R I" determines model 2l of r there is some ,$ < O in the subset RY a section well ordered by <" in type < yg. Hence, RY is well ordered by <" in type < yE, and t h e r e f o r e T < K . The sentences 151 "say" that in every model izI of r, RY contains elements which determine S1 sections of all types < K ; hence > K . Therefore r has no model. in I If Z C T and < K it is easy to construct a model of Z. Consider the following: l%xl=K;


R I.'= { 71 E K 1 3 X [a),(.) A i?

I(X)]

E c}.

Let 6, be the type of RY" in the natural ordering of its elements, i.e.,

R

I < a,),

= (TJ,p

where TJ, < q,., if p < p'

< 6,.

Obviously, < < K ; since K is a cardinal, 6, < K ; hence there is 5, < p such that So< yso. We assert that the particular disjunct of (41 obtained for the value t=t0holds in 21x. In fact, let { E R;"; then { = q,, for some p < 6, < ygo; hence, the type of S, n RY" is p. By Lemma 1.2.8

a,+ @3{].

Ch. 3, !j 11

163

INCOMPACTNESS OF ACCESSIBLE CARDINALS

In particular we have shown:

This shows that 141 holds in Zz. The sentences 15) occurring in Z (i.e. those with 9 E RFX) hold in U, by the definition of RFx. This completes the proof of the theorem. REMARK. The method of proof of this theorem is important because a more sophisticated version of it will render the proof of incompactness for a big class of inaccessible numbers K as well.

THEOREM 3.1.5. If

K

is a successor cardinal, then L,, is K-incompact.

PROOF.Let K be a successor cardinal, K = A + . The set K-incompact is the following: sentence 131 plus

161

vXYZ[

F ( X , y ) A F( X , Z ) + y % Z ]

r which makes L,,

AvXyZ[ F ( Y , X ) A F ( Z , X ) + J J e Z ]

A vx3Y[R,(x)+F(x,Y)l,

r has similarity type (2, 1,2). The sentences (61 “say” that F is a one-one function with domain including R , . r has power Now suppose r has a model, U E Mod(T). Let f= F ” . From (61 we have K.

c

RF Dom(f) and from 181 it follows that for every 77 < K there are elements in Range(fr RF) that determine a section of type 77 (Lemma 1.2.8). Hence Range(f1R:)

> K.

-

-

Since obviously Range(frRF) = R F ( f r R F is a one-one map between these sets) we have R F >/ K >A. On the other hand, 171 implies that each element of R: determines (in 1‘21() a <‘-section of type < A and therefore RF is well-ordered by <’ in type < A . Hence -

RF < A . This contradiction shows that

r has no model.

Let Z Y T , < K . We can assume without loss of generality, that sentences of form 181 occur in Z. Let p=supX, where X = { ~ E K /the sentence of form (81 corresponding to 7 is in 2 ) .

I64

Clearly < for the set 2 :

GENERAL RESULTS

[Ch. 3,

1

< A. Since K is regular, p < K . Let us construct a model 91z

We split the definitions of R,"' and F a x in two cases,: and Fux=gra ph of id,. Clearly in this case 161, 17) and the sentences of form 181 that occur in Z hold in 91,. (11) A < ~ < K . S i n c e y < A , the elements of X can be ordered in a sequence of type S = i.e., there is a one-one function g from 6 onto X.We set (1) p < A . Then R;''=p

x;

Hence Range(F"x) = Range( g ) = X . Clearly 161 holds in 3, by the definitions of g and F u ~(71 ; holds in 3, because S < A . If -q < K and the sentence of form 181 corresponding to 71 is in Z, then 71 E X . We write y =g-'(-q). Then y < S, i.e., y E RFT, and g ( y ) = 77. From this it clearly follows that

Thus all sentences of form 181 which are in Z hold in 3 ' ,. !X2€ Mod(Z). This completes the proof of the theorem.

-

The preceding results amount to the following:

COROLLARY 3.1.6 ( 1 ) K accessible cardinal + L,, is incompact. ( 2 ) K strongly accessible cardinal L,, is strongly incompact. In particular: (3) (Tarski) K accessible cardinal + K incompact, or: K compact * K strongly inaccessible. (4)K strongly accessible * K strong& incompact, or: K weakly compact K weakly inaccessible.

-

PROOF.Obvious. H

Hence

Ch. 3, f 21

INCOMPACTNESS RESULTS FOR INACCESSIBLE CARDINALS

165

REMARKS. (a). Under the generalized continuum hypothesis the classes AC and SAC coincide. Hence:

(b) In the Preliminaries we remarked that the assumptions “In= (a}” and “WIn= { w } ” can be consistently added to the axioms of set theory; Corollary 3.1.6 proves that C In and WC WIn. Hence the following statements can also be added to the axioms of set theory without destroying consistency: (i) “there are no compact cardinals >w” (C= {a}), (ii) “there are no weakly compact cardinals > 0’’ (WC = {a}). $ 2. Incompactness results for inaccessible cardinals

This section is mainly devoted to prove Theorem 3.2.1, Hanf‘s main incompactness result. It shows that the vast majority of inaccessible cardinals > w are strongly incompact, provided they exist. In fact, from this result and the information on the operation M contained in $2, Chapter 0 it follows that all inaccessible cardinals > a are strongly incompact with the possible exception of those that are hyperinaccessible of their own type; in particular, if {8tl<EON} is an enumeration according size of all inaccessible cardinals, each 8, is strongly incompact unless it is a fixed point of the sequence: E=8,. If {O;I[EON} is an enumeration in their natural ordering of these fixed points, all members are strongly incompact unless they are fixedpoints of the new sequence: E= 8;. This statement can be repeated ad infinitum. Therefore, if it is not inconsistent with the axioms of set theory to assume that weakly compact cardinals exist, they must be of galactic size. From the results in $ 3 it follows, then, that the same applies to cardinals defined by many other properties, which we will be able to prove weakly compact (measurable cardinals, for instance). Hanf‘s method seems to be applicable to ever increasing classes of cardinals beyond any limit, as long as they are (in some vague sense) “constructed” by means of (possibly infinitistic) operations on the classes MR(AC). The common feature to all these cardinals is that, roughly speaking, they are “describable” in terms of smaller ordinals. This concept of describability is explicitly formulated in HANF-SCOTT[l]; a more detailed exposition is given in LEVY [l]. Therefore Tarski’s assertion “we do not know any ‘constructively characterized’ cardinal >w of which we

166

[Ch. 3, I 2

GENERAL RESULTS

cannot prove that it is strongly incompact” seems fully justified (cf. TARSKI PI, p. 134). At the end of $ 2 we indicate how to modify Hanf‘s proof to apply to the more comprehensive class BC instead of AC. In this way it is provedwithout using the GCH-that many accessible and weakly inaccessible cardinals are not only incompact but also strongly incompact. Hanf‘s original proof was done for the languages LK,. Without additiona! pain we improve it slightly to L,,,. We have not succeeded to extend his proof to LK,. However, in $ 3 we show by different methods that if L,, is K-compact, then K is already weakly compact. Likewise, if L,, enjoys the strong compactness property, so does LKK.Thus, both compactness problems (i.e., weak and strong) for L,, and L,, are equivalent. This fact does not appear explicitly mentioned in earlier papers. After reading this section the student should proceed to read $3-its natural continuation-in order to gain better understanding of the results included here and their implications. we will use in our proof KEISLER[ I ] has devised a formula of L,,-that -which for A < K expresses the property “A is an accessible cardinal”. This requires some preliminary explanations. Suppose we have a total ordering < and two formulas cp, $ with exactly 3 free variables. Let Fn(x,y,cp) be the formula saying “cp(x;) is a map from a subset of S, into S,”: t/ZWV[cp(X,Z,W) ACp(X,Z,U)+WXU]

A\ZW[cp(X,Z,

W) +Z


A W

< XI.

The following formula Z(X,#) says: “there is y < x such that F,, where F x ( z ) = { u ~ $ ( x , z , u ) } for Z < X , is a one--one function with Dom(F,)=S, and Range( F,) $? ( S,)”, that is 3y { Y

<X A vZU($(X,Z,

U)+Z

<X A U < y )

A v Z W [ Z <X A W <X r \ v U ( # ( X , Z , U ) H # ( X ,

W,U))+ZZW]}.

The following formula @(x,cp) says: “there is y < x such that cp(x;) is a map from a subset of S, into S, and the range of cp(x;) is cofinal in S,”, that is

THEOREM 3.2.1. Let R be a weii-ordering relation of ON and Then K E MR(AC) implies that L,,, is K-incompact.

K

€In-

(0).

Ch. 3, $ 21

INCOMPACTNESS RESULTS FOR INACCESSIBLE CARDINALS

167

PROOF.Let 8 be the enumerating function corresponding to R. Let v be the least E such that K E M ( ~ , ~ ) ( A Since C ) . ~ ~ 1 {a} n -and M(R,o)(X)=X,we have 71 >0, and since M(R,h)(X)=UE.,AM(R~5)(X) for h limit, 77 is a succesSince sor ordinal, say q = 5 + 1; furthermore K M(R*S)(AC). M(R.C+

" ( x -) M(R,~)(x)= M ( M ( R . ~ )

(X))-(dc+

1 1 9

it follows that KEM(M(~?{)(AC)) and K > Or; hence there is a relatively closed set C such that C M(R*S'(AC)and K = U C. The set of sentences used to show the K-incompactness of LNu,will be constructed using R (restricted to ordinals less than K ) , 0, and the set C. The set r is of similarity type p ,

r

c

p=(2,2,2,2,3,3,1,1,1>. The meaning of the symbols involved is, approximately, the following: R, is the (well-ordering) relation < ; R , will represent R restricted to ordinals < K ; R , is used to represent the relation between (Y and defined by: (Y E M( ~ - ~)(Ac ); R , will represent relatively closed sets of ordinals in the form of a binary relation between its members and its supremum; R4 and R 5 are used to express accessibility (they play the role of q,4, respectively, in our previous considerations); R, will denote the set of all mdinals or singular cardinals of the model; R7 will denote the set of all cardinals that are not strong limit; R , is used in connection with the set C. consists of the sentences: 131 as in § 1,

r

168

[Ch. 3, 3 2

GENERAL RESULTS

1171

3W(@&)A@.,(Y)A l R , ( W ) )

1181

3 x ( @ , ( x ) ~ R ~ ( x )for ) all q E C,

1191

1( 3 v r 0 )

for t , V < K,(t,77)ER,

A (vn+l
new

Notice that all sentences of r are in L,, except for 1161-1 191; 1161-1 181 are in L,,, and the only sentence with an infinite quantifier is 1191. By 1191 every model % of r is well-ordered by <'; we can assume, then, that I%l=p and <'= E r p , for some ordinal p. By 1141, % has a last element; then p is a successor ordinal, say p = S + I . By the sentences 1181, % has initial sections of type q , for each q E C. Since K = IJ C, 94 has an initial segment of type > K . Since K is a limit ordinal and p a successor ordinal, we have K < p ; hence K < S and K E 1941.' We establish several partial results culminating ,in a proof that the existence of an 8 E Mod(T) is untenable. (I) RFUR:cAO (=(ON-CN)uAC). , By the remarks of p. 166 this implies that Let a E RF; by 191, % t= @ [ aRF]. S>'= a is not a cardinal or that a is singular. In both cases a EAO. Let a E R:; by IIO!, at=Z[a,R;']. For q < a we set

As we have remarked before, F, is a one-one function mapping a into

' 5 ( y ) for some y < a ; hence a < 2 y , and a EAO. (11) If a < K , 8, < K and ( a , 8,) E R;', then a E M'R7T)(AO).

This assertion is proved by induction on q. Assume that for all 5 < q

Let a and 8, be as in the hypothesis and

' This is the only point of the proof where the LwIwI-sentence1191 is used in an essential way.

Ch. 3, 9 21

INCOMPACTNESS RESULTS FOR INACCESSIBLE CARDINALS

169

(i) a = U 0,. Let y < a. By 1121 there is p E lrUl such that y < p < a and ( a , P ) E R f ; this means that p E 0,; hence a = U D,. (ii) D, is relatively closed. Let A c D, be a set bounded above by /3 E D , and without last element; obviously the following holds:

and since U A < p < a , we have by 1131, ( a , U A ) E R f ; hence U A E D , . By the assumption (a,B,)E RF and 11I ) , one of the following alternatives always holds: (a) a E R t or a E R : ; (b) there is 8 E IrUl such that 8 < a , (8,8,) E R F , and for every fl E I%), ( a , P ) E R : * (P,e)ER;'. In the first case, by (I) we conclude aEAOCM(R9")(AO), as desired. Assume then that (b) holds. Since 8 < a < K and 8, < K we have ( 8 , 8") ER; otherwise, by 1171 we would have (8,8,)E R F , which contradicts (b); hence 8 = 8 , for some [ < q , and by the last assertion of (b) we get: ( p, 8,) E RF

for all

p E 0,.

By the induction hypothesis (*) we conclude

D a -C M'R3S)(AO), and from (i) and (ii) we have a EM(M(~.<)(AO)); since 8 = 8 < a , we have a E 8, + 1 ; hence a EM(R,5+')(AO), and since [< 7,a EMcRqnS(A0). We are now ready to prove: (111) r has no model. Suppose that 3 is a model of r. By 1181, C C R f and by 1151, R f is <9'-closed (i.e., it contains the limit of any increasing sequence of its . 1161 we conclude that members); since K = U C , we have K E R ~ By (K , B { ) E R T , and applying (11) we get K E M(R3S)(AO), which is a contradiction. (IV) Every subset of r ofpower < K has a model. Let Z be one such subset of I'; let p be the supremum of all ordinals 7 appearing in the @,'s of those sentences of the forms 1161-1 181 which are in Z. Since K is not singular we have p < K .

170

[Ch. 3, 3 2

GENERAL RESULTS

We construct a model 91z by defining: I%I=p+l RrX=Rr(p+ 1 )

For each ordinal a < p, if a EM(R,?+

0 (AC) - M(R*q)(AC)

for some 11, we choose a relatively closed set D,GM(R.q)(AC)such that a = UD,; otherwise D, = a. R?'={(.,P>IP

ED,).

For each ordinal a < p, if a €(ON - CN) u Sing we choose a particular function f , with Dom( f a ) < a , Range( f a ) c a and U Range( f a ) = a ; otherwise let f a be empty.

R F = { ( a ,5 7 1 1 )

I

(59

11) E f ,

1.

For each ordinal a < p, if a < 2y for some y < a , let Fa be a one-one function mapping a onto some family of subsets of y ; otherwise let F, be defined by F,(q)= (7) for each 7 < a .

R:P= { a < p I a

< 27

for some y < a }

~:x=~n(p+i). It is now a routine matter to check that proof of the theorem.

E Mod@).

This completes the

COROLLARY 3.2.2. (1) All non-denumerable strongly inaccessible cardinals

K

which are not hyperinaccessibles of type K are strongly incompact. However many a's belonging to H"(1n) are also strongly incompact; for instance: ( 2 ) The first w cardinals that are hyperinaccessible of their own type ( i . e . , members of HD(In)) are strongly incompact. In particular, from (1):

Ch. 3, 3 21

171

INCOMPACTNESS RESULTS FOR INACCESSIBLE CARDINALS

(3) If (0, 15 E ON), ( 8 ; I 5 EON), (8: I E E ON), ...are the enumerations of In, fixed points of In, fixed points of fixed points of In, . .. mentioned at the beginning of this section, all B,, B;, B c , ... are strongly incompact, unless E = O,, E = 8;, [= 8;, ... . In particular the first, second, third,. .. ..., w , ‘ ~., . ,,wLh, , . . strongly inaccessible cardinals are all strongly incompact.

PROOF.(1) Consider aEM”(AC); in Chapter 0 §2.B, we have remarked that this implies a€ME(AC). Hence ( 1 ) follows from Theorem 3.2.1 by considering R = E . (2) We leave as an exercise for the reader to define a well-ordering relation R in ON that makes the first w members of HD(In) say, into elements of MR(AC)(cf. Chapter 0, 0 2.B). (3) We have shown in Chapter 0, 5 2, that all the cardinals Bc, 8;, 8 / , .. . such that [< O,, E < B;, E< e l , . .., are members of M(AC). Corollary 3.2.2 by no means exhausts the list of strongly inaccessible cardinals for which Theorem 3.2.1 establishes strong incompactness. The reader can amuse himself on sleepless nights trying to enlarge it. Theorem 3.2.1 can be modified to yield strong incompactness for a wide class of accessible and weakly inaccessible cardinals for which the theorems of 3 I only give incompactness. The idea is to start from a class of accessible cardinals different from AC. Let BC = SAC u { K E CN I K = A Y for some A, y < K } . We have shown in § 1 (cf. Theorems 3.1.3, 3.1.4 and 3.1.5) that all cardinals in BC are strongly incompact. I t is also clear that BCCAC, but these classes are not equal in general, unless additional set-theoretic axioms (e.g., the GCH) are assumed (in which case Theorem 3.2.3 coincides with Theorem 3.2.1.) THEOREM 3.2.3. Let R be a well-ordering relation on ON and K €MR(BC) implies that L,, is K-incompact.

K>W.

Then

PROOF.If K E B C we have already proved the result. In case K E B C the proof is very similar to that of Theorem 3.2.1. The main difference is in the choice of the relations R , and R,. In this case we want to ensure that each element of R , is either a successor cardinal or an ordinal which is not a cardinal. This requires the introduction of a new formula @’(x,rp),instead of 0, saying: “there is y < x such that for all z < x , rp(x,z;;) defines a one-one map from S, into S,”; R , has to be changed accordingly. We also want in R , all cardinals of form B y where y , ,l3
172

GENERAL RESULTS

[Ch. 3,

53

S,, and for w1# w 2 , w l , w,< x the maps + ( x , w , ; ; ) and + ( x , w Z ; ; ) are different”. R , will be, in this case a 4-placed relation symbol. We also have to add the infinitary formula (of L p y p y ) saying “every function mapping y+P is +( PY,6;;) for some 6 < py”. rn

We close this section by quoting, verbatim, the final remarks of Hanf‘s article [2] (pp. 322-323) concerning the range of applicability of his method to prove strong incompactness for all members of even larger classes of cardinals. Says Hanf: “Let M*(X) be the union over all well-orderings R of the sets MR(X). By the preceding theorems, every non-denumerable cardinal in M*(AC) is incompact and every non-denumerable cardinal in M*(BC) is strongly incompact. However, it is possible to show that the first non-denumerable cardinal, if any, not in M*(AC) (or not in M*(BC)) is strongly incompact. In fact, we can consider various ways of enlarging the class M*(AC) and show, for example, that all the non-denumerable cardinals in M(M*(AC)) or M*(M*(AC)) are incompact. We could also define various classes (M*)(R7T)(AC),(M*)R(AC) and (M*)*(AC) formed by iterating the M* operation or even the classes M(’(R.T))(AC),M(’R)(AC) and. M(**)(AC) formed by iterating the * operation itself. Unless one of these classes contains all cardinals, then each class we have mentioned is much more comprehensive than the earlier ones. Furthermore, we can show that the non-denumerable cardinals in each of these classes are incompact (in fact, strongly incompact except possibly for certain accessible cardinals). These facts can be proved in a manner quite analogous to the proof of Theorem 3.2.1. In the same way that the relation R , gave us an indexed family of reIatively closed sets, we make use, for example, of indexed families of well-ordering relations.” $ 3. Larger cardinals

In $ 2 we carried out the study of the incompactness properties of “constructively defined” inaccessible numbers. The present section is devoted to the study of characteristic properties and the “size” of cardinals K for which the language L,, enjoys some compactness properties. Three classes of those cardinals are dealt with here: (1) weakly compact, strongly inaccessible cardinals; (2) measurable cardinals; (3) (strongly) compact cardinals. We shall see that, unless they are empty, each of these classes is included in the previous one; for the first pair the inclusion is proper, whereas this

Ch. 3, $ 31

LARGER CARDINALS

173

problem is open for the second pair. Also the first member of (1) is smaller than the first member of (2), and the situation is unknown for the last pair. Thus, there are here two different ways in which the word “size” can be understood. We also describe (in some cases without proofs) the relationship of the aforementioned classes with large cardinals that occur naturally in other contexts of set theory and metamathematics (for instance, in partition calculus, describability by higher order sentences, etc.). This is still another sense in which the word “size” can be understood. Concerning characteristic properties of (1)-(3), we emphasize those that are related to the model theory of the languages L,... Properties with narrower metamathematical content are treated in a less detailed way. The main reason for doing so is that there exists at present a number of very readable, thorough and easily available papers and books dealing with the latter subject. We have in mind the following: KEISLER-TARSKI [I]; the first chapter of SILVER[l], [2]; significant parts of MORLEY[2], and DRAKE[l]. On the other hand, only a limited number of papers of rather restricted circulation deal with the metamathematical side of the question. A final word about the foundational status of our considerations. Since all the cardinals (1)-(3) are strongly inaccessible, it is consistent with the axioms of set theory to assume that these classes are empty. Moreover, it follows from Hanf‘s and other’s results that such assumption is consistent even if ZFC is enlarged by statements that ensure the existence of inaccessibles of the types considered in $ 2. Whether the emptiness of one or more of the classes (1)-(3) is provable in ZFC is still a matter of speculation, and by its very nature the problem might be bound to remain open. At present there are neither clues hinting at a proof that (1)-(3) are empty nor at a proof of the consistency that they are not. However, (2) is provably empty in Z F C + V = L [Scott] and (3) is so in ZFC+3a(V=L,), even if we add the (relatively consistent) statement that (2) is non-empty [VoptnkaHrbaEek]. In Appendix D we summarize (without proofs) several of the newer results on subjects that bear a very close relationship with the notions studied here. A. Applications of the ultraproduct construction to infinitary languages

In $3, Chapter 0 we introduced the ultraproduct construction. We prove here the fundamental theorem for ultraproducts-JLoi’s theorem-in a version that renders it useful for the applications that we have in mind. These applications are variants of the well-known proof of the compactness theorem for L,, using ultraproducts; they allow us to infer compactness properties of infinitary languages from the existence of highly com-

174

[Ch. 3, 8 3

GENERAL RESULTS

plete, non-principal ultrafilters and extendability of filters of that kind to ultrafilters, etc., in also highly complete fields of sets. The applications of the ultraproduct construction to the model-theory of infinitary languages are, otherwise, not very abundant. This is due to the fact that EoS's theorem applies to infinitary languages only when the ultrafilters involved have strong completeness properties. Part IV of Keisler's book [3] contains a version of the ultraproduct construction which may prove suitable for applications to infinitary logic, and discusses several of those applications. An interesting study on the preservation of infinitary properties by ultraproducts modulo not-so-complete ultrafilters is contained in BENDA[l]; one of his results is discussed in Chapter 5, $4. NOTATION.If f denotes the sequence ( 4 Ij E J ) of maps belonging to IT,&, (the sets I , J need not coincide!) and if i E I , then f ( i ) denotes the sequence ( J ( i ) Y E J ) of elements of A;. If 9 is a filter on I , f/ 9 denotes the sequence ( J / 9l j E J ) of elements of I I j E , A j / 5 .If g = ( a , l j E J ) is a sequence of elements of the set A , g denotes the sequence ( qIj EJ ) of maps 3 :Z+A with constant value equal to aj. THEOREM 3.3.1 (JLoS theorem for LJ. structures of type p and let 9 be a q ~ € F o r m ( L , ~ ,p)) ( a n d f E ( n j , , l % i / ) F v( 9 ) .

n xj/9

(*)

Let { % J i E l } be a famii) of ultrafilter on I ; let Then

< K-complete

t=cp[f/9,1 w { i ~ ~ l ' 2 1 , f = c p [ €61; f(i)]}

iEI

PROOF.It proceeds by induction on the structure of cp. If cp is atomic, (*) is true by definition of the ultraproduct construction. The case where cp is a negated formula is obvious: likewise, if cp is a conjunction of < K formulas, the theorem goes through straightforwardly by the < K-completeness of 6D . Finally, suppose that cp is (30r X ) $ , where X GVar, tion of the satisfaction relation, we have: (**)

fl a i / 9 t=cp[f/9] w thereisg:X+ fl I'ujl iEI

;El

such that Let

7< K . By defini-

fl iEI

Ylj/6D

t=$[f/9 *g/6D].

Ch.3, 0 31

175

LARGER CARDINALS

Since ( f * g ) ( i ) = f ( i ) * g ( i )for i E I , we infer that M g C N . Hence, if the left hand side of (*) holds, using (**) we get M g E 9 for some g ; then N E 9 and the right hand side of (*) holds. This proves the implication (d). Conversely, suppose that N E 9; by the form of cp, for each i E N there is a map g i : X + l i 1 such that % i k # [ f ( i ) * g i ] .Let g:X-+l-IiE,l%il be the sequence ( g, I x E X ) , where: g,(i)=

i

gi(x>

any element of

lail

if i E N ,

if i E I - N .

Clearly N Mg for the particular g just defined; whence Mg E 9. By the induction hypothesis this implies the right-hand side of (**), thus proving the implication (e) in (*). COROLLARY 3 . 3 . 2 . T h e canortical m a p r:%-+91J/9is an L K K monomorphism, whenever 9 is < K-complete. Hence, up to isomorphism, %< KK%'/9. PROOF.For a E I%[, r ( a ) = Z/ 6D', where ii : Z-+I%I is the map with constant value a. We have to show that for every cp EForm(LKK( p)) and every

By EoS's theorem we have:

%'P wo)l

(*I

@

(i~Zl%+cp[j(i)])E9;

notice that since the z ' s are constant functions, we have %pcp[f]

Since

* ( ~ E Z 21+cp[ I

j ( i ) ] )#B e { i E Z I~ / = c p [j c i ) ] )= Z.

9 is proper, the right hand side of

(*) is equivalent to %Fcp[f].

176

GENERAL RESULTS

[Ch. 3, $ 3

It follows from this corollary that the ultrapower of % modulo a
THEOREM 3.3.3. Let 7 = K and 9 be a < K-complete ultrafilter on I ; if either 9 is principal or < K , then T : a+%'/ q is an isomorphism. PROOF.By the preceding corollary it is enough to show that is onto. Let f : Z+l%l; we prove that in either case there is a,€ 131 such thatfequals almost everywhere, i.e. { i E I I f ( i ) = a,} E (hence ..(ao) = a,/ 9= f / 9). If 9 is principal, there is i, E I such that for every X c I : X

E e ~ i,EX.

Whence, defining a, =f( i,), we obtain the desired conclusion:

{ i E I I f ( i ) = a,} E 62. In the other case we have

u f-'(a)=I

E 9;

UEl'UI

< K , there is a,EI%( such that f - ' ( a o ) since 9 is < K-complete and E 9 . This is the desired conclusion. REMARK.We show later that if 9 is non-principal and < K-complete, %'/9is aproper extension of every structure % of power K .

-

JLoS's theorem has further consequences that interest us: THEOREM 3.3.4. K measurable

K

weakly compact.

PROOF.Let K be measurable, q a non-principal, < K-complete ultrafilter on K , and let Z c Sent(L,,), < K , be such that every subset of power < K has a model. Let < a, ( a< K ) be an enumeration of Z and for each be a model of p< K let We shall show that:

fl a p / 9E Mod(Z). P < K

Since 9 is non-principal and < K-complete it contains the complement of each interval (51 0 < ,$
Ch. 3, $ 31

177

LARGER CARDINALS

hence the right-hand side belongs to 9; by Kos’s theorem:

n

a f l / 6 D pax

for all A <

K,

P < K

as we wanted to prove.

COROLLARY 3.3.5. If R is a well-ordering relation of ON and K E MR(AC), then K is non-measurable. In particular, all the strongly inaccessible cardinals mentioned in Corollaty 3.2.2 of $ 2 are non-measurable. Therefore, if K is measurable, there are K strongly inaccessible cardinals less than K . PROOF.Trivial, from the preceding theorem and the results of §2. KEISLER [ 11 has given a direct proof of the last assertion in the preceding corollary using the ultraproduct construction (i.e., one not involving Hanf‘s [ 11, Theorem 14.5.1, pp. incompactness results); see also BELL-SLOMSON 298-299. The same conclusion follows from (seemingly) earlier work of Hungarian mathematicians in partition calculus, but was unnoticed until Hanf produced his result (see Corollary 0.5.20, p. 56).

THEOREM 3.3.6. Suppose

is an infinite cardinal with the following property:

For each cardinal A > K , every < K-complete, non-principal filter on a set of power A’ can be extended to a non-principal, < K-complete ultrafilter,

(EP)

Then

K

K

is strongly compact.

PROOF.Let ZcSent(L,,) be such that each of its subsets of power < K has = 0, and (a, la < 0 ) be a model (no restriction on the power of Z). Let an enumeration of 2 . We can assume that 0 > K , for otherwise Z has a model by assumptionl Given a set J c 0, J < K , let aJ be a model of A a E J u ,and let

J*={.rEs,(e)lJc.r) (recall that T K(0) = { X 0 1 7 < K } ) . Let (33 = { J * IJ E (0)). We show that 3 has the property:

sK

’ Or,what is equivalent, on A.

< K-intersection

I78

[Ch. 3, 5 3

GENERAL RESULTS

In fact, if (?L = (J,* I a < 8 ) ( I 3 < K), then U,<J, E f a<sJ,* l =f % l . On 93 =,0. As it was remarked in the other hand it is easy to see that Chapter 0, Q 3.A, these facts imply that there is a complete, nonprincipal filter 9 on Yf7, ( 6 ) extending :?I. By (EP) applied to this set, 9 can be extended to a < K-complete, non-principal ultrafilter 6D. As in Theorem 3.3.4 we show that

n

0

Z J / 9 EMod(Z).

J E LP,(B)

By twisting still another bit the proofs of the last two theorems it is possible to prove-using ideas from the theory of ultraproducts-that the following, weaker extension property implies that K is weakly compact: if C8 is a
.'

(Cf. SILVER[l], pp. 27-28 for the proof.) The two last mentioned properties are actually equivalent. However, we prefer to deal with these statements by means of the simpler techniques developed in the next paragraphs.

B. Measurable cardinals Under this heading we include several model-theoretical characterizations of measurable cardinals related to theorem 2 in KEISLER[l], and other statements implied by the measurability of K. Conditions 8-1 0 of Theorem 3.3.9 below seem to be new and are due to the author. NOTATION. Given an infinite cardinal X we denote by 91A the structure ( X , X ) , GA, called the complete structure over A. The similarity type p of 91A includes a unary relation symbol R, for every X ch, whose interpretation in 'u, is the set X . REMARKS 3.3.7. (1) In Q 5 below the notion of complete structure over X is defined in a slightly different way; there it denotes the structure having as relations all finitary (not just unary) relations on A. Conditions (5)-(10) of Theorem 3.3.9 apply as well to this structure; they also apply to the structure whose relations are all relations of degree (length) < A on A. The proofs of these assertions are the same as those included below.

'

We call the reader's attention to the fact that the prime extension may be principal even if is non-principal.

5

Ch. 3, 9 31

LARGER CARDINALS

I79

(2) It is clear that each a E A is definable in %, by the atomic formula (vo),in the sense that:

a=b

%AFR{a)[bI

holds for any /3 E [%,I. This leads us to the following general considerations; (3) If % is a structure of arbitrary type p each of whose elements a is L,,(p)-definable in % by a formula qo(vo), and BE,A%, the sentence 3!voqu(vo)holds in B. Therefore we can define a map i:%+B in the following way: a A the single element of q,".

We have, then: L E M M A3.3.8. (1) Zf %=,AB and every element of [ill is LKh(p)-definable in %, the map i : %+23 is an L,.-monomorphism (henceforth called the canonical monomorphism). ( 2 ) Under the hypotheses of (l), every L,.-monomorphism f : %+B is identical with i. (3) Zf, moreover, one of the following conditions holds: (i) Z < ,.B, or (ii) % G B and every element of 181 is defined in % by a quantifier-free formula (of L A V.>)> then i is the identity map from % to 23.

PROOF.For each a € l % Jlet qu(vo) be a formula of L,,(,u) defining a in 8. (1) Let qEForm(L,,(p)) and g:X=Fv(q)+I%l; since Fv(cp)
-

(ww[

x & , ~ g ~ x , ( ~ x ~ + ~ ] ~

Since the sentence on the right hand side is in L,,, and % E ~ A B we, infer that:

%bcp"gl

* Bbcp[iogl.

This proves that i is an L,,-monomorphism.

180

GENERAL RESULTS

[Ch. 3, 9 3

(2) Let f be the said mapping. Since both f and i are LKAmonomorphisms, from a E q f we get i ( a ) , f ( a ) E q i p . Since the sentence 3 ! v q u ( v ) holds in 23, we infer that i ( a ) = f ( a ) . Therefore i and f are identical. (3) In the first case the identity id: %+23 is, by definition, an LKAmonomorphism. By (2), i = id. The case (ii) requires a slightly more elaborated argument. The assumption 'ZI G23 implies that q'= q'n I2ll for every atomic or negated atomic formula q. Therefore the same equality holds for any q which is a (possibly infinite) Boolean combination of atomic formulas, i.e., for any quantifier) equivalently, of L K A ( y ) ]I .t is clear, then, that free formula of L K w ( y [or, aEqi', since qu is quantifier-free. By the same argument as in (2), we get i ( a )= a. We prove now the main theorem concerning measurable cardinals: THEOREM 3.3.9. For every infinite cardinal K the following conditions are equivalent : (1) K is measurable. (2) Every structure of power K (regardless the number of symbols involved) has a proper LKK-extension.' (3) For every structure 8 of power K there is a structure 23 such that 2329l,1B1212l1 and 23=KK21. (4) For every structure 9l in which every element is definable by a quantifier-free formula (of L,,), there is a structure 23 such that 23=xK91and Bs%. ( 4 ) For every structure % in which every element is L,,-definable, there is a structure 23 such that B?lr,,%and 23~8. ( 5 ) 21Khas a proper LKK-extension.' ( 6 ) There is a structure %>%, such that B=KK911, and ~ B ~ # ~ i ? l K ~ . (7) There is a structure 23 such that 23=KK%K and Bz%,. ( 8 ) 91Khas a proper LKw-extension.' (9) There is a structure 8 2 %, such that 23 =K,211, and 1'23)# 191KJ. (10) There is a structure 23 such that B=Kw21K and 2 3 ~ 2 ~ . ( 1 1) For each ordinal a > 0 there is a transitive set A and an elementary ( A , E r A ) such that embedding f of ( R ( K + ~ )€, r R ( ~ + a ) into ) f ( y ) = y for d l y < K , but K
' Up to isomorphism.

Ch. 3,

5 31

LARGER CARDINALS

181

PROOF.The scheme of the proof is indicated in Diagram 1. All the implications going from the first row to the second, from the second to the third rows, from the first column to the second, and the implications ( 4 ) * (4) and (1 1) * (12) are trivial, because each statement is a particular case of the preceding in the indicated directions. We prove the remaining implications.

Diagram 1.

(1) + (2). Let 9 be a non-principal, < K-complete ultrafilter on K, let f : K+I%I be a one-one, onto map, and let 7r : %+aK/OD be the canonical We show that f/ 9ERange(7r). injection into the ultrapower 'illK/9. Suppose, on the contrary, thatf/ 9= a / 9 =..(a) for some a E I%[; this = a } E 9. Since f is one-one, this set has power means that { aE K exactly one, and therefore it does not belong to bD because d is nonprincipal; a contradiction. (3) * (4),(6) * (7) and (9) * (10). Let Q stand for III in the first case and for a, in the last two implications; likewise, h stands for K in the first two implications, and for w in the last one. Let % by any proper, L,,-equivalent extension of 6. Suppose that 8 = 6 ,say byf:Q+%. f is an L,,-monomorphism. Every element of Q is defined by a quantifier-free formula [by assumption in the first case, and by Remark 3.3.7(2) in the two others]. By (2) and (3ii) of Lemma 3.3.8, f= id : a+%. Since 1231 #[El, f is not onto, a contcadiction. (2) * (4'). This proof is almost identical to the preceding one. Let rP be a proper L,,-extension of 0. The conclusion f = id is obtained in this case using (3i) instead of (3ii). (10) (1). Let 23 be as in (10). By Lemma 3.3.8 (I), the canonical map i : A,+B is an L,,-monomorphism. Since 23 r a,, i is not onto. Identifying 8,with its image by i, we assume that %,c8 (moreover, we can assume % ,, though this is not needed). that III, i Let b E 1231 - 191Kl.Define Q 9 ( K ) , as follows:

).(fI

-

182

[Ch. 3,

GENERAL RESULTS

It is easy to show that oi) is a non-principal ultrafilter on illustration we check the following two conditions: (a) for every X c K , either X E 9 or K - X E 6 % . Clearly

K.

53

As an

Since 'P = a,, this sentence also holds in 8 ; in particular, either b E RT or bER:?,; hence, X E $3 or K - X E 9. The proof that OD is an ultrafilter only uses %=aK.In addition we use 'u, C B and b E J'PI- \%,I to show: v (b) 6D is non-principal. Otherwise, there is a E K such that for all X C K : aEX e

(w b E R T ) ;

therefore b E R Z 1 . From %,CB, we get a € R;%,!. On the other hand ~ ! ~ I ~ R hence ~ ~ ~this ( Vsentence ~ ) ; also holds in 'H. Therefore, a = b , which is a contradiction. Finally we prove that 9 is < K-complete. Let '?i .D, LT
c

n

is obviously true in 21K; then it holds in 'P; therefore: R:%=

x

n

R;.

E 91

Since by assumption b E RT for all X E 91 , it follows that b E R:?,, as it was to be proved.' (1) 3 ( 1 1). Assume (1); let ~2be a non-principal < K-complete ultrafilter over K and let 'P= ( R ( K +a ) . € ~ R ( K a+) ) . Consider the ultrapower %"/OD. By Corollary 3.3.2 this is L,,-equivalent to 2 3 and is therefore a well-founded model of the axiom of extensionality. By the ShepherdsonMostowski theorem (see Appendix A) it is isomorphic to a structure 3 = ( A , € r A ) where A is a transitive set. Let f be the composition of the L,,-embedding of 'P into W / ? D with the collapsing isomorphism c of BK/'a into 'u. Then f is an elementary embedding of '3 into 91 which

'

The technique employed in this part of the proof will be used in several theorems later in this section and again in 5 5 below.

Ch. 3, 8 31

183

LARGER CARDINALS

meets the requirements of ( 1 1). Indeed, it is easy to prove by induction that f ( y ) = y for all y < K ; the proof of K E ~ ( Kis) somewhat more involved and proceeds by steps as follows:

(i) f(~) is an ordinal number, ie., (a) x EY E f ( K ) * x E f ( K ) ; (b)x, y E ~ ( K* ) x E y v x = y v y Ex; (ii) ~ c c ( i d / 9 ) ) E f ( ~where ), id:rc-tR(K+ 1) is the identity map.' From (a) and (ii) we conclude that K c ~ ( K ) ; this and (i) imply that The details of the proof are left to the reader.

K

E~(K).

(12) =+. (1). Assume (12); let % = ( R ( K + I), E ~ R ( K 1)) + and l e t f b e an elementary embedding of 23 into a transitive structure % = ( A , E T A ) which satisfies condition (12). Let 9 = { X E P ( K ) I K E ~ ( X ) } .We claim that 53 is a non-principal complete ultrafilter over K . It is straightforward to show that 9 is a non-principal ultrafilter over K ; the only delicate point occurs in establishing its < K-completeness. Let a < K and let { Y 6 1 [ < a } c 9 . We want to show that n,,,Y,EUi). Let Y = t < a Y , ; since Y,ELTl we have K E ~ ( Yfor ~ )all [ < a , so it will be enough to show thatf( Y ) = ,<J(Y,), for then we will have K E ~Y () and Y E 6D . Let S = { ( [ , y )Iy E Y,r\[< a } ; then S E R ( K+ 1). We have: } [ ={ Y I (E,Y) Ef(S)). (iii) and (iv) now yield the required conclusion. This completes the proof of Theorem 3.3.9.

n

n

DEFINITION 3.3.10. A cardinal K is said to be II:(C: resp.)-describable iff i t is strongly inaccessible > w and there are a set U C R ( K ) and a IIi(Z:resp.) sentence u such that ( R ( K ) , E , U ) l = u but for all A < K , AEON, ( R (A), E, U n R (A)) t= i u . Otherwise, K is II:(Z:, resp.)-indescribable. IIL denotes the first IIL-indescribable, strongly inaccessible cardinal > w (if it exists). The following lemma summarizes basic information on describable cardinals:

' If

9 is normal,

K

= c(id/

"2 ); for details cf. Exercise 3 . 3 .3.

I84

[Ch. 3,

GENERAL RESULTS

53

LEMMA(HANF-SCOTT[I]). K is IIt+'-describable w there is m E w such that K is II",describable. (2) If n > 0 and T; exists, then T: is J I ; , ,-describable. (3) If n > 1, m > 0 and T: exists, then T: is Z",describable. (4) T: = T ; = the first strongh inaccessible > a. (5) If 0 < n < p , T: exists and either n < p or m < q, then T: < 7~:. It has been proved that:

THEOREM(1) (HANF-SCOTT[l]). However, (2) (Vaught) K measurable 3 T:

K

measurable

-

K

is n:-indescribable.

< K for all n,m Eo.

The second theorem implies the result of Rowbottom mentioned after can be expressed by Corollary 0.5.20, p. 56. In fact, the condition A+(A),<' a Hi-sentence which, by the axiom of choice, is equivalent to a II:statement (cf. SILVER[l], pp. 22-23).

C. Weakly compact cardinals In this paragraph we discuss the mutual relationships among several mathematical and metamathematical statements, each involving a cardinal number K ; under the assumption that K is strongly inaccessible > w each of these properties is equivalent to the weak compactness of K . Without this assumption most of the properties in question d o not seem to imply all the others (cf. Appendix E). Among the statements mentioned we shall devote special attention to those that bear a closer connection to the subject matter of these lectures (i.e., involving model-theoretical properties of infinitary languages), and confine ourselves to less detailed remarks about the remaining ones. Some implications that require machinery beyond the scope of this book will not be proved at all. To begin with. we list all the properties to be considered (the relevant definitions can be found in Chapter 0). (1) L,, is K-compact. (2) For every relational system 9l of type p satisfying: (i) 7 ~ ~ 9 % ~ (ii) every element of 1911 is LKK( p)-definable in 91, there is a structure 'F such that: (iii) 'F > KK 9l; (iv) 2?~911;

31

Ch. 3,

LARGER CARDINALS

I85 -

(v) for every rpEForm,(L,,( p)) such that 'pa K there is a b E 181- 181 and b E 'pB. ( 3 ) For every relational system 2f of type p satisfying conditions (i) and (ii) of (2), there is a structure % such that 8 =,,% and 8 z 81. (4)Every binary tree of power K has a branch of power K . (5) LKUis K-compact. ( 6 ) If 9 is a ) K . (12) For any finitary relation S on K , the structure ( K , E ~ K , Shas ) an elementary extension of the form (A, E PA, T ) , where K
r

r

-

I86

GENERAL RESULTS

[Ch. 3, 4 3

modified here to yield (3) * (4). Recall that these implications hold under the assumption that K is a strongly inaccessible cardinal W they are equivalent to one another and are implied by any of (1)-(14); under the additional assumption that K does not exceed the first measurable cardinal each of (15)-( 17) implies any of (1)-(14). Beginning with the first set of statements, we remark that the implications (6) * ... * (10) (4) are a purely set-theoretical matter. (1) * (5) and (2) * (3) are trivial. We split the other proofs in theorems and lemmas.

(5)c

(7)

c (6)

THEOREM 3.3.1 1. (1)

-

Diagram 2

(2).

PROOF.Let i?l satisfy (i) and (ii) of (2). Observe first that since F < K and every formula of LKK( p) is a sequence of length < K of symbols from a set of power < K , we have: (*)

Form(LKK( p))

<

2

K’=K,

AECN

A<

K

the equality being true because

K

is strongly inaccessible (cf. statement (IS),

Ch. 3,

5 31

LARGER CARDINALS

187

p. 7). The inequality (*) and the assumption (ii) together imply that
-

The inequality (*) implies that 7'hK9"(%)< K ; since T, A < K , it follows < K. that We now show that every Z'c2, < K , has a model. Let X be the set of a's such that 'p, occurs in 2' and Y the set of Cs such that d, occurs in 2'. Both X , Y have power < K . For cardinality reasons, for every ( E Y we have:

4;-

u qfZ0. aEX

For each (E Y let z, be an element of this set. It is easy to check that the structure (121, zl)( (i.e., where z, is the denotation of dt) is a model of Z'. Since L,, is K-compact, Z has a model 23'; let B=B'rp. From the sentences (j) it follows that 8~~~1From 21. this and hypothesis (ii) it follows, invoking Lemma 3.3.8(1), that up to isomorphism we can assume 91 < K K % . By the sentences of the form (jj) it follows that d:'€l%]-11111, and by the sentences (jjj) it is clear that d?'E@((
THEOREM 3.3.12. (2)

* (12) and (2) * (13).

PROOF.(2) + (12). The structure % = ( K , E rK,S,a),EKclearly satisfies (i) and (ii) of (2); let 23 be a structure as asserted in the conclusion of that statement. BE,,% and K > O imply that K . If A = K , it would follow from '23 >% , that R"= S (where R is a binary predicate symbol denoting S in %), and hence 23 = 111, contradicting (iv) of (2). Therefore h > K . It is also clear that

Bl{ <,R}=(A,Erh,T)>

(K,ErK,S)

188

[Ch. 3, J 3

GENERAL RESULTS

(where T = R '). Notice that (v) was not used here. (2) * (13). Let ~ = ( R ( K )E, ~ R ( K ) , # , U ) ~ ~and ~ ( ,let ) , P be the unary predicate symbol denoting U in 21. Since K is strongly inaccessible, R ( K ) = K : therefore 21 satisfies the requirements of (2). Let B be as asserted in (2). From 23sKK% and K > W it follows that B is extensional and wellfounded. By the Shepherdson-Mostowski theorem (see Appendix A), there is an isomorphismf:%r { E } + ( A , E r A ) onto a transitive set A . A simple induction on p ( x ) proves thatf(c:)= x for every x E R ( K ) ' :thus A 2 R ( K ) , and if V = f [ P " ] one concludes at once that

-

(A,Er.4.V)>

(R(K),E~R(K),U).

We have to prove that K E A . Consider in (v) of (2) the particular (Luu-) formula q(co) asserting "uo is an ordinal": rp" is the subset K of R ( K ) : by (v) there is b E I B I - R ( K ) satisfying cp; then f(b) is an ordinal (in the real world), say (Y. Since b e R ( K ) ,(Y > K . Since A is transitive and a € A , also KEA.

THEOREM 3.3.13. (5) + ( 1 1). PROOF.Assume that L,, is K-compact, where K is a strongly inaccessible cardinal >a,and let A be a set satisfying the assumptions of (1 I). Let r be the complete L,,-theory of ( A , E r A ) , r=Th"*"((A, E ~ A , Q ) ~ For ~ ~ ) . each a E A let cu be the constant denoting a. Add a new constant c to the language and let Z consist of all the sentences in r together with the (first-order) sentences (i) "c is an ordinal", (ii) c < c,, (iii) cs < c for each [ < K . Arguments similar to those of Theorem 3.3. I 1 show that = K and that each subset of Z of power < K has a model. Since L,, is K-compact, it follows that Z has a model, which by the Shepherdson-Mostowski theorem may be collapsed onto a transitive structure B = ( B , E B ) . The map f : A - + B defined byf(a)=c: for each a E A is then an elementary (in fact, an LKu-)embeddingof ( A , E r A ) into B which meets the requirements of ( 1 I); indeed, by an induction argument similar to that of the second part

r

'

Assume p ( y ) = a < K and f ( c : ) = x for all x such that p ( x ) < c r . The inclusion yLf(c,?) is ) ; transitivity of A , z E A ; therefore z = f ( w ) for some W EIBI trivial. Conversely, let z E ~ ( C ; ~by and WE~C,?". Notice that J < K ; the L,-sentence Vu[uEc,,+vxEyv~cxJholds in 'u; therefore it also holds in 9;hence w = cia for some b E y ; then p(b) < a, and by induction hypothesis b=f(c?)=f(w)=z; thus z E y . This provesf(cia)Cy.

Ch. 3,

31

189

LARGER CARDINALS

of Theorem 3.3.12 using sentences (i) and (iii) above, it is easily proved that f(a)= (Y for all (Y < K ; K < f ( ~ is ) an immediate consequence of this result and (ii).

THEOREM 3.3.14. ( 5 )

-

(6).

PROOF.We employ here a technique that has been used in a simpler form (1) in Theorem 3.3.9 (abd that will be used again in in the proof of (10) Theorem 3.3.21). Consider the structure \ Z I = ( K , X ) , ~in ~ a language p having a unary predicate R, for each X E !fi. We shall construct an -: L,,-extension ‘s, of % in which there is an element c E 0 .$ Let Z be the following set of L,,-sentences in the language p u { cs 15 < K } u { d } , where the new symbols are individual constants: EKh (1) ThKVW((%5)[ (ii) R , ( d ) for each X E T. Since by assumption < = K , it is obvious that Z has power < K . Let Z’CZ, F < Klet ; ? X = { X E 9 1 1 R ,occurs in Z’}. Obviously !Y C 9 and 3 < K . Since 5 is < K-complete, !X E 9. Since 9 is proper, 0 ‘3,# B . Let x E 0 % ; it is clear that the structure (%,E,x),,, is a model of 2’. By K-compactness of L,,, Z has a model, say 0’; let Cr=Q’rp. By the l .FR,.6 sentences (i), 6 < ,,$; by the sentences (ii), dE‘E f Let us now fix Q and c with the mentioned properties, and define, for

,

-

n

,

xE0:

Exactly as in Theorem 3.3.9 we show that 4 is a prime, < K-complete filter in . The fact that c E 0 oTR: proves that 5‘C 4 , as desired. W NOTES.(1) I t is conceivable that c E 1911, in which case is principal. (2) The preceding argument also proves (5’) * (6’), where: (5’) L,, is K-compact. (6’) If 93 is a < K-complete field of subsets of K , of cardinality K , then every < K-complete, proper filter in ?h can be extended to a
THEOREM 3.3.15. (1 1)

*

(7).

PROOF.This is very similar to that of (12) * ( I ) in Theorem 3.3.9. Let ( : ~ be a < K-complete field of subsets of K containing all singletons such that

190

[Ch. 3, $ 3

GENERAL RESULTS

-7

= K . Let A be the least transitive set such that (i) K E A , (ii) % j A , (iii) { ( x , , $ ) I x E ~ ( , $ ) A ( < ~ } E A for all a < and ~ al1fE‘:V. Since K is inaccessible, a simple computation shows that A= K . By ( I l), there is a transitive set B and an elementary embedding f of ( A , E r A ) into ( B , E r B ) which moves K but no ordinal below K . Let O i l = { x €
90

GI;.

rn

THEOREM 3.3.16. (6)

* (7) * (8) * (9) * (10)

3

(4)

PROOF.(In sketch except for the fourth implication). (6) * (7). Let l.6 be a field of subsets of K satisfying the requirements of (7). It is easy to check that the set 5 = { Y E % I K - Y < K } is a < K-complete, proper filter in 93 . By (6) 9 can be extended to a < K-complete, prime filter G . G is non-principal because 9 contains the complement of each singleton, and is an ultrafilter because it is prime. (7) * (8). This is a variant of the proof of Ramsey’s Theorem 0.5.6 using ultrafilters. Suppose [ K ] ’ = C,+ C,. For each x E K , let M ( x ) = { y 1 ( x , y } E C,}. Let
xtE M(x,)

@

M ( x q )E 4.

Let S = { X , ~ ~ < K and } S’={~ESIM(~)E~’D It }is. an immediate consequence of (*) that x,y E S‘ implies { x , y } E C,, and x,y E S- S‘ implies { x , y } E C , . Hence the sets S ’ , S - S ’ are homogeneous for the partition (Co,C,); since ‘s; = K , one of S ’ , S - S ’ has power K . (8) * (9). This is proved by an argument similar to that used in Lemma 0.5.10. (9) * (10). Let % = ( A , T ) be a tree of power K in which every level has ’ 1 has only one root. For otherwise power less than K . We can assume that 1 we can chop the forest down to < K disjoint one-rooted trees, one of which has to have power K (by regularity of K ) ; any branch of power K of this subtree is the branch of 2l we are searching for.

Ch. 3, f 3)

191

LARGER CARDINALS

We shall define an ordering < on A extending T such that, for each x E A , the set R ( x ) = { y E A IxTy} is a <-segment, i.e.: u < u < w a n d u , w E R ( x ) imply

-

u€R(x).

Firstly, by induction on a, we define a total ordering with the following property ( x , y E La):


on the level La

f o r a l l y < a and allz,wELy,if zTx, wTy a n d z f w then z < u w . If < p has been defined for all p< a, and x,y E La,we define x <,y as dictated by (**): Case (1). If there are T-predecessors z,w of x,y resp. belonging to the same level and z # w, then we set x < uy if and only if the right hand side of (**) holds. Case (2). Otherwise, define <, in an arbitrary way. Condition (**) holds because in case (2) the right hand side is vacuously true. Secondly, we define < as follows ( x , y E A ) :

(**)

x
x
(i) h ( x ) < h ( y ) ' and there is w ELh(,) such that wTy or w = y , and x
We omit the tedious, but not difficult, verification that each R ( x ) is a segment with respect to <. By assumption ( A , <) has a subset C of power K which is well-ordered or conversely well-ordered by <. By chopping off a few elements from C we can assume (C, < YC) has type K or type K * ; we assume it has type K , the other case being symmetric. Let us define B = { x E A I R ( x ) contains a

< -final segment of C } ;

we prove: (a) B is a linearly ordered subset of ( A , T). Suppose not, i.e., that there are T-incomparable elements x,y E B. Clearly R ( x ) n R ( y ) = 0 . On the other hand, if D,,D, are tails of C contained in R ( x ) , R ( y ) respectively, D ,n D,#B because C does not have a last element. This contradicts the preceding sentence.

' Recall.that

h ( x ) denotes the height of x.

I92

[Ch. 3,

GENERAL RESULTS

53

= K . We show by induction that B n L, #fl for all a < K . Suppose (b) true for all ,8 < a. For each x, E B n L, (note: such element is unique by (a)) let D, be a tai1 of C contained in R ( x , ) . The regularity of K implies ,
DGL,U

u

R(y);

Y E La

-

-

since < K , there is y EL, such that D n R ( y ) = K ; for cardinality reasons this set is cofinal in D ; since R ( y ) is a segment, it contains a subtail of D . Therefore y E B , and B n L, #fl. (10) (4). Consider a binary tree % = ( B , T ) of power K ; we have to show that it has a branch of power K . It is clear that the length of a tree is the supremum of the length of all its branches. Hence, if ’3 has length > K , it has a branch of length (and hence power) > K . So we consider 23’s of length < K . Inductively we prove that all levels L, ( a < K ) of 23 have power < K , and apply (10). For each x E L,( ,8 < K), the set S, = { y E B ( yT x } is a branch of the subtree of 23 defined by {yEBIh(y)
u La. a
By induction hypothesis this subtree has power < K and therefore has branches, because K is strongly inaccessible. Hence < K . rn Our next result is:

THEOREM 3.3.17. (3)

-


(4).

PROOF.By way of contradiction, suppose that ( B , T ) is a binary tree of power K with all its branches of power < K . The structure ’11 = ( B , T ,b ) b E B obviously satisfies the hypothesis of (3). Consider the following set of sentences of L K K : ’ (A) “ < is a tree-like partial ordering” (cf. Chapter 1, $2. Example 1.2.20). (B) “< defines a binary tree”, i.e.. 3xVy(x
’

< is the symbol corresponding to T ; Suc(x,y) stands for “y is an immediate successor of x in the partial ordering < ”.

Ch. 3 , f 31

193

LARGER CARDINALS

and v x 3 y Z [ s U C ( X , y ) A sUC(x,z) A vw(sUC(x, w ) + w x y V wxz)]

(C) cb,< cb, for b,,b, E B, 6, Tb,. (D) i (cb,< cb,) for b,,b,E B such that not (6, Tb,). (E) 3!x Suc(c,,x) for each b E B having only one T-immediate successor. (F) i (3o)AbEC(o> cb) for each branch C of ( B , T).l (G) V a [ o < c b + V a T b ( l ; x c U )for ] each b E B.' We shall show that every model 0 of (A)-(G) is isomorphic to 2.This contradicts (3), thus proving (3) * (4). Let 1 0 1= C, <"= S ; it is clear from the sentences (C) and (D) that { cf I b E B } is an isomorphic copy of ( B , T); so we assume (C, S ) extends ( B , T ) . Let c E C- B ; we shall obtain a contradiction, thus showing ( C , S ) = ( B , T ) . Since c JZB , c is greater than the root of ( B , T ) ; therefore X = { b E B I bSc} # B . X is a totally ordered subset of ( B , T ) ; then either: (i) X has a last element, say b,. If there is b E B such that xTb for every x E X , let b, be the least of such b's. By (G), not cSb,. By (B), c and b, are all the immediate successors of b, in ( C , S ) : since c F B , 6, has exactly one successor, b,, in ( B , T ) . Using (E) we reach a contradiction. If, on the contrary, there are no b E B with the preceding property, it follows that X is a maximal linearly ordered subset (i.e., a branch) of ( B , T ) ; since xSc for all x E X , we contradict (F); or, (ii) X has no last element. Using (F) it is easily seen that X is a branch of ( B , T ) ; we argue as before and obtain a contradiction. We close our discussion of the statements (1)-(14) by proving the following:

THEOREM 3.3.18. (4)

c

*

(5).

PROOF.Suppose Z Sent(L,,( p)), Z has power K , and that every subset of Z of power < K has a model. Cleary we can assunie that ,Z < K , and taking into account that p # < K we can also assume that p is endowed with Skolem functions: by adjoining to Z all the defining axioms for these Skolem functions we can finally assume that Z is tidy. Let { u p I ,8 < K } be an enumeration of Sent(L,,(p)). For each ordinal a < we ~ define 2, = Z n {up I ,B < a } . Each Z, has power < K and thus it has a model.

' Notice that the assumption that all branches of ( B , T ) are of power < ( F ) and ( G ) are L,,-sentences.

K

implies at once that

194

[Ch. 3, f 3

GENERAL RESULTS

We define a set P C 2 < " . Note that e a c h p E 2 < " is a mapp:a-+{O,l} for some a < K ; such (unique) a is called E(p). For p E2<" we set: p E P w there is 3 EMod(Z,(,,) such that for every p < P(p), pp= 1

iff

%t-up,

and pp=O

iff

%F lap.

We order P by the relation of extension: for p , 4 E P :

We show that (P, < ) is a binary tree. (1) (P, < ) has only one root: the empty sequence A. (2) If q E P, the set S, = { p E P Ip < 4 ) is well-ordered by < . In fact, we note that if p E P and a < t ( p ) , then p ra E P; therefore S, = { 4 ra I a < t ( 4 ) ) and S, is well-ordered in type E ( 4 ) . In particular, the height of 4 in the tree (P, < ) coincides with e(4). (3) Every 4 E P has at most two immediate successors: 40 and 4 1. (4) If p E P and u!(,) E 2, then p has exactly one immediate successor. The converse implication holds whenever Z is closed under t= . In fact, if u,(,).EZ, it is clear that % t = ~ , ( ~ ) for every IZIEMod(Z,(,,+,). Conversely, if Z is closed under F and u , ( , ) g 2 then both 2 u { u , ( ~ ) and } Z U { iut(,)} have models, which implies that both p 0 and p l are in P; therefore p has two immediate successors. Since Z has power K , P has elements of length arbitrarily close to K ; therefore > K . On the other hand 2<"= K because K is a strong limit cardinal; hence P has power K . By assumption, P contains a branch C of power K . Let p , = UpEd : K+{O, l } be the sequence of length K determined by C. p , gives raise to a set Z, LSent(L,,( p)) in an obvious way:

for if up € 2 then pp = 1 for every p E P such that Notice that Z CZ, t ( p ) > p; therefore ( P , ) ~ = 1 and up E Z ~ In . particular, since Z is tidy, so is Z., The same argument shows that Z, contains all valid sentences. Observe also that Z, is complete, i.e., for every u E Sent(L,,( p)) either ~ € 2 or, TUEZ,. Now we construct a model for Z ,, using the same idea as in the model existence theorem for LUI, (cf. KEISLER[3], Part I, lecture 3) and as in Henkin's proof of the completeness theorem for L,,. Given t l , t 2€Term,

Ch. 3, f 31

195

LARGER CARDINALS

we define:

= is an equivalence relation, and we define a model 91 with universe Term,/ = by setting: ( t , / = ,..., t , / = ) E R '

'3

f "( t ,/ EE ,. . .,t m / =)=f ( t

R ( t l,..., t , ) E Z ,

,,...,t m ) / =

c'=c/-

for given relation, function and individual constant symbols R ,f,c, respectively. Using appropriate instances of equality axioms it is easily seen that R I, f I,c', are well-defined. The property

(**>

i?l F= cp [ t l / = ,. ..,t,/

-3

'3

cp( t , , . ..,t,) E 2,

(where cp E Formn(L,,( p ) ) ) shows, in particular, that 91 E Mod(Z,). (**) is proved by induction on formulas; the atomic case is immediate from (*); the connective cases are simple consequences of completeness and closure of Z, under infinitary conjunctions. So let cp = 3 x q . Since Z, is tidy, there is a term T ( o ~.., . u,) such that: hl...Un[3X\Il(Ul

Hence if cp(t,,..., t,)EZ,, hypothesis we have

,..., U , , X ) + \ k ( U , ,..., O , , T ( U l ,..., U , ) ) ] E Z , . then \ k ( t ,,..., t , , T ( t , ,...,t,))€Z,;

91k ' k [ t , / - ,..., f , / - , T ( t l

which implies 8 t= cp[ t , =,. ..,t,/ half is trivial.

-1.

by induction

,...,t,)/E],

This proves one half of (**); the other

REMARK. (4) + (1) is proved by a similar argument which, of course, requires the introduction of infinitary Skolem functions, terms with infinitely many arguments, etc. Since we have not developed in this book the theory of these infinitary notions, we leave as an exercise for the reader to reconstruct and prove the infinitary analogs of those definitions and lemmas of $2, Chapter 2 necessary to prove (4) + (1). We turn now to the statements (15)-(17) mentioned in p. 186. The implications (1) + (15) + (16) are obvious. The implication (16) 3 (17)

196

[Ch. 3, 3 3

GENERAL RESULTS

is proved by the argument of Theorem 3.3.14, as it has been remarked before Theorem 3.3.15. (17) + (15) requires an argument similar to that of SILVER’S[ I ] in proving ( 6 ) 3 (1);

THEOREM 3.3.19. (17) + (15). PROOF (In sketch). This is a variant of the well-known proof of the compactness theorem for L,, using ultraproducts. Let Z be a set of L,I,I-sentences of power K and similarity type p, such that every subset of power less than K has a model. In a manner similar to that of Chapter 2, g2.A it is possible to introduce (infinitary) Skolem functions and prove analogs to many of the results of that section. In particular, each existentially quantified variable can, in this way, be replaced by a proper function symbol. As in Lemma 2.2.5 we may assume, without loss of generality, that 2 is a set of universal sentences in the enlarged language p # . Since we need only eliminate existential quantifiers -

for sentences in Z, we may also assume p# = K . Let ??, ( 2 )denote the family of all subsets of Z of power < K . For each A E ??, (Z) pick a model %A of A; furthermore, choose 91A in’ such a way that each of the new function symbols denotes a total function (with denumerable many arguments). with the operations and relations Consider the product llAE -

epK

defined pointwise. Since p # = K and K is strongly inaccessible, there is a subset D of IIAE,N(,,21A,of power K , closed under all the operations of p # . For each pair (q,f), where rp is a quantifier-free formula of L,,J p # ) and f E D Fv(9’), we define:

For each A E c ? , ( Z ) let i = { A ’ E C ? , ( Z ) l A c A ’ } . b e t i.h be the < K complete field of subsets of 9,(Z) generat4 by all A’s and all J,;s. Since K is strongly inaccessible it turns out that G % = a ~ (cf. $ 3 , Chapter 0). Observe that the intersection of less than K A’s is again a sct of the form and thus non-empty. By the results of $ 3 , Chapter 0, {A ~ A E”?, ( 2 ) ) generates a < K-complete, proper filter 3 of ’$ . By hypothesis :T can be extended to a
i,

f-g

w {Alf(A)=g(A)}

Ed:

Ch. 3 , 3 31

197

LARGER CARDINALS

we write f/$ for f/- (fE0 ) .As in the ultraproduct construction we define a structure E by setting:

l%l=D/=: ( f I / g , . . . , f n / e ) E R D w (Al(f,(A) ,...,f,( A))ER’A}E for each n-ary relation symbol R of p * ;

Fa( f,/ 9,.. .,fm/ 9,. . . ) = (F’A( f,(A), .. .,fm (A), . . .) I A E ‘J?K(,X))/ $ for each function symbol F of p # . I t is easy to prove that for each quantifier-free formula and each f E D Fv(p)

(*)

Oi) ‘9,[f/‘]

@

{AEgK(Z)I,UAk9,[prAof]}€S

@

9,

of LUl,](p#)

Jp,fEG.

Given u E 2, let $ be the quantifier-free formula obtained by suppressing all universal quantifiers in u. Let f E D Fv(#). Since u contains only universal quantifiers, for each A E 9K(2) we get: UEA

*

~ A F U =$’

ZAk$[prAof],

c

or, in other words, { 6} J+,f. But { 6} E , by construction; therefore J+,,E &? ; by (*), Dk$[f/G]. Sincef/S is an arbitrary element of [%I, it follows that Oi) = u. Therefore 9 E Mod@). The most interesting of Bell’s results is that each of the statements K does not exceed the first measurable cardinal. This is proved by a variant of the argument used in the second part of Theorem 3.3.12; it also uses the implication (12) * (1) of Theorem 3.3.9.

( 15)-( 17) is equivalent to any of (1)-( 14), prooided

THEOREM 3.3.20. If K is strongly inaccessible and does not exceed the first measurable cardinal, then (1 5 ) * (1 3). PROOF.Consider the structure %=(R(K), € r R ( K ) ,U , a ) o E R ( and K ) the following set Z of L,I,I-sentences of type { E , P } u { co I a E R(K)}u { c } , where P is a unary predicate symbol and c is a new individual constant: (I)

Thwiw1(21),

(11)

“c is an ordinal”,

(111)

c,Ec

foreache<~.

I98

GENERAL RESULTS

[Ch. 3,

53

It is clear that every subset 2' of 2 of power < K has a model; namely ( R ( K ) ,E rR(K), U , U , [ ) , ~where ~ , Y is the set of all u E R ( K )such that c, occurs in Z', and ( is any ordinal < K larger than all the ordinals occurring in Y . 8, say by a map g (cf. By (15) 2 has a model, say q. Clearly % 5 01W1 Lemma 3.3.8). Since 8 { E } is extensional and well-founded (by (I)), the Shepherdson-Mostowski theorem shows that there exists a collapsing isomorphism h from 8 f { E } onto some transitive structure ( A , E r A ) . In , ~ R ( K )into ) particular, f= h o g is an elementary map from ( R ( K ) E ( A , E r A ) . Since K does not exceed the first measurable cardinal, condition (12) of Theorem 3.3.9 implies that !(a) = (Y for all (Y < K ; a simple induction argument proves, then, that f is the identity.' Therefore R ( K )G A ,

r

( A , E rA, V )

f

?

( R ( K ) ,E ~ R ( K u> ),

(where V = h [ P B ] ) and , h(c%) is an ordinal in A with at least K predecessors; this, together with the transitivity of A , implies that K E A . H

REMARK. (1) Notice that f is an LwIw,-map; only the elementary character off was used in the proof. It is plausible that the infinitary properties off could be used to push up even further the first ordinal, moved b y f ; this would improve Bell's result. In this respect we observe (Theorem 3.3.12) that i f f is an L,,-map, f is the identity without additional assumptions. (2) If V = L is assumed as an axiom of set theory, then there is no measurable cardinal, and the implication (15) * (13) holds for all strongly inaccessible K. But this may even be true in a universe of set theory with measurable cardinals; in fact, J. Paris has proved (unpublished, oral communication from J. Bell) that this is the case if V = L , , whenever, 9 is a normal ultrafilter on a measurable cardinal. (3) Note that (15) * (13)fails for many K'S larger than the first strongly compact cardinal K,. In fact, Lwlwlis A-compact for any cardinal X 2 K,. But (13) fails for the first strongly inaccessible after K,, say A,; by Hanf's results (§2), A, is not weakly compact; moreover, in the enumeration 0 of all strongly inaccessible cardinals, A,= 8,+ ,. This remark is due to P. Aczel. COMMENTS. (1) Property (2) considered in this section has a variant;

'

Suppose p(y)= a < K and f ( x ) = x for all x such that p(x)< a. The inclusion y Cf(y) is trivial. Conversely, let z E f ( y ) ; since the rank function is definable and f is elementary, p ( f ( y ) ) = f ( p ( y ) ) = f ( a ) =a ; hence p(z)< a and by induction hypothesis f ( r ) = r ; therefore f ( z ) E f ( y ) ,which implies z E y .

Ch. 3,

3 31

LARGER CARDINALS

I99

namely that obtained by changing L,, for L,, everywhere. This version seems to be weaker than (2). (2) Some of the preceding conditions imply inaccessibility; for instance (8) [by Theorem 0.5.11, Chapter 01, (14) [cf. SILVER[l], Lemma 1.14 ] and (9) [left as exercise for the reader]. Others do not, e.g. (10) [cf. Appendix El. (3) Property (8) and Theorem 0.5.19 of Preliminaries yield at once another proof of Theorem 3.3.4 above. This can be proved as well using (14) and the result of Hanf and Scott mentioned after Definition 3.3.10. The result of Vaught cited in the same page shows that the first weakly compact, strongly inaccessible cardinal > w is strictly smaller than the first measurable cardinal. However, KEISLER-TARSKI [ 11 (Theorem 3.5, p. 277) prove a much better result: T H E O R E MEach of the classes M(SI), M(m)(SI), M(m)(M(")(SI)), (M("))(")(SI),.. . consists entirely of non-measurable cardinals. In particular, any measurable cardinal K is the K ' ~ weakly compact, strongly inaccessible cardinal.

(For notation, see Definition 0.2.1 1, Chapter 0, and 0 1 of this chapter. Recall (Definition 3.1.2(d)) that SI denotes the class of all strongly incompact cardinals.) D. Strongly compact cardinals We discuss here a characteristic property and the size of the cardinals K for which L,, and L,, are (strongly) compact, i.e., A-compact for every infinite cardinal A. We have already shown, using the method of ultraproducts, that the following extension property for < K-complete filters implies the compactness of L,,:

(EP)

For every cardinal A > K , every non-principal, filter on A can be extended to a non-principal, ultrafilter on A.

< K-complete < K-complete

Conversely, the next theorem proves, with much the same technique as Theorem 3.3.14 above, that the compactness of L,, implies (EP). THEOREM 3.3.21. L,, compact + (EP). PROOF.Let A > K and 9 be a non-principal, < K-complete filter on A. Recall that a non-principal filter 9 is one where 96Z 5;in particular 9 is proper. Our first claim is:

200

[Ch. 3, 5 3

GENERAL RESULTS

n !7 =B.

CLAIM.We can suppose that For if F#B,we define:

3= ~ u { A - { ~n } ~s}, ~ E and check that 3 is a basis for a < K-complete filter on A. To see this we have to prove that GIL c_ 3 and < K imply 91 #B . Each such family 9L can be decomposed as follows:

n

% = ( % n 9)u {A-

{ u }l

u ~ x } ,

where X c n 9 and < K . Clearly we have: fl 91 = n (b n 9)n (A- X ) ; therefore, if we prove that n (91 n T)II X , it will follow that fl '? #B. l, Since % n 9< K and qL n 9c 9, by < K-completeness of 9, w e have: n ( % n 4 ) F~ ; since n 9 @ T , it follows t h a t X c_ n 9c ( % n 4). As it was proved in $3, Chapter 0, each such family %I can be extended to a proper,
n 93 =(n9)n ngEnq(A-{u})=(n \T)n(A- n 9. Since 91 C 9,we get n 9 =B. Thus the claim is proved, for if 9 can be

extended to an ultrafilter as asserted in the theorem, so does

9.

Let Iu, be the complete structure over A. We will construct a proper L,,-extension 'B of 'ZI, in which there is an element b E fl FR:. Let p=Sm(Iu,) and let C be the following set of sentences in the language p u { d } , d a new individual constant: (i) Th"."('ZIA), for X E 4. (ii) R,(d) Exactly as in Theorem 3.3.14, it is proved that every subset of Z of power < K has a model. By compactness of L,,, Z has a model, say 8'. By the sentences (i), 'B=B'rp is L,,-equivalent to %;, by the sentences (ii), d B ' E ,,,R;. Since every element of IIu,, is L,,( p)-definable in %A, by 9=il Lemma 3.3.8( 1) we can assume that Iu, < K,'B. Observe also that implies at once that:

n

b € fl RT X € 9

* bEl'BI-lflAl.

Let us now fix 'B and b with the aforementioned properties, and define, for X gA: X E 9 w bER:.

Ch. 3, 4 31

20 1

LARGER CARDINALS

As in the proof of (10) * (1) in Theorem 3.3.9 we show that 9 is a non-principal, < K-complete ultrafilter [note: the facts that 9 2 '21A and b E 1231 - 1u ' .J are essential to prove that 9 is non-principal. The fact that bE X E & : shows that 9C 9. This completes the proof of Theorem 3.3.21.

n

COROLLARY 3.3.22. L,, is compact w L,, is compact. PROOF.Obvious from Theorems 3.3.21 and 3.3.6.

-

It is easy to see that if K is a regular infinite cardinal there exist nonprincipal, < K-complete filters on K (for instance, 4 = { X c K I K - X < K } is one such filter). Therefore, the extension property (EP) implies that K is measurable; hence so does the compactness of L,,. (Note: the restriction " K regular" can be omitted from this assertion, since we know (Theorem 3.1.4) that L,, is incompact whenever K is singular). Below we give a proof of the same fact which is more in the spirit of this book insofar it only uses model-theoretical ideas.

THEOREM 3.3.23. L,, compact

-

K

measurable.

PROOF.We will show that L,,-compactness implies (8) of Theorem 3.3.9. By Theorem 3.1.4 we can (and do) assume that K is regular. Let 'u, be the complete structure over K , p=Sm(%,), and Z be the following set of sentences in LKw( p u { d } ) , where d is a new individual constant: ( 9 Th""W,), (ii) i R o ) ( d ) for 5< K .

Let Z'CZ, F < K . Let x = { < K I R { ~ ) occurs in z'>. Clearly
X

Thus, strongly compact cardinals are very large. The natural question that comes to mind is: how do compact cardinals compare with measurable cardinals? Obviously, the answer depends upon which set-theoretical [ 1J proved the following natassumptions are made. VOPENKA-HRBAEEK ural and elegant generalization of Scott's Theorem 0.4.29:

202

[Ch. 3, !j 3

GENERAL RESULTS

THEOREM (Vopenka-HrbaEek). I f there exists a strongly compact cardinal, then V Z L , for evety ser a.‘ Taking into account that if K is a measurable cardinal and 9 is a non-principal, < K-complete ultrafilter on K, we have

L,

+ V = LhDA

“K

is measurable”,

their theorem proves: If ZFC + “there is a measurable cardinal” is consistent, so is COROLLARY. ZFC “there is a measurable cardinal” +“there is no strongly compact cardinal”. I n particular: ZFC + “there is a measurable cardinal” k“measurable strongly compact”.

+

+

Certainly it is more significant to ask whether the implication “measurable * compact” holds in the set theory ZFC+“there is a strongly compact cardinal”; this problem is still unsolved (although Kunen has shown that the statement Consis (ZFC + “there is a measurable cardinal”)

* Consis (ZFC + “there is a strongly compact cardinal”) cannot be proved in ZFC). The related question whether the first strongly compact cardinal equals the first measurable has been partially solved by MAGIDOR111:

THEOREM (Magidor). I f ZFC+“there is a strongly compact cardinal” is consistent, then ZFC+ “the first strongly compact cardinal equals the first measurabfe cardinal” i3 also consistent. The question whether “equals” can be replaced in the above by “is larger than”, is still open.

’ Vopenka and Hrbacek use the following statement instead of the property (EP): (+)

Every non-principal, < K-complete filter on < K-complete ultrafilter.

K

can be extended to a non-principal,

This property seems to be weaker than strong compactness. Indeed, as Bell bas pointed out, I is, ‘L“-compact”. , the argument of Theorem 3.3.21 proves that (+) follows from ‘

Ch. 3, § 31

203

LARGER CARDINALS

Several other results along these lines have recently been discovered. They relate strongly compact and measurable cardinals with more special notions such as those of “supercompact” and “extendible” cardinals. These results and the techniques used to prove them fall beyond the scope of this book. See MAGIDOR [l], [2], MENAS[l], STERN[I].

EXERCISES

-

3.3.1. Let K be a strongly compact cardinal >a. Show that for each pair of ordinals a , ,8 such that /? > O there is a transitive set A and an elementary embeddingfof ( R ( K + ~ ) E , r R ( ~ + p )into ) ( A , E r A ) such thatf([)=[ for all 5 < K , but f ( ~ > ) a. [Hint: argue as in the proof of ( 5 ) (1 1) for weak compactness (Theorem 3.3.13).] 3.3.2. Define the predicate In(K,E) In(K,O) In( K , [ + 1 )

(“K

is inaccessible of type f ’ ) by:

* K is (strongly) inaccessible, (va < K)(gp < K ) [ a < p A In( p,5)]

In( K , A)

(W
Use condition (13) for weak compactness on p. 185 to give a modeltheoretic proof of the following weak form of Theorem 3.2.1: If K is a weaklv compact, strongly inaccessible cardinal > w , then In(K,K). [Hint: let A be a transitive set such that K E A and ( R ( K ) ,E ~ R ( K )< ) ( A , E r A ) . By induction on 5, show that In(K,() for all (< K as follows. We have In(K.0) by assumption, and the induction step for limit 6 is trivial. Assume In(K,,$) with ( < K . Show that ( A , E r A ) k In[K,[], and deduce that, for each a < ~ , the formula “there is an ordinal /? such that a < /3 and In( p.5)” holds in ( A , E A ) , and hence also in ( R ( K ) ,E 7 R ( K ) ) . This gives the induction step from ,$ to t+1.1 Change slightly the preceding argument to show: If K is a weakly compact cardinal (but not necessarily strongly inaccessible!), then K is a rK-number;in particular, K is the K ‘ ~weakly inaccessible cardinal. [Hint: use Corollary 3.1.6 to begin the induction. For the definition of the T~ numbers, see Chapter 0, $2.1

r

3.3.3 (Normal ultrafilters). Recall that a < complete, non-principal ultrafilter Lg over a cardinal K is said to be normal if w h e n e v e r f E ~ *and { a < K I f ( a )< (Y}E 4, then f-’[a] E O D for some (Y < K .

204

GENERAL RESULTS

[Ch. 3, 8 3

(a) Let 9 be a non-principal, < K-complete ultrafilter over a cardinal K . Show that 9 is normal iff for each transitive set A such that K C A , the ~ in the ultrapower ( A , E r A ) " / G I ) , element i d / 9 is the ' ' K ~ordinal" where id is the identity function on K. (b) Let f be an elementary embedding of ( R ( a ) , E r R ( a ) ) into a transitive structure ( A , E PA) which moves an ordinal, and let K be the first ordinal w w e d by f . Show that { x c K I K Ef ( x ) } is a normal ultrafilter over K . Deduce that each measurable cardinal supports a normal ultrafilter. (c) Let 9 be a normal ultrafilter over K and let A be a transitive set such that R ( K + I ) c A . Let f be the canonical elementary embedding of ( A , E A ) into the transitive structure isomorphic to the ultrapower ( A , E r A ) " / 9 . Show that 9= { x c ~ I ~ E f ( x )[Hint: }. use (b) to show that 9 c { X C K I K E f ( X ) } . ]

r

3.3.4 (Ramsey cardinals). Recall that a cardinal X satisfying h+(X),<" is called a Ramsey cardinal. Each measurable cardinal is Ramsey (Theorem 0.5.19). Let Ram(x) be the formula of set theory which asserts that x is a Ramsey cardinal. (a) Show that, if A is transitive and R(X+ l ) c A , then Ram@)

* ( A , E rA)i= Ram[h].

Let K be a measurable cardinal > a and let 9 be a normal ultrafilter over K. Let ( A , E A ) be the transitive structure isomorphic to the ultra+ and let f be the canonical elementary power ( R ( K l), E ~ R ( K l))K/62, embedding of ( R ( K + l), E ~ R ( K 1))+ into ( A , E r A ) . (b) Show that R ( K + 1 ) c A . [Hint: let j be the collapsing isomorphism of ( R ( K + I), E rR(K+ 1))"/9 onto ( A , E PA), let X = {x61E< K } c R ( K ) , and let t E R ( K 1>" be defined by t ( [ ) = {x, lq <[} for each 5 < K . Show that X = J ( t / 9 ) . ] (c) Let P = {A< ~lRam(X)}.Show that K E ~ ( P [Hint: ). use (a), (b), and the facts that K is Ramsey and f ( K ) > K.] (d) Show that P E 9. [Hint: use (c) and Exercise 3.3.3 (c).] Thus, in particular, each measurable cardinal K > O is a limit of Ramsey cardinals, i.e., there is a set of K Ramsey cardinals below K .

+

+

3.3.5 (Reinhardt). Let a EON, a > 0. A cardinal K is a-extendable if there is an ordinal fi > O and elementary embedding f of ( R ( K + a), E rR ( K a)> into ( R ( K + ~ )€ ,[ R ( K + ~ such ) ) that f ( [ ) = t for all ( < K , but f ( K ) > K . Show that any a-extendable cardinal is measurable, and that there are K measurable cardinals below K . [Hint: argue as in Exercise 3.3.4 (c), (d).]

+

Ch. 3, § 41

THE DOWNWARD LOWENHEIM-SKOLEM

THEOREM

205

3.3.6. (a) (Erdos-Hajnal) Let A be a cardinal satisfying 2'=Axo. Show that there is a function F:X"+A such that whenever A CA and 7 =A, we have F [ A " ] = A . ( F is called a scattering function for A".) [Hint: let ( ( A , , ( , ) [ a < 2') enumerate the set { X LA I =A} x A; by induction on a define a sequence such that s, €A," - { sp I /3 < a } , and set F(s,) = L.] (b) Prove Kunen's theorem: If 8 is an inaccessible cardinal >a, then the identity map is the only elementary embedding of ( R ( 8 ) , E l R ( 8 ) ) into itself. [Hint: suppose f is a non-trivial elementary embedding of ( R ( 8 ) , E r R ( 8 ) ) into itself. Then f moves an ordinal; let K be the least . ordinal moved. Let f , = f , f n + l = f o f f l and let A = s u P { ~ , ( K ) I ~ E u }Then A E R ( 8 ) and f ( A ) =A. Each f , ( K ) is measurable, hence (strongly) inaccessible, so 2'=AKo. By (a), let F be a scattering function for A"; then f ( F ) is also a scattering function for A", so that, if A = { f(()IE
§ 4. The downward Lbwenheim-Skolem theorem

In this section we prove the downward Lowenheim-Skolem theorem for the languages LKA.We begin by proving a version which generalizes all the particular forms of the theorem known from the literature; these forms are presented as corollaries. The proof is a refinement of the Tarski-Vaught [ 11). version of the theorem for finitary languages (see TARSKI-VAUGHT We go on to show by means of suitabIe counterexamples, without using the GCH, that the bounds given by Theorem 3.4.1 are best possible. We conclude the section with several mathematical applications of the downward Lowenheim-Skolem theorem. We shall use the notations and concepts introduced in Chapter 1, $3. A. The main theorem and its corollaries

THEOREM 3.4.1. Let 23 be an infinite structure of type p, let X 1231 and let r be a set of LKA(p)-formulas closed under subformulas ( i . e . , S b f ( I ' ) C r ) ; let p = sup {No, Fv( cp) I cp € r},v be a cardinal > 2 and r a cardinal > p. Assume

-

that one of the following alternatives hold:

206

[Ch. 3 ,

GENERAL RESULTS

for a N c p E r , p < c f ( ~ )and , max{ Then there is a structure 2l such that X

54

F,F , F } < v C T < % .

c 181, 9i < iA2 and: ?

in case (I), E = u T (in particular there is a model of power vp); in case (111, E = v<'. Furthermore, 8 can be chosen so that 8Fcp"gl

'3

for every cp E r and every g E 1911F"(v).

~~cpcpcl

PROOF.By the assumptions we can clearly suppose that Sm(T)= p, i.e., that every symbol of p occurs in some (atomic) formula of r. Let G be a fixed choice function for the family

U {C7"(DZ)-{kT}IZGVarandDG18~}. For cp E r, Z

c Fv(cp) and f E 1 8 1 ' it is convenient to define: cpB(Z,f, . ) = { gEliplF"'v)-Z IB t= cp [f*gl}.

Given Y C 1231 we denote by cl( Y ) the least subset of 1 8 1containing Y and closed under all operations in p ; notice that cl( Y ) contains all elements c', where c is an individual constant of p ; Y is closed if cl( Y ) = Y . In the course of the proof we shall use the following fact, which is proved by a simple computation:

CASEI. We shall inductively define a sequence ( D , I a < p + ) of subsets of 1 8 1with the following properties: (i) D, = v T for all a < p + ; (ii) D , c Dp for all a < , f 3 < p + ; (iii) D, is closed for all a < p + ; (iv) for every (Y < p +, cp E r, Z Fv(cp) and f E D,", the following holds: if cp'(Z,f, .) #B, then there is g E cpB(Z,f,.) such that Range( g ) c D,+ ,. Let Do be any closed subset of 1 8 1 , of power v T , including X . Assume D, has been defined for all (Y
c

yt= { ( c p , Z , f )Icp E

A Z C Fvcp r\f E DF-, } ;

Ch. 3,

5 41

201

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

Restricting the choice function G to '%Is{H} one gets a map (also called G ) defined on ?Tt and such that for all triples (rp,Z,f) E T6:

G(rp,Z,f) : Fv(cp) - Z + l ~ I , Bkcptf*G(cp, Z,f)l.

Let us now define:

Property (ii) follows immediately from these definitions and (*). Condition (iii) holds by the choice of D,. The following are used to prove (i): (a) for -every triple t E 5[,Range( G ( t ) ) < p, (b) z = v T . Combining these inequalities we get:

U Range(G(t)) <

IE

";E

2

1E

"Tr

=

Range(G(t)) < p . T [ < p . v ' = v '

(the last equality holds because v T 2 v p 2 2p>p). Now (i) follows immediately by using (*), (**) and the induction hypothesis. PROOFOF (a). Range(G(cp,Z,f))

<

Dom(G(rp,Z,f)) = Fv(rp)- Z

PROOFOF (b). By analyzing the definition of

5,it

<

Fv(cp) < p .

is clear that:

(***I cp E r and Z cFv(cp) imply

tion hypothesis:

< p and

9(Fv(cp))

<2P;

hence, by induc-

208

[Ch. 3, 5 4

GENERAL RESULTS

moreover for each rp ~r

C

=

DCP1 <

V ~ * ~ ~ = V ~ ;

Z E T(Fv(rp1)

from (***) and the last inequality:

< T.vr= vT. (iv) also holds; for if a = 5 - 1 and rp, Z,f are as specified, the triple (rp, 2,f) is in T6; if in addition rp'(Z,f, then g= G(cp,Z,f) is the map required in (iv). Case (B). 5 is a limit ordinal. Here we simply define D,= Us,,5D1..(ii) is obvious, (iii) is immediately verified, and in this case (iv) is vacuous. (i) holds as follows (recall that a)#@,

C
Thus, the inductive definition of the sequence (0,I a < R + ) is complete. D, and % =!I3 D. Clearly 181= D 2 X and: Let now D = Ua
r

Since D is a closed subset of 1931, % 23. It is only left to verify that %
(****)

%Frp[gl

Bt-dgl

(where gE(%\FV(v)) holds for atomic rp. Condition (****) is proved for all formulas rp by induction on the structure of rp, the connective cases being easily verified- Let q = ( 3 Z ) + , where Z LVar, 2 < A and Fv(+) = Fv(q) u Z ; we can further assume Z n Fv(rp) =@ . In this case the implication from left to right is trivial. As for the other, there is f, E 1231' such that 2' 3 t-+[ g*f,]; then f, E +'(Fv(rp),g, Observe that E I', since rp E r and r is closed under subformulas; moreover, since: a).

Range( g)

< Dom( g) = Fv(q)

+


Ch. 3, Q 41

THE DOWNWARD LOWENHEIM-SKOLEM

209

THEOREM

there is a < p + such that Range( g ) C 0,. By (iv), there is f, E $'(Fv(cp),g, such that Range(f,) C D,+ c 11' 11. Then 23 PI)[ g*f2], and by induction hy' 1t-$[ g*f2].Hence 9l t- cp[ g]. This completes the proof of case (I). pothesis 1

,

a )

-

CASE11. We shall only indicate the changes to be performed in the preceding proof. Notice that the condition p > Fv(cp) for all c p ~ T implies that p is a limit cardinal. The sequence of Da's is of length y, where y is a cardinal to be fixed later and y < 7 , p + . Condition (i) is replaced by

Do is chosen as before, but of power Y<'. If D, has been defined for all < E (where (< y) and ( is a successor ordinal (case A), D; and D, are defined as in (**). (a) and (b) are respectively replaced by: (a') for all t E Tc, Range( G ( t ) ) < p, (b') < <'. It is immediately verified, using p < Y<', that (a') and (b') imply (i'). The proof of (a') is exactly as that of (a). The proof of (b') is also similar to that of (b) if we keep in mind that Q, E T and Z SFv(q) imply 7
,

5

-

-

thus Range( g )
210

[Ch. 3,

GENERAL RESULTS

3

4

COROLLARY 3.4.2 (cf. DICKMANN [I], pp. 280-283). Let 2'3 be an infinite structure of type p , X 1231 and let A, K be infinite cardinals such that

c

-

Then there is a structure 'II such that 'II < %,X I'III and % = A . Moreouer, if K is regular the theorem holds under the weaker assumption A = A < ". KK

PROOF.Let

r =Form(L,,( -

F= where

p ) ) and A have the prescribed form.

Form(LKK( p)) =

p = max{ K , f } . Also: P=

{

p<" p"

if

K

regular,

if

K

singular ,

a!

ifK=a!+,

K

if

K

limit.

Therefore, if K = a ! + , then K > P and by case (I) of Theorem 3.4.1, there is 'II ji, X by assumption, we get A > p and A = A " Z p"= F. Then a model % with the prescribed properties exists, by case (I) of Theorem 3.4.1. Let K be limit and regular; A < " = A implies A 2 cf(A) 2 K . Thus again 2 p and A = A<" 2 p <"= F. Since p = K = cf(K), case (11) of Theorem 3.4.1 applies (with 7 = K , v = A), and we get a model % of power A < " = A with the prescribed properties.

_ I

REMARK. Notice that for the language L,, with A < conclusion of 3.4.2 because

K

we cannot improve

COROLLARY 3.4.3 (GCH). Let B be an infinite structure of type p , X LIB1 and K , A be infinite cardinals such that max { y ,F } < A < and K < cf (A). Then there is a structure % such that % < B,X C [%I and 3 =A. KK

PROOF. K < cf(A) implies A<" = A and K < cf(A) implies A " = A (GCH). So if K is regular the assertion follows from the second part of the preceding corollary. If K is singular, then K < cf(A) because cf(A) is always a regular cardinal; hence the result follows from the first part of Corollary 3.4.2.

Ch.3, 9 41

THE DOWNWARD LOWENHEIM-SKOLEMTHEOREM

21 1

COROLLARY 3.4.4 (Hanf-Karp; cf. L~PEZ-ESCOBAR [ 5 ] ) . Let A, K be regular infinite cardinals such that A < K and:

* LY~
~,~ECN’A\
Then every sentence rp of L,, which has a model B also has a model i?l of power < K . If, in addition f < K , then % can be taken as a substructure of B (euen as an elementary substructure).

PROOF,Let r = Sbf(cp). First we compute F and p. (a) i? < K . This is proved by induction on rp. It is obvious if rp is atomic, a negation or an existential formula. If rp = 7 < K , then

hence

-

since K is regular, I < K , and by induction hypothesis Sbf(+;) < K , the indicated sum is < K . (b) p
+

Next we show that without loss of generality we can confine ourselve_s to proving the theorem -under _the additional assumptions that 8 2 ~ , CpSm(cp) (so that f < and I;2 2. Indeed, if 5 < K , then the theorem is already proved. If f < K , by adding to rp an appropriate conjunction of formulas we can assume that all , 2 K , we prove the theorem for Sm(cp) and symbols of p occur in ‘p; if E define arbitrarily the denotation in i?l of symbols p-Sm(cp). Finally, if = 1, then r = { cp}, and rp is an atomic sentence; thus, cp E Sent(L,,), and since K > A 2 No, the downward Lowenheim-Skolem theorem for L,, ensures that cp has a model of power < K .

r)

212

[Ch. 3,

GENERAL RESULTS

54

Under these additional assumptions, apply case (11) of Theo_rem_3.4.1 to X = 0 , v = r and r=X, obtaining a structure IZI such that f#=T<' and having the last-mentioned property of 3.4.1 ; in particular, Zk'p

@

%i-'p;

then, it is clear that < K and Z k 'p. To prove the last assertion of the theorem repeat the preceding argument starting with r = Sbf('p) u Form(L,,( p)), which also has power < K. rn From the last result we obtain a downward Lowenheim-Skolem theorem [I]: for L,,, K regular, which includes Theorem 1.6, p. 283 of DICKMANN COROLLARY 3.4.5. Let 'pESent(L,,) have a model and let K be a regular cardinal > 0 ;then 'p has a model of power < K . I n particular, if K = h +,then 'p has a model of power < A . PROOF.K and w are both regular, and K > W by assumption; moreover, for all n E w and w < r < K. Hence the assumptions of Corollary 3.4.4 are fulfilled and the result follows instantly.

7'' = r

Likewise, for 8 strongly inaccessible > w we have: COROLLARY 3.4.6. Every sentence of L,, with a model, has a model of power < 8. PROOF.Every sentence of L, 3.4.4.

is in L,, for some

X < 8. Apply Corollary

B. Counterexamples The smallest cardinal for which Theorem 3.4.1 proves the existence of a model is 2p in case (I) and 2

q,+ -set. I

'p

be the LA+,+(<)-sentence axiomatizing the notion of

' See Example 2.1.15, pp. 109-1 10.

Ch. 3,

41

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

213

rp is the conjunction of:

(1) " < is a linear order";

(3)

rwb)A<X (. a

< 0,
Let I'=Sbf(rp); then F=A and p=A. On the other hand, result (C) on p. 1 10 implies that every model of rp has power 2 2" = 2p. Therefore, 2p is the lowest possible bound for case (I) of Theorem 3.4.1. rn Using the GCH, Example 3.4.7 can be modified so as to apply also to case (11) (Exercise 3.4.1). However, we present an example which does not use the GCH. EXAMPLE 3.4.8. Let A be an infinite cardinal >a. If A is not weakly inaccessible, the set H(A) of all sets hereditarily of power < A can be characterized up to isomorphism by a single sentence u, of LAA(E), where E is a binary relation symbol. For A = K + such a sentence was presented in Example 1.2.18. If A is a singular cardinal, let (X,(a
(I,,

(1)

3 x t l Y 1 (u E x ) ;

Clearly this sentence is in LAXIt easily follows from Definition 1.2.17 that H ( h ) ~ M o d ( u , ) .Conversely, if ( A , E ) ~ M o d ( u , , ) ,then by 8,. 8*, and the Shepherdson-Mostowski theorem, there is a transitive set M such that ( A , E ) = ( M , E 1M ) . Sentence (3) proves that every x E M has cardinality < A ; since M is transitive, it follows that M cH(A). Note that (1) implies that J3 E M . Finally, a simple induction on the rank of members of H(A), using that (2) holds in ( M , E r M ) , shows that H(A) C M . Hence H(A)= M , and ( A , E ) is isomorphic to (H(A), E H(A)). If A is weakly inaccessible > a , then there is a sentence uA of LA+A(E) which characterizes H(A) up to isomorphism. This is the conjunction of

r

214

GENERAL RESULTS

[Ch. 3, J 4

0,,O2 of Example 1.2.16, the sentence (1) above, and the following sentences, similar to (2), (3) above:

-

Notice that A < A < H(A). Indeed, it is easily verified that if K
fore A
Let A = K + ;then H(A) >h<’=(~+)k.=2‘. Therefore, by Theorem 3.4.1, case (I) (for v=2, ~ = p ) uK+ , has a model of power exactly 2“. Since any model of a,+ is isomorphic to H(K+), we conclude that H ( K + )=2“ and also obtain a second example showing that the result of Theorem 3.4.1, case (I), is best possible. Now let A be a singular, limit beth cardinal, i.e., A = aa,where a is a limit ordinal and cf ( a ) < a . Formula ll(ii) on p. 6 shows that A<’= X a + , =2A. Hence H(h) > 2A, and Theorem 3.4.1, case (11) (for v=2, T = A += ) , shows that uA has a model of power 2A. As before this implies that H(A) =2A and shows that the result of Theorem 3.4.1, case (II), is best possible when A is singular. Finally, when A is weakly inaccessible an argument similar to the above shows that the only model of uA, H(A), is of power 2
Ch. 3, $ 41

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

215

cardinality given by Theorem 3.4.1, and no model of smaller cardinality. However, we do not know whether this is the case without the GCH. Our next example shows that the condition < v T in Theorem 3.4.1 cannot be dispensed with. EXAMPLE3.4.9. Let Y be a cardinal > 2 and 7 a cardinal >No. Let =(v')+. Consider the sentence += 3ucp,<(u), where cp,<(u) is the LK+J{<})-formula of Example 1.2.6. Clearly + has a model of power K , namely ( K + 1, E I ( K + 1)). But + has no model of power < K , since ( A , <)tif and only if it has a proper initial segment isomorphic to (K , E K ) . In particular, this shows that if r =Sbf( +), then K

+

f_or any set A of power

< .'Y

Note that in this example p=N, and

F=K=(YT)+.

Setting K = ( v < ' ) + , the same example shows that the assumption < v<' in case I1 of Theorem 3.4.1 cannot be dispensed with. A similar argument proves that the downward Lowenheim-Skolem does not hold for class languages of the form L,, and, a fortiori, for those of the form LwA. Indeed, for every infinite cardinal K there is an La,sentence, namely 3vcp,<(u), having models of power 2 K but of no smaller power.

C. Applications In this paragraph we give several direct applications of the infinitary downward Lowenheim-Skolem theorem. The first is to group theory and [ 11. The later applications yield strong appears in KOPPERMAN-MATHIAS non-axiomatizability results in infinitary languages for a number of second-order notions, the common feature of which is that the (secondorder) quantifiers range over families of sets of unbounded cardinality. [l]; they generalize an earlier These results are due to COLE-DICKMANN [3]. (Notation and definitions used in the first result of KOPPERMAN application can be found in Chapter 1, §2.) DEFINITION 3.4.10. A class 4 of groups is bountiful iff for every infinite group G the following holds: if there is an H € B such that G C H, then thereisKE4 suchthatGcKcHand

z=z.

216

GENERAL RESULTS

[Ch. 3,

4

The following is a sufficient condition for a class G of groups to be bountiful. It is slightly more general than that mentioned in KOPPERMANMATHIAS [I], p. 132. THEOREM 3.4.1 1. Zf

E PC(L,I,(

g)), then

9 is bountiful.

PROOF.Let 9 = Mod(a)rg, where a E Sent(LmI,(p)) and g C p. Obviously we can assume < No. Let G be an infinite group for which there is an H E 9 such that G C H ; let =A. By assumption there is an expansion 6 of H of type p such that @Fa. By Corollary 3.4.4 there is R i $ such =A, Q€Mod(a) and G CIS?[. Let K = Q r g . Obviously K E G , and that from K i H , G H and G LlRl it follows that G is a subgroup of K . Hence 9 is bountiful. COROLLARY 3.4.12. The following classes of groups are bountiful: ( I ) The class T of all torsion groups (Example 1.2.10, $ 2, Chapter 1). (2) The class S of all simple groups (Example 1.2.1 I , $2, Chapter 1). (3) The class Cs of all characteristically simple groups (Example 1.2.13, $2, Chapter I). (4) The class Tn of all transitively normal groups (Exercise 1.2.2 (I), 9 2, Chapter 1). (5) The class of groups mentioned in Exercise 1.2.2 (2), 0 2, Chapter 1. PROOF.Trivial, since all these classes have been proved to be PC(LUI,(g))classes in 0 2, Chapter 1. As a second application of the downward Lowenheim-Skolem theorem we show that many classes of topological spaces are not RPC,(L,,) for any cardinals K,A, K > A . The argument is a refinement of a theorem due to KOPPERMAN [3]. We introduce first the proper definitions.

DEFINITION 3.4.13. We regard topological spaces as structures of the form ( X U T.X, T , E ) , where E C X X T and there is a set Y and a topology !.. on Y such that Y n 5=a and ( X , T , E ) - ( Y , !.., E), where E is the usual membership relation. Y = { Pt, 0, E } denotes the similarity type of topological spaces (‘‘Pt” for “point”, “0” for “open”). We assume familiarity with the definition of simple topological notions such as discrete, compact, metrizable, etc., spaces. All classes of topological spaces to be considered below are assumed to be closed under homeomorphism.

Ch. 3, 5 41

217

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

THEOREM 3.4.14. Let K be a class of topological spaces containing discrete spaces of arbitrarily large cardinality. Then K is not an RPC,(L,)-class, for any K,A, K > A. PROOF.Suppose K E RPC,(L,,) for some K,A, K > A. Then there are p 2 v, Z cSent(L,,( p)) and cp € Form,(L,,( p ) ) such that K = (Mod,(C))’P v ; i.e., for every structure of type v:

r

(*)

% € K w there is 2l’ of type p such that IM’EMod,(Z) and

8=

(%’rcpa‘) 1v.

Let T be a cardinal such that T > and T ‘ = T (e.g., ~ = 2 where ~ , a = max{ &,K}). Let % = ( Y u 9 ( Y ) , Y , 9 ( Y ) ,E) be a discrete space of power > T belonging to K; such an 91 exists by hypothesis. We can further assume that Y n 9 ( Y ) # B . By (*) there is %’EMod(X) such that % = ( % “ p ” ) r v . Let Y ’ c Y be such then 7 = 7. By Corollary that Y’ = T , and let X = Y ’ u { { y } Iy E Y ’ } ; 3.4.2 there is B’i Kh%’ such that X c_ 123’1 and 8’ = T . Therefore 8‘ E Mod(C), and B = (8’r qB‘)r v E K ; then B 1 { 0, E } = ( Z , 9,E) for some set Z and some topology 5 on Z . Clearly we can (and do) assume 2 CPt“, 5 GO’ and Z u ‘5 = 181. 3’ < *,%’ implies cpB’= rpa’n18’1; hence X ccp’ = 1231 and we get Y ’ c Z and { { y } Iy E Y ’ } c 5.Since 9 is a topology, it is closed under arbitrary unions of its member sets; then ’3’ ( Y ’ )c 9; hence:

contradicting the choice of 8 ’ . This contradiction shows that K E RPC,(L,,).

COROLLARY 3.4.15. The following classes of topological spaces are not RPC,(L,,) for any K,A, K > A : (a) the class of all topological spaces; (b) the class of all Hausdorff spaces; (c) the class of all discrete spaces; (d) the class of all T,-spaces (i=O, ...,4); (e) the class of all metrizable spaces. PROOF. Each of these classes satisfies the hypothesis of the preceding theorem. As a corollary to the proof of Theorem 3.4.14, we obtain

218

GENERAL RESULTS

[Ch. 3, 3 4

COROLLARY 3.4.16. The following classes of topological universal algebras are not RPC,(L,,), for any K,A, K > A: (a) the class of all topological groups; (b) the class of all topological rings; (c) the class of all topological modules.

'

PROOF(Sketch). Repeat the proof of 3.4.14, giving to each algebraic structure mentioned above the discrete topology; thus, it turns out that each of the classes (a)-(c) contains discrete spaces of arbitrarily large cardinality. (1) 3.4.15 (e) contradicts a remark of KOPPERMAN [3], p. 266. It REMARKS. may also seem to contradict Exercise 1.3.4(e), 9 3, Chapter 1 ; this contradiction is only apparent: metrizable spaces as topological structures of similarity type Y are essentially different from metric spaces as expressed in terms of the metric. Moreover, we conclude that the topology of a metric space is not first-order definable (even in the infinitary sense) in terms of the metric. [ 11 shows that the class of complete uniform spaces is not (2) KOPPERMAN an RPC,(L,,) for any K,A, K A. (3) The radical non-axiomatizability results 3.4.14 and 3.4.15 underline the higher-order character of the notion of topological space; closure under arbitraty unions of open sets is a second-order requirement in our formulation and third-order over the underlying set. Notice that certain fragments of second-order notions can be axiomatized in some L,,, A>w; namely, those (for example, well-orderings) that involve quantification of a set variable which need only range over sets of bounded cardinality (cf. Exercise 3.4.4). As further applications of this remark and of the technique of Theorem 3.4.14 we prove other non-axiomatizability results. The first of these theorems extends non-axiomatizability to several classes of compact spaces. The gist of the argument is to allow Stone spaces to play the same role as discrete spaces played in Theorem 3.4.14. We assume knowledge of some elementary facts of Stone's representation theory for Boolean algebras: (i) a Stone space is a compact, Hausdorff space with a base of clopen sets; (ii) any Boolean algebra B has a uniquely associated Stone space;

'

Construed in a similarity type containing symbols for the algebraic structure, as well as the symbols of Y.

Ch. 3, 9 41

THE DOWNWARD LGWENHEIM-SKOLEM

219

THEOREM

namely, the set of all ultrafilters on the algebra, with the topology induced by the product topology in 2’; (iii) any Stone space is the space associated to the Boolean algebra of its clopen subsets (with the ordinary set-theoretic operations); (iv) the power-set algebra of a cardinal (set) K has 22‘ ultrafilters; hence this is the cardinality of its Stone space; (v) an extremally disconnected space is one in which the closure of an open set is open; (vi) a Boolean algebra is complete (resp. atomic) iff its Stone space is extremally disconnected (resp. has a dense set of isolated points).

THEOREM 3.4.17. Let K be a class of compact topological spaces which contains the Stone space of power-set algebras of arbitrarily large cardinality. Then K is not an RPC,(L,J for any K,A, K 2 A. PROOF.Assume K E RPC,(L,,) for some K,A, K 2 A. With the same notation as in the proof of 3.4.14, let p 2 Y , Z G Sent(L,,( p ) ) and cp E Form,(L,,( p)) be the objects that make K an RPC,(L,,)-class. By assumption there is a cardinal 7 2 max{K,F} such that theStone space % of the Boolean algebra (3’ (7)is in K. By - the remarks above, 8 =2” and % has a base ‘3 of clopen sets; also q = 2 ‘ (even more, 05 with set-theoretic operations is a Boolean algebra isomorphic to 9 (7)). By hypothesis %=(%’rcp’’)rv for some %’EMod,(X). Enlarge %’ to 8’’by adding a predicate Clb whose extension is 9; i.e., a”= (%’, ”5 ). The following hold in %”: (i) Vu[Clb(u)-+O(u)] (every member of Clb is open); (ii) Vuo[Clb(u,)+3u,(0(u,) A Vz(zEu,t, izEu,))] (the complement of every member of Clb is open); (iii) Vu,[Clb( uo)+cp (uo)]; p P( (~~,, )) A~ O~ ( o~ ,~) +~ l (iv) ~ ~ o ~ , [ c p ~ ~ o ) ~ c A 3u2[Clb(u,) A u0Ec2A Vz(zEu,+zEu,)]] (Clb is a base);

( 4 Vu,~,[Pt(u,>A Pt(ul>~cp(u,)A T ( U , )A

-I (uo=ul)+ ~ ~ o ~ l ~ ~ ~ ~ o ~ ~ ~ ~ A I- ~ ~ ( z E w ~ A z E w(% ~ )is] ]a Hausdorff space).

~

,

~

We observe that (2’)”= 2‘. By the downward Lowenheim-Skolem theorem = (Corollary 3.4.2), there is B” < ,a” such that ?! 1B”land 23” = 2‘. Let 23‘ = 8‘‘ 1p and 8 = (23’ cp”) 1Y. Then 8 E K and we can assume that it is a compact space. By (v) 8 is also Hausdorff. Let $1=Clb”’; by (iii), ‘% cp”; by (i), (ii) and (iv), !X is a clopen base for 8. Since 23“ < KA91”, for every a E 1B’’l:

r

c

23” t= Clb[ a ] e 8’‘ t- Clb[ a ] ;

~

c

p

220

GENERAL RESULTS

[Ch. 3,

4

therefore: Conclusion: ‘13 is a Stone space with % as a clopen base. We prove now: CLAIM.

5>2

~ ~ .

By the remarks preceding the theorem, 23 is the Stone space of the Boolean algebra @ of its clopen subsets with usual set-theoretic operations as Boolean operations. Moreover, ?I c @ (as sets) and ‘3 is isomorphic to the power-set algebra 9 (T). First we show that 05 is a Boolean subalgebra of 4’;the verification of this fact is a routine matter except, possibly, for the complement operation; thus we prove:

(*I

- ,X=

-

@X for every X E 9 .

Let #(uo,uJ be the formula: V w [ w E u o - i ( ~ E u , ) Then: ]. % ” k + [ X , -,XI,

Since X , - ,X E %

‘13”k+[X, - @ X I .

c IB’’1, we get: ‘13”k+[X, - sX];

on the other hand we know:

hence (*) holds; so 94 is a Boolean subalgebra of @ . Each ultrafilter F of 05 is a subset of @ with the finite intersection property; let * be a map choosing an ultrafilter F* of @ containing F. We show that * is a one-one map. If F,G are ultrafilters on 94,Ff G , then there is X E 9 such that X E F and - %X EG; therefore X EF* and -,X= - , X EG* ; hence F*#G*. Since ‘13 is (homeomorphic to) the set of all ultrafilters of &‘ and 9 has 22’ distinct ultrafilters, we conclude &at > 22‘; this proves the claim. by the downward LowenheimHence we have on the one hand g <J, Skolem theorem, and on the other %>22T. Thus we have reached a contradiction, which proves that K cannot be an RPC,(L,,).

w

Ch. 3, 8 41

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

22 1

COROLLARY 3.4.18. The following classes of topological spaces are not RPCA(L,,) for any K,A, K > A: (a) the class of all compact spaces; (b) the class of all compact, Hausdorff spaces; (c) the class of all Stone spaces; (d) the class of all extremally disconnected Stone spaces. Our next result was originally motivated by the example of complete Boolean algebras,. but we have chosen to state it in terms of partial orderings; it has some overlap with Theorem 3.4.17. DEFINITION 3.4.19. A partial ordering is complete iff every bounded linearly ordered subset has a least upper bound and a greatest lower bound. THEOREM 3.4.20. Let K be any class of complete partial orderings containing the sets ($7 ( a ) , ) for arbitrarily large cardinals a . Then K is not an RPC,(L,J, for any K,A, K > A.

c

PROOF(In sketch). Suppose K E RPCA(L,J for some K,A, K > A . Let p, 2 , ~ have the same meaning as in Theorem 3.4.14. Choose 7 as in 3.4.14. By assumption there is a > T such that (?P ( a ) ,C) EK; hence (C? ( a ) , C ) =(%’“p‘’)l{ < } for some BI’EMod,(Z). Let X be a set of atoms of ??(a) of power r . Notice that the set of all atoms is defined by the following formula +(uo):

By the downward L6wenheim-Skolem theorem there is 8’ < = 7.Then: X C l%’l and (i) ’H’EMod,(B); ~ (ii) X c 4 % ’4%’. Putting these together we have:

and xEX

3

Yik “x is an atom”

such that

222

GENERAL RESULTS

[Ch. 3, 3 4

Therefore ‘P is a complete partially ordered structure with a set of atoms of power 2 7.Notice that the Boolean algebra axioms can be written in terms we also get: of < I ; hence, from 8’ < (iii) 8 is a Boolean algebra. A simple, well-known argument proves that-3 contains a subset isomorphic to the power-set of its atoms’; hence 2 2’, contradicting the choice of B’.

COROLLARY 3.4.21. The following classes of partial orderings are not RPC,(L,J for any K,A, K > A: (a) the class of all complete partial orderings; (b) the class of all complete lattices; (c) the class of all complete, distributive lattices; (d) the class of all complete Boolean algebras; (e) the class of all complete, atomic Boolean algebras. REMARK. Notice that a similar theorem (with even simpler proof) holds for structures formalized in the language for lattices (i.e.. containing the operation symbols A , v ). Our last theorem has been motivated by the example of complete linear orderings, a case not covered by Theorem 3.4.20. We shall need the following result, implicitly contained in SIERPINSKI [2], [3]:

LEMMA. For any limit ordinal E there is a linearly ordered set of power &?hoseorder-completion has power at+

,.

PROOF.For any cardinal K, let UKbe the lexicographically ordered set of all ,= 2“. and Sierpinski shows sequences of 0’s and 1’s. of length K . Then that UKis a complete linear ordering. Let H , be the sub-linear ordering of U, consisting of those sequences s : ~ + 2having a last 1: i.e., there is an a < K such that s, = 1 and for all p, a < ,l3 < K sp = 0. Then Sierpinski shows that H , is dense in U,, whence it is easily shown that UKis the order-completion of H,. . I t remains to determine the cardinality of H,. We may embed U, in U p if a < p , simply by adjoining a string of 0’s to each string of length a .

-

I

*

To express --x, use the formula saying “ul is the largest z such that z Aoo=O”.

Fix some well-ordering of the atoms: using it and completeness show that each set Y of atoms has a least upper bound. The map that carries Y into its least upper bound is one-one.

Ch. 3, 8 41

223

THE DOWNWARD LOWENHEIM-SKOLEM THEOREM

Further, if s is of length conclude that

K

and has a last 1, namely sa, then E <

HKc

K;

so we

u UA,

A
whence

H, < 2 u,= X
2"2<".

A
Hence the theorem is true for any K such that ~ = 2 < "and , this is clearly the case for K = at, where 5 is a limit ordinal. Taking U,& to play the crucial role in our method of proof, we obtain:

THEOREM 3.4.22.Let K be a class of complete partial orderings containing Ult for limit ordinals [ of arbitrarily large cofinality. The K is not an RPC,(L,,) for any K,A, K 2 A. PROOF.Suppose KERPC,(L,+) for some K,A, K > X . Let p,Z,cp have the same meaning as in the preceding proofs. Let E be a limit ordinal for which at>F, cf([)>K and U,&€K. Then 3F=aCand U,$=(%'rcpg')[{< } for some a' E Mod,(Z). Since Ha6 = at, by the downward Lowenheim-Skolem theorem there is ~

such that Hlc&181 ' and 3= aC Then B = (B' cp"') { < } is a complete partia'l ordering. Since 23' i KABl', 23 is also a linear ordering and H,t.Ccp"'=lBl; so B extends-Hli By completeness, B contains an isomorphic copy of USt. Therefore '% > at+,, contradicting the choice of B'. This completes the proof.

B'<

r

r

COROLLARY 3.4.23. The following classes of orderings are not in RPC,(L,J, for any K,A, K > A: (a) the class of all complete linear orderings; (b) the class of all complete lattices. EXERCISES

3.4.1.Using the limit cardinal hypothesis (LCH) and Example 3.4.7construct a sentence satisfying the conditions of case (II), Theorem 3.4.1, having no models of cardinality less than 2
224

GENERAL RESULTS

[Ch. 3,

5

3.4.2. Let X>w. Construct a sentence in LA+,(<), for X regular, and a sentence in LA,(<), for X singular, axiomatizing the classes of all wellorderings of cofinality > h and cofinality >A, respectively. 3.4.3. Let X>w. Show that the class W, of all well-orderings of power at <). If An CN < A , then W A EPC(L,,( <)). least 2<‘ is a PC-class in [Hint: add a ternary relation symbol and for each cardinal K < X one individual constant. Use the formula Z defined on p. 166.1 P

3.4.4 (Kopperman). (a) Show that the class of all topological spaces with a countable base (= separable) is an EC,,,I(v). Note. Recall that Y = { Pt, 0, E } denotes the similarity type for topological spaces; cf. Definition 3.4.13. (b) Using part (a) show: (i) the class of all compact, separable spaces is EC,I,I(v); (ii) the class of all metric, separable spaces is PC,I,I(v); (iii) the class of all complete, metric, separable (=Polish) spaces is pc,I,,(v).

Q 5. The upward Lowenheim-Skolem theorem and generalizations; Hanf numbers; Morley numbers A. The finitary case

The best known version of the upward theorem for L,, (due to TARSKI-VAUGHT [I]):

is the following

THEOREM 3.5.1. LetdB be an infinite structure of type p and K be a cardinal K 2 max { %, p } . Then there is an elementary extension ’u of 8 of

such that power K .

PROOF.Consider the type p’= p u { 6 1b E 181)u { c, I a < K } , where the ca’s are new individual constants. Let 2 be the following set of sentences of type p’: (i) cli)‘ (‘P), for a,p< ~ , a # p . (ii) 1(c,=cp) Every finite subset Z’ of Z has a model. In fact, if a I , ...,anare all the distinct a’s such that c, occurs in Z‘ and b,, .. .,b, all the distinct b’s in 181 such that 6 occurs in Z’, and if b’,, , . . ,b: are distinct elements of 1 91.

Ch. 3, 9 51

THE UPWARD LOWENHEIM-SKOLEM

225

THEOREM

is a model of Z' (where the interpretation of cq is b,! ( i = 1,. ..,n)). Hence Z has a model, say B. Clearly > K (by (ii)), and 0:r p > 8 (by (i)). By the 0 l p such that downward Lozenheim-Skolem theorem there is 8 i 12 ' 3 CI'iXl and g = K ; These facts, together with 23 i a l p , imply that 23<X.m This result gives a complete answer to the z o b l e m of existence of elementary extensions of a structure 23, when g 2 f , the power of the language. When < F, it leaves the question open for cardinals K such that 5 < K < ,E. A partial answer has already been given in Chapter 2, where we proved:

THEOREM 2.2.13. Let 23 be a structure of infinite power A and let there is an elementary extension of 23 of power K .

K

> A".

Then

There still remains the question whether the bound A" can be improved. The following example-due originally to Rabin (using the GCH), later proved by Keisler without the GCH (using limit ultrapowers) and finally [I]-shows that this is not the case, at least for simplified by CHANG non-measurable A. EXAMPLE 3.5.2. Let A be infinite and non-measurable. There is a structure 'ZI of power A in a language of power 2Asuch that every proper elementary extension of X has power >A". PROOF.Let fi; = A ; let the language p contain one n-ary relation symbol R, for every subset B & A "; let I'ZII = A and R: = B for every B C A " ( n E w ) . [This structure is called the complete structure over A ; cf. 0 3.B above.]

For the present proof p need only contain unary and binary R,'s. REMARK. LeLB > X, 1231II1811, be a proper elementary extension of X. We show that g>A". I t is a well-known result in set theory (see S I E R P I ~ ~ S[ IK] I or KURATOWSKI-MOSTOWSKI [ 11) that there exists a family (Pp I p
5

226

[Ch. 3, 5 5

GENERAL RESULTS

'9 is a non principal ultrafilter on A . The proof of this has been given in detail in $ 3 above (see Theorem 3.3.9). Since h is non-measurable, LID is countably incomplete, i.e., there is a strictly decreasing sequence (X,In E w ) C (9 such that ntwXn =O. For ,L?
n

F&z) =pp"

ts

a E X, - x,,

,

Since f Xl , = S , Fp is onto. Consider the relations R, ,Rpa,Rxncorresponding resp. to Fp, P p , X n (as subsets of A or A * ) . From (pr < 'P it follows that: 2+k

" R is a function mapping R, onto Rpa", 'a

Since xoER Z for all n Em, R;;(x,)#pp" for all n E w , i.e.,

R , ? ( S ~ ) €R:-

(*)

Po.

Now suppose (by (ii)) that P o n P, has rn elements, where P # a ; then:

T4 F Rpan Rpc,has exactly rn elements" "

Since Pp n P, C RP" n R; (by YI < '8)and both sides have rn elements, they are equal. From tgis and (*) it follows that

R$ (x,,) # R$ ( xo) whenever ,L? # a . Hence the set { R,?(xO)I,L? < A w } is a subset of 1231 of cardinality A"; i.e.,

%'>A".

B. The infinitary case: Hanf numbers: Morley numbers We have seen that the downward Lowenheim-Skolem theorem admits a quite straightforward generalization to the infinitary languages LKA.Do such generalizations exist for the upward theorem? This and related problems will be considered in the next chapter. Firstly, the compactness theorem that was used in the proof of the upward theorem does not hold in infinitary languages. Secondly, in L,,,

Ch. 3, I 51

THE UPWARD LOWENHEIM-SKOLEMTHEOREM

221

for instance, there is a sentence characterizing the structure (a,+. s . 0 , I ) (or any denumerable structure for that matter) up to isomorphism. Even

more generally, we shall see in $ 2 , Chapter 4 that there are infinitary sentences with a non-denumerable upper bound for the cardinality of their models.

DEFINITION 3.5.3. For each set X of sentences in an arbitrary language L, the Hanf number of X, h(X), is the least cardinal h such that for any sentence u of X the following holds: if u has a model of power >A. then u has models of arbitrarily high cardinalities [i.e., for every cardinal K there is a model of u of power > K ] . Whenever it is defined, h(Sent(L)) will be denoted by h(L). In a completely analogous way. given a cardinal Y one defines the Hanf number h”(L) replacing the words “any sentence u of L” by “any set Z of sentences of L of power < v ” . However these numbers will not be used here. It was HANF [ I ] who first noticed that h(L) exists for almost any language: LEMMA 3.5.4 (Hanf). For any set X of sentences in an arbitrary language L, h(X) exists. In particular h(L) exists for any language L in which Sent(L)the class of all sentences of L-is a set. PROOF.To each sentence u of X not having models of arbitrarily high powers, we associate a cardinal K, as follows: K, is the least h such that u does not have models of power > A . The least upper bound p of all such K ~ ’ Sexists because X is a set. and obviously p > h(X). This argument can be carried one step further to prove: LEMMA3.5.5. Let X be a set of sentences of some language L. Then cf(h(X)) <

F.

PROOF. With the notation of the preceding lemma, we observe that p = h ( X ) . Indeed. if h ( X ) < p , then there is u E X such that h ( X ) < ~ , < p ; obviously, u has models of power > h ( X ) (in fact, of power K,) but does not have models of arbitrarily large cardinality; this contradicts the definition of h(X). By the definition of p it is clear that cf(h(X))=cf(p)
228

GENERAL RESULTS

[Ch. 3,

5

REMARKS. (1) The class languages L,, and L,, are examples of languages L for which h(L) need not be defined. (2) For sets X , Y , X Y cSent(L) implies h ( X ) S h( Y ) ; if, in addition, each sentence of Y is L-equivalent to one in X , then h ( X ) = h( Y ) . In particular, for any similarity type p and cardinals K , ~ ' , h , h 'such that K < K',A < A', we have:

(3) By 3.5.5 we have, for example:

where p = max{ ,E, K } . (4) In general we shall not be concerned with Hanf numbers h(L,A(p) for aparricular type p (such as, for instance, the problem of computing the for the type p containing 5 binary relations, 16 Hanf number h(L,,(p)) ternary relations and 43 unary function symbols). As in the model theory of first order languages, we shall rather be interested in questions like: what is the best possible upward Lowenheim-Skolem type theorem for languages having up to so and so many symbols (of unspecified nature)? Thus, all Hanf number computations in this book will provide estimates for cardinals of the form

(*>

sup { h ( LKA(p)) I p is a similarity type of power


which, for notational simplicity, we shall still denote by h(LKA)(without reference to p ! ) . In each case it will be clear which particular cardinal of the form (*) we have in mind. A similar remark applies to the Morley numbers defined below. The argument of Lemma 3.5.4 applies to a variety of different situations. One of these is the existence of the Morley numbers m,,n, defined as follows. Let K be a fixed infinite cardinal. We consider only similarity types p of power S K . Notice that in this case Form(L,,( p ) ) < K and there are at most 2" types.

DEFINITION 3.5.6. (a) The Morley number nr, is the least cardinal A such that every set S of power < K of L,,-types which is omitted by a model of power > A, is omitted by structures of arbitrarily large cardinality.

Ch. 3, $ 51

THE UPWARD LOWENHEIM-SKOLEM THEOREM

229

(b) A second Morley number n, is defined as the least cardinal A such that every set of La,-types which is omitted by a model of power > A , is omitted by models of arbitrarily large power. It is clear that rn, < nK.We shall see in Chapter 4 that m, < n,; this is due to the fact that there are 2" sets of power < K of types, whereas there may be as many as 22' arbitrary sets of (non-equivalent) types. I t will also turn out that m, and h(L,+,) are identical. We shall consider the problem of exactly evaluating the numbers h(L,) as a generalization of the upward Lowenheim-Skolem theorem to the infinitary languages LKA.This evaluation is far from being achieved in general. Reasonable answers exist only for the languages LKa; most of Chapter 4 is devoted to compute h(LKa),m,,n,. In Chapter 5 we treat the case of languages with infinitary quantification. These computations will reveal how intimately connected Hanf numbers are with the expressive power of the respective languages.

CHAPTER 4

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

This chapter is devoted to the study of the languages L,,. In general we confine our attention to the case in which K is a successor cardinal, though many of the results presented here depend only on the assumption that K is a regular, uncountable cardinal. We begin by showing that the Hanf number b, for L,+, coincides with the Morley number rn, for omitting L,,-types (Definition 3.5.5). The example of the sets R ( a ) , which can be formalized by a single sentence of the appropriate infinitary language, leads us to a method whereby we can obtain lower bounds for 6, and 11,. Using rn, and a subtle technique of Morley involving indiscernibles and the Erdos-Rado theorem of partition calculus, we shall be able to compute upper bounds for b, and nK. The methods of $ 5 2, 3 are finally combined in 3 5 to obtain an exact estimate for the Hanf number of L,,,. As an application of the abovementioned results we show in $ 4 the impossibility of axiomatizing the class of all well-orderings in any infinitary language with finite quantifiers. Throughout this chapter, b, stands for h(L,+,), i.e., the supremum of h(L,+,(p)) for all similarity types p of power < K (cf. remark on p. 228). The Morley numbers in,, 11, are defined on pp. 228-229.

0 1. The Hanf number of L,,, and Morley numbers THEOREM 4.1.1 (CHANG[2]). For euey infinite cardinal

K,

rn, = b,.

PROOF.(I) rn, < 6,. Let S = {Z,I(< K } be a set of types of power suppose \u is a structure of power b, omitting each Z,. For 230

< K and (< K let

Ch. 4, $ 11

THE HANF NUMBER OF

L,+,

-

E, < K , the ‘ p i s and las of L,+,. Moreover it is clear that for any structure 3:

‘pc= i 3xAoEzeu and ‘p= /l\5.,K’p\5. Since

(HF-9, e (H omits

23 1

AND MORLEY NUMBERS

each 8,([<

‘p

are formu-

K).

Hence %t=’p. By definition of h, there are arbitrarily large a’s such that 2‘ 3 l= ‘p, whence arbitrarily large 9’s omitting S. (11) h, < m,. Recall that every formula of L,, has < A free variables. Let ‘p ESent(L,+,( p)); note that ‘p has at most K subformulas in L,+,( p ) and each subformula has finitely many free variables. For each aESbf(’p) with n free variables we introduce a new predicate symbol R, with n arguments. For each uESbf(’p) with no free variable, introduce a new predicate symbol R, with one argument.’ Let

Consider the following set r of sentences of L,+,( p’) , where u denotes a finite non-empty sequence of variables appropriate to the formula: for each atomic u ESbf(q), Fv(u)#0 for each atomic aESbf(rp), Fv(u)=B, u~VuR,(u) Vu[u(u)-R,(u)]

Vu[R+(u)++iR,(u)] for each $ESbf(’p), $ = i u , Vu[R+(u)++3yR,(u*y)] for each ESbf(’p), $ = 3 y u , Fv($)#B,

VuR+(u)++3yR0(y) for each Ic, E Sbf(q)$ = 3yu, Fv($) =0, Vv[R+(4-A5<.Ro,(”)l

for each $ ESbf(’p), $= A,<,UE.

VuR+(u)++3uR+(u) for each $ ~ S b f ( r p ) , F v = 0 , VUR, (u). The meaning of these axioms can be more easily grasped if we take into account that the axioms (i)-(iv) are equivalent to: (vii) Vu(Ic,(u)-l$,(u)) for $ESbf(’p),Fv(Ic,)#0, (viii) $++VuR+(v) for $ E Sbf(rp), Fv($) =O. We leave for the reader the task of proving these equivalences.

’

It would perhaps be more natural to consider R, to be a propositional variable-that is, a logical symbol taking on the values “true” or “false”-whenever o is a sentence. Since the languages considered in this book do not have propositional variables, we use instead unary relations which (by axiom (v)) are only allowed to denote the universe ( = “true”) or the empty set (=“false”).

232

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4,

1

Now we make three observations: (1) S K . This is obvious, (2) There is a set S of power S K of types in Law($) such that for any structure 23:

7

23t- r

H

23 omits S.

PROOFOF (2). S contains one type for each axiom of form (i), (ii), (iii) and (vi); namely the type whose unique formula is the negation of the corresponding axiom with the outer quantifiers erased. Thus S contains each of the types {-I (u(u)t,R,(u))} for 0 an atomic subformula of rp such that Fv(a)#B, etc., for (ii), (iii) and (vi). For each axiom of form ,)'i( (iii') and (v), S contains the type (in 0 variables) consisting of the negation of the axiorn. Finally, for each condition of form (iv) we throw in 5 the types {i (R,(u)+ R,((u))} for each 5 < K , and { R, ( u ) A i R + ( u )I E< K } , where # = / / c < K u 6 . Clearly S has power < K and each type in S has power < K . ' If 23 omits S and b is a finite sequence in 23 of the proper length, then for each atomic subformula u of rp we have: %l- 1 1(a*Ro)[b1

Hence Bl-V u [ o ( u ) ~ R , ( u ) Similar ]. arguments show that all axioms of form (i'), (ii), (iii), (iii'), (v) and (vi) hold in 23. Concerning conditions of form (iv), for every b in 23 we get: %I- ( R++R,()[ b] for every

i.e.,

E
and

23 i.e.,

'

1(RotA

IR+)[ 61 for some E < K ,

The number of free variables of types in 5 is unbounded. However, there are devices for transforming types of 5 into types with only one free variable in a suitable extension of the original similarity type. As an exercise the reader can smoke one of them out.

Ch. 4, 3 I ]

THE HANF NUMBER OF

L,+,

AND MORLEY NUMBERS

233

whence

Hence Y3k V u [ R , ( u ) ~ & , , R , , ( u ) ] , and we have shown that all sentences of form (iv) hold in B; hence Bk r. Conversely, if B k I’, then B omits S as it is easy to check. (3) Given a structure % of type p, Bl kcp w ‘u has an expansion 94’ of type p‘ such that Yl’F r.

PROOFOF (3). (e) is obvious by (vi) and (viii). (*) Define %‘ as follows:

8’l p = 21: RF’= #”= for every

{ b E 1911FV($’191 k + [ b ] } ,

+ E Sbf(cp) such that Fv($)#B

;

for every #ESbf(cp) such that Fv($)=B. It is obvious that %’t= r. Now let ,zI EMod(cp), 9I rn,; let 9” be an expansion of 91 obtained by (3). Since %’I= r, by (2) 91’ omits S, and since 91’ > rn, and 7’ < K there are arbitrarily large 8’s omitting 5. The reducts of these 23’s to p are, by (3). arbitrarily large models of cp. REMARK.The preceding theorem was first proved in the case K = W by Lopez-Escobar. He and Morley realized, furthermore. that nr, = Iiciice b,=3,1. Their method of proof. however, was based on the study of models of a-logic rather than the direct procedure used above. As a corollary of (the proof of) the preceding theorem we obtain an alternative proof of the downward Lowenheim-Skolem theorem for L,+, stated as Corollary 3.4.5:

THEOREM 4.1.2 (Alternative form of Corollary 3.4.5). Let K be an infinite cardinal, Z be a set of sentences of L,+,( p ) of power < K , and - 8 be an infinite model of Z. For every cardinal h satishing K < A < 93 there is

234

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

c

[Ch. 4, 0 2

2l E Mod(C) such that =A. Moreover, if X 1231 is a ser of power can be chosen to be an elementary substructure of 23 containing X.


PROOF.Since a set of sentences of L,+, of power < K is equivalent to a single sentence (its conjunction), it is enough to prove the theorem for this case, i.e., Z = { rp}. We also notice that cp involves < K symbols; hence we can assume < K . Since % + q , by (3) in the proof of the preceding theorem, 23 has an expansion B’ of type p’ such that B’F r. By (2), B’ omits the set S of types constructed there. Notice that < K ; thus, by the downward Lowenheim-Skolem theorem for L,, there is %’ i 23’ such that X 18’I and =A. Clearly 8’also omits 5; hence a’+ r, and by (3) of Theorem 4.1.1, 8’ p = 9l E Mod(rp), as desired. rn

F’

c

r

$2. Lower bounds for Hanf and Morley numbers The main results to be proved in this section are the following lower bounds for the Hanf number of the language L,+,: Q h, (Theorem 4.2.1). (I) For every infinite cardinal K , aK+ (11) If cf(K) > w, then 3,+< 6, (Theorem 4.2.4).

The technique for proving these results consists in defining a wellordering of the appropriate order type a by a sentence of L,+,, and then, by formalizing (a relevant part of) the cumulative hierarchy of sets R ( p), /I< a , concluding that all models of a certain sentence of L,+, have power =l,.,+

(Theorem 4.2.3).

THEOREM 4.2.1. For eoely infinite cardinal

K,

3,.

< $.,

PROOF.We shall exhibit a type p and for every a < K + a sentence rpa of L,+,( p) having models of power but of no larger power. Let a < K + and let { p,I(<max{ E , w } } be an enumeration of w + a without repetitions. The type p has: a binary relation E , a unary function r , individual constants cpd(5 < max( Z , a}).

Ch. 4,

21

LOWER BOUNDS FOR HANF AND MORLEY NUMBERS

235

The sentence g~~ is the conjunction of the following sentences: (1)

V x [ ~ E c ~ ~ t ) V ~ ~
(2) (3)

r(cpJxcpd

(4) (5)

vXVE<maxlh,w)r(X)xC~r~

v X Y [XEY

for each ,$<max{ Z , w } ,

(v11.

r (XI

+

vxy[vz(zExt)zEy)~xxy].

It is clear that the structure ( R (w+ a ) , E,p,PE)S<max(h,wl is a model of denotes the usual set-theoretical membership relation and p the rank function, both restricted to R (a+a). Notice also that R (a+ a ) =Xa. Now if Vl E Mod(rp,), we shall define a one-one map f:l%]+R (a+a ) (not necessarily preserving the structure of %!). Thus < am,proving Theorem 4.2.1. (i) E’ is well-founded. By (2), every element of J‘UI of the form r ’ ( x ) is equal to cp”, for some E< max{ Z , w > . Then, any element of X , where B # X [%I. whose image under r’ is a cs”, with PE minimal, is an E’-minimal element of X . By the results of 0 1, Chapter 0, the following principle of induction holds in %: ‘p,. Here E

c

v x [ v u (u E x ~ ~ ( u ) ) - * ~ ( x ) ] - + v X ~ ( x ) ,

allowing us to define maps between models of ‘p, by induction. (ii) Definition o f f : I%l-+R(w+ a). Using the principle of induction, if y E I%[ and f has been defined for all xElVl1 such that x E 5 , we define: f ( Y )= { f ( x )Ix E la1A 3 XEY}

- -

Notice that this is equivalent to:

(*>

f(x) E f ( y )

%k-@

xE7,

for all x, y E 121. We have to show that Range(f) R (w + a). This follows from: (iii) Y E 181 =+ P ( f ( Y ) ) f ( r Y y ) ) , and (iv) f(c;~=& for t < m a x { Z , u ) . Notice that (iv) implies that the right-hand side of the inequality in (iii) is an ordinal. Condition (iv) is easily proved by induction using (1) and (*). Now we prove (iii) using the induction principle mentioned above. Note

236

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, $ 2

that

Z k x E y + % t = r ( x ) E r ( y )+ f ( r ' ( x ) ) E f ( r ' ( y ) ) . By induction hypothesis we conclude that p( f ( x ) )
+

The following theorem captures the spirit of the technique used to find lower bounds for Hanf numbers. We formulate the result in terms of the Morley numbers m,, n, instead of the Hanf number b,; thus, we obtain a wider range of application (Theorem 4.2.3) and are able to give a somewhat more concise proof of Theorem 4.2.4.

THEOREM 4.2.2. Let [ be an infinite ordinal. Suppose there are a set T of LUJ p)-sentences, where 7 < K , a set S of types, an LJ p)-formula q ( v o , v , )

and an ordinal y such that: (a) for every iu E Mod( T ) omitting S, 'q well-orders I%/ in type < y ; (b) there exists 8 E Mod( T ) omitting S such that vB well-orders 181in type 2 6.

Then nK>at. If

s<

K,

then m, >Xt.

PROOF.Let p' consist of the symbols of p plus: two unary predicates M and Q,

a binary predicate E , a unary function symbol r, No individual constants c, ( n ~ w ) . Clearly < K . For each 4 EForm(L,,( p ) ) we define the relativization + Q of 4 to Q as usual, i.e.. by changing each subformula of the form Vx8 into Vx[Q(x)+O] and each subformula 3x8 into 3 x [ Q ( x ) r \ 8 ] . For each ZES, let C Q = { ~ Q I ~ E Zlet} ;

7

z'={ M ( v o ) }u { i(c0=cn) In EU},

Ch. 4,

21

LOWER BOUNDS FOR HANF AND MORLEY NUMBERS

237

and S Q = { Z Q [ Z E S ) u { Z ' } . Then g = T + l . Let TQ={I+QII+ET}and add the following axioms: = Vxy [Vz ( z E x ~ z E y ) + x=y] (extensionality), $2 = VXQ (~(x)), $3 =Vxr[xEy+cpQ(r(x),r(y))l. and I+3 assert that r is a "rank function" relative to E and the ordering cpQ, with values in Q. $'4=Vx[M (XIt, 1gyq Q(r(y)9r(x))1. 1c/, asserts that x E M iff x has minimal rank.

It is clear that if 'uEMod(T'), then

Let T ' = T Q u {$,,...,I+,}.

r

r

% Q" E Mod( T ) ; moreover, if % omits SQ, then % Q" omits S. Under this assumption Q " is well ordered by ( C ~ Q ) " ~ in Q type < y ; let ( a , [ a < { ) , l
a < ~ ( < { iff'ukcpQ[a,,a,]. ) Let us define

f ( a ) = { b E l % [ l b E % } foraEI%l,

-

and write, for simplicity, p = r a , a < b instead of %kcpQ[a,b],for a , b€QN. (1) Since 'u t- GI, a = b @ f ( a )= f ( b ) ; thus, f is a one-one map. (2) a €4 f(a) € Up<,A,. Let a E A , and b E f ( a ) ; therefore, p(a)=a, and bE"a. By $2, $3, 'Ir+(pQ[r"(b),r"(a)], i.e., p ( b ) i a,; then p(b)=ap, for some P < a , i.e., Up
From (1) and (2) we immediately infer:

A,=

(3) {f(a)laEA,} Notice also that:

< 2"fl<,Afl.

P

(4) < 8,. Indeed, it suffices to show: p(a) = a. * a = c," for some n E o. Suppose that there is an a E I%[ such that p(a) = a. and a # c," for all n €0; then 'up i 3ycpQ(r(y),r(u)); by I+4, ' u t = M [ a ] therefore ; a realizes Z', a contradiction. From (3) and (4) it immediately follows by transfinite induction:

A,

( 5 ) Up<,Ap < 3, for all

OL

< l.

238

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

Since p : 1iX2[)-+QZ', we get -

=

U ,.,{A,,

[Ch. 4, t 2

and thus:

u A, < c 3,<1,.

P

2l =

a<$

a<<

Hence, every model of T' omitting SQ is of power E. Clearly we can assume that 1231 =[0,5) and that a < fi iff B ~ v [ a , f i Ifor , a,p<5. Define the structure B' of type p' as follows:

where p is the rank function,

r

E"= E 123'1, c:'=ff

for all n EU,

M a ' = {ff },

S w = SB for any symbol S

E p.

Clearly B'I- qi (i = 1,. ..,4). By induction on the structure of 1c/ E Form,( p), , n E w , it is easily proved that for every a l , ...,a,< 5 :

23t-q[a1)...)a,]

@

%'CJ/Q["*

,...,a,];

it immediately follows that 23' EMod(TQ), and since 8 omits every I:E S , then 8'omits X Q ; moreover, it is clear that 23' omits 2'. Hence 8'omits SQ.

Thus, all models of T' omitting SQ have bounded cardinality (3*omitting SQ; hence n,>& The very last assertion of the theorem is trivially true, since SQ = 3 + 1. W As a first application of 4.2.2. we have:

Ch. 4, $ 21

LOWER BOUNDS FOR HANF AND MORLEY NUMBERS

THEOREM 4.2.3 (Shelah). For each infinite cardinal

K,

n,

239

>s(p)+.

PROOF.We show that for each ordinal 5, 2" < 5 < ( 2 " ) + , the hypotheses of 4.2.2. are fulfilled. For any such 5 we set y = 5 and construct p, T and 5. Since =2", there is an ordering iof 9 '(K) in type t, say 9 (K) =(a,la<(), with aa
H

a<&

Let p consist of a unary predicate symbol Q, for each a < K. a binary relation symbol < . T shall consist of the axioms stating that < is a linear ordering of the universe. 5 will be the set of the following types in two variables':

for each pair a , p such that a < p < K the type Z,,p defined by

x,p= { oo< 1 u { e,(u0) I Y E up>u { u { Q, (4I6 E a, 1 u { 1e,(4I 6 E

~Q,(V,>

0'

K

IY EK -up)

- a,

1.

We verify that the hypotheses (a), (b) of the preceding theorem hold. Let

Iu be a model of T omitting 5; we define a map f : 1%1+9( K ) by setting:

f( x) = { y < K I x E Q,! } for x E I%[. Since % omits Z, f is one-one. Since % omits all the types X,,p we obtain:

We prove (*). Let f ( x )= a,, f(y) = ap. f is one-one, x <5 implies f ( x ) # f ( y ) ; thus, a < p or Suppose p < a;since

(d) Since

-

~ € 0 ,

P < a.

xEQ,?,

6 E a p -yEQ,",

'

Theorem 4.2.2 has been formulated so as to apply to one-variable types. However, it is not difficult to see that 5 can be transformed into a one-variable type by suitably adjusting the set of sentences T. We leave the details to the reader.

240

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4,

2

the pair ( x , y ) realizes Z,,, contradicting our assumption that 2 omits S. Hence a < p, so f ( x ) i f(y). (+) If f ( x ) i f(y), then a < p and x # y ; hence x <% o r y <'x; if the last a contradiction; therefore relation holds, the pair ( y ,x ) realizes Z, X


Condition (*) clearly shows that the type of <' is less than or equal to [, the type of 9 ( K ) under < . This shows that 4.2.2. (a) holds. It is a routine matter to check that the structure 8 with universe T(K), where <'= iand aaEQy" w

yEa,

fory
is a model of T omitting S in which
THEOREM 4.2.4. Zf cf(K) > o,then

a,+ < m,= b,.

PROOF.We shall exhibit a similarity type p , a set T &Sent(L,,(p)) and a set 5 of types of power Q K such that the hypotheses (a), (b) of Theorem 4.2.2 are fulfilled for Y = [ = K + . The construction is based on a simple observation due to Chang. Assume that Cf(K) > a. The canonical well-ordering relation of K is the set-theoretical E. Recall that every ordinal a E K is also a subset of K , ~ C K Let . ~ C K XbeK a relation that linearly orders K (not necessarily a well-ordering). If r happens to be a well-ordering (not necessarily the canonical well-ordering of K ) then for each a E K , r a is a well-ordering of a (a subset of a well-ordered set is well-ordered). Since K is a cardinal and' a < K , then E < K , and (a,rp a) is isomorphic to ( p , E r p ) , for some 3/' E K . We shall show that the following are equivalent: (A) r well-orders K , ( B ) f o r e v e r y a E ~t h e r e i s p E K s u c h that ( a , r r a ) = ( P , E r P ) . PROOFOF (A) w (B). We have already shown that (A) * (B). (B) * (A). Suppose r does not well-order K , then there exists an infinite r-descending sequence

...r a , r a , - , r . . . r a 2 r a l r a o of ordinals a, E K . Since cf(K) >a,there is a E K such that { a , In Eo} 5 a . Thus r a does not well-order a and ( a , r a ) cannot be isomorphic to any

(P,E

r

rm.

r

Ch. 4, 3 2)

24 1

LOWER BOUNDS FOR HANF AND MORLEY NUMBERS

The idea is to use (B)-which can be "said" by sets of L,, sentences and types-to single out well-orderings of K amongst linear orderings. The similarity type p has the following symbols: K individual constants, ca ( a < K ) , a binary relation symbol, <, two unary predicates, K , L , a ternary predicate, R , two 4-placed predicates, F, G . We want to choose S and T i n such a way that any model of T omitting S be isomorphic to one of the structures = a), K < 5 < K + , defined by: for cr < K ,

6)

c," = a ,

(ii) (iii)

K'= { a Icr E K J = K , L'c[K,l)={ b I K < 131= K' u L',

(iv) (v)

<'=

E

b
rlq.

For every 5 E L' we choose a relation rEc K X K well-ordering taking care that in the case [= K , rK= E K . Then we define:

r

(vi)

R '= {<&V7Y)

I.$€

K

in type .$,

L" AY, ?1 E K A (%Y> E r t ) .

From the equivalence of (A) and (B) above, it follows that every rEhas K are (uniquely determined) the property (B). Hence for every ~ E there objects p E K and f: such that ( a , r c r a > 4 b ,E T P > .

We define: (vii) Let

F'

tl, . $ 2 L', ~

= { (&a,Y,S> .$I


L' A a E K A (Yvs> Efj).

relation rE, well-orders

K

in type

5,; hence

( K , r E I is ) isomorphic to a proper initial segment of ( K , r E 2 ) ; moreover, from

classical set theory we know that the corresponding isomorphism gtlE2is uniquely determined. Let (viii)

G'

={

(t~,Ez,Y,s)1[1"2E

L' A.$j<[2A

(Y,s) E&,E,).

Upon a moment's reflection it becomes clear that T should include the following sentences: for a < b < K ; (1) C a < C r (2) K(c,), for cr < K ; (3) " < is a linear ordering of the universe"; (4)Vx[ K (x) v 1 (x)l;

242

[Ch. 4, 8 2

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

( 5 ) vx3' [ K ( x )A L(Y)+x "]I; (8) vxJ"L(x) A L ( Y ) -+[x < y t," G ( x , y , . ;) is a n order isomorphism from ( K , R ( x , - , . ) )onto a proper initial segment of ( ,R(y;,.))"]].

I

5 should include the following types:

(9) Z = { K ( U o ) } U { 1( D o c C,) I a < K } , (10) ~ , = { u , < c a } u { ~ ( u o ~ c ~ ) ] for ~
r,

iff

Since a,+ is a model of T omitting S, 4.2.2(b) holds for t = K + . On the other hand, if 2l E Mod( T ) omits 5, then (by (lo), (1) and (3)) the set { c," a < K } is an initial segment of 12l1 well-ordered by <' in type K ; by (9) and (2), K ' = { c , " ~ ~ < K } . By (6), the set r , = { ( y , z ) l y , z E K ' and ( x , y , z ) E R ' } , x E L ' , is a linear ordering of K' which, by (7), satisfies the property (B) above; since (B) * (A), r, is a well-ordering of K". By (8), for x, y E L', x < % iff ( K ", r,) is isomorphic to a proper initial segment of (K', r,). Therefore, the map x-+ ( K ' , r , ) (=the order type of (K',r,)) is an isomorphism beL") and a subset of [ K , K + ) ordered by E. Heme <' tween (L', well-orders La in type < K + . By (4) and (5), (19l1, <') is the ordinal sum of ( K ' , <") and (L', <'); hence <' well-orders I%[ in type < K + . Thus we have proved that condition (a) of 4.2.2 is satisfied for y = K + . Since S has power K , m, >,+. rn

I

<'r

REMARK. In 0 5 below we shall give a characterization of the ordinal a such that m,=>,. From this it will be apparent that a is much larger than K + . HISTORICAL REMARKS. Theorem 4.2.1 is due to Morley. However the idea of the proof given here seems to go back to SCOTT[4]. Theorem 4.2.2 in its [ 11 but certainly was known to other present formulation occurs in SHELAH people. Shelah's paper also contains Theorem 4.2.3. A result similar to 4.2.4, for a different language, is due to MORLEY-MORLEY [l]; their proof

Ch. 4,

I

31

OMITTING TYPES; UPPER BOUNDS FOR MORLEY NUMBERS

243

used V = L . Later, Silver showed how to eliminate the assumption V = L, but used instead the heavy machinery of relative constructibility and absoluteness. The present simple argument based on the equivalence of (A) and (B) was given by CHANG[2]. The present proof is a slight simplification of his argument.

$3. Omitting types; upper bounds for Morley numbers In this section we prove the following results:

(I)

nK= (Theorem 4.3.1). (11) rn, < (Theorem 4.3.4). (111) (GCH) If cf(K) = w , then rnK= 3,+(Corollary 4.3.8).

(IV) m u =

au1(Corollary 4.3.9).

The method of proof is an extremely refined combination of the techniques of indiscernibles, Skolem functions, etc., developed in Chapter 2, $2, with the Erdos-Rado theorem of the partition calculus (Theorem 0.5.9(3)). The proofs given here use an extension by Chang of Morley’s original proof of (IV) given in his paper [I]. In view of the results of $ 2 above, we need only deal with the upperbound half of the equalities (I), (111) and (IV).

PROOFOF (I). As mentioned before we only consider languages with < K symbols. We observe first that it is enough to prove our result for the case that p is endowed with Skolem functions. For otherwise we form the type p # as in $2.A, Chapter 2 (which also has cardinal < K ) . The validity of (I) for p # then entails that (I) also holds for p. The verification of this assertion is left to the reader [Hint: If IZI is a structure of type p omitting the set 5 of types, the structure IZI# defined in $2.A, Chapter 2, omits all types in 5.1 It is convenient to enumerate the (at most K)terms of p in two different ways. The reason for this will soon be apparent. (1) ( T 1~[ < K ) will be enumeration (eventually with repetitions) of all terms of the language p. (2) For fixed n E w , (T,,,IS< K) is an enumeration (eventually with repetitions) of all terms in p having exactly n variables (for n = 0 it is an enumeration of all constant terms). NOTE.To stress notational simplicity, terms differing only in the names of their (free) variables will be considered identical. Once the preceding enumerations are fixed we can define the following

244

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4,3 3

functions: h ( 6 ) =the number of variables of

T~

(hence h :K+w);

since T* occurs in the list of form ( 2 ) corresponding to h(('), we define:

f ( 5 ) = the least 6 such that Thus 7[ is

7h(0,,(,, for

every

(hence f : K + K ) .

T ~ ( ~ ) , ~

(< K .

We shall prove a statement slightly stronger than that asserted in (I), namely: THEOREM 4.3.1. Let S be a set of types and T be a set of sentences, both in the language p. If:

(*I

for every { < ( 2 " ) + there is a model 913 of T of power 2 omitting S ,

a3

then : there are models of T of arbitrarily high cardinalities omitting S .

PROOF.The proof of Theorem 4.3.1 proceeds as follows. We will exhibit a series of statements [(i)-(v) below] such that (i) implies the conclusion of the theorem and ( k ) implies ( k - 1) for k = ii, . ..,u. Finally we will prove (v). Since < K , there are < 2" types in p. Let (A,IE<2") be an enumeration of all maximal types of p realized in some 213 ({ < ( 2 " ) + ) . We observe that if Z E S and (<2" then ZgA,. In fact, there is y < ( 2 ~ ) + such that 917 realizes A, and omits Z. (i) There are models Q of T of arbitrarily large cardinality realizing only maximal types which are among the Ais. (In other words, every type realized in such a 0 is 'included in some Ac) CLAIM1. (i) implies the conclusion of Theorem 4.3.1. PROOFOF CLAIM 1. Any such 0 realizes types included in some At. Therefore, by the preceding remark Q omits all types omitted in all the 9lU;s, in particular 0 omits all types in S. (ii) There exists a map g :~ - 2 " such that for every linearly ordered set ( X , < ) there is 23 E Mod( T ) , 12312X , such that for every (' < K and every increasing sequence x , < * * < xhto of length A((') in X : 4

Ch. 4, § 31

OMITTING TYPES; UPPER BOUNDS FOR MORLEY NUMBERS

245

Recall our prior notation that when r is a type, 8 a structure and a E I%l, then %I= T [ a ]means “a realizes r in Z”, i.e.,

CLAIM 2. (ii)

-

for every aET, % t = u [ u ] . (i).

PROOFOF CLAIM 2. Let X be an infinite cardinal and ( X , <) be a linearly ordered set of power > A. Let 8 be the structure corresponding to (X,< ) given by (ii). Let G=@(X,B). We show that 0 satisfies the assertion (i). = m a x { F , T,NJ> A (a) E = (b) By ( 2 ) of 2.2.10, Q < 8 ; hence 6 E Mod( T ) . (c) Every type realized in 0 is included in some A,. In the proof of Theorem 2.2.15 we remarked that by eventual changes of terms every element of @ ( X , 8 ) can be represented in the form ~ ‘ [ ,,..., x x,] where x I,..., x, is a increasing sequence in X of the proper length; in our present notation this means:

Suppose

r

is a type realized in 0 by the element c. Write c in the form E < K ; then

c=7?[x1, . . . , x h ( t )some ]

and since 0 < 8, BI=r[7F[xI,...,xh(,)]]. By the last assertion in (ii) we get As([), as desired.

r

(iii) There exists a map g : ~ + 2 ‘ such that for every linearly ordered set ( X , < ) the following set of sentences (in the language p’ = p u { X I x E X }) is consistent:

CLAIM 3. (iii)

3

(ii).

PROOFOF CLAIM 3. Since (**) is consistent, it has a model (of type p’). Take 8 = B Ir p . It is easy to check that 8 satisfies (ii). (iv) For every n E w there is a map g, : ~ - + 2 ‘ such that for every linearly

246

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

ordered set ( X ,

<)

(***I

TU

[Ch. 4, I 3

the following set of sentences is consistent:

u

-

... < x , i s i n ~ )

{ A ~ ~ ( ~ ) ( T ~ , ~x (, )~) ~ , x . ,. < . ,

q
This is just a restatement of (iii) in terms of the enumeration (2).

CLAIM 4. (iv) ==+(iii).

It is clear that g satisfies (iii), because

I-<= T~(~,,~([),

(v) For every n E w there is a map g, : ~-+2', a sequence {By 15 < (2")+} of structures and a sequence { (X;, =I, for every 5 < (2")+. (2) (3) X/cIByI for every 5 < ( 2 " ) + . (4) For every m < n, every 7 < K. every 5 < (2")+ and every increasing sequence x, < f. ..
8;t=

Ag,(q)

[ ~z.'q [xi* . ..,x m I].

CLAIM 5. (v) =+ (iv).

PROOFOF CLAIM 5. By compactness it is enough to show that every finite subset Z of (***) has a model ( ( X , <) is fixed). Let: Z={xEXlxoccursinC}, n,=max{nEwIn occursin Z}. Z is finite and no exists, because B is finite.

Consider the sequences { ( X t o ,
r

Ch. 4, 5 31

247

OMITTING TYPES; UPPER BOUNDS FOR MORLEY NUMBERS

holds for every increasing sequence x1
Z c T u U { A g m ( s ) ( ~ , , , r ( q , . . . , x , ) ) I m < n , , q < ~ and xl<

... <xm

isin X >

(which follows from the definition of no), implies that (By, e(z)),,== is a model of 2 .

REMARK.The reason that a sequence of %'s and a sequence of linearly ordered sets are needed-instead of just one B and one infinite linearly ordered set-is that B:+' and X;+l are constructed by an inductive procedure involving By,X,"for very high values of y. In this way we can ensure the validity of property (4). We turn now to the proof of (v). This is done by induction on n, the induction hypothesis being that go,. ..,gn have already been defined satisfying (v). It is convenient to introduce the following notation: DEFINITION 4.3.2. Let n E w , {<(2")+; c p : ~ - + 2 'be a function; suppose we already know go,.. .,gn- We say that the pair (cp, 5 ) is n-good iff there are a linearly ordered set ( X , <) and a structure %, such that: (1) B coincides with some a". (2) 77 >ac. (3) x c IBI. (4) For every m < n, q < K , and every increasing sequence x,< . * < xm in X , of length m:

,.

( 5 ) For every q < K and every increasing sequence x k< . . * n in X :

< xn of length

(In the case n = 0, (4) is vacuous.) In terms of the foregoing definition, (v) is implied by

THEOREM 4.3.3. For every n E w there is a map cp : ~ - + 2 'such that for every {<(2")? the pair ( c p , { ) is n-good.

248

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 5 3

In fact, in order to derive (v) from 4.3.3 it is enough to define:g,,=any map 'p : ~ + 2 ' such that for every 5 < (2")+ the pair ('p, 5 ) is n-good.

PROOFOF THEOREM 4.3.3. By induction on n. At any step n we will begin by proving the following, weaker result: STATEMENT I,,. For every lo < (2")+ there is a map rpo : ~ + 2 ' such that the is n-good. pair (cpo,l0) We work our way to 4.3.3 through the following successive steps:

STATEMENT 11,. There is q0:~ + 2 ' such that the set

STATEMENT 111,. There is 'p : ~ + 2 ' such that for all 5 < (2")+ the pair (cp,{) is n-good. (111,) is Theorem 4.3.3 for the value n. Thus, after (111,) has been proved we are ready to apply it in the proof of (I,,+ ,). The scheme of the induction is:

-

The proofs of the implications (I,) e (11,) nEw.

CLAIM6. (I,,)

PROOF OF CLAIM

* (111,)

are uniform for all

(II,,). 6. For g, : ~ + 2 " , let

-

(I,) asserts that (2")+= UpE(2x)rZp. Since (2")' =2", by the "pigeon-hole = (2')+,' For 'cardinality reaprinciple" there is 'po: ~ + 2 ' such that sons, Zqo is cofinal in ( 2 " ) + , i.e., (11,) holds.

'

This is, essentially, the reason that in the proof of (In).

+)+

is needed in the proof. The same fact is used again

Ch. 4,

5 31

249

OMIT~INGTYPES; UPPER BOUNDS FOR MORLEY NUMBERS

CLAIM7. (11,)

( 111,).

PROOFOF

7. The same rp=rpo as in (11,) will do, observing that

CLAIM

"(cp,{) is n-good" is invariant under descending {:

(rp,l,) is +good

A

l2G

+.

(rp,12) is n-good.

PROOFOF STATEMENT (I"). Case (1) n=O. Given lo<(2")+,set: 23=fl)21so+I (cf. (*) in the hypothesis of Theorem 4.3.1). X = a n y subset of 181 of power 2

< =any

afo+,(exists because

-'23

2 2zo+1).

linear ordering of X .

(1)-(3) of Definition 4.3.2 are true; (4) is vacuous in this case. Let r,(v < K ) be the type realized in 8 by the element T:,. Since rV is a maximal type realized in some flv, by definition of the A i s there is p<2" such that r, = A p . Write q0(q)= /3. We have just checked that (rpo,lo) is 0-good. Case (2) Induction step. The induction hypothesis is (111,). Once we have fixed 10<(2")+ we apply the induction hypothesis to l1= max{ lo,2") + n to get a linearly ordered set ( Y , < ) and a structure 8 such that (1) and (3) of Definition 4.3.2 hold and in addition: (2') > as,= 2 n ( 2rnax{lo,2'))* ( 4 ) For every rn < n , 17 < K, and y , < . - y m< in Y :

7

-

[Y I , .* , Y ~ I ] *

8 i= Ag,,,(v)[ V,:7

Fix [ < K . For p < 2" we define C; EC;

{ ~ ~ ~ * * . ~ ~ n + l } Y I

Let y , <

T,+ B

c[Y]"+l as follows:

<... < ~ n + l

and '23' A P [ T ~ + I , ~ [ Y I ) . . . , Y ~ + ~ I ] .

- . < y n,I.+ ,rbeisina Ymaximal . Let r be the type realized in 'B by the element type realized in some i?lv (by( 1)); hence

I , E [ y .I.,y,+ ,.

there is p < 2" such that r = A P ; hence { y I , .. .,yn+I}< E C;. Thus we have proved that [ Y]"+I= UP<2L'jfor any E < K ; hence [yln+'=

n u c;. [
By distributivity we have:

8<2'

250

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 0 3

Since 7>2,(3max(Eo,21)) and the index set in the right-hand side of the which has power 2" <3max(So2x), by the Erdospreceding equality is (Y)", Rado theorem there are X c Y and rpo E(2")" such that

and

We check now that the choice of the objects '23, ( X , < Y X ) makes (rpo,{o) into an ( n 1)-good pair. (1)-(3) of Definition 4.3.2 are clearly true. The formulas asserted in ( 4 ) hold for all m < n, all q < K and all y 1< . . .
+

+

c

PROOFOF (11). This follows from (I) by a simple cardinality argument. We need the following fact, which is a trivial consequence of the downward Lowenheim-Skolem theorem for Lww: (*)

Given a type Z of p and a structure \u omitting 2,there are structures omitting Z in every power X such that max{ 7 ,No) < X < The same applies to any set of types.

z.

THEOREM 4.3.4. m,< PROOF.Let (SEI[<2")

be a complete list of all sets of types of p of power (notice that there are at most 2" such sets, whereas without this restriction the number might be as high as 22'). Let a, be the least cardinal X such that if S, is omitted by a structure of power > A , then there are structures of arbitrarily high powers omitting S,. Note that m, < s u ~ , < ~ ~ a ~ . for each [<2". For if 5, is omitted by a model of power Also a,< 2 3 (2x)+, by virtue of (I) it is omitted by models of arbitrarily high powers, and by virtue of (*) it is also omitted by models of power < Now since cf( 3(2=)+) = (2")+ and (2")+ is regular, we get ~ ~ p ~ << ~ . a ~ Hence m, < 3 12s1+.


Our next goal is the

Ch. 4, 5 3)

OMITTING TYPES; UPPER BOUNDS FOR MORLEY NUMBERS

25 1

PROOF OF (111) AND (IV). We shall actually prove, without the GCH, a more general result of which both (111) and (IV) are immediate consequences; namely THEOREM 4.3.5 (Helling). Let K be a beth-number such that cf(K)=w; p denotes a similarity type of power < K ; let S be a set of power < K of Lam(p)-types and T a set of Law(p)-sentences. If for every { < K + there is a omitting S, then there are models of T of model 911 of T of power > arbitrarily large cardinality omitting S. PROOF.As in the proof of (I) we assume that p is endowed with Skolem functions. The enumerations ( T ~ ( ( < K )and ( 7 , , , ( 8 < ~ ) of terms of p, as well as the maps f and h, are exactly as before. Let 5 = {r,I a < K ) be an enumeration of 5. Enumerate each T,, r, = { a t 16 < K } , in such a way that each formula of ra is repeated K times. We will need the following relation R defined on K x K : ( a I , Y I )R(a2, Y2)

-

max { a Y I 1 < max { a2, Y2 1 19

v [max{ al,vl} = max{ a2,~

2 A)

(al< a 2 V

= a2 A YI< Y 2 > ) l 9

and the following properties of R : (a) R well-orders K x K in type K ; (b) for each P < K , the set Ys={(a,y)Imax{a,y)< P } is an initial segment of ( K x K , R ) with a largest element; in fact, Ys coincides with the set of all pairs of ordinals which are R-predecessors of or equal to ( P , P ) . Using the assumptions on K , we let (K,, In Ew) denote a strictly increasing sequence of cardinals with the following properties: (c) if K = W ,then each K,, is finite and K , , + ~ > 2%; (d) if K > w, then each K, is of the form at+ (so that K ~ + , 2%); (e) limnEwKn = K. After these preliminaries the proof proceeds by steps very similar to those of Theorem 4.3.1; therefore we shall only sketch it by indicating the changes to be made in Theorem 4.3.1. (i) There exists a map g : K x K + K such that for every linearly ordered set ( X , < ) there is 23 €Mod( T ) , 1231 2 X , so that for every a , y < K and every increasing sequence x l < . . . < xh(,) of elements of X :

(ii) There exists a map g :K X K + K such that for each linearly ordered set ( X , < ) the following set of sentences (of type p u { iI x E X } ) is consistent:

252

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 8 3

-

(*) T U { iu,u(,,,)(7,,(x l , . . . , X h ( y ) ) ) I a , y < ~and x , < . . - < x h ( , , , isin X } . Exactly as in Claims 2 and 3 of Theorem 4.3.1we prove that (ii) (i) and that (i) implies the conclusion of Theorem 4.3.5. (iii) For each n E w there is a map g, : K X K + K , a sequence {2?{ I { < K ' } of structures and a sequence { ( X , , <,>I{< K'} of linearly ordered sets such that: (1) Each 8,coincides with some PI,. (2) >>( for every { < K + . (3)X s c IBpI,I for every { < K +. (4)For every m < n, ,8 < K and { < K + ,

7

holds for every pair a , y of ordinals such that max{a,y} < j3 and every increasing sequence x I< { . . . < ,xm of length m, in X s . (ii) is proved as in Claims 4 and 5 of 4.3.1.(*) is consistent iff (iii) each of its finite subsets is. Let Z be a finite subset of (*); the consistency of Z is an immediate consequence of (iii) for the following particular values of n and j3: n=rnax{h(y)17,, occursin Z } P=max{a,f(y)Iua and r,, occurin Z}. As in 4.3.1,~ : K X K - + Kis defined by g(a,y)=g,,(,)(a,f(y)). The values g,(a,y) are defined by induction on n, each step of this process involving a secondary transfinite induction on the parameter j3 that appears in (iii). To carry this out we modify Definition 4.3:2 as follows:

4.3.6.Let n E w , ,8< K , { < K + , q : Y ~ KAssume . that the maps DEFINITION g, have been defined for all m < n and have the property (4)of (iii) above. The pair ( q , { ) is called (n,B)-good just in case there is a linearly ordered set ( X , <>and a structure 8 such that: (l)-(3) of 4.3.2hold. (4)For every pair a , y of ordinals such that max{a,y} < ,8 and every sequence x1< ' ' . < x, in X ,

Ch. 4, 5 31

253

OMIlTING TYPES; UPPER BOUNDS FOR MORLEY NUMBERS

( 5 ) cp is a strictly increasing map from ( Y p ,R ) into

* cp(%Y)<

(.,y)R(a',y')

(K,

E);

i.e..

d"'9Y')

holds for all ( a , y ) , ( a ' , y ' ) E Yp. Condition (5) is introduced with the purpose of pinning down the cardinality of certain sets of functions which appear in the proof, in view of the fact that.K is singular whenever K > W . The analog of 4.3.3 is:

THEOREM 4.3.7. For each n E w and /3 < K there is a map cp : Y p + ~such that for evety { < K + the pair ( c p , { ) is (n,/3)-good. In order to derive (iii) from this result, it suffices to construct a map g, : K X K + K such that the pair (g,r Y p , [ ) is (n,p)-good for evety /3< K and { < K + . This is done as follows. Using 4.3.7 pick, for each s E w , a map cp,: Y K s - - +such ~ that for every { < K + the pair (cp,,{) is (n,K,)-good. Let 8, = (P,(K,, K ~ ) ;8, is the R-largest member of Range (q,).Then set: g , ( a 9 Y ) = v o ( ~ , Y ) ,for ((-U.Y)EYKO7 s- I

g,(a,y)=

x

j= I

8,+cpJ(a,y) (ordinal sum)

for ( a , y ) YK,~ Y%-,, s>O. PROOFOF THEOREM 4.3.7. By induction on n . /3 is fixed throughout the proof. Conditions (1,)-(111,) are analogous to those of 4.3.3, Claims 6 and 7. CLAIM1. (I,)

-

(11,J.

1. Let I( Y p ,K ) denote the set of all strictly increasing maps from ( Y p ,R ) into ( K , E). Our first task is to show that I( Y p , ~<) K . Since the pair (/3,/3) is the R-largest element of Y p , it follows that for any f E I ( Y p , ~ )Range , (f) has an €-largest element: f( /3./3); hence since (K, 1 n E w ) converges to K , there is an r E w such that Range(f) c K,: thus, PROOFOF

CLAIM

I( Yp,K)C Clearly

=

7< /3 x /3 <

K;

hence

-

u

K?.

rEw

< K~~ for some r o E w . Therefore:

254

[Ch. 4, 8 3

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

But K~: < (2Kr)a= 2"'aX('hKro < K ~ ~ ~ I , ~ the~ last , ~ inequality ~ ) + holding by virtue of our assumptions about the cardinals K, (cf. (c), (d) on p. 251). So I ( Y ~ , K<)( ~ 0 + 1 ) . ~ r , , + l + Z

P

r>ro

Kr+I=K.

For each cp E I( Y p , ~ let ) Thus, (I,) asserts:

zp= { { < K + I ( r p , l ) K+

-

=

1s (n,P>-good}.

u

v E I(

zp;

Yp.K,

since I( Yp, K) < K, there is some rpo€ I( Y p , ~ )such that = K + ; hence Zpois cofinal in K + . (111,) is proved as in Theorem 4.3.3. The implication (11,) P

PROOFOF (I"). This is done by induction on n , using (111,) as induction hypothesis for the proof of (In+l). Case (1) n = 0 . The definition of cp is done in this case by transfinite induction on the well-ordering R of Yp. Given lo< K + , set 23, X and < as in the last part of the proof of Theorem 4.3.3. 8 omits each r,, a < K; so, for each y < K there is 6 such that: Moreover, since each u E r, appears repeated chosen larger than any ordinal p < K; let:

K

times, such a S can be

cp(a,y)= the least

S larger than sup{cp(a',y')I(a',y')R(a,y)} such that (*) holds.

Thus, conditions (4) and (5) of 4.3.6 are fulfilled, and cp is (O,p)-good. Case (2) Induction step. Fix lo< K + ; let = max{ lo,K} n . By induction hypothesis there is a linearly ordered set ( Y , i ) and a structure 23 such that ( l ) , (3) of 4.3.2 hold, and:

+

(2') 7> a<,= a n ( amax(lo,ti)); ym in Y and (4') for every rn < n , every increasing sequence y 1i .. i every a , y < K :

Ch. 4, § 31

255

OMITTINGTYPES; UPPER BOUNDS FOR MORLEY NUMBERS

For each triple ( S , a , y ) where S < K and ( a , y ) E Yp, define:

Let ( y l , . . . , y n + l } x E [ Y ] " + Iand ( a , y ) Yp. ~ Since 2' 3 omits ra there is a 6 such that ( y I , . . . , y n + lEC&; }~ moreover, such S can be chosen arbitrarily large below K ; therefore, by induction on R we can define a strictly increasing function f:( Yp,R ) + ( K , E ) (i.e., f€I( Y , , K ) ) such that E C$y,Y) for every a , y. Thus we have proved: { y , , .. .,yn+ .[,],+I=

Since I( Y p , ~<) K

x c Y,

<

>amax{co,K)

u

n

f(a,v),

Ca,y

J E I ( Y ~ , K( )a , v > E

3max{So,r), by the Erdos-Rado theorem there is an

>aS,and a 'p E I( Y p ,K )

such that

It is clear now that 23 and ( X , i r X ) make (rp,{,) into an ( n + l , P ) good pair. This completes the proof of Theorem 4.3.7 and, thus, also the proof of Theorem 4.3.5. COROLLARY 4.3.8. (GCH).If cf(K)=w, then m K= b, PROOF.By the GCH, K is a beth number; hence 4.3.5, and rn, > 3K+by Theorem 4.2.1.

a,+. m K < a,+ by

=

m, = b, PROOF.w is a beth number of cofinality o ;hence mu.< 4.3.5, and m, > 2 by Theorem 4.2.1. COROLLARY 4.3.9 (Morley-Lopez-Escobar-Helling).

=

Theorem

awl. by Theorem

lo,

(1) MORLEY [ 13 proved a weak form of Theorem 4.3.5 for K = w REMARKS. and only one type instead of a countable set of them. The equality mw=3wl is due to Morley and Chang. The observation that m, = b, was made by a number of people, including Chang, Helling and Lopez-Escobar (cf. MORLEY[I], p. 273). (2) The idea of using maximal types to prove Theorem 4.3.1 is due to Chang (see his paper [2], p. 50).

256

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, J 4

(3) Theorem 4.3.5 was announced in HELLINC[l], but the proof never published. Observe that only LCH (the limit-cardinal hypothesis) is needed in 4.3.8; it suffices to restate Theorem 4.3.5 in terms of strong limit cardinals instead of beth numbers with limit index. [ I ] ( without CCH) (4) MALITZ[2] (using GCH),and later BAUMGARTNER have improved Theorem 4.3.9 as follows: the Hanf number of the set of all complete sentences of LUlw is (recall that u E Sent(L,,,) is complete iff YI, %EMod(u) i?l=,+,%).

-

0 4. Applications: non-definability of well-orderings in L,, As an application of the methods developed in this chapter, in particular of the upper bounds for Hanf numbers given in $3, we shall show that well-orderings are non-axiomatizable. in a strong sense, in any of the infinitary languages L,,. The results included below are due to LOPEZ-ESCCBAR [2], [3]. A weaker result along the same lines was obtained by KARP [2] using a different method (see Exercise 5.4.5). We use the notation introduced in Chapter I , 692.3. Recall that W denotes the class of all (non-empty) well-ordered structures. Our first result is that W e PC,(L,,( <)) for any cardinal K ; assuming the contrary we construct, using additional predicates, a situation that violates the upper bounds for lj, given in $3. Using the downward LowenheimSkolem theorem, we improve the result to W tZ RPC,(L,,); we do not know at present whether this result is really stronger (cf. Theorem 1.3.16 and the subsequent remark). In Chapter 1, 02 we defined by transfinite recursion the formulas Oa,(c,) ( ,B EON), which have the following properties: (i) If ,B< K , @a, is a formula of LK,( < ) with one free variable. (ii) For every system i?l = ( A , < ,. . . ) (where < linearly orders A ) and every a E A : i?lt-Op[a]H { b ~ A l b < a hasordertypefi. }

THEOREM 4.4.1 (Lopez-Escobar). For every cardinal

K,

W @ PC,(L,,(

< )).

PROOF.Since each PC,(L,,) class is a PC in a larger language. it suffices to prove the theorem for PC classes. Suppose. then, that W is a PC(L,,) class for some cardinal A. Then there is a type (r. including < and a sentence 0 of L,,(p) such that W=Moda(0)r{ < } . In addition. we can (and do) assume that h is a successor cardinal, h = v + , and that 7 < T .

Ch. 4,

41

APPLICATIONS: NON-DEFINABILITY OF WELL-ORDERINGS IN

L,,

251

We enlarge p to p' by the addition of the following new symbols: binary relations R,, . . .,R,, a ternary relation R,, a unary relation R,, individual constants co, . ..,c6. The purpose is to define a sentence \k of Lhw(p')) such that: (I) There is a model of \E of cardinality aK,where K = 222": (11) I1 EMod(Br\\k) * %
er\

( LAu( I"')) = 6, G

a(2-)+ < a,.

Before writing down \k we describe the models we have in mind for (I).

PROOFOF (I). Let 6 = aK+1, ~ = 2 * ~ Since '. (6, E ~ S ) E W , there is a structure '73 E Mod,(B) such that '73 1 { < } = (6, E 1S ) , i.e., 181= S, < B = E 16. Let us extend '73 to a structure 6 of type p' as follows 8rpLB,

CF=o,

Cf=&J,

Cp=T,

C?=2,,

C?=2=',

C?=2*

2"

=K,

Cf=

a,,

Ry = { (a , a + I ) 1 a E 3 .}, RP={(a, a a ) 1 a G ~ } , R:=

{ a I O< a G a Kand a is a limit ordinal}.

By definition of the cy's. there is a one-one, onto map g, :c~,+:?(cF)

( i = 2,3,4); we define:

R,'D={(a,2)1a
For each a < K let f a : define:

aa+,+?? ( am)be

(i=2,3,4).

a one-one, onto map; then

The reader can easily check that the following sentences of LAw( p') hold in 8: (1) V x y [ R , ( x , y ) H x < Y r \ 1 3 z ( x < z A Z < . Y ) ] , (2) V x [ R 7 ( x ) ~ c o x< r \ V y ( y < x+3z(y < z r \ z

< x))],

258

[Ch. 4, 3 4

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

or

PROOFOF (11). Let %EMod(BA\); by 8, the relation <' well-orders I%l; hence there is an ordinal p such that (%, < ' ) = ( p , E r p ) ; thus without loss of generality we can assume I%[= p = {tI t E p } and <' = E p ; therefore each c: is an ordinal < p that we denote by ti( i = 0,. . .,6). By sentences (3)-(5): t o = O ; t l = w ; t2=r.From (18) we get: p = t 6 + 1 . Let

r

f

= { ( a ,X

> I a E&+I and X

={5

I ( a ,5 ) E R F } }

( i = 2,3,4);

using the appropriate instances of (6)-(11) we conclude that each f . is a one-one map such that 1. : &+ 9 (&) (i = 2,3,4); hence < 2", f , < 2.' and < 2 2 2 " = ~hence ; &
,+

.from (12) we learn that

(*I

g, is a one-one map within domain a.

From (13)-( 15) it follows that RF is a map with domain equal to & + 1 and

Ch. 4, 8 41

APPLICATIONS: NON-DEFINABILITY OF WELL-ORDERINGS IN

259

L,,

such that R f ( O ) = w , RF(&)=&. Using (16), (17) and (*) we prove by induction on a that

Therefore we get

RF(a) <

aa

for all a

< Es.

We prove now a refinement of the preceding theorem:

THEOREM 4.4.2. For every cardinal

K,

W

RPC,(L,,(

< )).

PROOF.As before, it suffices to prove the result for RPC instead of RPC,. < )) for sqme A. This means Suppose, by contradiction, that W E RPC (LAW( that there is a similarity type p involving <, a formula q E Form,(LA,( p ) ) and a sentence B of LAw(y) such that W=(Mod,(B))vr{ < } ; in other words: (i) For every ordinal p > 0 there is 23 E Mod,(B) such that

(ii) If ‘u EMod,(B), then there is an ordinal p>O such that (& <’”>-(P7 E rp). As in the preceding theorem we assume that X = T + and F < A . Let p’ be as before and \kv be the sentence (of Lhw(p’)) obtained from \k by relativizing quantifiers to v; let \k’= \kv A A ~ = ~ ~ I ( C ; ) . From (I) of the preceding theorem and (i) we obtain:

(i’) There is an ordinal 6 such that of type p‘ such that:

7 = 3, ( ~ = 2 ~ *and “ ) a structure illo

%I,EMod,f(Br\\k’), q)’llO= 6,

<‘O=

E

16.

Using (11) and (ii) we get: (ii’) If ill E Mod,.(B A q’), there is an ordinal p > 0 such that

Let F be a new binary relation symbol, p” = p u { F } and let

x

be the

260

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, f 4

sentence "F is a one-one map from the universe onto rp", i.e., V ~ ( YF ( ~ , bY

v (Y 1) A VY (rp ( Y )+3x

F ( x , Y ) )A Vx 3Y F ( x , Y )

A V x y z [ F ( x , y )A F(x,z)+Y = z ] A v x Y z [ F ( x , z )AF(Y,z)+x=Y]. - Let @ be 8 A 9'A X . Let % E Mod,,,(@); from it follows that rp% = % ,

x

and from 8 A \k' together with (ii') we get (11')

%€Mod,..(@) +

5
5
shown:

In order to contradict the fact that h(L,,( p " ) ) < 3 it is enough to Prove I(

(1')

@ has a model of power

aK.

PROOFOF (1'). We apply the downward Lowenheim-Skolem theorem 3.4.2 to the structure 2, of (i'). This requires some preliminary computations:

we also know (cf. Chapter 0, 0 1) that cf(2")> T , and a fortiori cf(K)>T; this implies ai=3, for every cardinal v < T (cf. formula 1 1 (ii). p. 6 ) ; hence 3,"=aK.Since A is regular, Corollary 3.4.2 applies; thus, there is 911 2lI ==I, such that 6 Ll2,l. Hence 911€Mod,,(8~\k'); using i , and , (1') we get

sc

and thus

q"o=

-- rp"1=

6

=

6, -

a K =3 , .

Let % be the structure of type p" defined by

r

9 p' =

F'=any

bijection from I3,l onto q * ' I .

Clearly 3 is the model of @ required in (1'). This completes the proof of Theorem 4.4.2. COROLLARY 4.4.3. For every cardinal K there is an ordinal 6 > 0 and a total& ordered but not well-ordered system ( A , < ) such that:

Ch. 4,

41

APPLICATIONS: NON-DEFINABILITY OF WELL-ORDERINGS IN

-

L,,

PROOF.Otherwise, there is a cardinal K~ such that for all ordinals S all structures ( A , <):

<)

(8. E r s ) = . , ( A ,

This immediately implies that

U

A is well-ordered by

Mod(Th"@'((G,

26 1

> 0 and

<.

E rS)))CW.

6 EON

6 >O

Since every member of W is isomorphic to some structure of the form (6, E YS), where 6 EON and 6 >0, the reverse inclusion also holds; whence: (*)

U Mod(Th""((8,

E rS)))=W

6 €ON 6>O

Let v > Sent(L,,(

<)); then /\TWO"((&

E rS))ESent(L,,(

<)). It fol-

lows that if A > Sent(L,,( <)), then u=

v /\Th"O"((S,

6 €ON 6>O

E IS))

is a sentence of LA,(<). It is easily checked that the left hand side of (*) is equal to Mod(u), and hence W = Mod(a). Thus, W E EC(LA,( <)) for some cardinal A, contradicting Theorem 4.4.1. KARP [2] has proved independently a refinement of the preceding corollary; she inferred from this result that W is not an EC,(L,,)-class for any K (see Exercise 5.4.5, p. 401). NOTATION. Given two totally ordered systems ( A , < ) , ( B , i ), we denote by ( A , < ) @ ( B , < ) the relational system ( A X B , where is defined as follows; for a j E A , b j E B (i= 1,2):

a),

Karp's result:

262

(Ch. 4, S 4

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

-

THEOREM. Let & be an ordinal such that 6 = sup,<&m a (ordinal exponentiation), let 8 = K and let ( A , < ) be a totalh ordered system with first element; then :

(*I

(8, E rs)=,,(s, E r 8 ) w ~ <>. ,

In particular, given a cardinal

K,

(*) holds for every cardinal 6

> K.

Corollary 4.4.3 has an interesting application related to the interpolation theorem for LUI, (Chapter 2, $3). This theorem can be stated in the following model-theoretic form: Let K , L E P C ( L U I , ( p ) ) be such that K n L = B ; then there is M E EC(LUI,(p)) such that K C M and M n L =B .

We show that-contrary to the situation in La,--this extend to PC,-classes in Lala.

result does not

EXAMPLE 4.4.4.Let K be the class of all isomorphs of (aI, E lal) and L be the class T- W of all total orderings that are not well-orderings. Clearly K n L = B . As mentioned in Exercise 1.3.1 (b), $3, Chapter 1, L is a PC in L,,. K is a PC,(L,I,(<)), in fact, K is the class of all reducta to { < } of models of the following set of sentences in the language { < ,F, c,, I n €a}, where F is a ternary relation symbol and the c,’s are individual constants: (1) “ < is a linear ordering”; (2) 3x@,(x) for cr
V x y z u [F ( x , y , z )/\F(x,Y,u)+z=Ul

and such that Suppose that there is a class M closed under KcM, M n L = J 3 . Then, (a,,E r a l ) E M . By Corollary 4.4.3 there is a system ( A , < ) which is not well-ordered and

Ch. 4, 5 51

FURTHER HANF NLTMBER COMPUTATIONS

263

By assumption ( A , < ) E M , and since ( A , <) is linearly ordered,

( A , < ) E L ; thus (A, < ) E M n L , contradicting M n L = B .

We have proved: there are PC,-classes K , L in L,,, such that no M satisfying K C M, L n M =B is closed under In particular, no such M can be an EC,(L,,,).

0 5. Further Hanf number computations No account on the Hanf number of languages with finite quantifiers will be complete without a description of recent work of Barwise, Kunen and, independently, Morley, which gets to the heart of the matter. This work [ 11; the results presented below are has been reported in BARWISE-KUNEN just a reformulation of some of the arguments in this paper. We shall first need some A. Preliminary concepts and results

We begin by recalling that a formula 9, of the language of ZFC (eventually containing certain sets as parameters) is called a bounded-quantifier formula (or a A,-formula) iff all of its quantifiers appear restricted, i.e., in forms Vx(x E y + . . .) or 3 x ( x E y A .. .), where y is a variable (or one of the parameters). Bounded-quantifier formulas are of central importance in the metamathematics of set theory owing to their absoluteness properties with respect to transitive classes. At only one point we will use an absoluteness argument and shall develop it in detail; so-we do not need to go into further details at this stage. Recall that H(K)denotes the family of all sets hereditarily of power < K, that is, for every set y : x E H ( K ) w foreveryzETC({y}), ? < K , where TC(x) denotes the transitive closure of x :

A simple induction argument on the rank of members of H(K) shows that H(K)C R(K)for K regular. A set of power < K of members of H(K)is again in H(K).

264

[Ch. 4,

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5

Some of the arguments developed below depend in an essential way on the existence of certain subsets 9 of Form(L,+,) with the following characteristics: (i) 9 should be small (of power < K ) and, as a set, well-behaved from the (metatheoretical) point of view of ZFC; (ii) however, 9 should be wide enough as to allow us to define certain notions and carry out certain arguments of infinitary logic (e.g., a version of the model-existence theorem mentioned in Chapter 2, $3). The natural notion satisfying these requirements is that of a fragment of L,+, (or, more generally, of LmJ. A systematic treatment of the ideas and methods dealing with fragments of infinitary logic goes beyond the scope of this book and has been omitted; fragments of L,,, are studied in detail in KEISLER [3]. Instead of fragments we will introduce the weaker notion of frame, which will be sufficient for our purposes.

DEFINITION 4.5.1. ( I ) Let p be a similarity type of cardinality (in p ) is a set 9 L Form(L,+,( p ) ) fulfilling the conditions: (i) 9 is closed under subformulas, Sbf(9) C_ 9 ; (ii) every symbol of p occurs in some formula of 9 ; (iii) 9 is closed under 1, 3, V and finite connectives; (iv) for any rc.k,/\rE\k VrE9; (v) 9 is closed under substitution of terms of p :

< K . A frame

-

0 E 9 A { o I , . . .,o m } c Fu ( 0 ) A t , , .. .,t ,

E Term,

+ 0( t I , , .t,)

E9 ;

(vi) T < K . (2) If for a given formula cp E Form(L,+,( p ) ) we impose the additional condition : (vii) cp E 9, then 9 will be called a frame for cp. (3) Given a frame 9,the set A ( * ) of (possibly infinitary) admissible conjunctions of 9 is defined as follows:

Notice the following, simple properties of A (9): (a) A (9) contains all finite subsets of 9 (by (iii)); (b) r E A ( 9 ) w V r E 9 (by (iv)); (c) A ( 9 ) < K (by (vi)).

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FURTHER HANF NUMBER COMPUTATIONS

265

REMARK. The notion of frame for rp is intended to portray “small” segments of LK+Jp) where one can talk about models of the single sentence rp. A (9)tells us which (possibly infinitary) conjunctions and disjunctions occur in 9.The crucial property in the preceding definition is (vi), the cardinality restriction. We need know the following two results:

RESULTA. Given any formula rp of L,+,( p) there is a frame for it. (In fact, notice that the set Sbf(rp) can be closed under conditions (ii)-(v) without blowing up cardinality beyond K.) RESULTB. By suitable coding of the symbols of p and formulas of L,+,( p), any frame \k can be identified with a set hereditarily of power S K; moreover, such a coding identifies A ( 9 ) with a set in H(K+). SKETCHOF PROOF. Code the symbols of p and the K variables of the language by members of H(K+),and the logical operations by set-theoretic operations preserving membership to H(K+).Since 9 is of power < K and all its members are coded within H(K+),its associated (coding) set belongs to H(K+).Likewise, all subsets of 9 will be coded by members of H(K+);

-< K,it also belongs to H(K+).4

since A (\k)

Thus, when necessary, we shall simply assume that 9,A ( ~ ) € H ( K + ) . Let 9 be a frame in the similarity type p. If v c p , 9 T v denotes the set 9 n Form(L,+,(v)), which also is a frame. Sent(9) denotes the set of all sentences in \k: Sent(*) = 9 n Sent&,+,( p)). \k-equivalence of structures is defined in the obvious way and denoted by

=*. -

It should be clear what is meant by a tidy( = Skolem) frame; notice that if there are enough function symbols to eliminate existential quantifiers from every formula of 9,by (iii) and (v) 9 necessarily contains all defining axioms for these function symbols. The notions of complete and of consistent sets of sentences relative to a frame 9 are obtained from the usual definitions by restricting quantifiers to 9. For purposes of immediate reference we include the following basic definitions:

266

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DEFINITION 4.5.2. Let \k be a frame of p. A set T C 9 (Sent(*)) is a consistency property (relative to 9)iff for every Z E r , E cp E Sent(*), r E A ( * ) and t , t,, t,ETerm,, the following hold: (i) either cpEZ or (icp)BZ; (ii) l c p *~ Z~u { c p i } ~ r ' ; (iii) AIXE * for all 8 Er, C u ( 8 ) E r ; (iv) cp=Vu\C/(u)~Z for all tETerm,, Zu{+(t)}EI'; (v) V r E C + for some BET, Z u { 8 } E r ; (vi) cp=3u+(u)EZ for some tETerm,, Z u { + ( t ) } E r ; (vii) (a) C u { t = r } E r ; (b) t , = t 2 E Z ZU{t2xtl}Er; (c) t,7=:2r $ ( t l ) E X * zu { 4 4 t 2 ) } E r .

+

*,

--

The reader is recommended to consult KEISLER [3], lecture 3, for further information.

DEFINITION 4.5.3. A set ZcSent(\k) is closed iff for every

cp €Sent(*),

+E*,

T E A (\k) and t , t , , t,ETerm,, the following hold: (i) either cpgC or (7cp)gZ; (ii) i q E Z + c p i EX; (iii) / \ r e * rcz; (iv) cp = Vu+ ( u ) E Z * { +(t ) I t ETerm,} C Z ; (v) * rnZ+fi; (vi) cp=3u$(u)EZ * { +(t)ltETermp } n Z # fi; (vii)(a) t x t E E ; (b) t l = t 2 E C + t 2 x t l € 2 ; (c> t , 7 = : 2 , +(tl)E): + $ ( t z ) E Z . Notice that a closed set is consistent (by (i)) and closed under modus ponens (by (v)). Condition (vi) can be made non-vacuous by adding enough individual constants to p ; in our case this will not be necessary since the sets of sentences used below will always be tidy. We will need the following fundamental:

Vrcz

RESULT C (Model-existence theorem).

(1) Every closed set of sentences of a frame has a model. (2) Every member of a consistency property can be extended to a closed set of sentences; therefore it has a model.

The proofs of these important results are analogous to the corresponding [3], lecture 3, except for minor and obvious changes. ones in KEISLER

'

rp 1 denotes the formula, defined by induction on the structure of rp, obtained by "moving a negation inside"; certainly, v i C * T ~See . KEISLER[3], p. 1 1 , for details.

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261

COROLLARY 4.5.4. Let Z be a tidy and complete set of sentences of the frame ‘k; then

Z has a model w 2 is closed. PROOF.The implication (e) is a particular case of result C. The other implication is a trivial consequence of:

for any 0 E Sent(*) and 8 E Mod(Z); this, in turn, follows immediately from the completeness of 2.rn The following observation-an immediate consequence of result B and the preceding definitions-turns out to be of crucial importance in the proof of the main theorem of this section; we state it as:

LEMMA4.5.5. Let p be a similarity type of power < K and ‘k a frame in p. The famib T( p, ‘k) of all complete, tidy and closed sets of sentences (relative

to ‘k) containing a fixed set r C Sent(*) is definable by a bounded-quantifier formula of ZFC with parameters in H( K ’).

PROOF.Use the following sets as parameters: ‘k, A (‘k), Sent(+), Term,, r. Examination of Definition 4.5.3 shows that each quantified variable is bound by some of the parameter-sets specified above. Likewise, “complete” and “tidy” (relative to 9)mean, respectively:

-

for every 0 €Sent(*), either 0 E X or 1 0 €2;

such that Fv(t)

for every n E o and 0 E ‘k, if Fv( 0) = n t €Term,

+ 1, then there is

= n and the sentence

belongs to ‘k. Therefore, also these definitions have all their quantifiers (th.at is, all those written in words) bound by some of the preceding parameters. The proof consists, then, in formally writing down all these definitions in the language of ZFC,using the sets in H ( K + )which code the parameters specified above. rn

268

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

LEMMA4.5.6. Let po be a fixed similariy type of power set of individual constants not in po; let p n = p o ~ { c ..., , , c,}

< K and

[Ch. 4, f 5

{ c, 1 n > 1 } a

for n > 1,

p m = p o u { c i I i > l}; let \k be some fixed frame in p,. For each n E w , let Z, be a complete, tidy and closed set of sentences in \I/ p,,. Furthermore, assume that the sequence has a model. of the 2,’s is increasing. Then E m = u,,,Z,

r

PROOF.For each A C Sent(*), let A,, denote the set of all sentences of A of type p,,. Let r be the family of all those AcSent(\k) such that: (a) only finitely many ci’s occur in each (PEA; (b) for every n E w , A, CZ,. It is easy to prove that I‘ is a consistency property; as an illustration we check conditions (iii) and (vi) of Definition 4.5.2. (iii) Let A E r and rp = AZ E A, where Z E A (\I/); by (a), rp-hence each uEZ-contains only finitely many ti's (say, with i < m ) ; so A u { u } satisfies (a). Clearly (A u {a}), =A,, if n < m. Let n > m ; by (b), rp E Z,; since rp+u is logically true and Z, closed, we infer u E Z,; hence (A u {a}), cZ,. Thus (b) also holds and A u { u} E r. (vi) Let A E I’ and rp = 3 v $ ( v ) € A . Condition (a) is verified as above. By (a) there is a least n E w , say m, such that r p E A , & Z , ; since Z, is tidy, since Z, is closed, there is a constant term to such that cp+$(t,)EZ,; + ( t o ) ~ Z , . Hence (Au

to>}>,^^, for n 2 m

and (Au {$(tO)}),,=AnCZn f o r n < m .

Then (b) holds, whence A u {$(to)}E r . Clearly Z, ET. By result C, 2, has a model, as it was to be proved. Turning to Theorem 2.2.14 the reader will find the so-called stretching theorem for sets of L,,-indiscernibles; roughly speaking it says that every structure with an infinite set of indiscernibles can be “stretched” so as to get another structure with the same true sentences (and even better) containing a set of indiscernibles of prescribed order type. The proof included in Chapter 2, $2, uses compactness. As in a number of other

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269

results, the compactness argument can be replaced in this case by one using the model existence theorem, thereby allowing a generalization to some infinitary languages. Thus, in spite of the lack of a fully developed theory of indiscernibles for infinitary languages', we can get a result good enough for our present purposes.

DEFINITION 4.5.7. (1) Given a frame 9 in the similarity type p plus enough / with a extra constants, a structure % of type p and a set X & J % together linear ordering <, ( X , <) is called a set of n-\k-indiscernibles ( n 2 1) iff for any two increasing sequences x 1< . . . < x,, y l < .. .
(2) If ( X , < ) is a set of n - 9-indiscernibles for each n 2 1, we say it is a set of \k-indiscernibles. The promised stretching theorem is:

RESULTD. Suppose % has an infinite set ( X , <) of 9-indiscernibles. Then ) of for every order type E there is a structure 58 containing a set ( Y , i 9-indiscernibles of type E such that for every n 2 1 and every pair of . .. < y , in Y , increasing sequences x, < .. . < x, in X , y l i (~,XI,...,X,)~Y(~,YI,...,Y,).

Moreover 23 can be taken of power max{ F ,K } . The proof is essentially given in KEISLER [3], lecture 13 (Theorem 19 (ii), p. 71). Using this result we derive the following:

LEMMA4.5.8. Suppose that for each n 2 1 a tidy structure H, has been given, which contains an infinite set (X,, <,) of n - ?TI-indiscernibles. Moreover, suppose the various %,'s and ( X " , <,).s are compatible with one another, i.e., for every x l < ,,... <,,x,, in X,, and every y , < , ... <myn in X,, where m > n, we have

'

CHANG(21 contains a discussion of various possible constructions of models generated by (infinitary) indiscernibles and their limitations.

270

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 9: 5

Then, given any infinite order type e , there is a structure % containing a set of 9-indiscernibles ( Y , <>of type c such that for any x , <,... <,x, in X , and a n y y , < ...
. . , x n )=%(B,y,, . . .,y,).

Moreover. % can be taken of power max{ 7 ,K } .

PROOF.Let p, be the type of the an's.9 is assumed, of course, to contain individual constants c,,. . .,c,, ,..besides the symbols in 1.1. For each n Eo, let: Z,=ThK+"((%,,x, ,...,x,>)nJr for some (hence, for any) increasing sequence x 1<, ... <,xn in X , (i.e., the constants ci are interpreted by the x,'s). It is clear that each Z, is consistent, complete and closed relative to 9 r p n (notation as in 4.5.6.)By assumption each anis tidy; hence so is 2,; clearly the sequence of the 2,'s is increasing. Therefore, by Lemma 4.5.6, Z= , UnEoXn has a model, B say. Using the compatibility assumption, it is immediately seen that the set { c? I i 2 l} ordered by

is an infinite set of 9-indiscernibles. Then, the assertion of the lemma follows at once by applying result D to the structure D and the set ({c"i> I}, <). B. Accessible ordinals

We return to Theorem 4.2.2. Consider the case when the set 5 of L,,-types has power < K . Theorem 4.1.1 shows us how to translate statements about sets of types of power < K into statements about a single L,+,-sentence. Performing this translation on Theorem 4.2.2 we obtain:

THEOREM 4.5.9. Let

1y be an ordinal; assume there is a sentence c p of L,+, involving a binary relation symbol and possibly other non- logical symbols, and an ordinal p such that: (a) in all models % of rp, <' well-orders I%/ in type < p ; (b) there is a model 8 of rp in which <' well-orders 1231 in type 2 0. Then I),> am.

Ch. 4,

51

27 1

FURTHER HANF NUMBER COMPUTATIONS

We show first how this result can be (slightly) improved by means of simple arguments already introduced.

DEFINITION 4.5.10. An ordinal a is L,+,-accessible (for short, ac accessible) iff there is a sentence cp of L,+,, involving a binary relation symbol <, a unary predicate U and possibly other non-logical symbols such that: (a) for each model 'u of cp, U" is the field of <" and <" well-orders U"; (b) there is a model 23 of cp such <' has type 2 a. We also say that the sentence rppins down the ordinal a. THEOREM 4.5.11. Let a be a ac accessible ordinal; then b,>

2,.

PROOF.Let cp be a sentence that pins down a. We show that there are a sentence $ and an ordinal fi fulfilling conditions (a) and (b) of Theorem 4.5.9. Take $ as c p ~ V u U ( u ) ;if ' u Mod(cp), ~ then 'ur U " E Mod(#); so Theorem 4.5.9(b) follows immediately from Definition 4.5.10 (b). It is also clear that in any model 'u of $, <" well-orders [%I. It only remains to prove that the ordinal fi exists. Suppose not; then there are models 'u of # in which <" takes on order types larger than any prescribed ordinal. Let Seg(r;) be the formula with a new constant c which defines non-empty initial segments of < : Seg(v) = U ( c )A 3 z ( z < c ) A u < c. It is a simple exercise to prove that

is precisely the class W of all non-empty well-orderings considered in $4. Thus we conclude that W is an RPC(L,+,(<))-class, contradicting the main result of the preceding section. This contradiction proves that there is an upper bound fi to the order types of <" in models 'u of #. Hence 4.5.9(a) holds. Therefore b, > 3,. rn The foregoing theorem proves part (2) of the following: LEMMA 4.5.12. (1) The class AC,+ of all K+-accessible ordinals is an initial segment of ON;

2 72

[Ch.4, f 5

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

(2) this initial segment is proper; in fact, by the results of $3, all its members are <(2'c)+; ( 3 ) AC,+ is closed under ordinal addition and multiplication. PROOF.( I ) Clearly, if p pins down a,it also pins down any p Q a. (2) is an immediate consequence of Theorems 4.5.1 1 and 4.3.4. (3) Let a. ,8 be K+-accessible ordinals and p, #, be L,+,-sentences pinning them down respectively. We prove that a.P is K +-accessible, leaving the case a + p as an exercise for the reader. In addition to the symbols in q, 4, consider the following predicates: U,, U2 (unary) < I ,< 2 (binary) and G (ternary). Let T be the conjunction of: (i) " < 1, <2, < are linear orderings with fields U , , U2, U , respectively"; (ii) p( c',, <#'I; (iii) +( U2, <2)? (here replace the old U , < in p by U l , < I , resp., and relativize all quantifiers to U , ; same for 4). (iv) "G is an order isomorphism between ( U , <) and the structure ( U , , < I )X ( U2,< 2 ) ordered by first differences":

Given a model 2 of as follows:

uB=u;,

T, define

structures 8,65 of type Sm(q), Sm(#), resp.,

u"= u;,

<%=

<,;


<2".

Since <', <" are well-ordering, (i) and (iii) imply that <' well-orders U'; thus, 4.5.10(a) holds. Conversely, given models 8,Q of 9,#, resp., we construct a model l!i of

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51

273

FURTHER HANF NUMBER COMPUTATIONS

by setting

(where b, c denote the first elements of UB, U", resp.), and defining the interpretation of the remaining symbols in the (only and obvious) way to make them compatible with (*); for example <; is

We leave to the reader the routine work of completing the definition of 'u and checking that it is a model of T. If 8 and B are chosen so that <' has type 2 a and <" has type 2 /3,

then

<'

has type 2 a./3, and 4.5.10 (b) holds. Thus,

T

pins down asp.

REMARK. More generally, AC,+ is closed under primitive recursive operations on ordinals (Karp-Jensen). Let a ( K + ) be the first ordinal not in AC,+. Trivially K + < a ( % + ) .Exercise 4.5.2 indicates how to sharpen this to a strict inequality whenever cf(K) > a . This, together with the preceding results, shows: (1) (11)

q("+, < b,,

K + u. Is it possible to improve these results? The rest of this section is devoted to show that (I) can be sharpened to an equality. Concerning (11); from Lemma 4.5.12 it follows that a ( K + ) is larger than K + - K + ,...,( K + )n... ; it is easy to see that it is also larger than ( K + ) ) ( + , ( ( K + ) " + ) ' + , . .. (all ordinal operations), etc. Extending these ideas, BARWISE-KUNEN [ 1] show-using methods from metarecursion theorythat a ( K + ) is the least ordinal not H(K+)-recursive, whenever cf(K) >a. However, in ZFC one cannot do much better than (11). Using the order a of Exercise 4.5.2 and forcing methods Kunen showed that if 2" is large with respect to K , one can consistently make a(K+) small (less than K + + , for example), or large (greater than 2", for instance). More precisely:

THEOREM (Kunen). Let K , 8 be regular cardinals such that o < K < 8. Assume ZFC consistent. Then there are models m, % of ZFC in which the values of

214

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 4 5

2’ are as foIlows:

2’=Xf 2‘=max(A+,O}

but:

for ail cardinals w < X < K , for all cardinals X > K ,

%ka(K+)>O.

Thus, for example, if ZFC is consistent, so are: (a) ZFC 2’0= N, + 2’1 = NEb3 + a@,) < N,, (b) ZFC + 2’0= N, 2’1 = N,,, + a(N,) > NL6,.

+

+

With this background on record we set to work for the equality 6, = a,(,+,.

C. Semi-accessible ordinals It turns out that the notion of h.+-accessible ordinal is still too coarse (and difficult to handle) to yield a direct proof of fjK= a,(,;,. The transition to a more manageable notion is carried out in two steps: firstly, by relaxing the well-orderedness requirement to merely well-foundedness; secondly, and more important, by adding the metatheoretic requirement that the wellfounded structures in question be reasonably well-behaved set theoretic objects; loosely speaking, “well-behaved” means “definable in ZFC by a condition that can be expressed in a single L,+,-sentence”. Recall that a weff-faunded structure ( X , R ) is one in which the binary relation R is well-founded. Each well-founded structure has an associated “rank” function p : X+ON, defined by induction on R (cf. Chapter 0, Q 1): p ( x ) =sup{p(y)

+ 1 IyRx},

for x E X .

By analogy with the tree notation, we define the height of ( X , R ) as follows: h((X,R))=sup{p(x)+l I x E X } .

DEFINITION 4.5.13. An ordinal a is K+-semi-accessib/e’ iff there is an (5,< ) such that:

x E H ( K + )and a well-founded structure

I BARWISE-KUNEN [ I ] call this notion “tree-accessible over H(K+)” (and, of course, one may (but need not) require (5,< ) to be a tree; indeed, one of height GU). Due to their loose use of the word “tree”, and lacking a better name, I use the prefix “semi”.

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215

(i) 9c 9 ( x ) ; (ii) the sets 5 and < are definable in ZFC by bounded-quantifier formulas using parameters from H(K+); (iii) a < h ( ( S , i )). The next two theorems prove that this notion is identical with that of +-accessible ordinal.

K

THEOREM 4.5.14. a is K+-semi-accessible + a is K+-accessible. PROOF.Let x , 5, < be as in the preceding definition and let cp(uo), +(uo, ul) be the bounded-quantifier formulas of ZFC with parameters a ,,...,a,, €H(K+),defining 9,<, respectively. Let w = T C ( { x , a , , ..,a,,}); . w is transitive and of power < K . Consider the similarity type p having the following symbols in addition to the binary predicate < : T, U , unary predicates, R , E , binary predicates, F, unary function symbol, an individual constant for each z E w. cz, Let 0 be the conjunction of: (i) 1( c , x c b ) for a , b E w , a # b , (ii) Vuoul[T(uo)~ u , E u ~ + u , E c , l , (iii) A , , w V u [ u E c , ~ V b , , u ~ ~ b l , (iv) “ T is the field of R ” , (v) Vu[T(u)t,p,(u,c,, ,...,C,” )I, (vi) ~ u o u , [ u , R u , ~ + ( ~ o , ~ ..,c,)I, ,,c,,,. (vii) “extensionality for E ” , (viii) “ F maps T onto U”, (ix) “ < linearly orders U ” , (XI V ~ o ~ , [ ~ o R ~<, F~( v~, )(l .u o ) In (v) and (vi), E is, of course, to be replaced by E . We prove that 0 pins down a. This requires an absoluteness argument which we treat first. Let 2l E Mod(f3); define a map f : X = T E u { c,” I z E w } 4

as follows:

uw

276

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch.4,

§5

Using (i)-(iii) one easily checks:

(*)

z E w A ‘u k T(c,) + { y E x I \II +c,,EcL}= z ;

so that f is actually a map. Using (i)-(iii) and (vii) it is easily seen that:

‘Ul=aEb o f ( a ) ~ f ( b )f o r a , b E X .

(**)

One has to consider four cases, according to the distribution of a, b in each of the two parts of X . As an illustration we do the proof in the case a E T’, b = c,” for some z E w. Suppose ‘u t=aEc,; by (iii), a = c,” for some u € 2 ; therefore f ( a )= u, f(b) = z , which implies f ( a ) E f ( b ) . Conversely, if f ( a ) E f ( b ) = z , then ‘u t= c,(,,Ec,, by (iii); next we see that V U [ U E C , ~ , ) , ~ U E ~ ] ; then by (vii), a = c&,, which implies W t= aEc,, i.e., ‘u != aEb, as required. We now generalize (**) to the following: CLAIM.Let u(u,, ..., on) be a bounded-quantifier formula of ZFC and s,,...,s,EX; then:

(i.e., f(s,),. ..,f(s,) satisfy u in the real world).

PROOFOF CLAIM. By induction. The case when u is atomic is precisely (**). The case when u is a Boolean combination of earlier formulas is obvious. Let us suppose u is an existential formula; since u is bounded-quantifier, u = 3 u ( v E o, A u’), where 1 < i < n and u’ has less logical symbols than u. (*) ‘uku[s, ,..., s,] implies that there is s E I % such that \ZIt=u’[s,s,,...,s,] and sE‘s,; by induction hypothesis and (**), (

hence

3‘

E ) +f(s) ~ f ( s iA)u’[f(s),f(s1 1 7 . . J ( s n ) l ;

(e) From ( V , E) Fu[f(s,), . . .,f(s,)] we get some u E f ( s i ) such that ( V , ~ ) k u ’ [ u , f ( s , ) ,...,f( s,)]; if si€T‘, then u E x and ‘ U k c ~ E s iif; si=c,” for some z ~ w then . u ~ f ( s , ) = z ;in either case cf E X and u=f(c:), whence, by the induction hypothesis, \II k u’[cu,sI,...,s,]; hence

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FURTHER HANF NUMBER COMPUTATIONS

211

i.e., )2[t=u[s,,..., s,]. Now apply the claim to cp,+,s,s,,s,EI’UI and cu,,..., cua, taking into , respectively, in account (v), (vi) and the assumption that cp,$ define 5,i the real world; we conclude:

-

‘Ut=T[s]* f ( s ) E ‘3

8ks,Rs,

f(SJ

i f(S2).

In particular, it follows that R’ is a well-founded relation on T’; using (viii)-(x) we immediately infer that <’ well-orders U’. Thus condition (a) of 4.5.10 holds. )); in fact, Likewise, 4.5.10(b) is a simple consequence of a < A ( ( ? , i if P is the height of (9, i ), the structure 8 with universe w , in which T, U , R , E , <,F and cz are respectively interpreted by ?,[O,P), i , E rw, E r P , p (the rank function of (5, < )) and z , is a model of 8 in which <’ has type > a . Thus we have proved that 8 pins down a , whence a is K+-accessible. W

THEOREM 4.5.15. a is K+-accessible a a is K+-semi-accessible. PROOF.Assume cp pins down a. Let po= Sm(cp); then po contains U and < ; without loss of generality we can assume po is endowed with Skolem functions. Let { c, I i > l } be fresh individual constants, p,, = pou { c , , . ..,c,} for n > 1, pm= p0u { c, Ii 1). Let \k be a tidy frame for cp in pm. We shall use Lemma 4.5.5 to define the well-founded structure (9, i ); we begin preparing the ground for it. Let ro={cp} and for each n > 1, let r n denote the set consisting of the sentences: (1) (ii) U(c,) for i = 1, ..., n, (iii) c,+ < c, for j = I , . . .,n - 1. Let T,, denote the family of all complete, tidy and closed sets of sentences By Lemma 4.5.5 relative to \ k r p , containing r,; let 5 = UnEU?,. (modulo a small change!) 5 is definable by a bounded-quantifier formula of ZFC with parameters in H(K+).Define the relation < on 5 by: cp7

,

Z’ i Z”

H

Z’, Z” E 5 and Z’ properly confains 2 ” ;

obviously, iis also definable by a bounded-quantifier formula with parameters in H(K+). i ) is well-founded. Otherwise, given a descending We show (5,

218

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4,

B

5

sequence

has a model, say 9l. we use Lemma 4.5.6 to conclude that Z,= UnEwXn Since all the En's contain 'p, 91 ~ M o d ( ' p ) However . <" does not well-order U", because each sentence ci+ < c, is true in 'u, contradicting the assumption that 'p pins down a. This contradiction proves our contention. It remains to be proved that a < h(( 9, < )). Recall that p denotes the < ). By induction on p(Z) we prove: rank function of (9,

,

(*I

if 9l k Z, then (for n > 1) Z E Tn + the predecessors of c," under <" have order type < p(Z), I:E To * <" has order type < p(E).

Suppose, for example, that at=Z , Z E Tn and c," has >p(Z) predecessors; let a be the p(Z)+ 1'' element of U'; set B=(91r,a) (i.e., c : + , = a ) ; let Z' = Th"+"(8) n (9rpn+ ,).Then Z' E Tn+,, and it properly extends Z, i.e., 2' < Z; hence p(Z')p(Z') predecessors under a , we conclude that a < h(( 9, i )). This completes the proof that a is a K+-semi-accessible ordinal. w D. Proof of the Banvise-Kunen-Morley theorem

We now prove that the last theorem.

b,

Q

a=(,+)by means of an argument parallel

to that of

4.5.16 (Barwise-Kunen-Morley). Let 'p be an L,+,-sentence, the THEOREM models of which have bounded cardinality. Then rhere is a K+-accessible ordinal a such that all models of cp have cardinality < am.

PROOF.Let po= Sm('p) u { X , <, F } , where X (unary), <,F (binary) are new predicates; without loss of generality we assume po is endowed with be as in the proof of 4.5.15; let 9 be Skolem functions; let (cil i > l}, pnnrpm a tidy frame for 'p in p,. For each n > 1 let the set r ncSent(9) consist of: (i) 'p? (ii) " X is an infinite set", (iii) " < linearly orders X " , (iv) " F maps X one-one, onto the universe",

Ch. 4, 0 51

FURTHER HANF NUMBER COMPUTATIONS

279

(v) " X is a set of n - \k-indiscernibles satisfying the same formulas as

CI,...,c,":

VV,

. ..u,

+

for all n E w and all formulas 0 E with n free variables, ( 4 A:= J(ci), (vii) cI < * . * < c,. Let To consist of (i)-(iv). Let 5, denote tI, family of all complete, tidy and closed subsets of Sent(\krp,) containing r,; as before T = UnEU5,is definable by a bounded-quantifier formula of ZFC with parameters in H(K+); the relation iis the same as in Theorem 4.5.15. We prove that (5, < ) is well-founded. Suppose not, and let Z,>Z,>

. - *

>En>

. * f

be an infinite descending chain. By Corollary 4.5.4 each C, has a model a,, in which ( X , , <,) = (X'm, is a set of n - \k-indiscernibles (by (v)); (ii) and (v) show that the assumptions of Lemma 4.5.8 are satisfied. Therefore, there are structures 23 containing a set ( Y , a ) of indiscernibles of prescribed order type (hence of arbitrarily large cardinality) such that <"n)

(%,,+,.

..&)=*(%YI,.

. .J,>

holds for any increasing sequence y l a . . . a y , in Y . Furthermore, by the last assertion of 4.5.8 for Y's of power > K , such 23's can be chosen so that g=7. Therefore, the structures ('23,Y , a , f , y l,...,y , ,...) (where f is a bijection between 1231 and Y , and y l , ...,yn,... is an arbitrary 4 -increasing sequence from Y ) are models of Em= UnEUE,-hence also of cp-of arbitrarily large cardinality. This contradicts our assumption that the mo) is well dels of cp have bounded cardinality, and shows that ( 5 , i founded. As before, p denotes the rank function of (5, i ).

-< 3W.(P+ l)(A), where P=p(C)

CLAIM.If Z E 9,and B t- C, then X " -

and

A=2'.

Assuming the claim, we show how to complete the proof of the theorem. For n=O the claim says that X " , and by (iv) also B, has power < 3 W . ( p + I ) (for h ) ,all models 23 of Z. But every infinite model \u of cp can be expanded to a model 23 = (%AX,< ,F ) of some Z E Yo by choosing X to be of power B, F any bijection between X and IBI(,< any any subset of

280

INFINITARY LANGUAGES WITH FINITE QUANTIFIERS

[Ch. 4, 5 5

order of X , and I: =Th"*"('H)n (+ rpo). (Note that p0 does not mention any of the ti's.) Therefore, we conclude that every model of cp has power < 3,.ts+I,(h), where .$=A(( 5,i )). We proceed now to a simple cardinality computation; by result B, ' P E H (K + ) ; since H ( K + ) & R ( K + )there , is Y < K + such that ' P E R ( y ) ; hence \k < a,, and h < 3,+ = a8;thus we get:

8, w, 1 are K+-accessible because they are < K + . By definition 5 is K+-semiaccessible; whence, by Theorem 4.5.14, K +-accessible. By Lemma 4.5.12, (Y = 8 +w. (<+1) is also accessible and, as we have proved, all models of cp have power < aa. We return to the:

PROOFOF CLAIM. This goes by induction on P=p(I:) using an argument > 3 w . ( p (A) + I for ) similar to those of $ 3 of this chapter. Suppose that < = <'. For each increasing some 23 such that 'HI- Z. Let X = X ' , let 'P, be the structure of type P,,+~ obtained sequence x = {xI,.. .,x,+ from 'pi by allowing x l ,...,x,,x,+ to denote cl,. . . ,cn,cn+ respectively, and let

the same cell if and Define a partition in [X]n+' by placing x . y in and only if Z , = Z y . Since Z, c\k, there are < 2' = A such Zx's, and therefore < A cells. By the Erdos-Rado theorem (0.5.9 (3), $5, Chapter 0), there is a Y C X such that > 3a.p(h),and every pair of members of [ YIn+' is in the same cell. 0

Let y o = { y l , . . . , y ~ + l }
r

(Brsmfcp).

..,x,

+ 1)

a ~urw+,(%yI

?..

..Y,0+ 1)

for every increasing sequence xI, .. .,x,+ in X . Let f be a bijection between

Ch. 4, 5 51

28 I

FURTHER HANF NUMBER COMPUTATIONS

Y and IB0l; clearly

2 3,.(p.+,,(A), the model 8’and the theory 2’ violate Since 7> the induction hypothesis. This contradiction proves the claim and, thus, the theorem.

COROLLARY 4.5.17.

b, = aa(,+).

PROOF.Theorem 4.5.11 and the remarks following it prove b, Theorem 4.5.16 proves b, < aa(,+,.

> a,,,+,;

EXERCISES 4.5.1. (a) Show that AC,+ is closed under ordinal sum. (b) Prove that AC,+ is closed under ordinal exponentiation. (c) Show K + c A C , + . 4.5.2 (Kunen). For f,g E K , define the relation:

fag

H

there is an a < K such that f ( p ) < g ( p ) whenever a
(a) Prove that if cf( K ) > w and a set L K is linearly ordered by a , then L is well-ordered by a. [Hint: use Chang’s trick explained in the proof of Theorem 4.2.4.1 Let 7, be the supremum of all possible order types of a -well-ordered subsets of K ‘ . (b) Show that 7, is K+-accessible. (c) Show that 7, > K + [Hint: construct a Q -increasing sequence of type K + below an appropriate fE K “I.

Problem. Can a ( K +) ever be a cardinal?

CHAPTER 5

LANGUAGES WITH INFINITE QUANTIFIERS

0 1. The Hanf number of languages with infinite quantifiers In this section we shall deal with the problem of evaluating the Hanf number of Lolu, and, more generally, of the languages LKh,for h >a. In Chapter 4 we dealt with the corresponding question for languages with finite quantifiers. The exact value of h(LK,) was shown to be, in general, independent of the axioms of ZFC (Theorem 4.5.17 and the theorem of Kunen pp. 273-274); nevertheless we obtained within ZFC reasonable bounds for h (LKu), i.e., bounds expressible by operations of cardinal arithmetic in ZFC. We shall see that even this expedient fails for languages with infinite quantifiers: there are no known bounds for h (L,l,l) unless additional axioms are attached to ZFC. The solution to the problem, then, depends upon the extra assumptions. The best investigated case is that in which strong infinity axioms (e.g., the existence of a measurable cardinal) are added to ZFC;most of the results proved in this section hold under assumptions of this type. Only one result (Theorem 5.1.16) is known under the hypothesis V = L which, as we know (cf. Theorem 0.4.29 and Appendix D), is contradictory with the existence of very large cardinals. The techniques used in this section fall naturally into two categories: (1) the use of partition calculus methods and of the techniques developed in $ 2 of Chapter 2, to trap the Hanf number of LulUlbetween two bounds (expressible in terms of certain partition cardinals); in particular we obtain lower bounds for h(Lulol). These techniques were invented and developed by SILVER[ 11, [2]; (2) the use of even larger cardinals-measurable, strongly compact-to give upper bounds for h(Lu,,l), or to show the impossibility of proving that 282

Ch. 5 , $11

THE HANF NUMBER OF LANGUAGES WITH INFINITE QUANTIFIERS

283

certain cardinals are upper bounds; the most significant of these results is Theorem 5.1.12, due to KUNEN[2]. A. Upper bounds

Our first results are next to trivial; they are a very simple-minded translation of first-order arguments that use compactness; however, they give the only known bound for h(Lulun).

THEOREM 5.1.1. Assume there exists a strongly compact cardinal K and that a < K ; let p be any similarity type. If 'pESent(L,,( p ) ) has models of arbitrarily large power < K , then 'p has models of arbitrarily large cardinality, In particular:

PROOF.Let X > K be an arbitrary cardinal; let p'= p u { c pIP<X}, where the cp7s are new individual constants; let:

Let 8' be a subset of Z power < K ; by assumption there is a model 2I of 'p of power > F;it is clear how to make (%,a),,,,, into a model of 2'. By strong compactness, C has a model, which obviously is a model of 9, of power > A .

COROLLARY 5.1.2. If

K

is strongly compact, h(L,,( p))=

K

for any p.

PROOF.Clearly h(L,,( p)) 2 K . On the other hand we have:

THEOREM 5.1.3. If K is strongly compact and both a and p have power then h(L,,( p)) < K . PROOF.Let ible: (*>

T=

max{

F ,a}; then a, 7 <

sent(L,,( p))

K

and since

K


is strongly inaccess-

< 7, < K .

Let X be the set of all L,,(p)-sentences large power; then, by (*),
not having models of arbitrary

284

[Ch. 5, 8 1

LANGUAGES WITH INFINITE QUANTIFIERS

By Theorem 5.1.1,

A, Clearly:


1

= sup{ p p

for all rp E X , where

is the cardinality of a model of rp f .

Since K is a limit cardinal, A: < K (rp E X ) , and sup{X; Irp E X } < K . Hence h(Laa(p ) ) < K .

K

is regular, it follows that

COROLLARY 5.1.4. Assume there is a strongly compact cardinal and call the first such number. If jT < K ~ then , h(Lu,ul(p ) ) < K ~ .

K,,

These simple results suggest the question whether it is possible to obtain better (smaller) upper bounds for h(Lu,ul).A partial answer is given by the following sequence of theorems due to SILVER [ 11 which, in particular, yield the upper bound i(o,) for the Hanf number of the set of prenex-universal sentences of Lulul(p), Q No (cf. Corollary 5.1.8, Lemma 5.1.9 and also Chapter 0, 0 5). THEOREM 5.1.5. Suppose Z is a tidy set of formulas of type p and {BE15 < K } a sequence of structures such that

X(2; (5, E); %J

for each

5<

K.

If ( Y , < ) is well-ordered and % ( 2 ;( Y , < );B), then every prenex-universal LKK( p)-sentence holding in each Bt holds also in 8.

PROOF.Let u be a prenex-universal L,,(p)-sentence such that B51=u for each 5 < ~ .u is (VvrP)rp where P < K and rp is quantifier-free. Suppose %F l a ; then there is fE1%IP such that Bt= i r p [ f ] . Since p< K there is X c Y , 7 < K such that Range(f) c@(X,B)=O. Since rp is quantifier-free, 6 F i r p [ f ] . Let y be the order-type of ( X , < r X ) . Then there is an order-preserving map g : ( X , < r X ) + ( y , E). By Theorem 2.2.15, g can be extended to a monomorphism h : &+By. Then By!= i r p [ h o f ] , contradictingBYl-u.1 THEOREM 5.1.6. Suppose that Sm(%)= p, C -

c

=K.

121,< well-orders C and

c

(1) Zf 7 Q A, (Y EON and K + ( ( Y ) $ ~ , then there is X C such that ( X , < r X ) has order-type (Y and ( X , < r X ) is a set of indiscernibles for 2.

Ch. 5, $11

THE HANF NUMBER OF LANGUAGES WITH INFINITE QUANTIFIERS

( 2 ) If 7 < KO, K-+(a),<" of ( I ) holds.

and

(Y

REMARK. The partition property Exercise 0.5.7.

285

is a limit ordinal, then the same conclusion

K+((Y)~*

for ordinals a was introduced in

PROOF.(1) There is no loss of generality in assuming C = in K define For { xo,. ..,xn-

K

and

< = Er

~.

f maps [ K ] < " into (:?(Form(LuJp))), which has cardinality <2'. By hypothesis there is X C K of type a which is homogeneous for f. By definition o f f this implies that ( X , < Y X ) is a set of indiscernibles for 3. (2) This follows at once from (1) using the variant of Lemma 0.5.14 * K+((Y);" considered in Exercise 0.5.7, p. 58 (which says that K+((Y),'" whenever (Y is a limit ordinal). rn

THEOREM 5.1.7. Let u be aprenex-universal senfence of LKK( (I), where A. (2) If cf(K) > 2' and u has a model of power u, where v-+(a);* for every then u has a model in each power > A . In both cases the structures so obtained can be taken to be elementarily equivalent to the given model of u. (Y


PROOF.(1) Let 3 be a model 'of u of power Y and Sm(3) = (I. By the < A, so we can assume that 3 is a considerations of Chapter 2, 9 2.A, Skolem structure By (1) of the preceding theorem, there are X C 131 and a well-ordering < of X of type K such that ( X , <) is a set of indiscernibles for 3.For each y < K , let X , consist of the first y elements of X and Ba= @ ( X y ,91). Since 23, is a substructure of 3 and u is prenex-universal, it is a routine matter to check that Byk u for each y < K . Let Z = Form,((X, < )). Since for each y < K,% ( 2 ;( X , , < ) ; !By),Theorem 5.1.5 shows that if ( Y , i ) is a wellordered set and % a structure such that %@; ( Y , < );%), then u holds in '$3. Given any cardinal p > A , take Y = p and < = E r p . By Lemma 2.2.14 there is a structure Cs such that X(C; (p, E r p ) ; Csj; then Cr=Q(p,O), and since p > h this implies that = p. Since u is true in 0,(1) is proved.Notice also that Z> Th(Y1). From Fh' (Z; ( p , E p ) ; 6)it follows that at-91 for each V E X . Therefore Q - 8 .

7

r

286

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5,

1

(2) Let 9l be a model of u of power v. By ( I ) of the preceding theorem, for each [< K there are an XScI%l and a well-ordering <, of X , of type [ such that ( X r
Since 7 < A , Range(h) < 2*
7 ,A < k ( A ) ,

PROOF,Clearly p < ;(A);

then h(PU,,( p ) ) < k ( h ) .

hence p+(A)Z<"; by (2) of 5.1.8, " < " holds. Also

where 7=max{ 7;,X}.Since ;(A) is strongly inaccessible > w (Corollary 0.5.17), r * < ;(A); hence cf(h(PU,,( p)))< ;(A) and the equality cannot hold.

Ch. 5, 311

THE HAMNUMBER OF LANGUAGES WITH INFINITE QUANTIFIERS

281

< No.Let % be COROLLARY 5.1.10. Let p include a binary relation < and a structure of power K , K+(w,)Z<~, such that <' well-orders 191.For each infinite cardinal A there is a structure 23 of power A such that 23 = % and <' well- orders I23 I.

'

PROOF.The result follows from (1) of Theorem 5.1.8 by taking into account that "< is a well-ordering'' is a prenex-universal LWl,l(p)sentence. Imposing more .stringent partition requirements on the conclusion of the preceding corollary;

K,

Silver improved

THEOREM 5.1.1 1. Suppose F < No and p includes a binary relation symbol < . Let % be a structure of type p andpower K , where K - + ( K ) ~ < * , such that <' well-orders ( a subset o f ) 191.For every A E C N , A > w, there is a structure '3 ofpower A such that B E %and <' well-orders ( a subset o f ) 13 ' 1 in type A. If A < K , 23 can be chosen so that '3 i 9. In particular, the same conclusion holds if K is measurable (Helling). Theorem 5.1.1 1 is actually a particular case of a series of existence theorems proved with a technique far more involved than the one used before. The same existence theorems, when applied in a suitable form to the constructible universe, yield the surprising conclusions that are summarized in Appendix D. B. Lower bounds

One may entertain hopes of generalizing the preceding theorems and get an upper bound for h(L+,( p)), F < No-instead of merely h(PU,I,I( p))in terms of partition properties of type K+(A)Z<~. These hopes are dispelled by the following result due to KUNEN[l], [2]. THEOREM 5.1.12. If ZFC + "there is a measurable cardinal" is consistent, then ZFC+"there is a finite language p such that h ( L W l Up))> l( the first measurable cardinal" is also consistent. A detailed treatment of this result falls outside the scope of this book. It suffices to say that using a binary predicate E , a function symbol of one argument and an individual constant, Kunen constructs an L,I,I-sentence The theorem holds even if <" well-orders a proper subset of 1 11.

288

[Ch. 5,

LANGUAGES WITH INFINITE QUANTIFIERS

1

whose models are isomorphic to transitive models 2V of the first m axioms of ZFC, for some m E w , containing a measurable cardinal K and satisfying, among others, the additional properties (2J?= K and 2V L, , where CC is some 'normal ultrafilter on K . He shows that x has no models of power > K + + but, if V = L , , x has models of power K .

x

c

In the same vein as Theorems 5.1.5-5.1.1 I above one can obtain lower bounds for certain Hanf numbers in terms of partition cardinals:

THEOREM 5.1.13. If there is a K such that K + ( w ) ~ < ~ ,then there is a denumerable ianguage p such that ;(a)< h(Lu,u,(p)). PROOF.Let the similarity type p contain one n-ary relation symbol R,, for each n > 1. Recall that S,, denotes the set of all permutations of n = {0,1,. . . ,n - 1 ). Let u be the conjunction of the following sentences of Lu,J I*.): =

A V V ~..u,.

I[

R,,( u W . . .,c,,- l)*R,,(

.. .

~ ~ ( 0 ) .,ow(,,- I ) ) ]

nEw

n E S"

j l < . . .
Given a set X and relations T,, LX'' (n E w ) such that ( X , T,,),,EwFul,the following defines a partition of [ X

-

{XO,...,X,,-,}E

Then we have: ( X , T,,),,Ewt-u2

C,

iff(xW...,x,,-,)€ T,,,

C,

iff(x, . . . . , x , _ , ) ~ T n .

there is no denumerable subset of X homogeneous for the partition (C,, C,).

The sentence u has models of all cardinalities less than ,?(a).In fact, if h < :(a), let f : [h]<"+2 be a function having no countable homogeneous set; it is clear that

(A, { ( X @ . ..,xn- 1)

If({

X,?.

...%-1 ) 1= 1 )),,Eu

is a model of u. On the other hand u has no model of power

> ,?(w). Observe

first that

Ch. 5 , $11

THE HANF NUMBER OF LANGUAGES WITH INFINITE QUANTIFIERS

K+(w),<"A~>

K

A-+(w),<,.

power 2 ;(a), define

289

Thus, if 91 is a structure of type p and

then, either rUk i u , , or the existence of an infinite homogeneous set for f precludes u2; in either case rUt= l u . Thus, we have proved that ;(a) h(L,pl( P)). As has been remarked before (cf. Lemma 3.5.5 and the remarks thereafter) :

Since ;(a) is strongly inaccessible > o (Corollary 0.5.17), it follows that cf(E(w))= ;(a) >2'@, and therefore h(L,l,l( p)) > k ( w ) . NOTE. Since u can easily be shown equivalent to a prenex-universal l ) be replaced by h(PU,,J. LuIuI-sentence,h ( L w l U can COROLLARY 5.1.14. There is a countable language p such that both h(L,,,,( p)), h(PU,I,I( p)) exceed the first weakly compact, strongly inaccessible cardinal larger than w.

PROOF. ;(w)+(;(w))i;

cardinal.

thus ;(a) is larger than or equal to the mentioned

Essentially the same argument as before shows:

THEOREM 5.1.15. If A E CN and there is a cardinal I( such that K+(X)Z< , then there is a similarity type p such that = A and

SILVER[ l ] improved the lower bound for h(L,l,l) given by Theorem 5.1.13 (cf. Exercise 5.1.3). In that paper he also investigated the size of h(L,l,l) in the absence of very large cardinals. He proves:

THEOREM 5.1.16. If V=L, then h(LulUl) is larger than any nrh order characterizable cardinal, for n Ew.

290

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, $ 1

DEFINITION 5.1.17. K E C N is nth order characterizable iff there is an nth order sentence u involving the symbol E such that ( 9 (R(K), €)Fa, (ii) (R(A),E ) F i u for any AECN, A # K . COROLLARY 5.1.18. If V = L and there is a II”,indescribable cardinal, then h(Lula,)exceeds the first such cardinal. EXERCISES 5.1.1 (KUNEN[2]). Prove that if A > w and p= h(L,,( p)), then p=aP.[Hint: let q be a sentence of LKA(p)without models of arbitrarily large power; add a unary predicate Q, a unary function symbol r and a binary predicate E. There is a sentence u in the language p’= p u { Q, E ,r} such that for every structure 8’of type p’, %’ is a model of u iff there is a model % of 9, and a well-ordering < of 131 such that if y is the order type of (131, <), then %’=R (y).] [ 11). Show that Theorems 5.1.9 and 5.1.13 hold if PUaI,I( p) is 5.1.2 (SILVER replaced by V ( p), the set of all conjunctions of Luu(p)-sentences with the sentence “< is a well-ordering’’ (7 Q No). [Hint: for Theorem 5.1.13 show that there is a finite set of axioms of ZF which, in conjunction with the defines an Law-sentence q assertion “there is no K such that K+(w)Z<~”, such that % E Mod(q) and (21 well-founded imply

add an axiom relating well-foundedness with

“

<

Q ;(a);

is a well-ordering”.]

[l]). Prove that there is a denumerable p such that 5.1.3 (SILVER

5.1.4 (SILVER[l]). (1) Let a be a limit ordinal, A < K be infinite cardinals ; [ K ] < ~ + A be an arbitrary map, and let such that K + ( ~ ) Z
Ch. 5, 8 21

THE LANGUAGES

L ,

FOR

e STRONGLY

INACCESSIBLE

29 1

( i ) % P { E , F , G } = ( R ( K )E, r R ( K ) , f * , g * ) , ( ii) there is a set X C K such that ( X , E r X ) has order type a and is a set of order indiscernibles for %. (b) Show that for any n > 1, 7 €Term,, Fv(7) = n, and any increasing the following sequence x of length n of elements of X such that T'[x]
7yx]= P [ X ' ] .

[Hint: if not, construct an increasing collection { y I I ( < a } of increasing sequences of length n in X such that { 7 ' [ y E ]I $. < a } is included in A, has order type a and is homogeneous for g . ] (c) Show that X is homogeneous forf. [Hint: use (b).] (2) Show that if a is a limit ordinal, ~+(a),<" and A+(a),<,, then K+(.),<,.

$2. The languages Lee for 8 strongly inaccessible

We have seen in $§ 1-3, Chapter 3, that Tarski's question whether the compactness property of L,, generalizes to L,,, whenever 8 is a strongly inaccessible cardinal > w , has a negative answer. I t is natural to ask whether there are other results from the model theory of L,, that generalize to L,. These languages are singled out as a subject of study on the (often unspoken) belief that the set-theoretical properties common to w and the larger strongly inaccessible cardinals may make the model theories of L,, and L,, a bit alike, at least closer than those of L,, and L,,, for weakly accessible K . Inverting the terms, what we are asking, roughly speaking, is: how much of the model theory of L,, follows from the strong inaccessibility of w? Naturally, there is a similar question for weakly inaccessible cardinals larger than w , but we do not know of any extension of non-trivial properties to those languages. In this section we shall see-in partial support of the abovementioned belief-that there are non-trivial results from the model theory of L,, that do extend to L,, and, moreover, that among the infinitary languages they generalize to these languages only. NOTATION. Throughout this section 8 will denote strongly inaccessible cardinals larger than w .

292

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 5 2

We shall focus our attention on the study of preservation of infinitary equivalence under sums and products. Recall that if %, % are structures of type p , then %@!I3 and iU X % (the simple cardinal sum and direct product of iU and %, respectively) are also of type p, whereas the full cardinal sum % + % is of type p‘=( p ’ u { P , } ) u ( p2u { P , } ) , where p 1 and p z are disjoint copies of p , and P , , P , are new unary predicates (cf. Definitions 1.1.20-1.1.22). Whenever sums are considered it is understood that the corresponding structures have only relation symbols. Note that some of the results below hold only under certain cardinality restrictions on the similarity types involved. Arguments due to FEFERMAN-VAUGHT [ 11 show that the elementary properties of any of these operations (and even of more general product operations) can be related in an effective way to (and therefore analyzed in terms of) the elementary properties of the factors.’ In particular this reduction yields the following:

THEOREM (Mostowski). Each of the operations defined in Defini~ions1.1.201.1.22 preserves elementary equivalence; in other words, let 0 denote any of

the operations simple cardinal sum, direct product, full cardinal sum; then

The same holds when

‘‘E”

is replaced by ‘‘ 5 ”.

The question whether this result generalizes to infinitary languages, and to which ones it does, has been considered by several people; for example, Lopez-Escobar posed the question whether it extends to LuLy.A full answer is given in MALITZ[I], [2], who showed that the properties under consideration generalize to L,,, K > w , if and only if K is strongly inaccessible. We shall begin studying the mutual connections among the problems of preservation of infinitary equivalence for the three main operations considered in this section: full cardinal sum, simple cardinal sum and direct product.

THEOREM 5.2.1. For any K , h such that h < K , if L,,-equivalence is preserved under full cardinal sums, then it is also preserved under direct products.

’

And, in the case of generalized products, also the elementary properties of other relational systems occurring in the construction.

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8

STRONGLY INACCESSIBLE

293

PROOF.Let Var', Var' be a partition of Var, the set of variables of the language, into two disjoint sets, each of power E;let f i : Var+Var' be a bijection, and for each vEVar let us write v i forf;(v), i = 1, 2. Let p be a similarity type; we define a map * from Form(L,,(p)) into Form(L,,( p')) such that for all cp E Form(L,,( p)): (1) Fv(rp*)=f,[Fv(rp)l uf,[Fv(cp)l; (2) for every pair 3,23 of structures of type p such that IrZIl n 181=a and 1 every g ~ ( 1 x~ IBI)~"(~), %~BI=cp[gliffZ++Fcp*[g'],

where g'E(l%l u IBI)FV(q*) is defined by g'(fi(v))=pr,(g(v)),

for all vEFv(cp), i = 1, 2

(pr, are the projection maps). The map is inductively defined as follows:

( R ( x ,,. . . ,x,))* = R ( x ; , .. .,X; ) A R ~x:, ( . . .,x,'), where R E p or R is w , and xI,.. .,xn are individual constants or variables;

( ( 3 X ) + ) * = ( 3 ( f I [ X ] u f 2 [ X ] ) ) + * , where X LVar,


By induction on formulas it is immediately checked that * has the desired properties. Suppose now that L,,-equivalence is preserved under full cardinal sums and that !Xi=,,%: ( i = 1,2); then using (2) and these assumptions we get

for all L,,-sentences cr. Hence L,,-equivalence is preserved under direct products.

THEOREM 5.2.2. For any K, h such that h 6 K, if LKA-equivalenceis preserved under full cardinal sums, then it is also preserved under simple cardinal sums. PROOF.For each rp E Form(L,,( p)) let @ be the formula of type p' obtained

294

[Ch. 5 , 5 2

LANGUAGES WITH INFINITE QUANTIFIERS

by replacing each atomic subformula S ( x , ,..., x n ) of x)by:

&(xi ,..., x,')vS,(x:

where (for i = 1,2)

x i= J

(

'p

(thus S E p or S is

, . . . , x,2),

x,

if xi is a variable,

c'

if xi is the individual constant c.

Clearly Fv(cp)= Fv(+), and a trivial induction on the structure of u 181)Fv(q): that for all g ~(131

'p

shows

% I 8 F q [ g ] w rZr+B++[g].

(*)

Suppose I?li - K ~ % i ( i= 1,2) and u E Sent(L,,( p)); since L,.-equivalence is preserved under full cardinal sums, using (*) we get:

This proves the theorem.

THEOREM 5.2.3. I f K is strongly inaccessible larger than w , w < X < K and L,.-equivalence of structures of similarity type of cardinality less than K is preserved under direct products, then it is also preserved under simple cardinal sums.

PROOF.Let p be an arbitrary similarity type of power < K containing only relation symbols. Let P,, P, be unary predicate symbols not in p. To each structure % of type p we shall associate structures I?l# and % # of type p u { P,, P,}. Let a 6? I%[. We define:

I%#l = Ia#l=WI u { a } ;

[

R "= R 'un = R u ( a , . . . , a )

)

for each n-ary relation symbol R E p ;

@ Pl"" = P-p = 121; py=p'u 1

"-

(4.

%e*$*v

-

F o r each 'p E Form(L,,( p ) ) let 'ps denote its relativization to P l ( v ) vP,(v). Then for all gE(lI?llu lBl)Fv(q): (*)

%@B+'p[g] ~ ~ X x # F q # [ g # ] ] ,

Ch. 5 , 8 21

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295

where for u E Fv(rp),

The equivalence (*) is easily proved by induction on the structure of rp. Assume now that %;=,.A%: (i= 1,2). We shall prove that %,@%~EKA%;@%~.

We consider two cases, according to whether A < K or A =

K.

Case (a). A < K . Since K is strongly inaccessible > w , A < K and < K , by an argument involving extensions of partial isomorphisms we can show that

The argument used here is a simple application of the back and forth method developed in 0 3 below; see Exercise 5.3.6. Let u ESent(L,,( p)). Using (*), (**), and the assumption that LKAequivalence is preserved under direct products, we easily infer

81 @%~=KA%;

@%;.

Case (b). A = K . Let oESent(L,,( p)). Since K is a limit cardinal, there is a < K such that uESent(L,,( p)). Since %j=KK%i, in particular we have By case (a) we conclude that @%2=Ka%; @a;. Hence

-

%,@%2ku

%;@%;t-u: ~


Since u is an arbitrary sentence from LKK( p), it follows that

This proves Theorem 5.2.3.

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LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5 , $ 2

REMARK.It follows from Theorems 5.2.5 and 5.2.13 below that the converses of Theorems 5.2.1-5.2.3 also hold. We prove now that L,,-equivalence, where 8 is a strongly inaccessible cardinal larger than w , is preserved under full cardinal sums. If p ' , p2 are disjoint copies of the type p , we write Form',, for NOTATION. the class Form(L,,( p i u { P i } ) ) ,X < K < 8; Form' stands for Form;,, and for the class of all relativizations cpQ of formulas cpEEForm' ( i = l,2). Given a class Z of formulas, Bool,(C) denotes the class of all Boolean combinations of length < K of formulas in Z; ie, the smallest class of formulas containing X and closed under negation and conjunctions (and/or disjunctions) of any length < K .

THEOREM 5.2.4. There is a map

* : Form(L,,( p'))+Bool,( (Form')'' u (Form')'') such that for every pair a, 23 of structures of type p, every cp E Form(L,,( p')) and every gE(l(lIlu IBl)Fv(vp):

Before proving this theorem we show that it implies:

THEOREM 5.2.5. L,,-equivalence is preserved under full cardinal sums. PROOF.For i = I, 2, define the transformations A

: (Form')f'-+Form(LeA(p ) )

as follows. If cpE (Form')', then all the symbols occurring in cp are in p ' u { P , } ; cpAi is obtained by replacing in cp each symbol of p' by the corresponding symbol of p , and each occurrence of P , ( v ) by u=u.

Note that Fv(cp)=Fv(cpAi) and that, given 911, 8 ' of type p and

g E Ia'lF"(P):

(**)

~ I , + W c p [ g* l ~'~cpA\"gl.

Suppose now that aI,=e~91[: ( i = 1, 2) and that 7EBoolg((Form')P1u (Form*)''), where 7 is an Lei( $)-sentence. Then: (***>

(lI'+i?12F7 e (lIZr;+(lI;WT.

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Condition (***) is easily proved by induction on the structure of the Boolean combination, using (**) for the case 7 E form')"^. If u E Sent(L,,( p')), then u* E Bool,((Forrn')'I u (Form2)"); thus, applying (*) and (***) we get:

Now we proceed with the

PROOFOF THEOREM 5.2.4. Let rp, +~Form(L,,( p')); we define rp-+J/ (read rp, J/ are -equivalent) iff Fv(J/) = Fv(cp) and for all Iu, 8 of type p and all g €((%I u JBI)F"(T):

+

%+B+rp[sl=

%++t=J/rc/[sl

Clearly is an equivalence relation compatible with all the logical operations of Lo,. We will show that for all rp E Form(L,,( p')): -+

(*I

there are rpv, cp" E Form(L,,( p')) such that (i) rpA is of the form / " \ I E L V j E J I r p / J ; (ii) rpv is of the form V/,=rAjEJ,,rpi,,; where rpl,j, rp;,j E(Form')'l u (Form2)'z for all indices 1, j , and = = L , L ' , 7,, for all [ E L , ['EL'; (iii) rp-+cpv-+rp".

T<e,

Then cp* can be taken as either rp" or rpv. First we observe that the existence of one of rp", rpv implies the existence of the other. For example, if cpA exists and satisfies (i) and rp-+rpA, then rpv can be taken as: rpv=

v

SE n

/EL

A

rpI,fiO* J,IEL

- r,

-

Since 0 is strongly inaccessible and 7 I < 0 ( 1 EL), then I l l E L J I< 8 (cf. formula (19), p. 7); hence rpv has the form indicated in (ii). Furthermore, the following chain of equivalences proves that rpv- +qA: Iu+Bkrp"[g] @ for every l E L there is j E JI such that

Iu + B t= rpI,.[ g ] = there is a map f such that Dom(f) = L, for all 1 EL, f(1) E J I ,

298

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LANGUAGES WITH INFINITE QUANTIFIERS

A similar argument shows that the existence of rpv implies that of rpA. Let r be the set of all rp EForm(L,,( p')) for which (*) holds. By a somewhat involved induction on formulas we will now prove that

r >Form(L,,( $1). (1) rp atomic * rpEr. Set cp = rp" = rp". Condition (*) holds because all atomic formulas are in

If

-

(214~17

(Form')'' l+a. +A=

then set

u (Form')''.

A V v / , j , qV= V Av,G,

/EL j E f l

wr=A v 1v;,j, IEL'jEf;

IEL' j E f i

v A lrp/,j.

(1+F=

/ELjEJ/

(3) r is closed under conjunctions and disjunctions of < O formulas. 7 < 0 ; set Suppose {rpjliEz}cr,

Since each QIP ( i E I ) satisfies (i), it is clear that the first formula also satisfies (i); similarly, the second formula satisfies (ii). The dual formulas (r\iE,q,)v and (VjEIrpi)" are obtained by the construction indicated above; thus, for example, if rpv = r\lEL;VEj,/,where Xi,,, Lj < O (i E I ) , then (r\iEIrpi)v will be the formula p

_

=

(4) r is closed under quantification over sets of variables of power
Ch. 5, 8 21

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+ E r, X C Var,

similar. Suppose cp = (VX)+,where #A=

where Let

e STRONGLY INACCESSIBLE

299


A V+/,j,

(EL j E 1 ,

_ L, J , < 8 for all I EL, and all the +,,,'s

are in (Form')PI u

-

and let xI, be the formula inside the first conjunctions ( I E L , Y E ?? ( X ) ) . Since < A < 8, then T ( X ) < 8; therefore cpl E Form(L,,( p')). Clearly cp- +cp,. We will prove that X / , ~ E ~therefore ; X / , ~ - + X ~ where ~ , is

xty

as in (i) of (*); hence

Defining q A to be the last-mentioned formula, we obtain cp-+cpA and q A has the form required in (i) of (*); the dual formula cp" is then given by the general construction explained above. Since, obviously, r is closed under + -equivalence, in order to prove that X / , ~ E ~it ,suffices to show: CLAIM. xI

PROOFOF

is +-equivalent to a formula in CLAIM.

r.

Let

By renaming bound variables we can assume that Fv(\k: v 9;) n X =0 for all I EL. Let 'k?, (\klf2,resp.) result from \k{ ($, resp.) by replacing each ~ ( F v ( ~ ) n ( x -Y)#B, resp.) atomic subformula 7 such that F v ( T ) Y#B ; by 3 u o i ( u o ~ u o )thus Fv(\k;,)n Y = 0

and

Fv(\k;,)n(X-

It is easily seen that the following holds (where

3

Y)=0. denotes logical equiva-

300

LANGUAGES WITH MFWITE QUANTIFIERS

[Ch. 5 , 3 2

lence):

To understand the equivalence between the first two lines observe that if is an atomic subformula of !P{ containing variables of Y-for example 7 = R , ( x , y ) with R , E pi, y E Y and x E X - Y-and if (as indicated by the antecedents of the first formula) the assignment 7

g :Fv(.I.k: v.I.:)+l%l

u 1231

\%I

is such that g ( x )E = P?+%, g ( y ) E \B\ = PT+B,then it is not the case that % + 8 t- R,[g ( x ) , g ( y ) ] ,because R?+'L 1x1X 1241. The proof of Theorem 5.2.4 is completed by observing that the first disjunct of the last formula above is in (Form2)P2 and the second in (Forml)P1; hence the formula is in r. The arguments that prove Theorems 5.2.4 and 5.2.5 also yield the following: K , h E CN,:K > A, and (I E Sent(L,,( p')) there such that for all structure3 %;, B; of similarity type p and = 1B11n lBzl =a,the following holds:

COROLLARY 5.2.6. (1) For eoery exists a v

l%il n

>K

( 2 ) For e v e v sums.

A E CN, La,-equivalence is preserved under full cardinal

REMARK.Note that in the proofs of Theorems 5.2.4 and 5.2.5 it is not essential that p ' , p2 be disjoint copies of rhe same similarity type p. Indeed, both proofs are straightforwardly adapted to the case where p i , p2 are arbitrary disjoint similarity types containing only relation and individual constant symbols.

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301

Thus, consider the operation extended full cardinal sum, 2 7 23, defined for any two structures %, 23 of (disjoint) simiIarity types p ' , p2, resp., such ) ( p2 u ( P , } ) that 131n 1231 =0, as the structure 0 of type p = ( pi u ( P I } u such that 101= tat u

Pl"= /%I, SF= SF

1%

Pz"= 1231, for each S , E pl,

S f = Sp for each S, E p,. Theorems 5.2.4, 5.2.5 as well as Corollary 5.2.6 apply to this operation. We state this fact as

THEOREM 5.2.7 (1) For every K , AECN and every uESent(L,,( p")), where p" = ( p 1u { P , } )u ( p 2 IJ ( P , } ) , there exists Y K such that for every pair of structures %;, 23; ofsimilarity type p, ( i = l , 2 ) , ~ % l ~ n ~ ~ 2 ~ = ~ 2 3the 1~n~232 following holds ; i~~=~ (i= ,% 1,2) ~

* (i~,7 =t,%

u

H

Bi T ~ , t = u).

(2) For strongly inaccessible I3 and infinite A such that A < 0, Lo,equivalence is preserved under extended full cardinal sums. ( 3 ) For every A E CN, L,,-equivalence is preserved under extended full cardinal sums. The proof of Theorem 5.2.7 is left as an exercise for those read.ers interested in gaining a deeper understanding of the arguments presented above. REMARK. Using a back and forth argument it is possible to give a simpler proof of the preservation of L,,-equivalence under full and extended full cardinal sums (see Exercise 5.4.6). However, this back and forth argument does not yield Theorem 5.2.5. We proceed now to prove the converse of Theorem 5.2.5: if LKAequivalence is preserved under simple cardinal sums, then K is strongly inaccessible. For each K not strongly inaccessible we will construct a counterexample showing that L,,-equivalence is not preserved. This involves some preliminary work. First, by induction on ordinals y, y > 1, we define the formulas $y (oo, u u,), involving a binary relation symbol .E:

,,

$ l ( u o , u l , ~ 2 ) = v o = ~i I (~ q w u , ) ~ V v ,(v,Ev,vv,Ev,), i

-

302

LANGUAGES WITH INFINITE QUANTIFIERS

(Ch. 5, 8 2

and for y > l :

Notice that $ y ~ F o r m ( L ( 9 ) (+{wE } ) ) ; for the sake of brevity it will be convenient to baptise the second and third conjunct of $y by $;, 4; respectively (y > 1). The following lemmas will help in understanding the meaning of these formulas.

5.2.8. Zj $. < y , then ~ # E ( u 0 , u , , u 2 ) ~ ~ ~ ( u 0 , u 1 , u 2 ) . LEMMA

PROOF.Assume ~ , L & u ~ , v ~ ,true. u ~ ) If .$=1, then V u 3 1(u,Eu, vu3Eu2); hence the antecedent in both 4; and is false; so $J,; 4; are true; since 1( 0 1 x u 2 ) also follows from then $+.is true. Let <>1 and u3Eul be true; using we infer

I);

+<

also hence 3u4( u4Eu2

A

v

I
u3, u4))

is true. Thus, we conclude that is true; we argue in a similar way to infer the truth of $,; which immediately shows that I/.+ holds.

$4

( ; xE'@%y, then x , y belong to the same REMARK.Let x , y ~ ( % ( u ( % if or both belong to 1'231). summand (i.e., both belong to

LEMMA 5.2.9. Let both

a,%satisfi

let z,zl,z2E\%I u 1231 be such that

the extensionality axiom

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Then z l and z2 never belong to the same summand.

PROOF.By induction on y. ( y = 1) The first hypothesis implies that z 1# z2. Since both zi's satisfy the 3u(uEw) and since both % and B satisfy extensionality, then formula i z1,z2cannot simultaneously belong to one of % or 8. Assume that y > 1 and the result is true whenever [ < y. Case 1. One of the zj's, say z I , satisfies the formula i 3 u ( u E w ) . By the first conjunct of Ir;, z 1# z2. Using 4; we easily infer that z2 also satisfies 3 u ( u E w ) ; clearly, two of z,z1,z2should belong to the same the formula i summand; then, by extensionality and the last hypothesis, we infer that either z = z l or z = z 2 ; hence either % @ B t - ~ l [ z , z l , zor2 ]%@Bt-+l[z,z2,zl]. Thus, we are reduced again to the case y = 1 and the result follows. Case 2. None of the zj's satisfies the formula i 3 u ( u E w ) . Then there are x,,x2 such that xjExeBzj ( i = 1,2). Using 4; we get wEI%IuIBI and [ < y such that wE"@'z2 and %@BF4,[z,xl,w ] ; by induction hypothesis x I and w (and hence z I and w ) do not belong to the same summand; by the remark above w and z2 belong to the same summand; hence the result is proved. H L E M M A5.2.10. Let %,B satisfy the extensionality axiom, and let z , x , x ' , y , y ' E ( % I u 123 be such that x,x' belong to one summand and y,y' belong to the other; let y E O N , y 2 1. Then (a) % @ t- 4y[ z ,x,Y I A 4y[ z ,x,Y ' I implies y =y ' ; (b) % @ B b [ z , x , y ] A [ z , x',y] implies x = x'.

4

PROOF.We prove (a); the proof of (b) is almost identical. Assume, for example, that x , x ' E )%I and y , y ' E lB1. We try to show: (*) whence

{ b E 1231 I bE'y}

I

= { b E 12 ' 31 bE'y'},

Bi= v u ( u E y ~ u E y ' ) and by extensionality, y =y ' . The proof of (*) is done by induction on y. ( y = 1) By the definition of 4 , it is clear that in this case both sets in (*) are empty. Assume y > 1 and that (a) holds for all [< y. Let bEI'231 be such that bE'y; from %@Bt-rC;[z,x,y], using ,'()I we get w ~ l %ul 1231 and t1< y such that % @B t- $,,[ z , w ,b ] and

W E'@'x;

304

LANGUAGES WITH INFINITE QUANTIFIERS

from %@Bi=+7[z,x,y']and wE'@'BX, using $,; < y such that:

[Ch. 5,

82

we get U E \ % ~ I J U and ~B~J

[2

% @Bk + J z , w, u ]

and uE '@'y'.

Clearly b , u E 1231. Let ma^('[^,[^}; by Lemma 5.2.8: (IICB 23

$JZ , W , b ] A $c[

Z,W

u , ].

By induction hypothesis we infer u = 6 ; hence bEHeBy'. Since both b, y'E1231, by the definition of EXeBwe get bE'y'. in (*). Similarly one proves the Thus, we have proved the inclusion other inclusion. Hence (*) holds and the lemma is proved. LEMMA 5.2.1 1. Let (II and B satisfy the extensionality axiom and let z,, z2, x , y E I%/u (B3(be such that z , , x belong to one summand and z2,y belong to the other; furthermore, assume that z,, z2 both satisfy the formula i 3u(uEw) in %@B.Then: @ 8I=

for all y EON, y

+,. [z ,,x,y 3

iff

% @ 8k 4,. [z2,y,X I

> 1.

PROOF.We prove, for example, the implication from left to right, the other being completely similar. We proceed by induction on y. ( y = 1) % @ B P $ , [ z , , x , y ]implies x # y , i 3 v ( u E x ) and 1 3 u ( u E y ) . Since '72, Y belong to the same summand, by extensionality we get z 2 = y ; thus 3 @ Bt-$,[Z,,YJl. Assume that y > 1 and the result holds for all [< y. We have to prove that

we prove, for example, the first, i.e., Let wEaeByy; by assumption %@B++;[z,,x,y];hence we get u E ( % ( u 3 J % ~ and [< y such that uE'@'BX and (II@Bk$t[zI,u,w]. Clearly u, x belong to the same summand, and the same holds for w ,y ; whence, by hypothesis, z l , u belong to one summand and z2, w belong to the other. By induction hypothesis it follows that:

Thus, formula (*) is proved.

rn

Summarizing, the results of Lemmas 5.2.8-5.2.1 1 amount to:

THEOREM 5.2.12. Let 8, B satisfy the extensionality axiom and let

z,, z2EI%\u IBI belong to distinct summands and satisfy the formula i 3u(uEw). Then, for every y > 1:

qY[z1,.;I=

{ ( x , y ) I x , y ~ 1 % [ 1 ~ 1 Band l

~ @ ~ ~ ~ y [ ~ l ~ x

is a one-one map from a subset of one summand to a subset of the other and 1 < 5 < y , then qY[z,,* , extends

i c ; [ z 2 , - , .] is its inverse. Moreover, if +JZP

- 9

a]

-1.

After these preliminaries we proceed to prove the promised converse of Theorem 5.2.5:

THEOREM 5.2.13. If K is not strongly inaccessible there are structures %, 8 , (5 of q p e { E } such that: (1) %=a, (2) 8 < wc (5% (3) %@B+"a91@E.

PROOF.There are two cases to consider according whether: (I) K < 2' for some cardinal h < K , or (11) K = UL<paL for some strictly increasing sequence (atI[
< g= it follows that Next we show:

+: :[; =

if y successor,

1)

if y limit

>o,

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LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 2

f-*r f N y ) l

if y successor,

f - i r f [ R ( y + l)]

if y limit >O.

Obviously, the second identity follows from the first using Lemma 5.2.1 1 (or Theorem 5.2.12). The first formula follows immediately from the following two statements which are easily proved by induction on y 2 1: (a) i) if y is a successor ordinal and x E R ( y ) , then

ii) if y is a limit ordinal

>0 and x E R ( y + l), then

(b) i) if y is a successor ordinal, x E

and

a $6 I- 4y[O, x,y 1, then x E R ( y ) , ii) if y is a limit ordinal >0, x E I%/ and

@at= @, x , y ] , then x E R ( y + 1). By way of illustration, (b) ii) is proved as follows: from %@Ot=$y[O,x,y], we get %I$OI-.J/C[O,x,y],i.e., for every z ~ there x exist WEIXIUIOI and [ < y such that wE'@'y and %@QI-$,[O,z,w]. By Lemma 5.2.8, 6 can be taken as a successor ordinal; moreover, z E 121;by (b) i), which holds by induction hypothesis, we infer that z E R(6). Thus, we have proved that x G Ut,,R(6)= R ( y ) , i.e., x E R ( y + 1). The next step consists in showing that there is a formula involving E that, upon fixing a parameter, characterizes each one of the summands of %a36and %@B. Let 6(u,,u,) be the formula

Then we show:

Ch. 5, 8 21

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307

u 101 and y E 131u 181, then: 9l@B k 6 [O, x]

iff

x~l9l1,

~ @ ~ ~ 6 [ f ( o ) iff , x ]x~lCSl, 9 l @8I- 8 [ O J ]

~EI%I,

iff

% @ 8 + 6 [ f ( O ) , y ] iff ~ E I ' E I . For example, we prove the last statement; the proof of the others is even simpler. Remark first that x E 131 implies xu (0) E 181; this means that 8 satisfies the formula:

(*I

13u I(u Euo)+V u23u3V u4( u4Eu3t,u4Eu2v u4= uo).

Since 9l =8 --Q, 8 and 0 also satisfy (*). (e) If y ~ l B 1 either , y = f ( O ) or 3 u ( o E 8 y ) . in the first case we have 9l@B!i-8[f(O),y].In the second case, set uo+f(0), u2+y in (*) and get a ~ ~ 1 2 such 3 1 that for all ~ ~ 1 8 1 : ZE'W - z E ' y v z = f ( O ) ;

by setting uo+f(0), u I + y , u2+w, the second disjunct of 8 holds; hence % @ 23 + 6 [f(O),yl. (a) Suppose there is y E I9ll such that 8698 P 8 [f(O),y];since 1 8 1n 1231 =a,then y #f(O); therefore, the second disjunct of 6 holds, and there are z , w E I%/ u 181such that: zE'@'y,f(O)E'@'w

and for all x~1'1IIu181, xE'@'y

-

+ xE'@'w;

hence z E ' @ ' ~ w and f(O)E"@%w; since z E I 9 l l and f(O)E181, we have derived a contradiction. Hence as was to be proved.

9l @8t- 8 [f(O),Yl

Consider now the following L,,-sentence

Y

6:

1 8 3 1

308

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5,

53

(A) and (B) together with Lemma 5.2.10, show that %@Qt=u.Since u asserts that $ h + 2 ( ~ o , ., - ) is a one-one map from one summand onto the other, and since # 5, it follows that %@BP l u . Thus the proof in Case (I) is complete. Case (11). K = Uz.,p(utfor some strictly increasing sequence of type fi < K of ordinals < K . In this case we take % = ( R ( K + ~ )~, r R ( ~ f 2 and ) ) 6 , 3 as before. Notice that $K Form(L,,). However, we can modify the definition of $a for a = K , K + 1, K + 2, so as to make the argument of Case (I) work. Let 4: be the formula obtained from $, by replacing the disjunction V,5E.,rtS by the formula Vs.,pfald,; likewise, I),,?+,and $:+2 are obtained respectively, instead of V IgS.,K$z in qK. by inserting $: and The same arguments prove that Lemmas 5.2.8-5.2.1 1 carry over to the present case. The statement corresponding to (A) is the following:

where f denotes the isomorphism between M and 6. Using (B) and the L,,-sentence u* obtained by inserting I):+2 instead of in u, it is proved, exactly as before, that %@Crku*but %@%I- la*. This completes the proof of Theorem 5.2.13. rn

$A+2

REMARK. Careful inspection of the arguments used in Theorems 5.2.1-5.2.5 shows that they also prove the analogs of Theorems 5.2.1-5.2.3 and 5.2.5 obtained by replacing i,,” for Notice also that Theorem 5.2.13 is already stated in terms of L,,-embeddings. Therefore, we conclude that LKA-embeddingis preserved under simple cardinal sums (extended full cardinal sums, full cardinal sums, direct products) iff K is strongly inaccessible “

“ = K ~ ” .

> 0. 5 3.

Extension of partial isomorphisms

This section contains a general exposition of an important technique known as the method of extension of partial isomorphisms or the back and forth method. This type of argument dates back to the very beginnings of set theory. G. Cantor was the first to use it, in his theorem that two countable dense linear orderings without endpoints are isomorphic. Langford used the same

Ch. 5, 5 31

EXTENSION OF PARTIAL ISOMORPHISMS

309

argument to show that any two such linear orderings (of arbitrary cardinality) are elementarily equivalent; this seems to be the first occurrence of a back and forth argument in a metamathematical context. Hausdorff used the argument again, in a somewhat more general form, to prove the analog of Cantor’s theorem for qA-sets.This type of argument occurs again in the proofs of uniqueness and of universality of saturated structures. Frai’sse and Ehrenfeucht extended Langford’s result showing that there is a purely set-theoretic characterization of elementary equivalence in terms of families of partial isomorphisms defined on finite sets, with the back and forth property. In particular, Ehrenfeucht gave a game-theoretic interpretation of such families of partial isomorphisms. However, it was KARP[2] who first pointed out that the natural metamathematical framework for the back and forth techniques was infinitary, rather than finitary, logic; more precisely, the class-logic Lma. Since Karp’s paper was published, back and forth type arguments have met with increasing application in model theory. Moreover, in the recent [3] and BARWISE-EKLOF [ I ] it has been shown the large papers of BARWISE extent to which this kind of argument also permeates other branches of mathematics, e.g., group theory. These works throw light upon a vast-and so far very little explored-field of research between algebra and metamathematics, the most significant result of which, up to now, is BarwiseEklof‘s extension of the theorem of Ulm in the theory of torsion groups. This evidence supports the feeling that the methods developed in this section are likely to become a major tool of research. We remark that back and forth arguments naturally fall into two main categories: the “one at a time” (or, equivalently, “finitely many at a time”) form, and the “less than h at a time” form, where X can be an uncountable cardinal. The former is not a particular case of the latter, but rather a special case of a restricted form of the latter, as we shall make clear in paragraph A. In keeping with the general concern of this book, our presentation below is somewhat slanted towards the latter form of the argument. The “one at a time” form, however, is given due attention in paragraph B. The exposition is organized as follows. In paragraph A we start from the example of qA-setsand work our way through a general form of the back and forth argument. We discuss the construction of isomorphisms from families of partial isomorphisms with the back and forth and the extension properties. In paragraph B we prove Karp’s theorem and discuss several of its variants; this illustrates the connection between infinitary languages and the back and forth argument; the case of finitary logic is also treated and we proye the Ehrenfeucht-Fralsse theorem.

310

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, § 3

This connection is taken up and studied in detail, and in a much more general form, in paragraph C. This general treatment-initiated by [ I]-grew out, and unifies the proofs, of several results in BARWISE-EKLOF the model theory of infinitary languages which, in different ways, used essentially the same back and forth argument: Karp’s theorem, Chang’s generalizations of Scott’s isomorphism theorem, etc. At the end of this and in the next paragraphs we analyze just this last kind of result. Paragraph D is devoted to an analysis of the connection between various logical relations (e.g., L,,-equivalence) and some of the usual algebraic relations (e.g., isomorphism) defined between relational structures; the back and forth characterizations of the former developed in paragraph C allow us to establish these connections without difficulty. Finally, paragraph E contains some counterexamples which put bounds on the possibility of generalizing some of the previous results. Our presentation below owes much to the following papers: KARP[2], [ 11 and BARWISE [3].’ CHANG[2], BARWISE-EKLOF Before starting paragraph A we fix some notation to be used throughout this and the next sections. For given structures 21, 8 of a similarity type p, we shall consider maps f of the following kind: (1) Dom(f) c l q Range(f) c 181; (2) Dom(f) is a substructure of 9l and Range(f) a substructure of 8, i.e., they are closed under all the operations, and contain the denotation of all the individual constants of p ; (3) unless otherwise specified, f is a monomorphism from 9l r Dom(f) into 8. Maps satisfying (1)-(3) will be referred to as partial isomorphisms. Recall that given X cIrzI1, the substructure of 21 generated by X is the smallest substructure of 21 containing X . If in addition to (1)-(3), f also satisfies: (4) Dom(f) is generated by less than X elements (where X is an infinite cardinal); then f will be called a A-partial isomorphism. An important exception is that the map 0 from the substructure of % generated by the empty set, into 3, whenever it is defined and is a monomorphism will be considered a partial isomorphism, regardless of whether its domain satisfies (2) or not. [Note that if 0 is defined, then it is unique, and in this case Dom(0) does not satisfy (2) iff Dom(O)=Hiff p does not contain individual constant symbols.] Throughout this section fcg will denote the relation of extension of maps; i.e., f g iff Dom(f) Dom( g) and g Dom(f) =f. Bold-face, up-

c

’

c

r

Special thanks are due to Professor J. Barwise, who made available this last mentioned paper before publication.

Ch. 5, $ 31

311

EXTENSION OF PARTIAL ISOMORPHISMS

per-case latin characters, I, F, G, etc., with or without subscripts, will denote families of partial isomorphisms. Finally, recall that formulas of L,, always contain less than A free variables (formula property). A. The back and forth argument As basic motivation, we present Hausdorff‘s extension of Cantor’s theorem on denumerable, dense linear orderings. The result to be proved is:

5.3.1. If A is a regular, infinite cardinal, any two v,-sets ofpower A THEOREM are isomorphic. PROOF.The argument falls naturally into two parts: Part I. Let A be a regular infinite cardinal and let a, ‘23 be v,-sets (of arbitrary, possibly distinct, power). Define the following set of partial maps from 3 into 93:

I = { flf

is an order preserving map defined on a subset of of power
Then I has the following properties: (i) (A-extension property) Given a set F c I , totally ordered by the extension order c , of power less than A,’ there is a g E 1 which extends all the f E F. (ii) (One at a time back and forth property) (a) (“Forth” part) for every f E 1 and a E 181 there is g € I such that f c g and a E Dom( 8 ) ; (b) (“Back” part) for every f € 1 and b E 1’231 there is g € 1 such that f c g and b E Range( g). PROOFOF (I). (i) is obvious, for g = U,,,f is already in I, since Dom( g) < A by regularity of A. (ii) We prove only part (a); the proof of (b) is similar, and is left as an exercise for the reader. Of course, we can assume a 9Dom(f). Let ( Y ,Z ) be the cut defined by a in Dom(f):

y = { Y EDom(f) IY c E a } , Z = { z E Dom( f ) I a <‘z }.

’ In the case of singular A, the following should also be required:

=

Dom(

UIEFf)
312

[Ch. 5, 5 3

LANGUAGES WITH INFINITE QUANTIFIERS

Since f is order-preserving and Y <"Z, we obtain f[Y] <'f[Z]; since Dom(f)
foGf1c ... cf,c ...

(a
of maps in I as follows: (A) if a is an even, successor ordinal (i.e., a = p+ 2n, where ordinal and n E w , n > 0): f,=some map g E I such thatf,-,Gg

p

is a limit

and ~ ~ + , - ~ E D o m ( g ) ;

(B) if a is an odd, successor ordinal (i.e., a = P + 2 n +

1,

fl limit and

n Eo):

f, = some map g EI such that fa

-, c g and bp+, E Range( g).

In both casesf, exists in virtue of the back and forth property. (C) If a is a limit ordinal < A : fa = some

map g € 1 extending all the maps fs, 6 < a.

A simple induction shows that the fs's form a chain under extension; by

the extension property, there is a map satisfying (C). It is easily checked , is order-preserving and Dom(f) = 1911, Range(f)= 181; that f= U& hence it is an isomorphism. It is apparent from the preceding proof that part I1 has very little to do with linear orderings: the construction of the required isomorphism depends exclusively on the two properties of the family I. Indeed, it is easy to turn part I1 into a general theorem.

Ch. 5, 5 31

EXTENSION OF PARTIAL ISOMORPHISMS

313

NOTATION. Whenever there is a non-empty family I of partial isomorphisms from 'u into 8,with the A-extension and the one at a time back and forth properties we shall write 'u -teB; likewise, I : 2I -te8.

THEOREM 5.3.2. Let 'u, 8 be similar structures of power < A or, more generally, generated by sets of power < A. Then: ( l ) ' u = B * 'uqe23. Moreover, the following holds: ( 2 ) i f I :'u =fe8,then every fo EI can be extended to an isomorphism between 'u and 8 . PROOF.(1) (=+) Let 'u -fB. Set I = { f}. It is clear that I has the required properties. The other implication follows from (2). ( 2 ) Fix f o E I . Let X , Y be sets of power < A such that: (i) Dom( fo)u X , Range( fo)u Y generate 'u, '21 respectively; (ii) X n Dom( fo) = Y n Range( fo) =O. Let X = ( a , I a
of maps in I. An easy induction on a proves: (A) if a = p 2n is an even, successor ordinal, then

+

{ a , 16 <

(B) if a = p + 2n

p + n - l } c D o m ( fa);

+ 1 is an odd, successor ordinal, then

(C) if a is a limit ordinal, then

therefore, since Dom( fa), Range( fa) are substructures of 2,8 respectively, they contain the substructures generated by the sets on the left hand side of the respective inclusions (A)-(C). From this fact it readily follows that if f = Ua<xfa,then Dom( f ) = 181 and Range( f ) = 1231. Moreover, since each of the fa's is a monomorphism, so is f; whence f is an isomorphism. The detailed verification of these facts is left as an exercise for the reader.

314

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 9 3

Further analysis of the foregoing argument reveals that the extension property of I is only needed to construct the mapf, at limit stages. Thus, if A = w , there are no limit stages to take care of, and we get rid of the extension property; in other words, the union of a finite chain of elements of I coincides with one of its members, and therefore is in I. It turns out that this is also the case for structures of power < A , whenever A has cofinality w . In order to carry out the proof of this assertion and also for other reasons that will become apparent in paragraph B, we find it convenient to introduce the following:

5.3.3. Let A be a cardinal > 1; we shall write I : %=fB to DEFINITION indicate that I is a non-empty family of partial isomorphisms from % into $3‘ with the less than A at a time back and forth properties: (a) (“Forth” part) for everyf € 1 and every A c [%I, 7
-

Tu =fB means that there is an I such that I : % =fB. Clearly, if I : % ={B and A 2 K, then I : %=CB. The reader can check by himself that I : % =;B 1:%=23. Another exercise is to check that =f is an equivalence relation. Notice also that % =peB implies % +B; actually something much more general is true: LEMMA5.3.4.I : % =FeB

=+ I : 2 =fB.

PROOF.We prove the “back” part; the proof of the other is similar. Let ~ E and I BclBI, F
focf,r ... cfsc...

(6
with the following properties: (9 fo=f; (ii) f, whenever 6 < y < p; (iii) { b, 16 < LX}c Range(f,), for LX < ,B; (iv) f, €1, for all a < p. Applying the A-extension property to the chain {fa I a < j3 } ( ,L?
rf,

Ch. 5, 9 31

315

EXTENSION OF PARTIAL ISOMORPYlSMS

We notice that the converse of this lemma need not be true, except in the trivial case A = w . Indeed, Counterexample 5.3.41 shows that the right hand side may hold for some I, without the left hand side holding for any I. Next we prove the promised:

THEOREM 5.3.5. Let A be a cardinal with cofinality o and 8,'8 be structures of power < A, or generated by < A elements. Then: %=!I3

@

8=p.

Moreover, the analog of (2), Theorem 5.3.2, also holds in this case.

PROOF.The implication (+) is exactly as in 5.3.2. As for the other implication, let I :8=fB. To simplify notation let us assume that fo = 0 and 8, B are of power < A ; the general case is obtained by slight changes, following the pattern of the proof of Theorem 5.3.2. Let ~ 8 1 = ( a a ~ a < A )I,B [ = ( b , ( a < A ) ; let ( K , ( ~ E U - - ( O }be ) a strictly increasing sequence of cardinal numbers converging to A. We define a sequence

fOCf1C ... Cf,L ... ( n e w ) of maps in I, as follows: (A) if n is even, n > 0 ,

f,=some map g E I such that { a s 16 < K , } C Dom(g) andf,- 1 Cg; (€3) if n is odd,

f, =some map g E I such that { bsl S < Kn} C Range( g ) andf,-

I

Cg;

Maps g with these properties exist by virtue of the less than A at a time back and forth property. f, is an Since each f, is a monomorphism, it is clear that f= UnEw isomorphism between 8 and 23. H

COROLLARY 5.3.6. If cf(A)=o and 8,23 are generated by sets of power then

PROOF.Immediate from Theorems 5.3.5 and 5.3.2. H

< h,

316

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 3 3

These results naturally suggest the following questions: (1) Does 5.3.5 extend to cardinals other than those of cofinality a? (2) How does -f relate to -f" in the general case? Counterexample 5.3.41 shows that Theorem 5.3.5 does not extend to regular, uncountable A's; the situation for singular A of cofinality larger than w is not known, but the answer is likely to be negative. The connection between =ph and = f e has not been studied beyond the preceding simple results. While the relation -f has a very clear metamathematical meaning (it is equivalent to =m ~ ) ,no connections between -f" and infinitary languages are known. However, = f e has an interest of its own since, in many examples, natural families of partial isomorphisms turn out to have the extension property. For example the families, of all partial isomorphisms, of all A-partial isomorphisms (if A is regular), and of all partial elementary maps, of % into B, all have the A-extension property. Hence, if any of these has the one at a time back and forth property, then %=fe23 is true. An immediate consequence is that %=$B holds, so % = 5 = A implies %= 23. Another example of this situation is given in the following, well-known extension of Theorem 5.3.1 : THEOREM 5.3.7. Let 3, 23 be elementarily equisalent, A-saturated, infinite structures; then: (1) % =feB; in particular: (2) % -123 (hence, by the results of subsections B and C , % =m~23); (3) i f %, 23 have power A, then % = B .(cf. Theorem 2.1.14, Fundamental Theorem, part (4)).

PROOF.T o simplify notation, let us assume that the underlying language p has no function or individual constant symbols; as an exercise the reader can do the proof of the general case. We take I to be the set of all elementary maps defined on subsets of 131 of power
Since %=a,then OEI; so I # H . It is obvious that I has the A-extension property, since the union of an extension chain of elementary maps is elementary. We illustrate the proof that I has the one at a time back and forth property by showing the "back" part. Let f € 1 and b €181 be given. Let ( a , I a < y ) be an enumeration of Dom(f), where y = Dom(f) ; let C, be

Ch. 5,

5 31

EXTENSION OF PARTIAL ISOMORPHISMS

the type over

4= p u { c, 1 a < y }

'b=

317

realized by b in (23, f(u,)),,,:

{~PEForml(L~~(~y~)~(\W~f(a~))~
Since f ~ 1 (%,a,),.,, , E (%,f(a,)),<,, whence Z, is compatible with (%,u,)a
2 , in

therefore,

('%U,a,>a),<,= < ~ ? f ( ~ , ) , b ) , < , .

so that the map g = f u { ( u , b ) } is an extension off which belongs to I, and such that b E Range( g). Thus, we have proved that 1' 1 ~pX'"23.Statements ( 2 ) , (3) follow from Lemma 5.3.4 and Theorem 5.3.2, respectively. B. Partial isomorphisms and infinitary languages

To illustrate the connection between infinitary languages and the back and forth method-a connection that will be further studied in subsection C-we begin by proving the equivalence between E: and =A, in the simplest case: X=w. This is Karp's theorem. Its proof also gives, in a particularly transparent way, the gist of the main techniques used in this [3]. section. The formulation presented here follows that of BARWISE

THEOREM 5.3.8 (Karp). For any structures \u, '23. the following are equioalent: ( 1 ) '%=,a'H;

(2) %=L'H; (3) there is a set I such that I : \u=z,P'23, whose elements are w-partial isomorphisms. PROOF.Clearly ( 3 ) * (2). (2) + (1). Let I be such that I : 91 =P,'lt. By induction on the structure of = L,,-formulas, we show that if f E I , Fv('p) < n and a,, ...,u,EDom(f), then: (*)

9i + 'p [ a , , .. .,a,,]

.S

'23 + 'p [ f(a

. . . ,f( a,,)].

For 'p atomic this is true because f is a monomorphism. The case of propositional connectives is trivial. So, assume that the equivalence (*) is valid for 'p(o,,. . .,u,,o); we prove it for 3ocp(c,,.. .,u,,,u). The following are equivalent:

318

LANGUAGES WITH INFINITE QUANTIFIERS

for some g €1, f c g and a E Dom( g), IIIt- q [a,, . ..,a,,a] e for some g €1, f c g and a E Dom( g), 23 + c p [ g(a , ) , ...,g(a,), g(a)] for some b~ 1231, ~ t - ~ ’ [ f ( a...,f( , ) , a,),b]

*

-

[Ch. 5 , 8 3

23 + (3ucp)[f(a,),. ..,f(a,>l. The second and fourth equivalences follow respectively from the forth and back properties of I; the middle equivalence holds by induction hypothesis. Thus, (*) is proved; (1) follows instantly. (1) * (3). For notational simplicity assume that p, the similarity type of IIIand 23, has no function or individual constant symbols; thus, “a-partial isomorphism” just means “partial isomorphism with finite domain”. The proof of (2) j (1) tells us that I should be the set of all L,,-maps defined on a finite subset of 131, with values in 1231; that is, f E I just in case Dom(f> is finite and for all rp E Form,(L,,( p)) and all a,, .. .,a,, E Dom(f): t-’p[a,,...,a

,I * B b t f ( a , ) , .. .f(a,)l;

or, equivalently: (IZI,X)xEDorn(fi

-m~(23j.f(X))xEDom(f)‘

Since %)1~,23, then OEI; hence I f B . We show that I has the “forth” property. Let f E I and aElIIIl. Let Dom(f)={a, ,...,a,-,}; obviously we can assume that a # a,, i = 1,. . .,n - 1. We wish to find a b E 1231 such that for every cp E Form,,(L,,( p)):

(**I

% + q J [ a ,..,a,,. ,,a1

-

8t-.cp[f(a,),. ..,f(a,- ,),bl.

Suppose there is no such 6; then for every bEI%l there is a formula such that

,,

clearly, IIIk $ [ a,, . . .,a, - a ] , so

(p6

Ch. 5, $ 31

319

EXTENSION OF PARTIAL ISOMORPHISMS

and

B

‘

(‘On)

14[ f ( 0 1 >,. ,f(an- 1)I* * *

This contradicts the definition off. Hence, there is a bE1Bl such that (**) holds. Condition (**) implies:

..

( a , ~ l *,an.

I,a>-rnu(BJlal), ...,f(an- l),b)y

whence the map g =f u { ( a ,b ) } is in I. The proof of the “back” property is similar. Therefore I : 8 %B is proved.

and (3)

A slight modification of this proof yields the following

THEOREM 5.3.9. For any two structures 8, B and any embedding h : 8-23 the following are equivalent: (1) h is an L,,-embedding: 8 5 LuB;

( 2 ) there is a set I such that I : 8 =zB and for every n Eo and every a ,,...,anE181 thereisan f E I s u c h that f(ai)=h(a,),f o r i = l , ..., n. In particular, if 8 G B and h is the identity map, we get a criterion for We leave the proof of Theorem 5.3.9 as an exercise. There is a refinement of Karp’s theorem suggested by a very natural logical notion in the language L,,: the relation. &, of logical equivalence of sentences of quantifier-rank ’< p ( p EON); 8&,B holds just in case for all cpESent(L,,) such that qr(cp)< p, 8t-cp a Bkrp. The back and forth characterization of rb, is as follows:

< ,,B.

THEOREM 5.3.10. For any two structures 8 , B, and any ordinal p, the following are equivalent: (1) z=&B; (2) there is a sequence =(I, I a < p ) with the following properties: (a) each I,, a < p, is (I non-empty set of partial isomorphisms from 8 into 23; (b) I, GI, for y < a < p; (c) One at a time the back and forth property). If a 1 < p, then there is a g €1, such that f G g (i) for every f E I,, and a E and a E Dom( g ) ; (ii) for every f EI, and b E 1231 there is a g E I, such that f g and b E Range( g ) ;

+

+

c

320

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 3

(3) there is a sequence as in (2), with the additional property that all members of U, are a-partial isomorphisms.

NOTATION. If has the preceding properties we shall write 4 : 3 ~2p.pB. The symbols 9 , $, etc., with or without indices, will stand for sequences of sets of partial isomorphisms. PROOF(In sketch). We just indicate the modifications to be made in the proof of Theorem 5.3.8. (2) * (1). One needs to prove that if a < ,8, fEI,, qr(cp)< a and a , , .. .,a,, E Dom(f),

This induction is a routine variant of the corresponding one in Theorem 5.3.8. (1) 3 (3). For a < ,8 define I, to be the set of all partial maps f defined on a finitely generated substructure of 2l such that (*) holds for all 9’s of quantifier-rank < a . Then (a) is obvious and (b) is easily checked; in proving property ( 2 . ~ 4 ,say, choose the formulas cpb to be of quantifierrank < a ; then qr(+) < a and qr(3u,,+) < a + 1; then use the fact that fEI,,,.

<cu

Clearly, a characterization of can be obtained from Theorem 5.3.10 in the same way that Theorem 5.3.9 was obtained from the proof of Karp’s theorem. Notice also that “21=au%” is equivalent to “\zI-~uBfor all ,8 EON”; hence, clause (2) in each of Theorems 5.3.8 and 5.3.10 are related in a similar way; indeed, this can be proved directly (see Lemma 5.3.26). A back and forth argument involving a sequence of sets of partial isomorphisms and, indeed, a proof similar to that of Theorem 5.3.10, were already used by Ehrenfeucht and Frai’sse in their characterization of elementary equivalence. The only difference rests on the following observation, the verification of which is left as an exercise (cf. Exercise 5.3.1).

FACT.Let p be afinite similarity type; for each n e w and each finite set X of variables there is a finite number of logically inequivalent, finitary formulas cp of type p such that qr(rp) < n and Fv(cp) 2 X . More precisely, if for Luu(p) formulas cp, such that Fv(cp), Fv(+) C X , we define:

+

Ch. 5, 8 31

321

EXTENSION OF PARTIAL ISOMORPHISMS

then there are finitely many equivalence classes cp/= that qr(tp) < n. Using this fact we shall prove:

of formulas cp such

THEOREM 5.3.1 1 (Fralsse-Ehrenfeucht). Let 91, 2? be structures of finite similarity type. Then the following are equivalent : (1) %=B; ( 2 ) there is a sequence Pi = (I, I n E w ) of sets of partial isomorphisms from 91 into 23 such that for every m E w : (I, I n < m ) : Iu ~ ; , ~ 2 3 ;

-

(3) there is a sequence as in ( 2 ) with the additional property that all its members are w-partial isomorphisms.

PROOF(In sketch). (3) (2) is obvious, and (2) * (1) is proved exactly as in Theorem 5.3.8; this part works also for similarity types of infinite cardinality. (1) * (3). Let p be the similarity type of 2 and B. For n E w define I, in exactly the same way I, was defined for a < in the proof of Theorem 5.3.10. We prove that each (I, In < m ) has the “back” property. Let f E I , + I ( n + I < m ) and bEl231. Let Dom(f)={a,, ...,a,-,}. We wish to find an a E 181 such that for all cp EForm,(L,,( p ) ) , qr(cp) < n, (*)

% q[

.. . a, - I’a] ?

-‘

?8 T [ f

*

. .,f(ak - I), 61‘

Suppose there is no such a ; then, for each U E I % / there is a formula cpa such that qr(pu) < n , Fv(qa) L { c , , .. .,u k } , 91 k q a [ a , ,....a,- l , a ] and 23 I= i r p u [f ( a l ) ,. . .,f (ak- 61. Let #,,. ..,#s be representatives of the finitely many equivalence classes of formulas with free variables in { v l , . ..,u k } and quantifier-rank < n ; these exist by virtue of the fact stated above. Let j , , . . .,j, E { 1 , . . .,s} be the indices corresponding to representatives of the classes cpu/ =, for a E /%I; i.e., there is an r, 1 < r < t , such that tpa = G,,; (i) for every a E (ii) to each r such that 1 < r < 1, there corresponds an a E 121 such that q a = 4j; Let

Since each qJ, has quantifier-rank < n , so does 4. Moreover, by (i) and holds for all a E l % l , it is clear that the fact that Iuk-cpa[a1,...,ak-,,a]

322

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5 ,

53

# [ a i , ...,ak- ,,a] also holds for all a E 11211, whence

sincefEI,+, and qr(Vck#) < n + 1, it follows that

hence, there is some s, 1 Q s Q t , such that

using (ii) we obtain some a, E 1 2 1 such that quo=#js; whence

contradicting the choice of quo. Hence, there is an a E 11211 such that (*) holds; since qr(q)= qr( i q ) , this immediately implies the equivalence in (*). This proves that the map g = f u {(a,b)} (or its extension to the substructure of 121 generated by Dom(f)u { a } ) is in I,,. Since f c g and b ERange(g), the proof of the “back” property is complete. The “forth” property is similarly proved. It is important to notice that the restriction to finite similarity types is essential:

EXAMPLE5.3.12. Let p consist of denumerably many unary predicate symbols P , , ( n ~ w )Let . I % l = w , P:=[n,o), I % ~ = w + 1, P : = [ n , a ] , for each n Eo. It is clear that there is no partial isomorphism f from 121 into 23 such that w E Range(f); for if f ( a )= o and a E 1 21,pick an n > a ; so 2l F i P , , [ u ] ,but 23 P P,,[a]. However, % ~ 2 3 indeed, ; Th(%) admits elimination of quantifiers, i.e., every formula is equivalent to a quantifier-free formula; from this and the obvious fact that % 23, it follows immediately that 121 < %; in particular,

%=a.

c

Owing to the fact that in section 4 we shall use this example again,’ we indicate with some detail the steps involved in the elimination of quantifiers.

‘ Strictly speaking, a simple modification of it.

Ch. 5 , $ 31

323

EXTENSION OF PARTIAL ISOMORPHISMS

Note first that we must have two logical symbols T, F denoting truth and falsity, in order that our assertion be true for closed formulas. Our second observation is that the problem of proving that an arbitrary formula rp is equivalent to a quantifier-free formula $ such that Fv($) CFv(rp), reduces to proving that: (*)

every formula of the form 3u@, where @ is a conjunction of atomic and negated atomic formulas containing the variable u, is equivalent to a quantifier-free formula.

[I], Chapter The details of this reduction can be seen in KREISEL-KRIVINE

4.

Now since an atomic or negated atomic formula containing the variable u is of one of the forms P,(u), l P m ( u ) ,x x u , 1( y x u ) , a formula 3u@ as in (*) has the general form:

(**I

12

11

14

13

A P , < U >A A1 lPm,A kA= l ( x k x o ) r=l J=

A

A (Y/*u)]> I= I

where n , , .. ., n i l , m,, . ..,m12Ew,and one or more of the conjuncts may not appear (in which case we put t, = 0, for the corresponding i). Moreover, it is obvious that we can restrict ourselves to the case in which all the variables x k ,y I are distinct from u. We remark the following simple facts: (a) if m < n, then Th(%)t=Vu[P,(u)+Pm(u)]; (b) if n l , ..., nk E w and n=max{n, I < i < k } , m=min{n, I I < i < k } , then (1) Th(Wt=~ ~ [ ~ , ( ~ ) - / " \ ; k = ~ ~ , , ( ~ > l , (ii) Th(cZI)+ V v [ i P , ( u ) - A ~ = = ,1Pfl,(u)I. It follows that in (**) we can reduce each of the first two conjuncts to a single member, whenever the corresponding t, is #O; in other words, we can assume t , , t , E ( 0 , l}. Likewise, if t,#O, we can similarly reduce the third conjunct; for example, if t3 = 2, by transitivity of = we can write this , ; the fact that 3 u ( r p ~ $ ( u ) ) term in the form x , x u ~ x , ~ x using - r p ~ 3 u ( $ ( u ) ) whenever u does not occur free in rp, we can bring the conjunct x 1 x x 2 outside the formula. Observe that no reduction of this kind applies to the last conjunct. Summarizing, (**) can be written in the form:

I

[I

(***>

12

13

14

A P,(u) A A1 l P m ( 0 ) A kA= 1( X x u ) A IA (YI*')], = 1

r=l

J=

where t, = O means that the corresponding conjunct does not appear and t , , E ( 0 , I}.

f*, I3

324

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 3

To reduce (***) to an open formula we analyze several cases: I. t,#O. Then (***) is obviously equivalent to: 12

‘I

14

A1 1 p r n ( x )/A ~ Y,*~. = I

AI

;=

I=

Henceforth we assume t3 = 0. We have the following cases: By statement (a) above, (***) is always false if m < n ; obviously it is always true if n < m ; all of this when t , , t 2 > 0 . If t 2 = 0 , (***) is true; if t , =0, (***) is true if m > 1 and false otherwise. Henceforth we assume t,>O and continue our analysis as follows: 111. t , , t,#O. If n > m , (***) is false. So suppose n < m; we observe that in this case (***) asserts that the elements y , , . . . , y f ,do not “fill up” the interval [n,m- 11; hence: if n + t , < m - l , (***) is true; if n + t , > rn- 1, (***) is equivalent to: 11. t,=O.

m-l

f4

7(‘r(V/) V IA =I

r=n

A

l‘r+

I(Y~)).

Thus, we can assume that one or both of t , , t2 is 0. IV. If t 2 = 0 , then in either one of the cases t , = O or t , >0, (***) is true. Thus, we arrive at our last case: V. t,#O, t,=O. Here, as in case 111, (***) is asserting that y , , . . . , y f 4do not cover the interval [O,m- I]. Therefore, if m > t , (***) is true, and if m < t , it is equivalent to m-l

14

V A (pr(Y/>+pr+l(u)). r=O / = I

Thus we have completed the analysis of all cases, and proved the assertion of Example 5.3.12. W

REMARK.Consider the relation between similar structures obtained by replacing in condition (2.c) of Theorem 5.3.10 the one at a time back and forth property by the corresponding finitely many at a time version (where any function and any finite number of elements can be simultaneously extended at each step). Call =-Wp,p the relation thus obtained. I t is easy to prove either by elimination of quantifiers in a manner similar to that of Example 5.3.12, or directly, by using Theorem 5.3.1 1, that (a,<) = ( w + ( a * + w ) , <). However, for no m > 2 does the relation (w. <)=-Wp, ( w + (a*+ a),< ) hold. This shows that-unlike -the relation 8 =;,bB does not necessarily imply ‘u =Wp,p‘H.

Ch. 5,

31

EXTENSION OF PARTIAL ISOMORPHISMS

325

C. The back and forth argument in a general setting We move now into a higher level of abstraction and consider various generalizations of the concepts and techniques employed so far. The gist of these generalizations is as follows: instead of the relation =,A-which expresses a model-theoretic property defined in terms of the class of all L,,-formulas-we will consider more general classes of L,Aformulas, and will try to obtain a “back and forth” type characterization of the model-theoretic relations naturally defined by them. Of course, the classes under consideration will have to satisfy some reasonable restrictions, if the method of extension of partial maps is to carry over to them. In this way we will obtain a uniform treatment of the back and forth argument which applies to the most outstanding classes of L,,-formulas: universal, existential, positive, etc. Under appropriate assumptions, the model-theoretic relations between structures induced by these classes of formulas, relate to the algebraic notions of extension, embedding, homomorphic image, etc., in the same way as L,,-equivalence related to isomorphism in the case treated in subsection B. A word of caution is in order at this point. The results of this paragraph -as well as those of the preceding one from Theorem 5.3.10 on-are sensitive to the way the notion of quantifier rank is formulated. As we know (see remark after Definition 1.3.12), there are two convenient ways of defining the quantifier rank in L,, and its sublanguages, depending on whether the condition qr((3X)cp)=qr(rp)+ 1

(where X CVar, ? < u )

x=

is accompanied or not by the proviso I . Let us call these two notions restricted and wide, respectively. The restricted definition of quantifier rank corresponds to a one at a time back and forth property, while the back and forth argument corresponding to the wide definition takes thefinitely many at a time form. The results from Theorem 5.3.10 through the end of $4 hold under either of these definitions with two important exceptions: (i) the EhrenfeuchtFrai’sse Theorem 5.3.1 1 only works under the restricted definition (as shown in the final remark of paragraph B), and (ii) Theorem 5.4.21 below and its corollaries are valid only under the wider definition of quantifier rank. We begin by introducing some notation and setting forth the basic requirements that our classes of formulas shall obey; the definition of normal class of formulas is due to BARWISEAND EKLOF[I].

326

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 3 3

NOTATION.5.3.13. Let A be an infinite cardinal and @ be a class' of L,,-formulas of a fixed, but arbitrary, similarity type. For each ordinal a, we put: qr(cp) < a},

Qa= Q,

= (9, E @ I qr(cp) = a},

@ [ I ] ={cpE@plcpbegins with 3},

I

@[V]= { cp E @ cp begins with V } .

For the technical reasons explained above we also introduce the following convention:

A*={

2

if A = w and the restricted definition of quantifier rank is used

X

otherwise.

DEFINITION 5.3.14. A class @cForm(L,,) following requirements2:

is normal iff it satisfies the

(Nl) ( u ~ ~ u ~ ) E @ ; (N2) cpE@ c p * ~ @ ; (N3)cpEQ + Sbf(cp)c@; (N4) for every set q c @, f,'P and v'k are both in @; (N5) if cp E @ and rp' results from cp by replacing some occurrences of a free variable of cp by a term, then cp'E@; (N6) for every a EQN, if Qa+,[3]#.0(resp. @,+,[V]#fl), then for every and every X LVar, O);

' Recall this means that CP, as a set-theoretic object, is defined by a formula of ZF. ' Throughout this section rp* will denote a reduced form of rp; see Definition 1.3.5 and Lemma 1.3.6.

Ch. 5, 0 31

EXTENSION OF PARTIAL ISOMORPHISMS

(4) the class Exmxof all existential L,,-formulas

321

(here CPa[V]=B for all

a);

(5) the class Unmhof all universal formulas; ( 6 ) the class POS,, of all positive formulas; As counterexamples consider the following: (7) the class Pren,, of all prenex formulas ((N4) fails); (8) the class of all negative L,,-formulas (i.e., negations of positive formulas; here (Nl), W3), fN6) fail); (9) the set Form(L,.), where K ECN ((N4) fails). REMARK5.3.16. By (N2) it is possible to prove properties of formulas in a normal class CP by induction on reduced formulas. More precisely: Let P be a property of L,,-formulas, invariant under logical equivalence.' If (i) every atomic and negated atomic formula in CP has P; (ii) if 'p E CP, 'p = V 9 or 'p = A9, and every formula in 9 has P, then 'p has P; (iii) if 'p E CP, 'p = ( 3 X ) # or 'p = ( V X ) # , and # has P, then so does 'p; then every formula in @ has P. Indeed, since the class of reduced formulas is the smallest class satisfying conditions (1)-(3) of Definition 1.3.5, from (i)-(iii) it follows that every reduced formula in CP has P. Now, if 'p is an arbitrary formula in a, by (N2), ' ~ * E C P ; since 'p* is reduced, 'p* has P ; since P is invariant under logical equivalence, k'p++'p* and Fv('p)=Fv('p*), we conclude that 'p also has P. All properties of formulas considered in this section are invariant under logical equivalence. The following lemma (cf. BARWISE-EKLOF [l], pp. 29-30) gives the basic information that we will need about normal classes: LEMMA5.3.17. Let CP be a normal class of Lmx(p)-formulas, %, Y3 be structures of type p, a n d f : [Xl-+IBlbe a CPO-map., (a) CP0=@' is the class of quantifier-free formulas in CP; (b) CPo contains all atomic formulas of the form t,%ttz, where t , , t, are terms of p ; (c) i f 'Pa=B , then QD=B for all p > a, and CP = Ur
' This means that whenever Cf. Definition 1.3.17.

'p

has P , Fv(q)=Fv($) and P'pc*$, then Ji has P.

328

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch.5, J 3

(f) for any a, @* is normal; (8) B , = B t R a n g e ( f) is a substructure of 8 ; (h)f is a @-map from % onto B,.

PROOF.The proofs of (a)-(f) are very simple exercises that we leave for the reader. Notice that (a) and (d) do not use normality of @; (b) uses (N1) and (N5)'; (c) and (e) are proved by induction. To prove (g), use the fact that (by (b)) uowg(ul, .... u,) and u,=c are in a, for every individual constant c and function symbol g in p: therefore, they are preserved by f, which implies that Range(f) contains c' and is closed under the operation g'. (h) By induction on (reduced) formulas, we prove that for arbitrary q E @ and gEI'UIFv('):

when cp is atomic or negated atomic, (*) holds by the assumption that f is a @,-map; the other cases are simple and left as an exercise for the reader. We proceed now to define some model-theoretic relations induced by a normal class of formulas, which we shall use instead of

The relations equivalence

and

= o o ~ =$A

have the peculiar property that one half of the

implies the other half. Therefore, when generalizing to (usually) asymmetric normal classes, it is natural to demand only one side of the equivalence.

DEFINITION 5.3.18. Let @ be a normal class of Lmh(p)-formulas and 8,'$3 be structures of type p. (1) The relation %(@)B holds just in case @nTh"X('U)cTh"h('$3); in

'

The conjunction of (N I ) and (N5) is actually equivalent to the conjunction of the following two clauses: (NI') o o ~ w , E @ ; (N5') if cp € @ and cp' results from cp by substituting all occurrences of some free variable of cp by a term, then c p ' ~ Q .

Ch. 5, 8 31

EXTENSION OF PARTIAL ISOMORPHISMS

other words, iff for every

'p E @

n Sent(L,,( p ) ) ,

%F'p

(2) For a cardinal

K

2 A,

329

=3

Bk'p.

%(@),B holds iff

@n ThKA(%) cTh"'(B). REMARK 5.3.19. (1) It should be recalled that the relation % <&8 (some@%, see p. 94) is defined to mean that I 9 l C 1B3( times also denoted by % i and for all 'pE@and all gEIXIFV(v),

-

%t=cpg,[gl ' H F d g l .

The relation % 5 $8,f is a @-morphism from 91 into %, is similarly defined (see Definition 1.3.17). (2) The relation "(@)" just defined is reflexive and transitive, but not necessarily symmetric or antisymmetric. (3) When both %(@)B and B(@)Xhold, it is customary to write %=@B; however, we shall hardly have occasion to use this relation. (4) Let p EON; by Lemma 5.3.17(f), @ p is a normal class; therefore the is clear from Definition meaning of the relations %(@p)Band %(@p)KB 5.3.18. Since every formula of L,, has quantifier-rank < K ( < K , if K is regular), for p 2 K , B(@p),B w %(@),B. It is also clear that %(W)B * but the reverse implication need not hold. Having generalized the logical relation =,A, we need now to introduce the appropriate analog of the relations =: and =:,p of partial isomorphism considered in paragraphs (A) and (B). The necessary definition is:

5.3.20. Let @, p, 8 , 8 have the same meaning as in 5.3.18. DEFINITION ( I ) The relation % =&B holds just in case there is a sequence of sets of maps, 9 = (Iz I 5 < p ) with the following properties. (i) each f € Ugs is a (partial) @,-map defined on a substructure of %, with values in IBjl; this is, for every quantifier-free 'pE@and every g E Dom(f)FV(v):

(ii) each I,, (< p, is non-empty; (iii)q<E< /3 3 I t ~ I v ; (iv) (forth property) if 5+ 1 < /3 and @,+,[3]#D, then for every

330

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 0 3

f € I t + I and every A c [%I, 7
-

I:%=p3. We proceed now to prove the analogs of Karp's theorems; we begin with the analog of Theorem 5.3.10. THEOREM 5.3.21. For any normal class @ of Lmh(p)-formulas, structures 8 , 23 of type p and ordinal p, the following are equivalent: (1)

WWB;

(2) 8=&23; (3) 4 : %=&23 for some 4 =(Is I [ < p ) with the additional property that PI, is a A-partial @,-mao (i.e., Dom( f ) is generated by a set of each f E UC< power less than A).

-

-

PROOF.Obviously (3) (2). (1). As in 5.3.10 we prove by induction on reduced formulas (cf. (2) Remark 5.3.16) that if q E @ , qr(cp)=[< p, f € I t and gEDom(f)Fv(vf, then: (*>

%t=v,[gI3 Bt=cp[fogl.

If 9, is quantifier-free, then cp E @;, so (*) follows immediately from the fact that f is a @,-map. The case where v = A'k or cp = V q is trivial (use (N3)). So, assume cp=(3X)# and Wcp[ g]. By (N3), #E@ and (= { + 1, where qr(\C.)={; therefore @[+ l[3]#B. By the second assumption there is an a E J81xsuch that at=#[ g*a]; since F
Ch. 5, 5 31

EXTENSION OF PARTIAL ISOMORPHlSMS

33 I

tion 5.3.20(iv) there is an f’ €IS such that Range(a) cDom(f’) and fcf’, whence g * a EDom(f’)FV(4). By induction hypothesis, BP$[f’ 0 ( g * a ) ] ; since

f’

O

( g * a )=

(f’ g)*(f’ O

O

a ) = (fog)*(f’ O a ) ,

it follows that B F (3X)$[fog], i.e., Bkcp[fog]. The case cp = ( V X ) $ is treated similarly, using condition (v) of Definition 5.3.20. (1) 3 (3). The preceding part of the proof together with (3) dictate what the 1,‘s ought to be:

f €1, iff f is a partial @‘-map such that Dom(f) is a substructure of Ill generated by
Let 4 =(I,/,$< /3). It remains to prove that 4 :%=&B. One by one we verify the conditions (i)-(v) of Definition 5.3.20. (i) Since iP,C@ for each [, every f € UCapI, is a @,,-map. For the same reason we have: (ii) q < , $ < p * iPvc@** I ~ G I ~ . (iii) Since YI(@%)B,the (unique) partial map 0 defined on the substructure of Ill generated by the empty set is well-defined and is a @-map; hence, it is in each I,, ,$
there is a b:X+\BI such that for allcpE\k<,Bt=rp[b*(foc)].

Assume (**) is false; then for each bEIBIX we can choose a formula cpb E \k‘ such that Bl- icpb[b*(foc)].Let

by (N4) it is clear that $E\k‘; since iPt+,[3]#B,by (N6), ( 3 X ) $ E @ r + ’ ; moreover, we have:

Ill+ (3X)$[c] and Bk (VX>-~#[foc],

332

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 5 3

which contradicts the assumption that f is a @(+'-map. Hence (**) holds. Pick b :X+IBl as in (**); let g ' : A u C + R a n g e ( b ) ~ f [ C ] be defined as follows: g'r C=f,

g'rA=boa-'.

Since A n C =B , g' is well defined. g' extends in the obvious way to a map g defined on the substructure of 9l generated by A u C:

(***I for every term 7 of p and every z ',. ..,z, E A u C. Obviously we have g 2 g'>f, A Dom( g ) and Dom( g) is a substructure of 12[ generated by < A elements. To complete the proof of (iv) it only remains to show that g is a @-map. Let cpE@ and hEDom(g)FV(q)be such that 9lFcp[h];we wish to prove that Bkcp[ g o h ] . Since Dom(g) is generated by A u C, each element h ( u ) in Range(h) can be expressed as the image of some finite sequence z l ,...,zg in A u C by some term 7" of p : h ( u ) = T,"[z,,...,z,] Denote by

7; the

term

7"

for t, E Fv(cp).

with the following variable in the ith place:

if zi E A the variable a - ' ( z , ) ; if zi E C - A , the variable c - '( t i ) ; let cp' be the result of substituting each uEFv(cp) for 7; in rp. Clearly qr(cp) = qr(cp') and, by (N5), cp' €@; hence cp' E cP4; moreover,

9l -F cp" a* c], whence cp'E\kl. By (**) we conclude

(****)

23 P cp" b*(f

0

c)].

Recalling the definition of cp' and performing in (****) the changes indicated by the equality (***), we immediately conclude

B-Fdg oh].

Ch. 5,

8 31

EXTENSION OF PARTIAL ISOMORPHISMS

333

This completes the proof of (iv). The proof of condition (v) of Definition 5.3.20 is dual to the preceding one, and is left as an exercise for the reader. H The preceding argument also proves, mutatis mutandis, the analog of Karp's Theorem 5.3.8. COROLLARY 5.3.22. For any normal class @ and any structures 'II, 8,the following are equivalent: (1) 'u(@)B; (2) ' u = p ; (3) there exists a set I of @.,-maps such that I : 'u =%B and the domain of every member of I is generated by less than A elements. There is, of course, an analog of Theorem 5.3.9 for the relations i { and

i . the, proof of which can easily be supplied by the reader.

THEOREM 5.3.23. For any normal class @, any ordinai /3, and any two structures 'u, B such that I'ul 1231, the following are equivalent: (1) 'u +23; (2) there is a sequence 4 = ( I t I E < /3) such that 4 : III =&B and for every set A I'ul of power < A and every 5 < /3 there is an f € I t such that ACDom(f)andftA=ida. The following are also equivalent: (3) 'u <.B; (4) there is a non-empty set I of partial maps such that I : 'u =gB and for every set A /!? of power I/
c

c

~ B "for " every We turn now to prove that the relations " ~ ~ and coincide for normal families @ satisfying an additional closure condition which depends on the cardinalities of 'u, 23 and A.

/3 EON, IZI =$,;Y'

DEFINITION 5.3.24. Let @ be an arbitrary class of LWA-formulas'and let a be an ordinal >O. @ is a-quantifier-closed if and only if: whenever there is a Z such that Z+ 1 < (Y and @<+ ,f3]#kl (resp. at+,[V]#kJ)), then @<+ ,[3]#kJ (resp. @<+ ,[V]#kJ) for arbitrarily large 5 below a .

'

need not be normal.

334

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 0 3

REMARKS. (1)If CP satisfies condition (N6) of the definition of normality, then @ is a-quantifier-closed iff whenever CP contains some formula of rank < a beginning with one of the quantifiers 3,V, it also contains all possible quantifications of the same type, of formulas in (D of rank < a . (2) The class of all quantifier-free L,,-formulas is a-quantifier-closed for all a >O; given an infinite cardinal K the class of existential, positive and universal L,,-formulas are examples of K-quantifier-closed classes (which are not normal); given an ordinal /3 >0, the classes of existential, positive and universal L,,-formulas with quantifier-rank < p are examples of j3-quantifier-closed classes. LEMMA 5.3.25. Let CP be a normal class of L,,-formulas and let a be a limit ordinal > 0. Then the following are equivalent: (a) CP is a-quantifier-closed; (b) if @ , + l [ 3 ] # 0 (resp. CP'5+I[V]#0)for some ( E O N (not necessarily < a ! ) , then @l+l[3]#0 (resp. CPl+,[V]#O) for all l < a .

-

PROOF.(b) (a) is obvious. (a) * (b). First we establish the followingi (*)

77, Y E O N A V GY A @ ~ + I [ ~ I*# ~@ q + ~ [ 3 1 # 0 .

By (c) of Lemma 5.3.17, CPq #0 ; let rp ECPq and let X CVar, 0 < < A (if Fv(rp)#0, take 0 c X C Fv(v)); by (N6), (3X)rp E CP; since qr((3X)rp) = q + 1, we conclude that CPq+ J3]#0. Assume that @+, J 3 ] # 0 , where ( need not be < a. Choose an 77 < ( such that q < a . By (*), CP,,+,[3]#0.Since CP is a-quantifier-closed it follows that @[+,[3]#0 for arbitrary large { < a . Using (*) again we conclude that CPl+J3]#0 for all { < a . The proof for the universal quantifier is similar. LEMMA 5.3.26. Let Q, be a normal class of L,,-formulas and ler %, 8 be =
-

PROOF.(1) (2). This implication is immediate and does not use the closure assumption; set I,= I for all (EON, where I : 3 and verify , every p EON. that (I, 1 ( G j3) : % E $ , ~ Bfor (2) * (1). First observe that 5.3.22 (3) implies, for each B EON, the existence of sequences = (It I ( < 0 ) such that 4 : % =&$, and every map belonging to each I, is a A-partial map, i.e., its domain is generated by < A

Ch. 5,

31

EXTENSION OF PARTIAL ISOMORPHISMS

335

elements; for the rest of this proof we only consider sequences with this additional property. Note, then, that any two monomorphisms with the same domain which coincide in a set of generators of their domain are equal. This shows that

Let y be as in the hypothesis, and consider a sequence 4 = (Ic][< y) of length y 1 with the stated properties. Since 4 is a decreasing sequence (Definition 5.3.20 (1) (iii)), a simple argument of cardinality shows that there is a t 0 < y such that I,=It0 for any [ satisfying y >[> to.Let I=It0. By Definition 5.3.20 (l)(ii), 1 # 0 . Since @ is y-quantifier-closed, Lemma 5.3.25 shows that if @c+1[3]#0(@t+I[V]#0, resp.) for some [EON, then 1 [ 3 ] # 0 (ato+ '[V]#B, resp.). Hence, (iv) and (v) of Definition 5.3.20 (1) immediately imply (i) and (ii) of Definition 5.3.20 (3), respectively, which shows that I : 'B =:8.

+

Counterexample 5.3.40 shows that the closure assumption in the preceding lemma cannot be omitted. Careful examination of the argument proving Theorem 5.3.21 reveals that some extra information concerning the fength of the formulas used in the proof, is missing in the statement (see, especially, the proof of condition (**) in Theorem 5.3.21). In the sequel we will try to use this additional information. Theorem 5.3.27 was first proved by CHANG[2] in the cases where CP is the set of all existential and of all positive L,,-formulas, respectively. The present version is a natural extension of that result to the context of this paragraph.

THEOREM 5.3.27. Let CP be a normal class of L,,-formulas, y#O be afixed cardinal and let' K = y>'; suppose in addition that @ is K-quantifier-closed. Then, for any two similar structures 3,23 of power < y the following are equivalent : (1) 'BI(@)% (2) W @ ) 3 -

PROOF.It suffices to prove (2) * (1). By Corollary 5.3.22 and Definition 5.3.20(3) we need to construct a non-empty family I of partial @.,-maps such that I : % =$%. Owing to the form of the relation (2), the natural

' The operation y > h has been introduced in Definition 0.1.2(l1).

336

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5,

83

candidate is the family of all partial (anForm(L,,))-maps defined on a substructure of 3,with values in 23; in other words, for a map f defined on a substructure of %, let f E I iff x>x,Do~n(@.),(23~f(x))x

EDom(f)’

This choice of I works. I # @ , since the hypothesis %(@)K23 implies that I contains the (unique) map 0 defined on the substructure of % generated by the empty set. The rest of the proof is devoted to showing that I has the < A at a time back and forth property relative to @. This is an immediate consequence of the following:

CLAIM. Let p EON, /?
(11) if @P,+,[V]#@for some (EON, then for every d:P+ISs)l there is a c:P+IEI such that

It is apparent that (I) and (11) of the claim correspond, in a slightly different notation, to the forth and the back properties for I, respectively. For notational simplicity let us assume that the language of 3 does not contain function nor individual constant symbols. Assume that QC+,[3] #B for some [EON, and that a n f € 1 and an A Cl%l, 7
x, ‘a)xEDom(/);

a<

p(a)~(23dx)7

b a ) x E Dorn(f); a < $

then the map g : Dom(f) u A -+I81such that g 1Dom(f) =f and g(a,) = b, for a < /3 is in I, extends f and A Dom( 8). The “back” part is similarly proved using (11).

c

PROOFOF CLAIM. (I) Suppose this is false, i.e., that there is some c:P+IEI such that for all d : p+l%,l there is a formula qdE @ n Form(L,,) such that

Ch. 5 , 8 31

337

EXTENSION OF PARTIAL ISOMORPHISMS

-

by (N4), $E@; since I B l P < y o < y>‘= K, we also have $EForm(L,,). Since @ is normal, K-quantifier-closed and at+,[3]#Hfor some [EON, Lemma 5.3.25 and ( N 6 ) imply that (3u rp)J/E@n Sent(L,,). Obviously @I-( 3 v r f l ) J /since ; a(@),%, it follows that %I- (3urfl)J/, which implies that B I - q d [ d ]for some d € l B I P . This contradicts the definition of the formulas tpd; hence (I) is proved. (11) Assuming that this is false, there is a d:fl+IBl such that for all c :p+IQI there is a formula qcE @ nForm(L,,) such that Fv(rpc)L {u,lS
one proves, exactly as in the preceding case, that I/J E @ n Sent(L,,) and together with the assumption O(@),B, imply that Bk+, whence B k q c [ d ]holds for some ~€161; this contradiction proves (11).

a+$; these facts,

REMARK.Notice that in the foregoing proof (I) is independent of the cardinality of 0.and (11) is independent of the cardinality of %. Moreover, if Q U [ V ] = 0 for all (or, equivalently, for some) a>O, (11) is not used; likewise, when Q U [ 3 ] = Bfor all a >0, (I) is not used. Thus, we obtain the following corollary to the proof of Theorem 5.3.27:

--

COROLLARY 5.3.28. Under the assumptions of Theorem 5.3.27, if

% has arbitrary cardinality, we have: (1) O , [ V ] = B implies %(@)KB %(@)8;

(2) @,[3]=B implies B(@),%

5
and

B(@)%.

Applying Theorem 5.3.27 and Corollary 5.3.28 to various normal, quantifier-closed classes of formulas, we obtain

COROLLARY 5.3.29. Let y be an arbitrary cardinal, X an infinite cardinal and K =y”; assume also that 8, 8 are similar structures and that 5 < y ; then (1) (CHANG[2]) %(Ex,&lB+ %(ExmA)23; (2) W J n K A ) %* WJn,J%; iJ; in addition, < y, then (3) (CHANG[2]) %(Pos,,)B * %(Posm,)B; (4) (CHANG[2]) % EKAB * mx8.

338

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 3

REMARK.In spite of its apparent generality, 5.3.28 does not seem to be much more general than the first two cases of 5.3.29. For example, a normal, K-quantifier-closed class @ with the property that @JY]=a,must already be close to one of the classes EX^, for some p > K , or EX,,; the difference may possibly lie in the quantifier-free formulas. We also mention the following: COROLLARY 5.3.30. Let A be an infinite cardinal, and K > A be strongly inaccessible. If @ is a normal, K-quantifier-closed class of LwAformulas and a, 23 are structures of power < K , then %(@)!I3 and %(@),% are equivalent. PROOF.Let y = max{ % , 23 }; then y < K . Since K is strongly inaccessible, for any cardinal Y
5

The result follows immediately from 5.3.21. and =,A, Owing to the symmetry of the relations improve the last statement of Corollary 5.3.29:

LEMMA 5.3.31 (Kueker). Let y, A and then for arbitrary 2323: %=,A%

K

it is possible to

be as in CoroI!ary 5.3.29; i f

5 < y,

* %Em,X%.

PROOF.It suffices to prove (I) and (11) of Theorem 5.3.27 under the < y alone. We have already observed that when < y the assumption formula constructed in the proof of (11) is in L,,; but this need not be true in the case of (I), since the formula constructed there depends on the cardinality of iD. We modify this part of the proof as follows. Let cEIQ(b; we wish to find a d € [ B I Bsuch that

Ch. 5, 0 31

EXTENSION OF PARTIAL ISOMORPHISMS

339

6 k irpc.[c’]. Let

< y , then where c’ ranges over all sequences satisfying (*). Since 4JEForm(L,,); it is also clear that 6 k ( 3 0 r P ) $ . We observe that (**)

for every 8 E Form(L,)

such that Fv(8) c {us 16 < p },

6t=8[c] implies 6 k (Vurp)[+B].

For if c ’ : P-.l61 is such that Bkrl/[c’], then (O,C~)~,,,~,A(&,C~)~<,, (otherwise (*) would hold and rpc, would be one of the conjuncts of 4, which implies that both 6t=rpc.[c’] and Bk lrpc.[c’] hold); in particular, 6 k 8 [ c ] implies 6t=O[c’]. From these facts and 6 ~ we ~ infer: ~ 9 (0 B k ( 3 V r P ) 4 J 9 (ii) if 8 E Form(L,A), Fv(8) { u, 16 < ,f3 } and 6 k 8 [ c ] , then we have B +(VU rP”+el. Using (i), choose a d:P+IBl such that Bk+[d]; from (ii) it follows that

(6, ca>a

=KA(’,

da)a
w

Note that Theorem 5.3.27 and all its corollaries and improvements are independent of the number of symbols of the underlying language p. If the cardinality of y is taken into account, some of the preceding results can be made more precise in at least two different directions: (1) for a given structure 9l of type p, the relation %(@)8is characterized -independently of 8-by a single infinitary sentence of L,A(p), where K depends on the cardinalities of 9l and y; indeed, K = max{ ycA, F } +,where % < y. Thus, Theorem 5.3.32 below is a generalization of Scott’s isomorphism theorem for L,,, (cf. p. 135). This result applies to a restricted variety of normal classes @ of formulas; indeed, the only natural example of such @ known at present is the class Form(L,,). Note also that even for y of small cardinality this value of K may be slightly larger than that obtained in Theorem 5.3.27 (see formulas (20)-(23), p. 7). (2) one may relax the size of the bound K , trying instead to obtain one which is independent of the cardinality of the structure % and “3. A result in this direction is mentioned without proof on p. 344; it applies to the class of all L,,-formulas of quantifier-rank < 0, for any fixed ordinal p; the bound K thus obtained depends only on A, /3 and

-

340

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5,

53

THEOREM 5.3.32 (CHANG[2]). Let ip be a normal class of Looh(p)-formulas containing at least one logical& true sentence; for a fixed cardinal y Z O , let K =(max{ y<', p})+;let % be a structure of type p and power < y . Then: (1) there is an LKA(p)-sentence va (not necessarily in i p ! ) such that for any structure of type p: BkqlX

* %I(@)%;

( 2 ) if, in addition, ip is closed under -+ and contains the universal closure of any formula in ipn Form(L,,( p)), then for any structure 23 of type p: %t="Pa e %I(@)%.

PROOF.In this proof we shall use a, a', 6 , b', etc., as (metamathematical) ,<xl%(y. Notice that this set has variables ranging over \%I<'= U power < y<'< K . For any two such a, b with Dom(a)=a, Dom(b)=P, say, a * b denotes the natural concatenation of a and 6 : and

Do m(a* b ) = a + P ,

For each a < A we set X u = { vl < a } , and shall also use the notation Xu * X p with the same meaning as above. Our first aim will be to define, by induction on 5, certain formulas qi, where (< K and a E I%u(<', in such a way that the following holds: (i) q~jE CP n Form(L,,( p)), for [< K and a E J%IJ<'; (ii) if Dom(a)= v, then Fv(rp2) CX,. (If Dom(a')> v, %krpi[a']means, as usual, that q$ is satisfied in % by the assignment vl+ai(l < v).) 4 = 0). If a =k3 (the empty sequence), let (p," be any logically true sentence in ip; we can choose it in ipn Form(L,,( p)) and qr(cp,0)< 1.' If a #a, let q~: be the conjunction of all formulas 8 such that: (iii) 8 is atomic or negated atomic; (iv) Fv(0) C X , (where v = Dom(a)); (v) 8 E i p ; (vi) %t-#[a].

I{

'

For if @=Qo, the assumption that @ contains a logically true sentence implies that p must contain some individual constant c ; by (N1) and (NS),czcECJ. If CJ#@,, Lemma 5.3.17. (c) implies that Q, #,0 ; by (N6) one of 3 u o ( u o ~ u oor) Vuo(uo~uo)is in CJ.

Ch.5, 8 31

EXTENSION OF PARTIAL ISOMORPHISMS

34 1

Since T; < K and each such 0 contains a fixed, finite number of variables chosen in X,, then there are < K formulas satisfying (iii)-(vi), whence cp: E Form(L,,( PI); by (N4), rp: E CP. Assume (p," has been defined for all r] <[ and all aEJ%J<'. ([>O, a limit ordinal). Take rpj=A,pp:. Notice that (i) and (ii) hold for rpf, by the induction hypothesis and (N4). ([= -q 1) Consider the following sets of formulas: i(vii) if CP',+1[3]#,0, the set of all formulas

+

for & < A *

(3Xa)cpJ-u,(Xu- X u )

and a'EI%[I';

if CP,+,[3]=0, the empty set; (viii) if Oq+,[V]#O, the set of all formulas

(VX,)

v

a'EI%l'

rp$-u.(Xu X , ) A

for each a < X * ;

if CP, ,[V] = 0 , the empty set. < K , and the induction hypothesis, each of In virtue of (N4), (N6), the formulas defined by (vii) and (viii) satisfies conditions (i) and (ii). Let rpi be the conjunction of all formulas in the sets defined by (vii) and (viii), where it is understood that no conjunct appears for the empty set of formulas; since I%[<' < K , it is clear that rpi satisfies (i) and (ii). A simple induction on [ < K proves the following additional properties: (ix) if a #0 or 6 > 0, then qr(cpi) = [; qr(rp3 < I ; (x) %t-cpj[a]for all UEI%[I<', [
-

As an illustration we shall prove (xi). Whenever a # 0 or [>O, the implication from right to left follows immediately from (ix) and (i), which tell us that cpf Eat, and (x). So we confine ourselves to prove the reverse implication. ([=O) If a # @ , then the left-hand side asserts that b satisfies (in B) all of the atomic and negated atomic formulas satisfied by a (in %); whence, the result.

342

[Ch.5, 9 3

LANGUAGES WITH INFINITE QUANTIFIERS

(5 a limit ordinal > 0) In this case the result follows immediately from the definition of q$, since (by (N2)) any formula in @ of rank 6 is equivalent to a conjunction or disjunction of formulas of smaller rank. (5= 77 + 1) We have to prove that %+$[a1 * BFrCl[bl, for each J/ E @€. Without loss of generality we can assume that $ has one of the forms ( 3 Y ) O (if @,+1[3]#i3)or ( V Y ) 8 (if @,+1[V]#i3) where 8 EW. Moreover, by renaming bound variables, we can further assume that Y = X , for some 6(a ^ a’)<)l<*+&(”’)
whence

at-(3X&)8[bl as we wished to prove. The case where +b =(V Y ) 8 is similarly treated. We proceed now with the construction of qa. Let us define a map j: IHIJ<*-+K as follows: f(a,a’) =

-

1 $‘

least ordinal 5 < K such that % I i & [ a ’ ]

if one such 6 exists, otherwise.

Since I%[<’ < y<’
(otherwise to+1 would be the smallest ordinal some a, a ’ E ( % ( < Xwhencej(a,a’)=to+ , 1). Now let

5 such that %t-

i ~ : , , E [ a ’for ],

Ch. 5, 8 31

EXTENSION OF PARTIAL ISOMORPHISMS

343

it is clear that qa E Form(L,,( p)). Moreover, if @ satisfies the hypotheses (2), then 9%E @; by (x) and (xiii) we have %I= qn. Therefore, if %(@)B, we conclude that B F qa. This establishes one-half of the conclusion of (2); the other half follows from (1).

PROOFOF (1). Assume BPq,; the desired conclusion %(@)B follows from the statements (1') and (11') below, by exactly the same argument used in Theorem 5.3.27 to derive the conclusion %(@)Bfrom the corresponding statements 5.3.27 (I) and 5.3.27 (11). Let a
(*I

b{>{<"i

( 3 9

then: (1') if @'Co+,[3]#fl, then for all P
A

23 + ( ( 3 X p >v?-d )[ b

17

which informs us that there is a b':P+lBl such.that

whence, by (xi) we immediately derive the conclusion of (1'). This completes the proof of Theorem 5.3.32.

COROLLARY 5.3.33.'Let A be an infinite and y a non-zero cardinal; let p be an arbitrary similarity type and K = (max{ y<', 7 })+. Then for each structure 9l of similarity type p and power < y there is a sentence qa of LK,(p) such that for every 23;

If, for example, we put y = A = w , < w-i.e., if we consider countable structures % in a countable language p-, then rpa is in Lw,w. As we shall see below, this is precisely the Scott sentence of 3 .

344

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, $ 3

REMARK.The ordinal 5, defined in the proof of Theorem 5.3.32 is an important invariant of the structure %, probably containing a wealth of metamathematical information about 3.For more details, cf. BARWISE [3], p. 23. As an example of a different type of extension of Theorem 5.3.27, we mention without proof the following result of BENDA[2], in the proof of which the characterization of elementary equivalence given by Theorem 5.3.1 1 is extended to infinitary languages. THEOREM (BENDA[2]). Let X be an infinite cardinal, /3 EON and p be an F).* Then, for any two structures of arbitrary similarity type; let K = aP(X--' type p, the following are equivalent: (1) %U&AB; (2) %&B. The same equivalence holds if the relations i and < respectively.

LA

fix,

32,

and =,$ are replaced by

This result can probably be extended to other normal classes of infinitary formulas.

D. Logical and algebraic relations between models Having dealt extensively with back and forth characterizations of various logical relations between structures, we now turn our attention to the connection between these and some of the usual algebraic relations between structures, such as embeddability, homomorphic image, etc. The connecting link is the following simple extension of Theorem 5.3.5: THEOREM 5.3.34. Let @ be a normal class of L,,-formulas, where cf(X)=w, and let %, 23 be structures such that % is generated by < A elements and %(@)B. Then: ( 1 ) if @,[3]#B, there is a @-mapf:%+B; (2) $, in addition, @,[V]#B and '23 is also generated by < X elements, then f can be chosen to be surjective. PROOF.By Corollary 5.3.22, a(@)!€!implies 'ZI = % : whence, by Definition 5.3.20(3), there is a non-empty set I of @,-maps from substructures of 9l into 8,such that I : % =gB. Let X = {auI a
'

A - = K if X = K + , and h - = h if X is a limit cardinal (cf. Definition 0.1.2, where the beth numbers a,(K) are also defined).

Ch. 5, 5 31

345

EXTENSION OF PARTIAL ISOMORPHISMS

increasing sequence converging to A. Let foEI. Since @‘,[3]#8, I has the less than A at a time “forth” property relative to Q. If @i[V]=8,use this “forth” part to construct, by induction on n, a sequence

(*I

where f,, ’, =some map g E I such thatf, C g and { a , I a < K,}

C Dom( g).

Put f= UnEJ,.As in Theorem 5.3.5 one verifies without difficulty that Dom(f)=IIuU( and that f is a @,,-map; an easy induction on (reduced) formulas shows, in addition, that: ~ + d g l BPrp[fogl

for each rp E @ and every g E Dom(f)F’@+“.Hence f is a @-map. Thus (1) is proved. In the case of (2), I also has the “back” property. Therefore it is enough to modify the construction of the sequence (*) as follows; choose Y = { b, I a
f, = some map g €1 such that f,- I C g and { a , 1 a < K,-

C Dom( g),

if n is even, n > 0, and f, = some map g E I such that f,- ’, G g and { b, I a < K,-

1c Range( g ) ,

if n is odd. As before one checks that Range(f)= 181, i f f = U,,J,. Hence f is a surjective @.,-map. By Lemma 5.3.17(h), it follows that f is a @-map.

For A=o, a similar theorem appears in BARWISE-EKLOF [l], p. REMARK. 32. Recall (Definition 1.3.19) that if @ is a class of (finitary or infinitary) is a @-map, then formulas of type p a n d f : ~Iu~--+~B~ ( I ) whenever @ contains the set At( p) of all atomic formulas of type p, f is a homomorphism, (2) whenever @ contains all atomic and negated atomic formulas of type p, f is an embedding. Using these well-known facts we obtain: COROLLARY 5.3.35 (CHANG[2]). Let A be an infinite cardinal such that cf(A)=w, and K = A > ~ ; for any two structures Iu, B of power < A , the

346

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, $ 3

following holds :

*

8E'B; (2) %(ExKJ'B * III6 'B; (3) %(UnKA)'B* 'B ,c 8; (4) III(Pos~.)'B + 'B is a homomorphic image of 8. (I)%GKA'B

-

PROOF. Since the negation of a universal sentence is existential and vice-versa, III(ExKA)'B 'B(UnKA)III;hence (2) and (3) are equivalent. Use Corollary 5.3.29 to infer from each of the assumptions the corresponding relation with K replaced by co. Apply Theorem 5.3.34 to each of the relations thus obtained to get the corresponding conclusions. H REMARK. The converse of each of the implications (1)-(4) is true. Putting A = w (whence K = ol)in (1) of the preceding corollary we obtain Chang's Theorem 2.3.12; likewise, in the case h = w , ( 2 t o r , alternatively (3)-is known as the countable embedding theorem (cf. BARWISE [2]). Combining Corollaries 5.3.33 and 5.3.35, in the case where h = w and the language is countable, we obtain Scott's isomorphism theorem 2.3.1 1. E. Counterexamples

The aim of this paragraph is to exhibit examples that show the necessity of the assumptions in some of the foregoing theorems, thus putting a limit on the possibility of further extensions. Our first counterexample is related to [2], deals with (2) and (3) of Corollary 5.3.33. The next, due to BARWISE Corollary 5.3.35; it will also serve as a counterexample to possible extensions of Theorems 5.3.32 and 5.3.26. The restricted notion of quantifier rank is used in Counterexamples 5.3.37 and 5.3.40. COUNTEREXAMPLE 5.3.36. Corollary 5.3.33 and Scott's isomorphism theorem cannot be improved. Let p = { c, I a

for every quantifier-free cp E Sent(L,,,( p ) ) and every 9f such that %t-qthere are countable sets E,, Z , ~ [ w l 1 2 such that: (1) for every a , ,B<wl, ( a , , B } € E , 3 % k c , z c p ; (ii) for every y, & < a , ,{ ~ , & } E Z , * %I= i ( c , = c 8 ) ; (iii) for every structure 'B of type p, if 'Bkc,zc, for every {cr,,B}EE,, and 'Bk 1( c y z c 8 ) for every { y , S } € I , , then BPcp.

Ch. 5, 8 31

(**)

341

EXTENSION OF PARTIAL ISOMORPHISMS

-

Let Y Go,, Y < No. For every rpESent(L,,,(p)) there is a quantifier-free rp* E Sent(Lw,,( p)) such that Fv(rp) = Fv(cp*) and for every 3 of type p where each element is denoted by a c, with a E Y , we have %krp++cp*.

(For rp=3u$(u) set rp*=VaEyrp*(ca).) Now let I2ll=w and every element of 2l be denoted by constants. Let y c w I be so that 7=No and every element of \u is denoted by a c, with ( Y EY . Let 2Ik-q; by (**), %l=rp*. Let X = u(E,.uI,.); then
x

c2

'-

w

As an exercise the reader can extend this example to the case h >a.

COUNTEREXAMPLE 5.3.37. For any linear ordering ( A , <) and countable ordinal P, ( A , <)(ExLw)(w,, E

rol).

PROOF.By Theorem 5.3.21 it suffices to find a sequence $ = ( I E [ ( < P ) of order-preserving maps, defined on finite subsets of A , with values in a,, having the one at a time back and forth property relative to Exmw. To this end we make the following definition. Let v < w , ; a (not necessarily order-preserving!) map f defined on a finite subset of A , with values in w , , is v-wide just in case (i) if a is the first element of Dom(f), thenf(a)> v ; (ii) if a , b E Dom(f) and a < b, then f ( a )+ v < f ( b ) . Thus, the empty map is v-wide for all v < o , , and f is 1-wide iff it is order-preserving and 0 E Range(f). Let us now define a sequence ( v , l a < w l ) of countable ordinals as follows:

va+

1=

v,

+ va,

vs = sup{ vt I [< 6 }

if 6 > 0 is a limit ordinal.

348

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5 , 0 3

For a < o , , we define I, to consist of all v,-wide maps with finite domain. We shall prove that $ = ( I t [ [ < p ) has the properties stated above (the numbers ( i t ( v ) below refer to the various clauses in Definition 5.3.20 (1)). If (Y < .$,then v, < vt; therefore any vt-wide map is also v,-wide, whence ItGI,; thus (iii) holds. By the same argument, each member of Ut
+

if a < a,, if a,-

,)+ v, f(a,-,)+v,

, < a < a, for some k ( k = I , . . . , n - I),

if a,-,
Since f is v,+ ,-wide and v,+ g is the required map.

= v,

+ v,,

it is obvious that g is v,-wide. Thus,

COROLLARY 5.3.38. For any linear ordering ( A ,

<),

( A , <)(EX,~,)(W,, E r w , ) .

PROOF.Any formula in Exuluhas quantifier-rank less than wI. In particular, we obtain (a*,<)(Exulw)(wl,E r w 1 ) I ; since ( w * , <) is not embeddable in any well-ordered set, this counterexample shows the necessity of the cardinality requirement on B in Corollary 5.3.35(2). Chang pointed out that the above also shows that the relation %(Exwlu)%cannot be characterized by a single LuIu-sentence. COUNTEREXAMPLE 5.3.39. There is no Lulu(<)-sentence u such that for any linear ordering ( A , + )

PROOF.Suppose such a sentence u existed. By the preceding corollary, By the downward Lowenheim-Skolem theorem for LwIw

(o,, E rwl)+u.

' Recall that x* denotes the inoerse of the order-type x.

Ch. 5, f 31

EXTENSION OF PARTIAL ISOMORPHISMS

349

(Theorem 4.1.2) there is a countable % ? c ( w , , E r w , ) such that Bt-u, whence (a*,< )(ExUJ%; by Corollary 5.3.35 (2), we would conclude (a*,<) ,C 23, contradicting that 8 is well-ordered. Counterexample 5.3.37 can further be used to get a counterexample showing the necessity of the closure assumption in Lemma 5.3.26.

COUNTEREXAMPLE 5.3.40. There is a class { < ) such that (*) but (**>

(a*,<)=&(aI,

Q,

of L,,-formulas

of type

for all ,8 EON.

E rw,)

not (a*,<)=:(al,

E yw,).

PROOF.Fix an ordinal Po such that 0 < Po < w I and set @ = does not satisfy the closure assumption of Lemma 5.3.26. Indeed, is not y-quantifier-closed for y > Po; in particular, it is not w,-quantifierclosed (as would be required by Lemma 5.3.26 in this case). By Counterexample 5.3.37, (a*,<)(Exk,)(q,

ETq);

in virtue of Theorem 5.3.21, this implies at once that (*) holds for P < Po. If P >Po, fix a sequence (I,I[<, Po) realizing ‘the relation (*) for the ordinal Po; then, the sequence (I5 15 < P). where I,= ID, for Po < 5 < P, realizes the relation (*) for the ordinal p; indeed, this holds in virtue of (Ex?U),+ ,[31=(E&)E+

I[VI =B for 5 > Po,

a fact which makes conditions (iv) and (v) of Definition 5.3.20 (1)

~ucuousIytrue. This establishes (*).

Now we show that (**) holds. Suppose not, and fix a non-empty family I of partial @.,-maps realizing the relation -5. Since it contains all atomic and negated atomic formulas, each member of I is an order-preserving map. Since PO>O, (EX$,),+~[~]#B for some t E O N ; then an easy induction using the “forth” property of I shows the existence of a sequence of members of I such that (0,. . . , - n + 1 } Dom(f,) for all n > 1. I t is clear th a t f = U,J, is an embedding of (a*,<) into (a1.E yo1), a contradiction. This contradiction shows (**), thus establishing that the proposed example has the required properties.

c

350

[Ch. 5, f 3

LANGUAGES WITH INFINITE QUANTIFIERS

Our next counterexample shows that Theorems 5.3.5, 5.3.34 and 5.3.35 fail when the restriction cf(A) = w is removed; more precisely, it shows that these results do.not extend to h = w , , and, in general, to regular uncountable A.

COUNTEREXAMPLE 5.3.41. Two structures 91, 2‘3 of power N, such that ‘13. The first such counterexample seems to have been given by Morley (unpublished; cf. CHANG [ 2 ] ,p. 45); the one given below was discovered by W. W. Tait and is included here with his permission. The structures X, ‘13 are trees; we describe their construction. (I) Construction of 8.By induction on n E w we define first a collection %, = ( A , , <,) of partially ordered sets, as follows: % = mu,Bbut %

A , = { (a,P,Y>10 G a G P < w l AY < G I } , (a,fl,y)

Assuming that



a,, has been

* Y=Y’AP=P’Aa
defined, we put

,

A,+ =A,U A:,

where A;=

{(a,b)laEA,-

U A , and b € A ,

k
I

,

and for a, b € A , : (1)

a<,+lb

* a<,b;

for a E A , and (a’,b’)EA;: (2) a<,+l(a’,b’) for ( a , b ) , (a’,b’)EAA:

(3)

-

a=a’;

(a,b)<,,+I(a‘,b’) e a = a ’ A b
By (1) it is clear that < , + I extends <,, i.e.. 9[,c%,+,. From (2) and (3) i t follows that the elements of A ; occur (in the order < , + I ) after those of A,,; more precisely: (i) if a E A , , then

{b

16 G .+la}= { b El%,] 16 G “ a } ;

if U E A ; , say a = ( x , y ) , x E A , - U k < , A , , y E A O , then {bEl%n+IIIb ~ w ~ o Y ) .

{zEI%,IIZ
Ch. 5, f 31

EXTENSION OF PARTIAL ISOMORPHISMS

35 1

From (i) we infer, by an easy induction on n : (ii) for any n E w and a € A , , { b E I%,lb < ,a} is well-ordered. Figure 1 illustrates the first steps in the process of generation of the %,'s; is obtained by just placing a copy of %, on top of each of those points of a,, having at most one successor.

.. . .

D

vs1 copies

H 1 copies

s 1 copies

. .

0

.. . .

H 1 copies

-

H 1 copies

of YI 0

0

-

-------,

(each stem represents a copy

.. .. . .

O

6 , copies

)

Figure 1.

352

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 4 3

Now put 'II=UnEw%,,. For simplicity we shall write < instead of <' if no confusion arises. From the definition above it is clear that = N, for each n Eo, whence 'ZI = N,. We prove some further properties of the Zn's and 9: (iii) b ~ l ' u/ \l a E l'u,,l /\b < a * b €l\ll,l /\b <,a. Notice that bEI'II,l implies b<,a, because < extends <,. Since bEI9l1, there is a least m ~o such that b E l'IIml; assume, by way of contradiction, that n < m ; hence n < m - 1 and aEI'U,_,I. From b < a it follows that b <,a, whence, by (i), b E ,I, a contradiction. This proves (iii). From (ii) and (iii) we get: (iv) for any n E w and a E A,,,

and this set is well-ordered. Hence 9I is a tree. The order type of the set { b E 1 2 1I b < a } will be called the type of the element a. The order of a , in symbols o(a), will be the least integer n such that a E A , . An element a€lYl[ will be called terminal iff it is a maximal element of the set Ao(a); thus, a is terminal just in case o(a)=O and a = ( a , a , y ) for some a , yO and a = ( x , y ) for s o m e x of of order n - 1 and somey € A , of the form ( a , a , y ) . Notice that: (v) if a is terminal, b < a , b < x and o ( b ) = o ( x ) ,then x < a. We also make the following definition: two elements a, b of 9l are said to be connected just in case there is some c E I%( such that c < a and c < 6. This relation is clearly reflexive and symmetric; (iv) implies that it is transitive, whence an equivalence relation on 91. A set is connected iff any two of its elements are connected; a connected component is a maximal connected subset (or, alternatively, an equivalence class under the relation of connectedness); thus, the connected components form a partition of 91. We observe that: (vi) if X is a connected component of Yl, then there are uniquely determined ordinals P. y < w1 such that

Whenever this condition is satisfied, we say X is a connected component of type ,l3 (or, even, if necessary, that X is the y t h connected component of type

P 1.

We prove now: (vii) any two connected components of the same type are isomorphic.

Ch. 5, 3 31

353

EXTENSION OF PARTIAL ISOMORPHISMS

PROOFOF (vii). Let X, Y be connected components of type ,B, say. By induction on n we construct a chain focf, c ... cf,, c ...of orderpreserving maps such that f,,:X n A,,+ Y n A,, and f,,is onto. For n = 0, we note that for some yo, y I < a ,

If f,,is already defined, we put f,,, =f,,uf,', where f,' : X n A;+ Y n A ; is defined by the condition f,'((a, 6 ) ) = ( f , ' ( a ) ,b )

for ( a . b ) EX n A;.

Clearly Range(&)= Y n A;, whence Range(f,,,,)= Y n A , , + I . Now set f = U n t vf,,; one easily checks that f is an isomorphism between X and Y . The isomorphism that we have just defined will be called the translation of X onto Y and denoted by t x , From (vii) we easily infer (viii) Let X cl%lbe a union of connected components of \u containing at most countably many components of each type.Then % -2I- X.

,

PROOFOF (viii). Let X ! be the yth connected component of type

P ; let

-

By assumption Y p < No. Let g : ol-ml - Y be a bijection The required isomorphism between % and 1' 1 - X is defined by:

if Z is a connected component such that Z n X =if, ifZ=X,PcX. Recall that 9lCI3'11 denotes the cardinal (disjoint) sum of two copies of '2L (cf. Definition 0.3.4). By a trivial modification of the preceding proof it is shown: (ix) 9l@%=%. For bEI%l we define: T,= { x E 1911 I for somey €A,,. ( b , y ) < x } .

354

[Ch. 5, $ 3

LANGUAGES WITH INFINITE QUANTIFIERS

Then (x) Tb=i?l. The proof of (x) is somewhat similar to that of (vii). We first observe that TbC LJk>o(blAk (by (iii)); then by induction on m we define a sequence fo Cf, C . . . (If,c . ..of order-preserving maps such that fm : Tbn A m + O ( b ) + l + i ? l and f = Um,,fmis an isomorphism. Let f o ( ( b , y )= y and

fm + 1 (( x > Y > ) = (

f,(x)?y>

for (x,y )

xEAm+o(b)+l-

Tb,

u+

k4m

y

A 0,

o( b )

It is clear thatf is order-preserving; by induction on k it is easily seen that for each y E A , there is an x E T, n A k + o ( b ) + such that f ( x ) = y ; hence f is onto. The isomorphism defined in this manner will be called canonical and denoted by f,. If b < a we define T,, a = { x E

Ib <x

and there is no z E 1 2 1 such that b < z < a a n d z 6 x ) .

It is clear that T,, a n T6',a =B for 6 , 6' < a, b # 6'. We prove: (xi) (a) If a is terminal and b < a: T b ,a ={ x El'%[I ( b , y ) 6 x for somey E A , such that ( b , y )

a},

whence T,, a C T,. (b) If, in addition, b is non-terminal, then T,. = T,,

PROOFOF (xi). (a) Let x E Tb,a;by (iii), o(x) > o(b). If o(x)= o(b), by (v) we conclude that x < a, contradicting the second clause of the definition of T,,,. So o(x)> o(b), whence x > ( b , y ) for some y € A , ; by the last part of the definition of Tb,ait is clear that ( b , y ) 6 a. This proves the inclusion c . Conversely, assume x > ( b , y ) for some y € A , such that ( b , y ) $ a, and, contrary to the condition x E T,,,, that z 6 x for some z such that b < z < a. By (iii), z and ( b , y ) are comparable. Since z < a and ( b , y ) $ a, the relation ( b , y ) < z is impossible. Thus b < z < ( b , y ) ; by (iii), either o(z) = o ( b ) , whence (by condition (2), p. 350) z = b , a contradiction, or o ( z ) = o(b)+ 1; in this last case, using condition (3), p. 350, we obtain I = ( b , w ) for some w E A , , w < o y ; since z < a and o ( z ) = o ( ( b , y ) ) , using (v)

Ch.5, 5 31

EXTENSION OF PARTIAL ISOMORPHISMS

355

again we conclude ( b , y ) < a, a contradiction. This proves the inclusion 2 . (b) This follows from (a) if we prove that for no y € A o , ( b , y ) < a. Suppose, by way of contradiction, that (b,y,) < a for some yo € A o . Since b is non-terminal and a is, there is some z such that o ( b ) = o ( z ) and b < z < a. By (iii), z and ( b , y o ) are comparable; (b,y,) < z would imply that o ( b )+ 1 < o ( z ) , a contradiction; then z < (b,y,), whence by clause (2) on p. 350, z = b, which also is a contradiction. This proves (b). From (xi, b) and (x) we immediately obtain: (xii) If b < a, a is terminal and b non-terminal, then Tb,u= %. It is natural to ask what isfb[ T,,,] when b < a and both a, b are terminal; the answer is: (xiii) If b < a and both a, b are terminal, then f b [ Tb,u]equals 9l minus exactly one connected component. In particular, using (viii), we immediately conclude that Tb,u= 9l.

PROOFOF (xiii). By the assumptions it is clear that there is a (maximal) totally ordered subset Yo of A , such that for a l l y € A o (*>

( b , y ) < a 's y E Yo.

Let Y be the connected component of % containing Yo; we prove

From (*) and (xi,a) we get

Conversely, if z E I%/ - Y , pick a y E Tb such thatfb(y)= z . We show that y E Tb,u; clearly y 2 ( b , w ) for some w € A o ; if ( b , w ) < a, then w E Yo, whence z = f b ( y )2 f b ( ( b ,w)) = w ; since w E YoC Y and Y is a connected component, z E Y , a contradiction. Then ( b , w ) $ a ; by (xi,a) we get y Tb.0. (xii) and (xiii) together show that Tb,u= %, whenever a is terminal and

356

LANGUAGES WITH INFINITE QUANTIFIERS

b < a . We define:

sa={a}u u

Then (xiv) If a is terminal, then

[Ch. 5 , 3 3

(Tb,au{b})'

b
T,={xE(%Ila<x),

S , = X a - T,,

where X , is the connected component of \zI containing a .

PROOFOF (xiv). The first equality is easily proved. For the second we observe that for any x E X , x@T, e a p x

e~

(foreveryb,b
- bfa)

xES,.

The second equivalence follows from the fact that x and a are connected. The third is proved as follows: if x < a, then x € Sa; if x and a are incomparable, then x E Tc9,,where c is the largest b such that b < a and b < x, whence x E S,; the converse is obvious. Before proceeding with the construction of 23, we remark two simple properties of % whose proof we leave to the reader: (xv) For given a < o l , in any connected component of % there is a terminal element of type a. (xvi) Any linearly ordered subset of 9-l is countable. (11) Construction of 23. 23 is obtained by adding to % a new connected component of type wl. Formally we proceed as follows:

Ch. 5, I 31

EXTENSION OF PARTIAL ISOMORPHISMS

357

The proof that 23 is a tree can easily be derived from (iv); we leave it to the reader. We shall call P the principal component of 23. Since P contains a chain of type ol,we immediately conclude from (xvi): (xvii) % P '23. Define the index of elements x E P as follows: i(x)=

{

x

if x ~ o , ,

a

if x = ( a , a )

for some a ~ l % l .

For a < a l the athapproximation to P is the set: B,= (

x P l ~i ( x ) < a } u { a}.

Obviously, P = U,<,,B,. We prove that (xviii) If a is a terminal element of Z of type y , then S,- By. PROOFOF (xviii). By assumption, { b E 1311 b < u } has type y ; then there is a unique order isomorphism g : { b E \%I I b < a } + y + I . For each b < a choose an isomorphism h, : Tb,,-+3; such an isomorphism exists by (xii) and (xiii). Let:

Clearly f : S, = By. (xix) For every y
358

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 3

(j) Dom(f) is either a countable union of connected components of %, or it is one such countable union plus one set of the form S,, for some terminal element a E /%I.

P’-“

’

4 YI

i;

L----

P

Figure 2. (jj) If X is a connected component of 2 in Dom(f), frX=id,; if S, c Dom(f) for some terminal a E I%[ of type y , S, is an isomorphism between S, and By. Notice that (xx) if f € J , t h en \zr-Dom(f)=\zr and %-Range(f)=%. Indeed, if Dom(f) is just a countable union of connected components, then % - Dom(f) = % by (viii), and Range(f) = Dom(f), so that

fr

‘23 - Range(f) = (%@ P ) - Dom(f) = (% - Dom(f))@ P - \zr@ P = ‘23.

If Dom(f) contains S, for some terminal a of type y , then Dom(f) = Dom(f’)@ S,, Range(f) = Range(f’)@ By, where f’is the restriction o f f to some countable union of connected components of 3,and therefore % - Dom(f’) = % and ‘23 - Range(f’) = ‘23. By (xiv), S, = Xu- T,, where X , is the connected component of % containing a , and, by (x), T a x % ; using (viii) and (ix) we obtain % - Dom(f) = (% - Dom(f’)) - S, = % - S,

Ch. 5, 5 31

359

EXTENSION OF PARTIAL ISOMORPHISMS

Likewise, by (xix), ’

B-Range(f)=(B-Range(f’))-

B y = B - By=B.

J has the following property which we will use in the back and forth argument: (xxi) Given countable sets A C 11211, B C 1231, there are maps f,g E J such that A Dom(f) and B C Range( g). The first assertion is obvious. The second also, when B lies outside the principal component of 23. Then suppose B n P#,0 and let C= B n 11211. Let Dom(g) include for each c E C the connected component X, containing c, and g r UcECXc = id. Let y =sup{ i ( x ) ( x E B n P}; since B < No, then y
r

(f,f’f”) E I,

w

f EJ and f’,f ” are isomorphisms such that f’ : %+121 - Dom(f) and f ”: B+B - Range(f).

If I, has already been defined, we define I n + l as the set of all triples ( h , h’, h ” ) of maps such that there are (f,f’,f”) E I, and ( g,g’,g”) E I, satisfying the following conditions: (A)

Dom(h)=Dom(f)uf’[Dom( g)], Range( h ) = Range(f) uf”[Range( g)].

(B)

r

h Dom(f) =f,

h ( x ) = (f”og)(r>7

where x ~ f ’ [ D o m ( g ) ]and y is the unique element in Dom(g) such that

x = f ’ ( y ) . (In other words, h is defined by the equation h of’=f”

(C)

h’ =f’0 g(

og.)

h” =f” g ” , 0

By the induction hypothesis it is clear that h , h‘ and h“ are orderpreserving partial maps and that h’ and h” are defined on % and 23, respectively. It remains to show that h’, h” map‘onto %-Dom(h) and

360

LANGUAGES WlTH INFINITE QUANTIFIERS

[Ch. 5, § 3

23 - Range(h), respectively. Indeed, h’[Z]=j’[g’[%]]=j”% - Dom( g)]=f’[Z] -f’[Dom( g)] =

(a - Dom(f)) -f”Dom( dl

= 2l - (Dom(j) uj’[Dom(

g)]) = 2l - Dom(h),

and similarly for h ” . Finally, we define I to be the set of all partial maps f such that (f,f’,f”)E UnEwIn, for somef,f”. Clearly I#B and this I has the f”-’[B]; by (xx) there are isomorphisms g’, g” such that ( g , g ’ , g ” ) €Iw Let h , h’, h” be constructed from f,j’,f”,g , g’, g” in the manner indicated in (A)-(C); then ( h , h ’ , h ” ) E I , + , , whence h E I ; by ( B ) , f c h , and by (A): Range(h)>f”[Range( g)]>f”[f”-’[B]] =B. The “forth” property is proved in a similar way. This completes the construction and proof of the properties of 2l and % asserted in Counterexample 5.3.41.In addition we observe that by definition, J contains the identity map defined on any countable union of connected components of a; since J C I , by Theorem 5.3.23 we immediately conclude that the following stronger relation holds between 2l and B: (xxii) 2l < mw,B. rn REMARK. (1) It is a routine matter to extend Counterexample 5.3.41to‘any regular uncountable cardinal, instead of w,. For singular cardinals X of cofinality larger than w, the status of the implication

is unknown; there does not seem to be any simple way to adapt to this case the ideas used in Counterexample 5.3.41. (2) The preceding construction has been slightly simplified by J. Paris, using essentially the same example. It is natural to ask whether there are simpler examples of the same situation; one may ask, for example, whether there are total orderings 2l,8of power 0, such that 21=mw,Bbut 2lzB. J. PARIS[ l ] gave a positive answer to this question, but his construction is

Ch. 5, $ 31

EXTENSION OF PARTIAL ISOMORPHISMS

36 1

too involved to be presented here. Note that for well urderings,

without cardinality restrictions. It is clear that the structures "r,Bconstructed in Counterexample 5.3.41 have many automorphisms. It is natural to ask, then, whether it is possible to find structures with the properties of Counterexample 5.3.41 which in addition are rigid, i.e., have no automorphism other than the identity. GREGORY [ l ] gave a positive answer to this question, and used this theorem to prove that Beth's definability theorem fails for various infinitary languages. His proof gives a general method for constructing rigid structures B,6 of any given infinite power K with the properties B z 6 and 2 3 ~ ~provided ~ 6 , we are given structures "ro,"rl of power K such that 910z911 and 910-mK"rl.The rest of the present section is devoted to prove Gregory's result. Our first lemma shows that there are many structures of a given regular uncountable power with the properties of Counterexample 5.3.41.

LEMMA5.3.42. Let A be a regular uncountable cardinal. There are 2A structures {%,If ~ 2 ' }of similarity type of power < A, such that (1) = A for a 1 1 f ~ 2 ' ; (2) "r, s ag for all f, g E2', f + g ; (3) % , ~ f ~ , x % ~ for aNf, g EP. PROOF.Let "ro, 911 be structures of power A such that "ro=m~911and " r o ~ " r lFor . notational simplicity we shall assume that the similarity type of 910, "rl consists of a single binary relation symbol R (as in 5.3.41), but the proof below works for any similarity type containing only relation symbols. "r, is a modified form of the full cardinal sum of A copies of "r, and 911. For a
R

{ ((a,ao>,(a,al>>I (ao,a , ) E R

'%a)

and a
362

[Ch. 5, 8 3

LANGUAGES WITH INFINITE QUANTIFIERS

Clearly =A. Let f,g E2', f#g. If a is such that f(a)#g(a), we have

1.

( % , ~ ~ ~ ) r { ~ > ~ % , t a ) ' % g t a ) ~ ( % g r ~ ~ ) ~ { ~

This clearly implies that',3 ag.Thus (2) holds. We now prove (3). Since %O=m~%,, by Corollary 5.3.22 there exists a non-empty family I of partial isomorphisms of %, into %, such that I: For each h €1 define a partial map h* from 8, into gg as follows:

%o=I%,.

(a,h(a))

iff(a)=Oandg(a)= 1, iff(.)

(a,h-'(a))

iff(.)=

= g(a),

1 andg(a)=O.

The verification that h* is a partial isomorphism is a routine matter and is left as an exercise for the reader. Let I * = { h * l h ~ I } . Obviously 1 * # 0 . It is not difficult to check that I*:%,=@,. By way of illustration we prove the back property. Let
< A ( i = O , 1). Clearly BiCpri[B],and therefore Now, using the back property of the family I get a map h , €1 such that B,CRange(h,) and h ~ h , and , using the forth property of I get a map h, E I such that B, C Dom(h,) and h , C h,. Since

Ch. 5, $ 31

363

EXTENSION OF PARTIAL ISOMORPHISMS

it follows that BcRange(h,*); indeed, if ( a , a ) E B and g(a)=O, then aEB,CDom(h,,), so that (a,a)ERange(h,*); if g ( a ) = 1, then a E B , C Range(h,), so that (a,a)ERange(h,*). Likewise it is proved that h* c h,*. Thus I+ has the required back and forth property, and this shows that (3) holds, completing the proof of the lemma. REMARK. Note that the regularity and uncountability of A were only used to get two structures a, Yl, with the properties of Counterexample 5.3.41. Thus, the preceding argument also shows:

--

COROLLARY 5.3.43. Let A be an infinite cardinal and let Ylo,Yll be any two structures such that %,sY and l, %o-m~%,.l Let ~ = m a x {Yl,, Iu, }. Then there are 2’ structures {aaIa<2’} of similarity type of cardinality < max{ Sm(%,) ,A} such that: (1) K = K for a < ~ ; ( 2 ) B a s B B for a, P
We now proceed to construct Gregory’s counterexample. COUNTEREXAMPLE 5.3.44. Two rigid structures 23, C 5 of uncountable regular power h such that B (5 and 23 By Lemma 5.3.42 there are 2’ structures of a certain similarity type p = { R, I a
and n#O}.

Henceforth We use f , g , h, ...to denote members of D , and F, G , H , ...to denote maps from subsets of D into D. The structures B, (5 are of similarity type p ‘ = p u { S } , where S is a binary relation symbol, and are defined as follows:

\Bl= 161= D, No assumption is made about the cardinality of 910,91,.

364

LANGUAGES WITH INFINITE QUANTIFIERS

(f,g)ESB=S‘ (fl,

...,f,)E R:

@

[Ch.5, 5 3

there is m>O such that D o m ( g ) = m + l and f=gIm, there are f E D and m E w such that Dom(f) = m + l , (f,&)€SBf o r j = 1 , ...,n, and ( f i ( m + l ) ,...,f,( m+ l))ER>*),

where R, is an n-ary relation symbol in p ( a
I

y(O)=fiO)

i

Figure 3. This intuitive image is formalized as follows. For fE D,put: Or={ g E D I ( f ; g ) E S B ) ,

Srn(Bf) = Sm(6,) = p,

1q= I6,I

R?=

~ p D;n,

= D,?

R>=

if R, is an n-ary relation symbol ( a
~ $ D;n,

Ch. 5 ,

3 31

EXTENSION OF PARTIAL ISOMORPHISMS

365

Clearly Bf= and 0,s B,(m,+ whenever Dom(f) = rn + 1. Indeed, the map Ff:B,+%,(m) defined by F'(g)=g(rn+ 1) is a canonical B,1 n lBgl =,0 and isomorphism between these structures. Note also that 1 n I0,l = B , whenever f#g. = Obviously =A. It is easily seen that 23~6. If, on the contrary, we had B z H 6 , then for every f E D there would be a g E D-namely g = H(JFsuch that B,= Q,. But iff= {(O,O)}, then B,=i?lo, whereas every 0, is isomorphic to some i?lp with /? >O; since the Bu's are pairwise non-isomorphic, it follows that 8,s 6, for any g E D, and therefore B % 6. We show next that 23 is rigid. The same argument shows that 6 is also rigid. Let G be an automorphism of 23. By induction on Dom(f) we show that G(j) =f. Case (I). Dom(f)= 1. Then Bk i 3uoS(uo,ul)[~];hence

which implies that Dom( G O ) = 1. Note that G [ D,] = DG(,), and therefore that G D, is an isomorphism of 23, onto BG(n.Hence,

r

'f(0)

=Bf= B G ( f f

%G(f)(Ob

and this implies that f(O)= G(j)(O), since the Bu's are pairwise nonisomorphic. Therefore G(f) =f. Case (2). Dom(f)= rn+ 1 > 1. Clearly (frrn,f)ESB. Since G is an automorphism, ( G ( f r r n ) ,G O ) E SB. By induction hypothesis, we get G(frrn)=frrn, whence (frrn,G(j)>ES@. Therefore Dom(G(j))=rn+ 1 and G(A rn =f m.The argument of case (1) shows that B,(m, = Bf= B ,,, = BG(n(m),and this obviously implies that f ( m )= G(j)(rn).Therefore G(f)

r

=f.

We turn now to the proof of 23=,~6.In virtue of Corollary 5.3.22 it suffices to find a family F of partial isomorphisms such that F : 23 sf0. The rest of the proof is devoted to the construction of such an F. By assumption 21u=m~Bpfor a, /?
+

~ { f , g= )

Then

( FL

0

+

I, ' 0

h E I { , c n ) , g ( m ) + 1I ]

366

[Ch. 5, 8 3

LANGUAGES WITH INFINITE QUANTIFIERS

We shall consider sequences ( X , In EW) of collections of maps with the following properties: (i) O < %
F € J ( j , g i A F' EJ ( Y , ~ , A) F Z F'

* f +f' A g +g' ;

(this condition says that X , contains at most one map from each family J{,,gJ,and that the domains and ranges of distinct maps in X , are disjoint); (iv) for each F E X , + , there are G EX, and f E D o m ( G ) such that J(f.G(Jl).

Let 4 denote the family of all sequences satisfying (i)+iv). We shall denote members of 4 by ( X , ) instead of ( X , In Em). For each sequence ( X , ) E 4 we shall define a partial map Hex,> from (a subset of) 1'231 into 101. Dom( HCxn))= { f E D

I

there exists a sequence (Gk I k E Dom(f)) such that G k € X k , f r ( k + l ) ~ D o m ( G , )for EDom(f)%and 'k+ I J { f f ( k +l).Gk(fr(k+I))] for kEDom(f)- l}.

Note that (ii) and (iii) above imply that for each f E D there is at most one sequence (Gk I k E Dom(J)) with the properties just mentioned. Hence, every f E Dom(H(xn>)has a uniquely determined sequence with these properties. For f E Dom(HCxn>) we define H ( , m ) ( f ) as follows:

(*)

{

DOm(H<x,>(f))= Dom(f), (H(Xm)(f))(k)=

Gk(fr(k+

for kEDom(f),

where (Gk I k E Dom(f)) is the sequence of maps specified above. Note that if gE)'23,r(k+,,1and k€Dom(f)- 1, then

so that ( G k + l ( g ) ) r ( k +I ) = Gk(fr(k+1)). An easy induction on r shows that if Dom(f> > r 1, then

+

(**)

G r ( f r ( r + l ) ) ( k ) = G k ( f r ( k + l ) ) ( k ) for all k < r .

Ch.5, 0 31

367

EXTENSION OF PARTIAL ISOMORPHISMS

In particular, if Dom(f) = r + 1, then

G,(f)(k)=G,(fr(k+ l))(k) for all k < r.

(***)

Next we check that H(,") is a partial isomorphism of 2' 3 into 0. (a) Hcxn) is a one-one map. Indeed, if f,,f2E D and H ( , n ) ( f , ) = H ( , )(f2), then the first equation in (*) shows that Dom(fl) = Dom(f2), and thi second equation in (*) together with (**) shows that

r

G,'( f l (j + 1 )) = q2(f2 ( j+ 1 )) forj

+

r,

where r 1 = Dom(f,) = Dom(f2) and (Ck I k < r) is the sequence of maps corresponding to (i = 1,2). By induction on j we conclude from the last equality that (****)

q2 and

f,r ( j + 1)=f2r(j+ 1) f o r j < r.

Indeed, if j = 0, then using condition (iii) above we conclude that Gd = Gi; since each of these maps is one-one it follows that f, 1 1. If (****) has already been established f o r j < r, then

r =f2r

,

so that taking (iii) into account, it follows that C,!+ = Gi+I ; since these are one-one maps, we infer that f, ( j + 2) =f2 ( j + 2). Thus (* * * *) is proved, and for j = r we conclude that f,=f2. (b) H,, ) preserves the relation S ; if f l , f2 E Dom(H(,")), then (fl,f2>

Indeed,

*

(fI9f2)~SB

E

sB

-

r

r

(fqX")(fl)*

H(XJf2))

+

E

sG.

r

there is r > 0 such that Dom(f2) = r 1 and f,=f2 r e there is r > O such that Dom(f2)=r+ 1 and H(xn)(fl)=H(,,)(f2rr) e ( q x " ) ( f A H(XJf2)) E sG* The first and last equivalences are true by the definition of the relation S , the second because Hex,> is a one-one function, and the third because H(,")(frr)=(H(,")(f))rr (r < Dom(f)), as it is easily proved from (*). (c) H(,,) preserves the relations R,; i.e., if f,,. ..,f,E Dorn(H(,")) then

368

[Ch. 5, 4 3

LANGUAGES WITH INFINITE QUANTIFIERS

If G E J(f,g)(with, say, Dom(f) = Dom( g) = r + l), then G = Fg-' 0 h F, Recalling that F,(f')=f'(r+ l), we obtain for some h 0

G ( f ' ) (r

+ 1) = Fg(G (f ' ) ) = Fg

0

Fg-I 0 h o,F,( f')

= h ( F f ( f ' ) ) = h ( f ' ( r + 1)).

Since h is a partial isomorphism from 'il,(,) & ( r + l)EDom(h) ( j = 1, ...r ) , we have (fl(r ts

+ l), ...,f,(r + 1))

ER2r)

into 'illg(r)+l, if each

* (h(f,( r + l)), ...,h(f,( r + 1)))

E R:.(r)+'

(G(fl)(r+ I), ..., G ( f , ) ( r + l))ER?r)+l.

Applying this remark to G = Gr+,, using (b), the definition of HCxn), and recalling that g ( r ) + 1 = G,(f)(r)=H c x n > ( f ) ( r ) ,we obtain the following equivalences: (fl,... ,f,>E R," there a r e f E D and r E w such that D o m ( f ) = r + l , (f,&)ES' ( j = 1 ,...,n) and ( f l ( r + l ) ,...,f , ( r + l ) ) € R P * there are f E D and r E w such that Dom(H(xn)(f))= r , (H(xn)(f), H < ~ . ) ( $ >E) S ( j = 1,. .. n) and ((Gr+ t(fi))(r + I), * * . (G,+ I ( f n > > ( r + 1)) E Ra'~r(f~r)* there are g E D and r E w such that Dom( g ) = r, (g,H(xnJ&))ES" ( j = l , . . . , n ) and W(x")(fl)(r+ I), ...,H(,">(f,)(r+ 1)) E R 3 ( r ) + l (H(x")(f~), . . . , H c x " ) ( f n ) ) € R ~ . 7

-

Let us now define the notion (X,) S (Y,), for elements of 4 , to mean that for every n E w and every f,g E D the following conditions hold: (1) X,,nJ(f,g) +fi * Yn n J(,,gj #f12 (2) if F, G are the unique maps in X, nJ(,,+ Y, nJ(f,g), resp., then FGG. Note that (d) (Xn) (Yn> * H ( x m >H <~Y , ) ' Indeed, if f E Dorn(H(,")), then there is a uniquely determined sequence (GklkEDom(f)) such that GkEXk,f r ( k + I)EDom(G,) (kEDom(f)) and G k + 1 EJ{ .rr ( k + i ) , w y r (k + I ) ) ) .

The preceding clauses (1) and (2) obviously imply the existence of a sequence ( H, I k E Dom(f)) such that H , E Yk, G , H, (whence

Ch.5, 8 31

EXTENSION OF PARTIAL ISOMORPHISMS

369

fr ( k + l ) E D o m ( H , ) ) and

Let us now put

Since 4 #0 , then F # 0 . Also F has the less than X at a time back and forth properties. By way of illustration we prove the "forth" part. Let F E F and A CIBl, 7
Hk€YkG,f r ( k + l ) E D o m ( H k ) for k E D o m ( f ) , Hk+IEJ{fr(k+1),H*(,T(k+l))}

for k E D o m ( f ) -

By (d), (I) and (11) instantly prove that F has the required property. The case n=O is dealt with exactly as below, and we leave it as an ...,Y, have already been exercise for the reader. Thus, assume that Yo, defined satisfying clauses (l), (2) and (11) for every f E A such that Dom(f) < n + 1. We first divide the set of all f E A such that Dom(f) = n + 2 into disjoint classes by the equivalence relation

(i.e., iff there exists g E D such that Dom( g) = n the equivalence class off is the set

+ 1 and f,,fi E'B3,).Thus,

Since 2
370

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 5 3

Applying the induction hypothesis to fa r ( n + 1) we obtain a sequence ( H k I k < n ) such that H k E Y k , f , r ( k + l ) E D o m ( H k ) , for k < n , and Hk+lEJ{/.t(k+l),H,(/,f(k+l)))

fork'

For each a < p we shall construct a map Ha.There are two cases to consider. Case A. There exists G E X , + I such that Dom(G)CIB,al(,+l,). By (iv) of the definition of 4 there are G'EX, and.gEDom(G') such that G E J(g,G,(g,). Since if# D o m ( G ) C 18fmt(n+ n we conclude that g =f,r(n+ 1). Since J{g,G,(g)j has the < A at a time forth property, there such that G c Ha and A a ~ D o m ( H a ) . exists Ha EJ(g,G,(g,) Since X,, Y, have properties (1) and (2) of (I), there is H'EY, such that G' C H '; then g E Dom( H,) n Dom( H'), and by (iii) this implies that H, = H '; hence Ha EJ ( r ( n + i ), r ( n + I ))I. -Case B. There is no G E X , , , such that Dom(G)C(8,-r(,+l,l.Since A ,
,

such that A , C Dom(H"). Now set Y,+ = { H a I a < p}. Y,+ satisfies (11) for anyf E A such that Dom(f) = n + 2. Indeed, in this casef E A , for some (Y < p, and H L = Hk for k < n, HL+I = Ha is the required sequence. (I) is also satisfied because if G EX,+I, then there exists a < p such that fa EDom(G), so that we obtain G c Ha by case A. It is obvious that Y,+l satisfies conditions (i), (ii) and (iv) of the definition of 4 . To check (iii), let a, P < p be such that H " E J ( , , ) , HP EJ(gl,h.).Then g=f,r(n+l),

r

h = H, (f ( n + 1

1 9

g'=fpr(n+l),

r

h' = H, (f p ( n + 11).

If g = g ' , then h = h' and fa-&; therefore a = p and H a = H p . If h = h ' , since H, is one-one, then g = g', and we also conclude that H a = H p . This completes the inductive definition of (Y,) and the proof that F has the < A at a time forth property. The back property is similarly proved. Thus B and 0 fulfill the required conditions.

EXERCISES 5.3.1. Prove the Fact stated on pp. 320-321.

Ch. 5, § 31

EXTENSION OF PARTIAL ISOMORPHISMS

37 I

5.3.2. (a) Prove that A-saturated * A-universal. [Hint: use a “forth” argument similar to that of Theorem 5.3.34 (l).] (b) Prove that if 3 is a weakly A-homogeneous structure (cf. Definition 2.1.13 (4)), then the family I of all partial elementary maps from a subset of power
I%/

5.3.3. (a) Prove that any two atomless Boolean algebras TI,, B2,have the relation 23,-,PBz. (A Boolean algebra 23 is atomless iff for every xEl231, x #O * there is a y E 1233)such that y #O and y < x.) (b) Does the relation 8=$23 hold when 8,23 are models of the same complete, No-categorical, first-order theory T? 5.3.4. Let T be a complete, first-order theory without finite models, in a countable language p . A model % of T is called atomic iff for every n E w and every n-tuple a,, ...,a, E (81there is a formula ‘p f Form,(L,,( p ) ) such that for every other formula 4 E Form,(L,,( p)):

(in other words, every n-tuple of % realizes a principal n-type of the language p). Using back and forth arguments prove: (a) If 8,23 are atomic models of T, then 8-”op. (b) (Vaught) If 3 is countable and atomic and B€Mod(T), then % 5 23. Using the theorem of Ehrenfeucht-Mostowski (Chapter 2, 8 2), prove: (c) If T has an atomic model, then it has atomic models in each infinite cardinality. [Hint: Skolemize the language and add an appropriate set of indiscernibles to an atomic model of T; going down use the LowenheimSkolem theorem.] 5.3.5. (All arithmetical operations below are understood in the ordinal sense.) (a) Prove the following results (originally due to Mostowski and Tarski):

372

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 3

[Hint: use the variant of Theorem 5.3.11 for the relation < .] (b) Prove that each of the models in the right-hand side of (a) is iff for every a ~ l % there l is an L,,La,-definable. (X is L,,-defnable formula rpo of the language of a, with one free variable, such that for every

x E la[,

'uF=Cp,[x] w x = a . )

(c) Describe the ordinals a for which (w'", for the other structures ocurring in (a).

E) i ( a , E),

and similarly

5.3.6. Given a structure X without function symbols, let % # and 3, be the structures defined in the proof of Theorem 5.2.3, that is, Sm(%")=Sm(X,)=Sm(X)u

{ P,,P,},

where P , , P, are new unary relation symbols;

where a fZ \XI; R~*= R% * =R%'U { ( a ,...,a)}, n times

for any n-ary relation symbol R of Sm(X);

for any individual constant c E Sm(X);

Let h be an infinite cardinal and /3 EON. Prove: (1) X E ~ A * B ' u # ~ ~ i and B # X,=mAB#, (2) % &AB 3 X* &B# and u ' , &A%#. [Hint: use the back and forth characterizations of and -&.I (3) If K is strongly inaccessible larger than w , h < K , and 2,23 have less 3 % # =,A%* and %# E ~ A B * . than K symbols, then = , ~ 8 [Hint: use the theorem of Benda stated at the end of P3.C.l (4) Prove the analogs of (1)-(3) obtained by replacing the relations =mA, =&A, E K iby 5 m,i 5 5 KA, respectively.

pi,

Ch. 5, 3 41

APPLICATIONS OF BACK AND FORTH METHOD

373

$4. Applications of the back and forth method The best way to appraise the import of a mathematicaI idea or technique is by examples and applications. This section is intended to achieve this goal, as well as to illustrate the assertion-at the beginning of $3--that the method of extension of partial isomorphisms is destined to become an important tool of research; we do this by indicating some of its applications in algebra and model theory. Many more applications, among them some of the most significant, have had to be omitted, since they go beyond the scope of this book. The algebraic applications are: first, to the study of some metamathematical properties of the direct product and direct sum constructions; second, to the theory of free abelian groups. Regarding model theory, we apply the back and forth techniques to the analysis of infinitary properties of direct products reduced modulo regular filters and of models generated from indiscernibles.

Application I. Direct products and sums The type of question that we will attack first, is of the following kind: given a normal class of infinitary formulas, how does the relation %(@)B behave under various algebraic constructions performed on the factors 9l and %? For the cases of direct products and direct sums (of infinitely many factors) we will obtain particularly pleasant answers. Recall that IIiEI'iXidenotes the direct product of the structures a;, i E I ; as usual, this is the structure whose universe is the Cartesian product of the sets I%;[, i E I , and whose operations, relations and distinguished objects are defined coordinatewise. For each j E I , pr, : IIiE191j+%, denotes the projection on the jth coordinate: pr,((a, I i~ I ) ) = a,; clearly each projection is a homomorphism. In various parts of algebra, notably group theory, the direct sum construction is at least as important as that of direct product. Thus, to define direct sums, we assume that the underlying language p has a special individual constant symbol 0. The direct sum G3iE,21j is the substructure of IIiEr%; generated by the set of all elements a EIIjE,91j such that a;= 0'1 for all but finitely many i E I ; in particular, operations, relations, etc., in are coordinatewise defined. Note that not all elements of $iE191j need be of the form ( a ; 1 i E I ) where a; = 0 for all but finitely many i's;

374

[Ch. 5, 5 4

LANGUAGES WITH INFINITE QUANTIFIERS

this will be the case whenever each operation G in p satisfies the equation G(O,..., 0)=0 (e.g. in group theory).'

THEOREM 5.4.1 (BARWISE-EKLOF [ l ] for X = w ) . Let @ be a normal class of LWh-jormulas;(2; I i E I } , { B iI i E I } are two collections of similar structures such that %i(@)'23i for all i E I . Then (1) rI;E,~;(@)rI;E,'23;; (2) @ i E I'Ii(') @ i E 1'23,. PROOF.Applying Corollary 5.3.22 to the hypothesis, for each i E 1 we obtain a non-empty family Fi of @,-maps such that Fi : 'Ui =gBi. Now we proceed coordinatewise, as it is customary when dealing with direct products. For each sequence ( L l i E I ) , wheref;.EF,, we denote by I J i E I f ; the mapf with domain II,,,Dom(f;.) defined by the following set of equalities: pr, 0f = f i

0

pr,

for each i E I ,

or, in other words,

f((a; I i E 1 ) )= (AX.;)

IiE0

for each ( a i I i E I ) E r I i E I D o m ( f ; ) . If some of the

rIiEIf; = 0.

4's

are 0, we put

Since each Dom(f;) is a substructure of a;, Dom(f) is a substructure of rIiE19li; notice also that Range(f)=rIiE,Range(4). Moreover, f is a a,map; indeed, if r p ( t ~ ~ , . . . , t ~ , J is a quantifier-free formula in @ and a' ,...,a" EDom(f), then we have

The first and last equivalences hold by the definitions of direct product and o f f ; the middle one by the fact that all theJ;'s are Q0-maps. I The direct sum of two factors was introduced in Chapter 1, 5 I.B, in a somewhat different formulation, under the name cardinal sum (cf. Definition 1.1.20);we have decided to reserve the name direct sum for the present formulation, closer to current algebraic usage.

Ch. 5, 3 41

APPLICATIONS OF BACK AND FORTH METHOD

375

We define now

the desired concluPROOFOF (1). We will prove that F : II, =:H, sion, then, follows immediately from Corollary 5.3.22. Since F Z B , it will suffice to prove that F has the < A at a time back and forth property relative to @. As an illustration we do the proof of the “back” part: if @6+,[V]#,0 for some [EON, then for every f € F and B C 1231, B
BC

Range(g,)=Range(g).

B, C IEI

lEI

Therefore g has the required properties. This finishes the proof of (1). PROOF OF (2). It would seem natural, in this case, to “restrict” the preceding construction to the corresponding direct sums, i.e., to consider the following family of partial maps:

This choice works, but first we must check that this family of maps is well defined. (i) E a c h f E I I f E , F , maps a subset of @iE,jZIj into @fE,B,. Suppose f=IIjE,f;.,where J € F i for each i E I . Since Dom(l;.) is a substructure of jZI,, O’iE Dom(1.). Since l; is a @.,-map and uowO is in @, f;(O’i)= 01 ’ . Then, if a,= for all but finitely many i’s, the equation h(a,)= 0’1also holds for all but finitely many i’s. This proves (i). (ii) I f f = IIjE,f;., then Range(f1 CBiE,jZI,)= @,EIRange(f;.). This can easily be checked by the reader. From (i), (ii) and (l), it follows immediately that 0 ’ 1

Thus (2) is proved. R A simple extension of the preceding proof yields the following:

376

THEOREM 5.4.2. If i E I , then: (1) rI;,,X;

(2)

[Ch. 5, f 4

LANGUAGES WITH INFINITE QUANTIFIERS

is a normal class of L,,-formulas

and X i <$3,

for all

<&iEI~:;

@ ; € f a< ;o @ ; € f B ; .

PROOF.By Theorem 5.3.23, the collections Fi of partial maps of the preceding theorem can be chosen with the additional property:

(*I

7

for every A i ~ l X i l ,
The result will be proved if we can show that the collections IIiEfFiand also enjoy this property; since the maps in the latter set are restrictions to of maps in the former, it obviously suffices to prove this property for I I i E f F i . Let A CrI,EflXil, 7
( flJ;)r fl A,=id; iEI

iEI

since A CrIiEfAiCIIiE,Dom(f;),we have shown that there is an f €IIiEfFi such that A C Dom( f ) and f PA = id,. rn As a special case of Theorems 5.4.1 and 5.4.2 we have:

COROLLARY 5.4.3. (BARWISE-EKLOF [ I ] , CALAIS[ 11). If X,=-,Bi for each i E I , then n;Ef%;EmAH;EfB; and @ i E , % i = m ~ @ i E fThe B ; .same holds for the relations (EX,*), (Un,,),

(POS,,),

i ,,,

<

etc., instead of

-,A.

In the case where all the structures X i , Bi coincide with a fixed structure stronger conclusion:

X,we can obtain a

5.4.4. If I , J are infinite sets of power > A , I L J , and ‘2l is any THEOREM structure of similarity type appropriate for direct sums, then

PROOF.Let 0 denote 0” and % = Q = @,%. For IoC I let B[I,] denote the substructure of B generated by those sequences aE(B1 such that a,=O for i E I , ; obviously B[I,] is canonically isomorphic to Define B[J,] similarly for J , C J .

Ch. 5, 8 41

APPLICATIONS OF BACK AND FORTH METHOD

377

Each bijection p : 104J0, where I , C I , J , c J , determines a unique isomorphism f, : B[I,]+O[J,], as follows. If b E 1'3[Io]1 is a sequence such that b,=O for all but finitely many i's, putfp(b)=(c,(jEJ), where cj=

{

bP-'(,)

ifjEJ,,

0

if j E J,.

Since the set of such b's generates 8[10], f, extends uniquely to the whole of '23[ I,]. We observe that (i)f, is an isomorphism. Since the set of all c's such that cJ=O for all but finitely many j ' s and c,=O for j E J , , generates O[J,], it follows that f, is bijective. If 9, is an atomic formula and b ' , , . .,b" E l'23[Io]l,assuming, without loss of generality, that b,k=O for all but finitely many i's, we obtain the following equivalences:

B[I,]t=Cp[b',.. .,b"]

-3,

for all i ~ l , ,%t=9,[6,',..., b,"]

-

for a l l j E J , , %~cp[bd-~,,, ,..., bp"-,(,)]-3

a t=d(f,(b')),, .. .,(fp(b")),I o.IJOl+ 9,rfp .. .?f,( b")I. for a l l j EJ,,

-

The first and last equivalences hold by the definitions of B[I,] and O[J,] respectively, the third by the definition of fp, and the second by the assumption that p is bijective. Also one can easily derive from the definition of f p : (ii) p CP' fp We will also need the following observation: (iii) For every B ClBpl, there is a set I , c I such that B C1'23[ZB]1;if B = A similar statement holds for 0. is finite, so is ZB; otherwise To prove (iii), we deal first with the case B = { 6). If, in addition, b, = 0 for all but finitely many i's, take I , = Ib = { i E I Ib, Z O } ; if b does not have this form, taking into account that all operations of i? are l finitary, we conclude that there is a term 7 and elements b',...,b" such that b = ~ ~ [ b.l.,b"] , . and each b k has almost all coordinates equal to 0; by the above, for each k , 1 < k < n, there is a finite set I k such that b k E 8 [ I k ] ; clearly { b ' ,...,b " } ~ %u z[ = l I k ] ; since ~ ' ( 0..., , o)=O, we conclude that b E%[U;!lZk]. The case in which B is finite follows immediately from the preceding case. Finally, if B is infinite, for each b E B pick a finite set Ib such that E B [ I , ] ; setting I , = UbEBZb, we obtain B c B [ I B ] and, - bclearly, B =

-

G.

c-4'.

c.

378

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 8 4

With these facts established, we define a collection F of isomorphisms from substructures of 8 of the form B[I,] onto substructures of 0 of the form O[J,], as follows:

F=

(& Ip

is a bijection from a subset of I into J , and

Dom(p) < A ] .

Using Theorem 5.3.23 we establish the relation B i wAOby showing: (iv) F : B =.hp6, and (v) for each B c 1231,
fr

REMARK. The collection F has the stronger property F:B=:*e6; we leave the proof of this as an exercise for the reader (cf. Exercise 5.4.1). Notice also that if the assumption I C J is replaced by 7 < 7, the corresponding conclusion is 5 wA@J%. Structures of the form @$ occur frequently in algebra. We mention two examples from group theory: (1) every free abelian group’ with K generators is isomorphic to @,Z, where = K ; (2) every torsion-free, divisible abelian group’ is isomorphic to @[Q; every such group can be construed as a vector space over Q, and its dimension is precisely 7 .

’

An abelian group G isfree iff there is a set X of generators of G such that any map from X into any abelian group H can be extended to a unique homomorphism from G into H . * G is forsion-free iff for every n E2 and x E G, nx = 0 x = 0. G is diuisible iff for every n € 2 and x E G there is y E G such that ny = x .

Ch. 5 , 9 41

APPLICATIONS OF BACK AND FORTH METHOD

319

Applying Theorem 5.4.4 to these examples, we obtain:

COROLLARY 5.4.5. Let X be an infinite cardinal. (1) If GI, G, are free abelian groups on K , , K, generators respectively, and if X < K~ < K,, then GI 5 ooh G,. ( 2 ) If G I , G, are torsion-free, divisible abelian groups of dimension K , , K~ over Q , respectively, and i f h < K~ < K,, then G, S m h G,. In our next application we take a closer look at the infinitary properties of free abelian groups. Application 11. Free abelian groups

Most of the results in this paragraph are contained in BARWISE[3], especially in § 5. Let us begin by recalling a couple of simple facts about free abelian groups. Fact A. For an abelian group G, the following are equivalent: (a) G is free (on K generators); (b) G - B I Z (where 7 = K ) ; (c) G is generated by a family { x , l i E I } (of power K ) and for every k E w , n ,,..., n , E Z and i ,,...,i k E Z , k

n.xJ I , =O j= 1

n,=

... = n , = o .

Fact B. Every subgroup of a free abelian group is free. (A) can be proved by the reader; for a proof of (B), see FUCHS[I], p. 74.

NOTATION. In the sequel ( x ) will denote the infinite cyclic group generated by x ; of course ( x ) = Z . Likewise, if g E G , then ( g ) will denote the subgroup of G generated by g. No confusion will arise from this double usage. The letters m , n , etc., with or without indices will denote variables ranging over Z. DEFINITION 5.4.6. (1) For an infinite cardinal K , an abelian group will be called K-free, iff all its subgroups of cardinal less than K are free. (2) A subgroup Go of an abelian group G is called thin in G iff it is finitely generated and G - Go@ GI, with G I = G. For example, a free group is K-free for all K (Fact B), and conversely; but there are N,-free groups (of power 2’0) which are not free, as we shall see

380

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 0 4

below. As an example of a thin subgroup we mention that if I is infinite, then each ( x i ) is thin in @ f E , ( x i ) ; furthermore, using Facts A and B the reader can easily see that any finitely generated subgroup of a free group G on infinitely many generators is thin in G. The main result of this section shows-using extension of partial isomorphisms-that for an infinite index set I ,

We shall see that this result contains a fair amount of information about the metamathematics of free abelian groups. As a preparatory result we prove the following:

THEOREM 5.4.7 (Baer-Specker). Let I be an infinite set and H = I I t E , ( x f ) . Then (1) every element of H belongs to a thin subgroup of H ; ( 2 ) eoety finite& generated subgroup of H is included in a thin subgroup; ( 3 ) H is not free. PROOF.(1) For h = ( n , x , [ i E I ) E H , h#O, define the norm of h, llh[l, as follows: llhll= min{ [nil1 ni # O and i E I }.

Statement (1) is proved by induction on IlhlI. Case (A). llhll= 1. Get i o E I such that nio=? 1; let

G = { ( m i x i I i E I) I mi,=O}. Clearly G = I I I E f , , z f o ( x l ) and, since I is infinite, G- H . Moreover, H = ( h ) @ G . Thus, ( h ) is a thin subgroup of H . Case (B). llhll> 1. Assume ( I ) true for all gE H such that IlgII < Ilhll. Dividing each n,#O by llhll we get integers 4,, r, such that n , = q , . ( l h l l + r , , 0 < rf < llhll and 4, # O . Putting y , = ( q f x ,I I E I ) , y 2 = ( r , x , I i E I ) (where qf = r, = 0, whenever n, =O), we obtain h = / / h / ( y+, y 2 . Let ioE I be such that ~ n , o ~ = \ so ~ hthat ~ \ , 4,,= ? 1 and rfo=O. Let

K = { ( k , x ,1 1

EI

) I k,,= 0 ) ;

so y 2 E K and, as in Case (A), K - H ; also H = ( y , ) @ K. Since Ily211 < llhll and K - H , by induction hypothesis there are subgroups F,, F2 of K such that F , is finitely generated, F2- K ( = H ) , y,E F , and K = F , @ F Z ; hence H = ( ( y , ) $ F , $ F , . Since ( y , ) @ F , is finitely

Ch. 5 , 3 41

APPLICATIONS OF BACK AND FORTH METHOD

38 1

generated and contains h, ( I ) is proved. (2) is easily derived from (1) by induction on the number of generators. (3) By Fact B, it suffices to exhibit a subgroup of H which is not free. First, select a fixed infinite denumerable subset I , of I ; we shall work within the subgroup H , of H generated by { x,l i E I , } . By enumerating I , we can (and will) write H , in the form n,,,(xi). Fix a prime p . Let Go be the set of all ( n , x , )i E o ) such that for each m >0, p" In, for all but finitely many i's; clearly Go is a subgroup of H,. We prove that Go is not free. = 2"o. Indeed, if ( m i I i E w ) E w w , then (i) (mi 1 i ~ w ) + ( p'mixi1 i ~ w )

is an injective map from w o into G,,. (ii) CB i E w ( ~ i ) L Go. This is clear. (iii) Each coset of Go modulo p G , contains an element of CBiEw(xi). Let x = ( n , x , l i ~ w )Go, ~ and let i,, ...,ik be all the indices i such that p "Jn,; finally, let y = (mix, I i E a), where mi = n, for i E { i,, ...,i k } and m,= 0 otherwise; clearly y E CBiEw(xi) and p " divides all coordinates of x - y , whence x - y EpC,; so (iii) is proved. An immediate consequence of (iii) is that G,/pG, is countable. The proof that Go is not free follows immediately from this and: (iv) if 7 = 20' and G is free, then G / p G = 2"~. If { y j I j E J } is a set of free generators of G , it is clear that 7 =2*O and that yY-y,..EpG, for j ' , j " E j , j ' # j " (otherwise yi.-yj-=p. Ei=lnjky,k which, by Fact A(a), contradicts the freedom of G). Hence { y , / p C ) j E J } is a subset of G / p G of p0wer.2'~. This completes the proof of (3). rn We turn now to the proof of our main result: THEOREM 5.4.8. For any infinite set I ,

PROOF.Let G = @ i , l ( x i ) , H = I I i , , ( x , ) . By Theorem 5.3.23 it suffices to show that there is a non-empty family F of partial isomorphisms such that F : G &H with the additional property: (*)

for every n E w and a , , ...,u,EG there is a n f E F such that u ,,..., u , ~ D o m ( f )andf(a,)=u, for i = I ,..., n.

382

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, 5 4

As F we choose the family of all (partial) isomorphisms from a thin subgroup of G into a thin subgroup of H . Clearly FZB . Let a,, ...,a, E G ; by Theorem 5.4.7 (2) the subgroup Go of H generated by {a,,. ..,a,,} is included in some thin subgroup Ho of H ; moreover, GoC G. Following Definition 5.4.6 we have remarked that Go is a thin subgroup of G . Therefore, the inclusion i : Go-+Ho is in F, whence F has property (*).

To illustrate the proof of F : G=$H we show that F has the “back” property. Let f E F and b E H ; we assume, of course, that b ERange(f). Choose an a E D o m ( f ) ; Dom(f)@(a) is finitely generated, hence a thin subgroup of G ; likewise, by Theorem 5.4.7 (2), Range(f)@(b) is thin in H ; defining g : Dom(f)@(a)+Range(f)@(b) by g ( x + nu) =f(x)

+ nb

(x E Dom(f), n E Z),

we obtain a map g EF which extends f and b E Range( 8). The proof of the “forth” property is similar; thus 5.4.8 is proved.

As a corollary we obtain the following non-axiomatizability result, in the same style as the results of Chapter 4, 94:

COROLLARY 5.4.9 (Kueker-Keisler). The class Fr of all free abelian groups is not axiomatizable by any class of Lmu(g)-sentences.’ PROOF.G <

H , but G is free and H is not (Theorem 5.4.7 (3)).

This result calls for some comments: (1) Using Fact A(c) it is easily seen that Fr is a PC(L,,,)-class. Indeed, it coincides with the class of reducts to the language g of all those abelian groups that also satisfy the following sentences, which include an additional predicate G : (a) ( G is a set of generators).

‘g = ( + ,0,- ) is the similarity type for (abelian) groups.

Ch. 5, 5 41

APPLICATIONS OF BACK AND FORTH METHOD

383

(b) ( G is an independent set).

(2) Replacing the symbol G by a quantifier of the appropriate length, it follows that the class of all free abelian groups on < K generators is EC(L,,). This suggests the question whether Fr itself is axiomatizable in some L,, with h > w . Probably the answer is negative, but the arguments of Chapter 3, P4.C, do not seem to work in this case. In the same vein we pose the following:

PROBLEM. For X > w , is it possible to find a free group G and a non-free m, H ? group H such that G i We carry Theorem 5.4.8 one step further to show:

THEOREM 5.4.10 (Kueker). Let F be the free abelian group on Nogenerators. For any infinite abelian group G, FEm,G w G is N,-free.

PROOF.(+) Let Go be a countable subgroup of. G; we wish to prove that Go is free. Let qFbe the Scott sentence of F (cf. Theorem 2.3.1 1); since qF is in L,,,, G qF. By the downward Lowenheim-Skolem , theorem for L,,, (Theorem 4.1.2) we get a group GI such that GoC G, C G, =No and GI + q F ;then GI = F, so G, is free. By Fact B we infer that Go is free. (e) Suppose G is N,-free; by Corollary 5.3.33, it suffices to prove G F q F in order to conclude F = , , G . If G is denumerable, then G is free, so G = F and GFq,; thus we assume that G is uncountable.

+

Let A be a denumerable set of independent elements of G. By the downward Lowenheim-Skolem theorem, we get a subgroup G I of G such =No and that A c ( G , I ,

Since G is H,-free, GI is free, and so G, = F; therefore GI kipF, and G + q F as desired. rn

384

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5,

54

In particular we conclude from this theorem that the group n j E , ( x i ) of Theorems 5.4.7 and 5.4.8 is N,-free. Incidentally, we mention that N,-free groups admit an axiomatization in Lu,u; indeed, q+, the Scott sentence of the free group F on No generators is one such axiomatization (albeit a somewhat esoteric one); but there are others with a simpler and clearer [3], 5 5). group-theoretic meaning (cf. BARWISE We close this paragraph by remarking that the most significant application of back and forth techniques to group theory is a beautiful extension of the basic structure theorem for torsion groups: Ulm’s theorem; this result of Barwise and Eklof has been published in their paper [I]. We turn now to applications of back and forth arguments in model theory. Application 111. Direct products reduced modulo regular filters As our application in this area we have chosen to prove a remarkable theorem of BENDA[2] to the effect that the reduced product construction modulo filters of a special kind, not only preserves logical equivalence in certain infinitary languages but-much better-strengthens the relation of elementary equivalence to that of infinitary equivalence in certain languages of the form LwA. The importance of this result lies in the fact that-unlike our earlier applications of ultrafilters in Chapter 3, $3-it applies to a wide class of filters; indeed, in the case of a denumerable index set I this class coincides with the set of all w-incomplete filters on I and, except when I is measurable, the class of all ultrafilters to which the result applies coincides with the class of all non-principal ultrafilters on I . Thus, this result should, in principle, offset the impression that the reduced product and ultraproduct constructions are of no use in infinitary logic. Further remarks in the same direction can be found in KEISLER[3], part IV. We proceed now to define the filters we shall be dealing with:

DEFINITION 5.4.1 I. Let Y be a filter on an infinite set I and A be an infinite cardinal number. 9 is called A-regular just in case there is a set 5X C 5 of power A, such that every infinite subfamily of 5% has empty intersection; we shall also say that 3E. makes ‘3 A-regular. (1) A-regular filters are usually called ( w , A)-regular; since we REMARKS. shall not consider (K,A)-regular filters for K’S other than w , we have omitted the reference to K . (2) Recall that a filter $7 is w-incomplete iff there is a denumerable set

Ch. 5, 3 41

% ~ such 9 that

385

APPLICATIONS OF BACK AND FORTH METHOD

n

%=0. The following lemma gathers all the elementary information we will need on regular filters:

LEMMA5.4.12. Let ‘3, 8 be filters on I . (1) ‘3 A-regular and ‘3 g * B A-regular; ( 2 ) ‘3 A-regular and o < K < A j $7 K-regular; ( 3 ) ‘3 A-regular * ‘3 w-incomplete j ‘3 non-principal; (4) ‘3 w-incomplete ‘3 w-regular; Thus, the families of w-incomplete and w-regular filters coincide; in particular; ( 5 ) if 7 is less than the first measurable cardinal, then every non-principal ultrafilter on I is w-regular; (6) 9A-regular 7 > cf(A).

-

-

PROOF.(1) and (2) are obvious. (3) By (2) 9 is o-regular; say % makes it so; then % also makes 9 w-incomplete. (4) Let % make ‘5 a-incomplete, i.e., Y =No and n % = O ; we may assume that ! Y is descending (for, otherwise, we replace Y by a cqllection of suitable finite intersections of sets in ‘3 ). It is clear, then, that if $4 Yy is infinite, n ?I = ‘3, whence n 9 = i f . ( 5 ) For such - I , every non-principal ultrafilter is a-incomplete; apply (4). (6) Let 7 = K and suppose K < cf(A); let :X = { X,l a < A } be any subfamily of 9 of power A; we assume that the enumeration a+X, is one-one; without loss of generality we can also assume I = K . Suppose for each a € K there is some /3, < A such that a E X , ,$< &; since K < cf(A). 0 = sup{ /3, I a < K } < A ; this implies that Xp+ = 0 , and thus 0 E 4.contradicting the fact that L.T is proper. Therefore, there is some CI~EK such that Y = { a I a o ~ X a is} cofinal with A: Y is infinite and a,€ n P E ,XP. This proves that :T cannot be A-regular, a contradiction.

n

,

-

REMARKS. (5) proves, in particular, that if Z = w , then “a-regular”, “oincomplete”, “non-principal” and “uniform” are all equivalent for ultrafilters. This naturally suggests the question how big is the family of A-regular ultrafilters on A, when A > w . Pfikry has proved that under the assumption V = L every uniform ultrafilter on w , is a,-regular (cf. PRIKRP [2]). From (6) we obtain, e.g. that there are no a,-regular filters on o. Next we establish some elementary facts needed in the proof of the main theorem; as a corollary we get the existence of A-regular filters on any set of power A.

386

LANGUAGES WITH INFINITE QUANTIFIERS

LEMMA 5.4.13. (1) Given two arbitrary sets I , J and a map f : J+ define a map f* : I + 9 ( J ) as follows:

[Ch. 5 , 3 4

9 ( I ) let

US

f*(4 = { jE J I i E f ( j ) } . Then : (a) f** = f ; (b) f onto * f* one-one; ( c ) for all i E I , i E (A. (2) Suppose !X makes A-regular afilter 5T on I ; then: (a) Range (f) % * for ecery i E I , { f ( j )( jE f * ( i )j is finite; (b) if, in addition, f is one-one, then for each i E I , f * ( i ) is finite; in other words, f* : I-+ ??JJ). (3) Conversely. iff: J+??JI), then ? = Range(f*) has the following properties: (a) every infinite subfamify of % has empty intersection; (b) if, in addition, f is onto, then X has the finite intersection property.

qEPJ

c

PROOF.( 1 ) (a) and (c) are immediate consequences of the definition off*. Regarding (b), let i, i’E I be such that f*(i)=f*(i’); then, for all j € J , j ~ f * ( ie ) j ~ f * ( i ’ )hence ; i Ef(j)w i‘Ef(j). Since f is onto, there is some j , J ~such that f (j,) = { i } ; since i Ef (j,), we infer i‘ E { i}, whence i= j’. (2) (a) Since Range(f)C Gx and f ( j ) # 0 , it follows that the set { f ( j )Ij E f c ( i )j cannot be infinite. (b) follows immediately from (a). (3) (a) Let { i k l k E w } be an arbitrary infinite subset of I ; if for some j € J , j~ f l k E / ( i k ) , then { ik I k ~ w Cf}( j ) , and f ( j ) cannot be finite. kEJ(ik)=O. Therefore (b) To show: for any n E w and any { i l ,..., i n j C I , n ; = l f * ( i k ) # O . Since

n ,,

n

{ i,, ..., i n } E 9JZ)

= Range( f ) ,

there is a j E J such that f(j)={i,,..., i n } ;thereforeje n;=,f*(i,). rn

COROLLARY 5.4.14. On any set I of irzfinite power A, there is a A-regular filter. =

PROOF.Since I is infinite, s,(I)= 7 ; let f : I + T o ( I ) be a bijection. By (3.b) above there is a filter 9 on Z such that 5 2 3i = Range(fc). By (3.a), Gx makes 9 A-regular. rn The main result in this paragraph is:

Ch. 5, 3 41

APPLICATIONS OF BACK AND FORTH METHOD

387

THEOREM 5.4.15 (Benda). Let I be an infinite set and 9be a A-regular filter on I . Let p be a similarity type of power < A . Let {Zi/i€Z}, { 2 3 ; / i ~ Z be } two collections of structures of type p such that %;=Bi for each i E I . Then:

II %i/5=,x+ iEI

l j23;/9. iEI

COMMENTS. These introductory remarks are intended to motivate some of the technical constructions occurring in the proof. ( I ) We would like to use the back and forth characterization of elementary equivalence given by Theorem 5.3.11. The first difficulty in doing this lies in the fact that the similarity type p need not be finite. We based on ( 2 ) of overcome this by an argument using the regularity of 9, Lemma 5.4.13, which allows us to replace %; and 23; by suitable reducts 2lj= Eirs*(i), 23; = Bi rs*(i), where the s*(i>’s are finite subsets of p ; the s*(i)’s are chosen so that no information is “lost” in passing from 3; to %;, i.e., I I j E I % ; 5 / = rIjE,%;/9, and similarly for the 23;’s. ( 2 ) Let (FL I n E w ) be the w-sequence of sets of partial isomorphisms from %; into 23; given by 5.3.1 1. Using these we want to construct a family F of partial isomorphisms from %=IIiE,%;/9 into 2 3 = r I j E I B j / 9such that F : 2l &+23. The proof of Theorem 5.4.1 suggests to take quotients modulo 9 of partial isomorphisms of the form I I j E l f ; (where each f.E U,,,FL) as the members of F. Unfortunately, this choice of F does not work; indeed, we will meet difficulties in trying to extend partial where { i € Zif; EF;} E 9. The notion isomorphisms of the form IIjElf;/9, of a family ( f ; l i E I ) chosen along certain maps d : w + w (for a fixed denumerable partition 9 of I with special properties) is precisely designed to overcome this difficulty; it expresses the intuitive idea that, as i varies, the J’s have to be chosen to belong to FL’s with increasing index n . This idea allows us to construct an F whose members are certain maps of the 5, which enjoys the required back and forth property. form II,,,J/ This formulation-intended to clarify Benda’s original proof-is due to the author.

PROOFOF THEOREM 5.4.15. We begin by defining all the objects and notions that will be used in the proof. As usual, we shall assume that a correspondence has been fixed, which, assigns an element x EIljEIl%;1 such that to each y E 12lI = (IIjs,21j/51 y = x / 9; thus, when referring to an element x / 5,we mean that x is the representative selected by this map. ! C 9 that makes 9 A-regular, and will We will also choose a fixed X apply (2) of Lemma 5.4.13 to various sets of power < A, as follows. Firstly,

388

[Ch. 5,

LANGUAGES WITH INFINITE QUANTIFIERS

4

let s :p+ % be a one-one map; then for each i E I , s*(i) is a finite set of symbols in p ; we set %;=9liTs*(i), 23i=B1l s * ( i ) . Secondly, if A GI%[ is a set of power < A and a : A + % is a one-one map, then a*(i) is a finite subset of A ; let xl, ...,x k , be representatives for each of the distinct members of a*(i); then {x:, ..., x?} is a finite subset of /%,I, which we denote by A (i). This notation is incomplete, since A (i) depends also on the map a ; since the choice of a is immaterial for our arguments we will omit reference to it. Similarly, we define B ( i ) for any B C 1231, < A . Next we show how to construct a partition 9 = { J k I k Ea} of I with the following properties: (i) for each k€ a , Un>kJnE 9; (ii) i E J , 3 a * ( i ) < k . Let j : A > % be a bijection. By Lemma 5.4.13(2.b), each p(i)is a finite

= k } . It is clear that 1 = UnEwJn and that subset of A. Put gk = { i E I I J , n J , =B for n # m. To prove (i) we observe first that for i E I :

f"0

=

f*(i)

2k

-sthere

is z € [ X I k such that i E

nj ( x ) ; X E Z

hence

u ~ , = ( i ~ ~ i r ( l ) > k j = u u nfw n>k

n>k

z ~ [ ~ x ]E nr

Note that each set of the form nx E r f ( ~ ) where , z E U,,,[A]", is the intersection of a finite, non-empty family of sets f ( x ) E % c 9, and therefore belongs to 9. Hence Un>kJfl E '3. Finally, we prove (ii). Given a one-one map a :A+%, (where A Cl%l and let b : A + A be the map b = f - l o a ; then b is one-one and a = f o b . For i E I we have:

z
a*( i) = { x E A I i E a ( x ) } = { x E A I i Ef ( b ( x ) ) ) = { x € A I b ( x )E f c ( i ) } = b - ' [ f c ( i ) ] .

-

-

Since b is one-one, we have that a*(i) = b - ' [ f c ( i ) ] < f * ( i ) . Hence (ii) follows. Note that (ii) implies that i E J , * A ( i ) < k . By assumption, %i=23i, whence LU;=Bi; since 23; have a finite similarity type, by Theorem 5.3.1 1 (Fraisse-Ehrenfeucht's characterization of elementary equivalence), for each i E I there is a sequence (Fk I n E a) of sets of a-partial isomorphisms from 'zr: into 23: such that for each m E a , (Fi 1 n < m ) :2l -%,,B. ~

Ch. 5, 5 41

389

APPLICATIONS OF BACK AND FORTH METHOD

Recall that the product IIjEff, of a family I i E I ) of partial maps is defined coordinatewise (cf. p. 374). Fix an element b o E rIjE,lBjl. We define the quotient I I j E , J / % in the natural way, as the map f such that:

(L

Dam(/)={

IIIIjl

x/9lxE iEI

and

I

{iEZlxjEDom(f;.)}E9

and (1) if IIjEfDom(J;)#B, then f ( x / ' 3 ) = ( ~ . ( x j ) I i E I ) / 5 for x E

n Dom(f;,),' iEI

(2) if IIiEfDom(J) =B , then either (j) {iEIIDom(J;)=B}E'%, and we setf=O; or (jj) {iEIIDom(f;,)}=B F 9,and in this case we define f ( x / 9 ) as ( l ( x j ) I i E I ) / 9, where

1.'( X i ) =

i

J(xj)

if xiEDom(&),

b:

if xjeDom(f;).

The reader can easily check that f ( x / '3) does not depend on the choice of representatives in the class x / S . We prove that the map thus defined is a partial isomorphism from III into B. Let R be an n-ary relation symbol in p and a ' / '3,. ..,a"/ '3 E Dom(f); for simplicity we assume that up EDom(&), for all i E I , k = 1, ...,n. For each i E s ( R ) we have R Es*(i); so R belongs to the language of 23:. Sincef; is a partial isomorphism from Xj into B;,we have:

III; F R [a,', ...,a:]

w 23;t- R [f,(ui'),...,&(a?)

whence

1,

I I I P R [ a ' / S , ..., a " / % ] w B F R [ f ( a ' / 5 ) ,...)f(a"/9)] 'In this case, for every y E Dom(f) there is an x EHi,,Dom(fi)

such that y = x / 5 .

390

[Ch. 5, 5 4

LANGUAGES WITH INFINITE QUANTIFIERS

In the remaining cases (i.e., those of (2) in the preceding definition) the proof is analogous and is left as an exercise for the reader. Recall that a linear map I : w+w is a map of the form l(m)= am + 6, where a, b E w. Let 5 denote the set of all maps d : w+w which increase ; is, faster than any linear map in,some interval [ n , ~ )that d E '7 m for each linear map I : w+w d ( m ) > l ( m ) for all m n.

there exists n E w such that

-

Note that T ' #0, since the map m+m2 belongs to ':i . Note also that if d E ' T and 1 is a linear map, then d - - / E \ T (here ( d - l ) ( r n ) = d ( r n ) - l ( m ) if d ( m )2 [(rn), and is 0 otherwise). We introduce now the main notion in this proof. A sequence ( A I i E I ) is said to be chosen along a map d E '7, in symbols ( J I i E I ) (along d), iff whenever i E I and k E w i E Jk

*

E Fa,,,.

The required family of partial isomorphisms is: F=

( n f;/$7I there is

d E 9 such that ( J I i E I ) (along d )

iEI

Clearly F # 0 , and, by the above, the members of F are partial isomorphisms defined on substructures of a, with values in 'P. Next we will show that F has the < A + at a time back and forth property; to illustrate the procedure we only do the proof of the "forth" property, since that of the "back" part is similar. F ACIYll, 7 < A , be given; then f=n,,,f;./\?,for some Let ~ E and family (f;1 i E I ) (along d ) , where d E 5 . Let d'= d- id, where id :w+w is the identity map. Then d ' E 9. Let i ~be larbitrary and r n E w be such that i € J m . By property (ii) of

-

the partition '3' we have A ( i ) = k S m, say A ( i ) = { x : , ...,x,"}. We analyze the following two cases: (a) d ( m )> m. Since (FL In ,< d ( m ) ) has the one at a time back property, and f i ~ F > ( , , , ) there , is a map g,'EFa(,)-, such that f;.cg! and x!~Dom(g,'); applying the property to g) we obtain a map g:EFh(m)-z such that g! Cg,' and x , ? ~ D o mg,'). ( Repeating the procedure k times we obtain a map gk E F',,,,- such that g,"-' g," and x," E Dom( g,"). We put gi = g,". Clearly, f;. C g,, A (i) C Dom( gi) and, since d'( rn) < d (rn) - k, we also have g, E F>,(,,,). (b) d(rn)< rn. We define g, to be any map in Fb. We check now that if g = I I i E , g i / 9 , then g E F , f c g and A c Do m ( g ) .

Ch.5, 3 41

39 1

APPLICATIONS OF BACK AND FORTH METHOD

(c) g E F. This is clear, since ( g , 1 i E I ) (along d') and d' E '5. (d) f C g , By the preceding construction, Dom(f,) C Dom( g,) for i E ufl,,Jn, where r is such that d ( m ) > m for all m E[r,w). Let x = ( x i 1 i E I ) be such that x / 9E Dom( f); clearly X E T. Since d > id in [r,w), then d ( k ) < k implies k < r . So, by the choice of g, (Case (a)), for i E X u n n,rJfl we have f ( x , )= g(x,). Since n U ,,>,J, E 5,we conclude that

whence f ( x / 5)= g(x/ 5). (e) ACDom(g). Let x = ( x , l i E I ) be such that x / ' y € A . We want to show that x / F ~ D o m ( g ) . Let a:A+'?Y be the one-one map which 9,i.e., appears in the definition of the A ( i ) ' s ; then a ( x / T ) E !y { i E I \ i E a ( x / 9)} E 5,whence (cf. definition of a* in Lemma 5.4.13) { i ~ I l x / F t E a * ( i ) } ~ Fby t ; definition of the A ( i ) ' s , x / F E u * ( i ) ej x , E A ( i ) , whence { i € I l x , E A ( i ) } E ? ? . Since A(i)CDom(g,) for all i E I , it follows that { i E I [x, E Dom( g , ) } E 9, whence x / ?o E Dom( g ) . Thus (e) is proved. We have proved that F : % =f+B; by Corollary 5.3.22 we conclude that % =m ~ + B as ,we wanted to prove. Among the various corollaries of Benda's theorem, we mention the following:

COROLLARY 5.4.16. Let 9be a A-regular filter, p an arbitrary similarity type and a;, %; ( i E I ) structures of type p such that %,=B,for all i E I . Then

-

r

PROOF.Let cp E Sent(L,+,+( p)); clearly Sm(cp) < A. Let %: = %, Sm(cp) and Bj= Bi Sm(cp). Theorem 5.4.15 applies to these structures, whence F = ,A+ H i I%;/ 9. Since II,

r

and the same holds for the BI's, we obtain

392

LANGUAGES WITH INFINITE QUANTIFIERS

Since rp is arbitrary, we have proved the conclusion.

[Ch. 5,

4

rn

Using (4) and (5) of Lemma 5.4.12 we obtain, for the case A=w:

COROLLARY 5.4.17. Let % be an w-incomplete filter on I and { Iui I i E I } , {Bi I i E I } be families of structures of (an arbitrary) type p such that Iui-Bi for all i E I . Then (1) ~ i ~ , ~ i / ~ ~ ~ ~ ~ , ~ i ~ , B i / ~ ; (2) if, in addition,
4 is a-regular; since cf(2'0)>~,, by 5.4.12 (6), L)D is not 2xo-regular. Let 'u = ( A ) , 23 = ( B ) be structures of type 0 such that 7 = 2'0 and = 22n0; clearly %=B: also

jijp7

3 = 22-0 > 2x0.

Therefore, the sentence of the language L p ) + p ) + ( 0 )stating "there are at but not in 23?"/ L'i, . rn most 2"o elements" is true in 'u/ This shows the necessity of the condition that % be A-regular in Theorem 5.4.15. EXAMPLE 5.4.19 (concerning the restriction on the cardinality of p). This example is a simple variant of Example 5.3.12. Let p consist of w 1 many predicates P, (( < wl). Let 121= w I ,

IBl=wl+ 1,

Pf=

[t;,W,),

P?=",wl].

Ch. 5,

41

APPLICATIONS OF BACK AND FORTH METHOD

393

By exactly the same kind of procedure as in Example 5.3.12, we prove that Th(%) admits elimination of quantifiers; since % c B ,it follows that %i B,whence %=B. Let 9 be a non-principal ultrafilter on w ; then 9 is w-regular. The LW2Jp)-sentence u = 3xA6<,,P6(x) holds in / ' B 9 but not in a"/ 9. Indeed, W,/ 9 E nt
n

Theorem 5.4.15 is the principal and neatest of a series of results along the same lines obtained by BENDA[2]. In his dissertation 131 he proved later that for filters 9 enjoying additional properties, the conclusion of 5.4.15 Without can be strengthened to assert equivalence in the language L,,,,. major difficulties, the reader will be able to find variants of FrGsseEhrenfeucht's Theorem 5.3.1 1 which apply to logical relations between structures induced by more general classes of formulas: by doing this and subsequently adapting the proof of Theorem 5.4.15, he can prove variants of it in the style of Theorem 5.4.1. The author has recently used a variant of the construction carried out in Theorem 5.4.15 to prove the non-finite axiomatizability of certain (first order) theories categorical in power N,;cf. DICKMANN [2].

Application IV. Infinitary properties of models generated by indiscernibles As our last application of extensions of partial isomorphisms, we consider the following type of question. Suppose that the (ordered) structures YL, '% are sets of order-indiscernibles (in Lu,) for the structures %, 23 respectively; moreover, assume that infinitary relations of a special kind hold between % and '% (say, e.g., %(Ex&,)'% for some ordinal p); are these properties reflected in some way in the structures @( %), @( '% ) generated by %, '% ? We shall see that the answer is almost unconditionally positive. The first result of this kind-concerning the relation < &A (AECN, /? €ON)-appears in CHANG[3]; by simple application of the methods developed in 0 3, we will obtain even more general results in a considerably simpler way: indeed, all the results proved below, except Chang's, seem to be new, as well as their proofs. Notice that in Theorem 5.4.21 and its corollaries quantifier ranks are understood in the wide sense for A = w . The general theory of (first-order) indiscernibles has been extensively developed in Chapter 2, 42, and the reader is referred there for details; besides very general ideas, the only technical point of this theory to be

394

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch. 5, I 4

used here is the notation X(Z; %;a), introduced in Definition 2.2.12, which means: (i) 9l is a structure of similarity type y endowed with Skolem functions; (ii) % = ( X , < ) is a set of (L,J order-indiscernibles for 3 ; (iii) Z is a tidy set of Luu(y)-formulas and cp€Z iff there is (or, = equivalently, for all) an increasing sequence x of X , of length Fv(rp) such that 9lt=cp[x]; (iv) % =@(%,a)(=th e Skolem hull of 'X in 9l). Throughout this paragraph y, @, @* will be as follows: (v) y will stand for similarity types endowed with Skolem functions; (vi) @ will be a normal class of L,,-formulas of type { < } containing the formula uo < u , ; (vii) @* will be a normal class of Lmh(y)-formulas. Notice that by Lemma 5.3.17, normality implies that each equality of the form 71

,

(u;, . ..,u,' ) =T 2 ( u2 . ..,U i ) , 9

where T,, term,, is in @*. By the clauses (N3) and (N5) of the definition of normal class (Definition 5.3.14), the requirement (vi) is equivalent to the assertion that @ contains at least one formula involving the symbol <. This requirement also implies that every partial @,-map is order-preserving (caution: throughout this section we use the irreflexive notation for linear orderings). The kind of connection we need between @ and @* is expressed by the following DEFINITION 5.4.20. We say that @ and @* are correlated iff for every aEON: (i) @:[3]#D 3 @e[3]#D0, (ii) @.,*[V]#D * Qe[V]#fl. Our main result is

THEOREM 5.4.21. Let A be an infinite cardinal and /3 an ordinal; let y be a similarity type and @, @* be correlated, normal classes of L,,-formulas (of types { < } and y, respectively), all satisjjing the requirements (u)-(uii) aboue. Let Ex = ( X , <) and 9 =( Y , -4 ) be ordered sets and %, 23 structures of type p such that x(Z;Ex; %) and %(XI; 9;23) hold. If Z n a,*C Z', then

y

&L?i

j

Ch.5, J 41

APPLICATIONS OF BACK AND FORTH METHOD

395

PROOF.Let 4 = (1, I a < p ) be a sequence of sets of partial @,-maps from % into 3 such that 9 : Usi -&??I. We want to define a sequence g* =(I,* la < p ) of sets of partial @.,*-mapsfrom III into 23 such that !f * : III =g.,p2?. We can do this in a rather simple manner since there is a natural way to associate to each partial @,-mapffrom ‘X into 3 , a partial @.,*-mapf” from ‘u into 23. We have already used a construction of this kind (in a slightly different context) in the proof of Theorem 2.2.15. We will briefly describe it and its properties. Since %=@(%),each element aElIII1 can be represented in the form a = ~ ~ [ x..., , ,x,], for some term 7 of p, Fv(T) = n , and some sequence x i , . ..,x, E I ’X 1; moreover, by eventually changing the term 7, we can assume that the sequence is increasing: x i < . .. < x,. This representation is not unique in general. A similar representation exists for 9 4 and 8. The map f” is defined in terms off as follows: and

Dom(f”) = Q(Dom(f), Z)

for each 7ETermr and x , , ...,x,EDom(j). Notice that O* : @(B,%)+Q(B, 23) is uniquely defined by the preceding equality, and that Range(f”) = @(Range(f),23). Exactly as in Theorem 2.2.15 we use the fact that !X is a set of L,,-indiscernibles for III to prove that i f f is an order-preserving partial map from 5X into 9, J, is an L,,( p)-formula, Fv(4) = n , T~,...,T,are p-terms and x i , . ..,x, are increasing sequences of elements of Dom(j) of the appropriate lengths, then:

-

(*>

‘uI-J,tT:ixllY .,.3%ll

* ~ ~ J , [ 7 ~ [ f ( X, . .I. ), ~l ~ [ f ( x , > I l .

( Uproves ~ ) , that f” is When li/ is an equation of the form T ~ ( U ~ ) X T ~(*) well-defined, and when li/ is a quantifier-free formula in @*, (*) provesusing the hypothesis Z n @; CZ’-that f” is a @.,*-map;the details of this are as in Theorem 2.2.15; therefore, we omit them. Having established these simple facts, we set:

I,*={PI~EI,}

fora<

p,

9* =(I,*Iff < p ) . Clearly, I,*#afor each a < D, and Y * is decr is decreasing since 9 is so. The argument above proves that each member f” of U, $2 is a @:-map

396

[Ch. 5,

LANGUAGES WITH INFINITE QUANTIFIERS

54

defined on @(Dom(j), %), which is a substructure of 8.Thus, in order to we need only establish the back and complete the proof of Y * :% &,#3 forth property; we illustrate the argument by proving the “back” part. We assume here that a fixed representation of elements aEl%l in the form T ~ [ X has ] been chosen in a uniform manner; in other words, that we have fixed maps a + T a and a+xa such that xa is an increasing sequence in %, and a = T;[X,]. Similarly for ’?I and 8. Suppose [ + 1 < p and @F+,[V]#kl; l e t ~ E I ~and + l Bc(81, be given. We want to find a g* €1; such that f* L g* and B C Range( g * ) . Using the representation b+rb, b+yb chosen above, define the set yB= U b E B Y b ; since each yb is finite, then < A ; note that B C Q ( Y ~ , B ) . Let f be the map in I,+ which defines f”. Since and a* are correlated, at+, [ V ] # 0 . Using the back property of 4 , we get a gG1, such that f c g and YBCRange(g). Then, it is obvious that f* C g* and that

z
Range( g * ) =@(Range(g ) , B)2 6(YB,B)2 B . This proves the “back” property. The “forth” property of $ * is similarly proved; thus Theorem 5.4.21 is proved. i Using Theorem 5.4.21, we immediately conclude:

-

THEOREM 5.4.22. Under the assumptions of Theorem 5.4.21, we have %(@)%

%(@*P)B,

%(a)%=+ %(@*)B. PROOF.The first assertion is an immediate consequence of Theorems 5.4.21 and 5.3.21, and the second of Theorem 5.4.21 and Corollary 5.3.22. REMARK.Note that if l % l ~ l ‘ ? II and f(xi)=xi ( i = l , ...,n), then

f*(T”[ X I , . ..,x J ) = 79l[ x,, . ..,.I,

z
Therefore, if 4 has the property that for every Z C I ?X 1, and (Y < /3 there is an j € I a such that Z c D o m ( f ) and f rZ= id,, then 4 * has the same property. Thus we can also conclude:

-

THEOREM 5.4.23. Under the assumptions of Theorem 5.4.21 we also have % <$?!

% <@?! *

%<$*B,

a<@*%.

Ch. 5 , 8 41

397

APPLICATIONS OF BACK AND FORTH METHOD

PROOF.By the preceding remark, using Theorem 5.3.23. Applying the preceding results to some of the best known normal classes, we obtain: 5.4.24. Let EON. Let 8 , B be any two tidy structures COROLLARY generated by sets % , ?! of order-indiscernibles,respectively. Suppose that for any atomic formula cp with n free variables and any pair x, y of strictly increasing sequences in % , % , respectiuely, 3kcp[Xl

* Bt-cp,[vl.

Then we have (1) % & A % * %=&B, (2) %(Ex&,)% * %(Ex&,)B, (3) %(Un&x>% * iU(Un&,)B, (4) (Chang) % < &,% * 8 < &,B. Similar results hold for the relations obtained by omitting well as for the relations i$, etc. ,,

p

in the above, as

PROOF.Apply Theorems 5.4.22 and 5.4.23 to each of the following pairs of correlated, normal classes of L,,-formulas:

a Form Ex,,(

a*


Form(',,(

Unm,(<),

PI).

Ex,,(

P).

Unm,(

PI.

REMARK.The reason we have not listed the corresponding result for positive formulas is that %(Po&)%

*

% &A%

(and similarly omitting p). Indeed, each formula cp of the language { < } is equivalent to a positive formula of the same quantifier-rank. To see this, it suffices to express cp in reduced form cp* and observe that each negated atomic component of cp* is equivalent to a positive formula: 1(x
7(xxy)++x
In order to give examples of some infinitary properties of models generated by indiscernibles which follow from the foregoing results, we list

398

[Ch.5, 4

LANGUAGES WITH INFINITE QUANTIFIERS

below several known infinitary properties of ordered sets.

PROPERTY 5.4.25 (Infinitary properties of ordered sets). (1) If !X and % are qA-sets (A an infinite cardinal), then 3E. = a ~ %; moreover, if !X c 9, then % 5 m,63 . In particular, (2) If % and 3' are densely ordered sets without endpoints (or, in general, such that % has first (last) element iff % does), then % ; if, in addition, % '9 , then % < . (3) (Barwise) If % is any ordered set, 9 =(a,, E a,), and p < wl,then

c

r

%(Ex&,) 9. I (4) (Karp) If S EON, S = Zt<8wE (ordinal sum and exponentiation), %, = (6, E S ) , and '9 is any ordered set, then

r

%, =&%,

@ 9.

For well-orderings we have: (5) (Chang) (a) For any /3 EON ( A B + ' , E) <&,(Ap+'.((+ (AB+I, E )

< &A(AP+'*(, E)

l), E )

for any [EON,

whenever ( is a limit ordinal and cf(t) > A .

(b) If /3 is a limit ordinal such that cf( p ) > A, then ( A P , E) i & x ( A B - ( [ + l), E ) ( A p , €5) i &A(Ap.(, E) In particular, (c) For any cardinals A, cf(v) > A,

for any (EON,

whenever [ is a limit ordinal and cf(() > A . K,

v such that w < A < K , A"

< v,

cf(K) > A,

(1)-(3) have already been proved, or nearly. Indeed, the first assertions of both (1) and (2) are immediate consequences of Theorem 5.3.1, Part I, and Theorem 5.3.8 (if in (2) both 5% and 9 have, say, first but not last element, modify the corresponding case of Theorem 5.3.1, Part I, by considering partial maps which send the first element of % onto the first

'

This property holds only under the restricted definition of quantifier rank, while (4) and (5) also hold under the wide definition.

Ch.5, § 41

APPLICATIONS OF BACK A N D FORTH METHOD

399

element of 9). The second assertion of (1) and (2) can be easily proved by the reader, using Theorem 5.3.23. Property (3) is Counterexample 5.3.37. We shall omit the somewhat tedious and purely technical proofs of (4) and (5). See KARP[2] and CHANG[3], respectively. Instead, we shall give an illustration of what can be done by using these infinitary properties of linear orderings. The following is a version of a result which, in various particular cases, was observed by several people (see, e.g. MAKKAI[4], pp. 293-94, and CHANG[2], p. 18).

THEOREM 5.4.26. Let K be an infinite cardinal and cp E Sent( L +.),. If cp has models of arbitrarily large cardinality, then for each power A > K , cp has a model of power < 2‘ which has L,’+-extensions of arbitrarily large cardinality. PROOF.We begin by recalling three facts that have been proved in various parts of this book. (A) There is a sequence ( X u1 v ECN A v > A) of linear orderings with the following properties: (i) !XP is an -q,+-set(hence all ‘3,’s are q,,+-sets); (ii) Xuc X P for A < v < p ; (iii) = 2”.

z,

The proof of (A) has been indicated in Example 2.1.14; indeed, it is enough to put %, = H,,+ for each v > A, v ECN, and take into account statements (B) and (C) of that example (cf. also Exercise 2.1.6). (B) Given a sentence cp of L,,,, where p = Sm(cp), there is a language p’ of power < K and a set 5, of power < K , of La,( p’)-types such that for any structure 8 of type p :

8 k cp w there is an expansion 8’ of 8, of type p’, omitting 5,. This is, indeed, the main fact in the proof of Theorem 4.1.1. (C) Let T be a (first-order) theory in a tidy language of power < K and 5 an arbitrary set of types in this language. If T has models of arbitrarily large cardinality below B(2a)+omitting 5, then for every linearly ordered set X there is a model 2? of T omitting 5 and generated by the set of indiscernibles X ! . This result combines the statement of Theorem 4.3.1 with (one of) the results actually obtained in its proof. We blend (A), (B) and (C) to prove our result. By the hypothesis and (B) we conclude that there are structures of type p’, of arbitrarily large cardinality, which omit simultaneously all types in 5,. Consider the Skolem type p’# associated with p’, which was constructed in Chapter 2, s2.A;

400

-

LANGUAGES WITH INFINITE QUANTIFIERS

-

[Ch. 5, 8 4

since p’ < K, also p’# < K. Introducing Skolem functions in the way indicated by Lemma 2.2.6 in each of the structures mentioned just above, we can further assume that all the structures omitting 5, are of type p’# and are also tidy (ie., models for the set Z of all defining axioms of the Skolem functions of p’”). Applying (C) to T = Z and S=S,, for each of the qA+-sets’X,” given by (A), we obtain a tidy structure B, of type p‘# such that

23, omits S,, (jj) %” is a set of (LJ order-indiscernibles for B,, (jjj) 23, is generated by X,. (j)

By Property 5.4.25( l), we conclude ‘3, i ?YP for A < v < p, where v,

p E CN; by Corollary 5.4.24, B, -

By ( j j ) and

7<

K

<

< h < v, we obtain

Now, put IUV=23,rp. Since each 23” omits S,, using (B) again, we infer that IU, E Mod(rp) for all v ECN, v 2 A. Thus, a, is a model of cp of power < 2, having the 8,’s (for v >A, v ECN) as L,,+-extensions of arbitrarily large cardinality. This is precisely what the conclusion of our theorem asserts. rn If in the preceding theorem we replace Lmx+by L,,, sharper conclusion:

we can obtain a

5.4.27 (MAKKAI [4]). If a sentence of L,+, has models of arbitrarily THEOREM large cardinality, then it has a model of power K which has L,,-extensions in each power > K.

PROOF.Instead of the sequence ( %, I v E CN A v > A) of q,+-sets, we use now: (A’) There is a sequence (9” I v E CN A v 2 K ) with the following properties: (i’) each 9”is a densely ordered set without endpoints; (ii’) % , ~ f 9 o r K~< v < p ; v,pECN; (iii’) % = v. Such a sequence exists by the upward Lowenheim-Skolem theorem (for Law)applied to the theory of densely ordered sets without endpoints. The remainder of the proof is exactly as that of the preceding theorem, replacing X, by ”7, throughout. Instead of Property 5.4.25 (1) apply now mwUSpfor K < p ; v, pECN. rn Property 5.4.25 (2) to conclude ?4” i

Ch. 5,

I 41

APPLICATIONS OF BACK AND FORTH'METHOD

40 1

EXERCISES 5.4.1. Prove that the family F of partial isomorphisms defined in Theorem 5.4.4 has the property F : '23 =P,'"Q. 5.4.2. Is it possible to find a group H such that every h E H belongs to a thin subgroup of H but ( h ) is not thin in H ? 5.4.3. (CHANG[2] for LuJ. Prove that any sentence 'p of L,,, with models of arbitrarily large cardinality also has in each power A > K a model of power X with LA,-extensions of arbitrarily large cardinality. [Hint: recast the proof of Theorem 5.4.26, using Property 5.4.25(5.c), instead of (A).] 5.4.4. Prove the following "poor man's" upward Lowenheim-Skolem theorem for L,, (due to CHANG [2]): Let p be a countable similarity type, K an infinite cardinal, and suppose there exists a cardinal A such that A-+(K")$" (ordinal exponentiation). Then, for every structure % of power A there is a structure '23 such that: (i) 23 < %; (ii) 5 = K ; (iii) '23 has L,,-extensions of arbitrarily large cardinality. [Hint: Skolemize the language p . Using the partition property satisfied by A, find a set % of (LuJ indiscernibles of order type K ' ; take '23 to be the Skolem hull of % in %.I 5.4.5. Prove the relation 5.4.25 (4) (due to KARP[2]). [Hint: let ,$<6, ,$,6 EON. An &interval of Xa is a set of the form [&.a, &.(a I)), for some a < 6; likewise, an a,-interval of €3 3 is a 3I. A set of the form [(w'.a,a), (&.(a + l), a)), for some (Y < 6 and a E 1 translation from an &interval of onto a corresponding w%nterval of %a €3 3 is defined in the obvious way (p+( P,a)). For each 5 < 6 let I, be the family of all maps which are the union of finitely many translations or pairwise disjoint &-intervals. In proving that = (It 16 < 6 ) has the back and forth property, take into account that every &interval is a union of d+'-intervals. Note that the result holds even if quantifier ranks are understood in the wide sense.]

+

5.4.6. (a) Using the back and forth characterization of = m ~prove an analog of Corollary 5.4.3 for full cardinal sums: if ai, Bi ( i = 1,2) are Ig21= 1B,( n \B21=0 and % i = m ~ B( i = 1,2), then structures such that [%,In + 912= - A B ~ B2.The same result holds for extended full cardinal sums (cf. p. 301).

+

402

LANGUAGES WITH INFINITE QUANTIFIERS

[Ch.5, 8 4

(b) Study the possibility of generalizing (a) to arbitrary normal classes of L,,-formulas. (c) Define the full cardinal sum operation for infinitely many summands, and extend (a) to this case.

5.4.7. Prove that if Q, is a normal class of L,, formulas, I a set, 68 an ultrafilter on I , and { 8;I i E I } , {B;I i E I } collections of similar structures such that { i E I1 %;(Q,)Bi} E '3, then

n %/9(Q,)II B ; P .

iEI

iEI

[Hint: look at the proofs of Theorems 5.4.1 and 5.4.15.1 The same result holds if the relation %(Q,)Bis replaced by 8 <@B throughout. Note. The proof of this result suggests several possible generalizations to the case where 9 is just a j l t e r , depending either on special properties of '3, of Q, or of the partial O0-maps which realize the relations 9Ii(Q,)Bj. Investigate some of these generalizations.

APPENDIX A

INDUCTION FOR WELL-FOUNDED RELATIONS AND THE SHEPHERDSON-MOSTOWSKI THEOREM

THEOREM A.l (Induction principle). Let R be a well-founded relation on a class K, and let 0 be a formula. If V u [ u E K r \ u R x ~ @ ( u ) ] ~ @ ( for x ) every x E K,' then Vx(x E K-+@(x)).

PROOF.Assume that the conclusion is false. We observe first that the class u E K A 1@ ( u )has an R-minimal element, i.e., that there is an xoEK such that 1@(xo) and

(*I

V u ( uEK ~ u R x ~ + @ ( u ) ) .

Indeed, by assumption there exists a yo E K such that i @(yo).By Definition 0.1.1 (b,ii), the class 1@ ( u )A u R ~ is, a set. If it is empty, take x,=yo; otherwise, by Definition 0.1.1 (b,i) it has an R-minimal element xo, which obviously has the required properties. By (*) and the hypothesis of the theorem we infer @(xo),contradicting the choice of xo.

THEOREM A.2 (Definition by induction). Let R be a well-founded relation on the class K, and let H be a class (map) such that H : V+ V. Then there exists a unique (class) map F : K+ V such that for all x E K : (**)

r

F ( x ) = H ( F { Y E K IYRX)).

'Recall that if the class K is defined by the formula cp(o), then u E K stands for cp(u).

403

404

INDUCTION FOR WELL-FOUNDED RELATIONS

"4PP. A

PROOF.(1) Uniqueness. Assume that the maps F, F' satisfy (**). Let @ ( u ) be the formula F ( u ) = F'(u). If @ ( u ) holds for all uRx, then F r { u € KluRx}= F ' r {uEKlu Rx } ; by (**) we obtain that F ( x ) = F'(x), i.e., @(x). From Theorem A. 1 we conclude that Vx(x E K+@(x)), which shows that F= F'. (2) Existence. We define first the ancestral relation R induced by R in K:

,.

* xRyv(3n)

X ~ Y

l)(3so...sn-l)[

n-2

1

,A( S , R ~ , + , ) A X R S ~ A S ~ - ~ R ~

r=O

Note tkat (i) 4 2 R and is transitive. (ii) R is the smallest transitive relation on K Sontaining R. For if S C K x K is transitive, R C S and xRy, then either xRy (and therefore xSy), or for some n 2 1 and some so,. .., s n - ,E K,

i

xRsoR .. .Rs,,- I Ry, whence, x s s o s . . .ss,: ,Sy, and by transitivity of S, xSy. Hence k C S. We also_ have: (iii) R is well-founded. IFdeed, suppose that there exists a non-empty subset A of K without R;minimal element; then, for every a € A there exists a b € A such that bRa; thus, either bRa, or there are n 2 1 and so,. ..,s,,-, E K such that bRs,R ...Rsn- ,Ra; in either case there exists c E K such that cRa, showing that A has no R-minimal eleqent, contradicting that R is well-founded. For each a E K, { x E KI xRa} is a set. This is a consequence of Z F s replacement and union axioms. Let

i ( y ) = { u E KI uRy }

( = R - '[ y]);

Since R is well-founded, for each y E K, i ( y ) is a set. Let

Then,

U i"(a)={xEKlxI?a}, n Eo

APP. A1

which shows that { x E KI x d a } is a set. Now we show: (***)

405

INDUCTION FOR WELL-FOUNDED RELATIONS

,.

for every x E K there exists a unique map f, : { y E Kly<x}+ V such thatf,(y)=H(f,r{zEKIzRy}) for eachy E K , yRx.

Before proceeding with the proof of (***) we show:

CLAIM. (***) implies that F = U,,,j, sion of the theorem.

is the map required in the conclu-

This is proved by induction on R. Let y EDom(f,!n Dom(f,.); then yRx and y i x ' ; thus, if zky, then by transitivity of k , ZRXand z i x ' ; hence

{ z E K l t R y ) c { z E Klzdy } cDom(f,) n Dom(f,.). Therefore, if f,(z)=f,.(z) L(Y)=

for all zRy, then we have

~ ( f {r z ~ ~ i z R y )H(f.r )=

{~EKI~RY))=LLY).

Thus, F is a map. Moreover, it is clear that (**) is an immediate consequence of (***).

PROOFOF (***). This goes by intuction on Assume that (***) has been proved for all x E K such that xRu; we prove it for u. An argument similar to that of the claim shows that g = U x E K ,kJx , is a map; moreover:

i.

Dom(g)= U {D o m(f,)lx EKAx &)=

U {ytKlydx}= U i"(u). XEK

?I 1>

xRu I

Setfur U,,!R"(u)=g. Since we wish to get a mapf, such that Dom(f,) = { x E K ( x R u } , we need to definef, on R"o(u)={xEKlxRu}; this should be done as dictated by (***): j,(x) = H ( g r { y E ~ l y ~ x )for) x E R o ( u ) . It is clear that the mapf, satisfies the requirements of (***). Uniqueness of such f, is shown as in (1). This completes the proof of Theorem A.2. H

406

INDUCTION FOR WELL-FOUNDED RELATIONS

“4PP. A

COROLLARY A.3 (The Shepherdson-Mostowski theorem). Let R be a well-founded and extensional relation on the class K. Then there is a unique transitive class M such that ( K, R = ( M , E M >. Furthermore, the isomorphism is unique and satisfies the equation

>

r

F(x)={F(u)luRx} forxEK.

PROOF.Apply Theorem A.2 with H = id, : V - t V , the universal identity map, to obtain a map F : K+ V such that F ( x ) = F 1 { u E KI uRx} = { F ( u ) I uRx}

Let M (****)

= Range(F).

for x E K.

Since

yRx w F ( y ) E F ( x ) fory, x E K ,

then, ( K , R ) = ~ ( M ,E

r~).

Since any other isomorphism between ( K , R ) and ( M , E 1M ) necessarily satisfies (****), F is uniquely determined. Note that M is transitive; let y E x and x E M ; then x = F ( t ) for some t E K, and

x = F ( t )= { F ( u ) I uRt}, whence y = F ( u ) for some u such that uRt, and therefore y E M . Let us finally prove the uniqueness of M , i.e., if X is a transitive class and ( K , R ) ( X , E Y X ) , then X = M . Let G be an isomorphism between ( K , R ) and (A’, E Y X ) ; by induction on R we show that F ( x ) = C(x) for x E K. If x is R-minimal, then F ( x ) = 0 ; if G(x)#0, let y E G(x); since X is transitive, y E X , and y = G ( u ) for some u E K; since G is a n isomorphism, uRx, contradicting that x is R-minimal. Now assume that { u E K 1 uRx} #0,and that G ( u ) = F ( u ) for all u E K, uRx; then y E F ( x ) w y = F ( u ) for some u such that uRx w y = C ( u ) for some u such that uRx w y = C ( u ) for some u such that G ( u )E C(x) Y E G(x);

=

the first and last equivalences hold by the definitions of F and G ; the second by induction hypothesis, and the third because G is a n isomorphism. In particular we have X = Range( G )= Range( F ) = M .

APPENDIX B

AN L,,&, FORMULA WITH NO PRENEX FORM

We present here an Lu,u, formula which is not logically equivalent to any prenex formula of that language. This example is due to Martin Jones and is reproduced here with his permission. Roughly speaking, the formula codes the disjunction of all prenex Lu,u, formulas with one free variable; the coding is done by means of finite maps with values in R(w,); a Piagonal argument then shows that the formula constructed has no prenex form. We begin by proving some absoluteness results needed in the construction.

I. Some absoluteness notions and results DEFINITION B.l. Let M be a class and let cp(ul, ..., u,,) be a formula of the language of set theory with free variables amongst u l , . ..,u,,. cp is said to be M-absolute (or absolute relative to ( M , E r M ) ) just in case for every xl,. . .,xnE M , ( M , E rM)t=cp[x,,..., x,,] w ZFCF.(X, ,..., x,,).

REMARK. (1) We shall write cp(xI,.. .,x,,) instead of ZFCFcp(xl,. ..,x,,). (2) ( M , E r M ) F c p [ x ,,..., x,] is equivalent to q M ( x,,...,x,,), where q Mis the relatiuization of cp to M , inductively defined by: if cp is atomic,

cpM=cp cpM =4

if 9 ,= $ 1 A#,,

5-M

if cp= i+, ~ + I ~ = V U ( U E M + + if~ )cp=Vu+, c p M = l(+">

T ~ = ~ U ( U E M A + if~cp=Iu+. ) 407

408

AN

Lwlwl FORMULA

WITH NO PRENEX FORM

"4PP. €3

(3) The first three clauses show instantly that any quantifier-free formula is M-absolute for each class M . Note that the definition ensures that formulas logically equivalent to M-absolute formulas are also M-absolute. Our next step is to show that bounded quantifier formulas are M absolute for any transitive class M . Recall that a bounded quantifier formula is a formula any subformula of which of the form 3u$ (Vuq, respectively) is such that + = ( u € y ~ O ) (#= ( u Ey+8), respectively). As usual, we shall write ( 3 u Ey)O and (Vu Ey)O instead of 3u(u E y AO) and V u ( u Ey+O), respectively. LEMMA B.2. Zf M is a transitive class and cp is a bounded quantifier formula, then cp is M-absolute.

PROOF.By induction on the structure of cp. Since a subformula of a bounded quantifier formula is also bounded quantifier, it suffices to prove the result when cp is of either form ( 3 u ~ y ) or # (Vu E y ) q . Note that in this case y E Fu(cp), say y = ui for some i , 1 < i < n. Let xl,. . .,x, E M . The induction hypothesis tells us that for every

xEM:

#"(x,,. .*,x,,x)

@

#(XI,

.. .,X,J).

Thus, if cp = (Vu E ui)#, cpM(xI,...,x,) e for all x ~ x ;I , C / ,..., ~ ( x,,x) ~ ~ w for all x E xi, +(x,, . . .,x,,x) @ cp(xp...,x,).

Likewise, if cp = ( 3 u E ui)#,

-

cpM(xI,..., x,) e there is x E x i such that # M ( ~,,...,x,,x) w there is x E xi such that #(x,, . ..,x,,x) cp(xp...,x,).

In both cases the first and last equivalences are trivially true; the second holds by induction hypothesis and transitivity of M ( x E x i * x E M , since x i E M ) . Now we analyze some cases of unbounded quantification. LEMMA B.3. Let M be a transitive class and let cp, 4 be M-absolute formulas. Assume in addition that cp has the following property:

APP. BI

SOME ABSOLUTENESS NOTIONS AND RESULTS

409

For every x , , ...,X , E M , I= V v [ c p ( x , , ...,x,,v)+v

(*)

€

MI.'

Then 3 ~ 9 13,u ( c p ~ + )Vv(cp++) , are M-absolute.

PROOF.(1) Absoluteness of B(v,, ...,vn)=3ucp(u,,. ..,unru). Let xl,. .., x n EM. Then

e M ( X,..., ,

xn)

w ~ u ( u E M A ~ ~ ~ ..., ( x x,,u)) ,, ,x,,,u)) uT ( x 1, .. xn7 u, @ e(x ,,...,xn).

* 3 u ( V E MAP(x1, *

. 9

The first and last equivalences are trivial; the second holds by M absoluteness of rp, and (the relevant part of) the third follows immediately from the assumption (*). (2) Absoluteness of 3 u ( c p ~ + ) .This is a particular case of (1) because cp A has property (*) whenever cp has it, and is M-absolute. (3) Absoluteness of Vu(cp+#). Since is M-absolute, so is 14;by (2), Jo(cp A 14)is M-absolute, and therefore so is i 3u(cp A 14);clearly,

+

+

Vu('p+#)el

3u(TA

14);

hence, Vu(cp+#) is M-absolute. We shall apply these and other absoluteness properties to the case M = R(a2). In the next lemma we give some properties of R(w2) and a list of R (a,)-absolute formulas used in our construction. LEMMA B.4. (1) i) * R(a)CR(p)AR(a)ER(p); ii) x ~ R ( a , * ) 9 ( x ) ~ R ( a ,u) x~ ~ R ( a ~ ) ; iii) ( x c R ( a 2 ) v x ~ R ( a 2T) )<~a , ~ f : x + R ( a ~*) f E R ( a 2 ) . (2) The following formulas are R (a2)-absolute: 2=

{x,y};

t=(x,y);

z= u x ;

Fnt(x) ("x is a function"); t = Dom(x);

t = Range(x);

'Formulas enjoying this property are frequently called M-complete ouer u.

410

AN

LOly,FORMULA WITH NO

PRENEX FORM

z = x ( y ) (“z is the image o f y under x”);’ 2 = x [ y3

(“z is the image set u f y under x”);

Ord(x) (“x is an ordinal”); z = n,’

for each n Ew ;

z=w; -

z =No;

z=R(x);3

z=wi; 2


zER(x);3

( 3 ) (Absoluteness offunction quantifers over R(o,)). If q ( v , ...) is R(w,)absolute, so are the following formulas:

i f ~ ( v , ,... ,vn,...) is R(w,)-absolute, so is:

where Q i E { 3 , V } ( i = l , ...,n ) .

PROOF.(1) (i) and (ii) are trivial. (iii) p(y), p ( f ( y ) ) < w , for each y E x ; since 5? < w I and w, is regular, Y =SUP{P(Y) IY E x } < w , and P = s u ~ { ~ ( f ( y1 ). ~)E x }
’ As a predicate of x , y , z .

’As a predicate of ’ As a predicate of x, z.

z.

Note that A(o) defines R(w,).

APP. BI

CONSTRUCTION OF THE FORMULA

41 1

whencefE R ( a + 4 ) C R(w,). (2) We confine ourselves to a few remarks, leaving the bulk of the proof as an exercise for the reader. For most of these formulas absoluteness follows immediately from Lemmas B.2 and B.3, and the absoluteness of previous formulas. For example, “ 7 = N,” and “ 7 < N,” use property (iii) and B.3. “t = R ( x ) ” and “t E R ( x ) ” require to prove first absoluteness of maps defined by transfinite induction on absolute operations (cf. Theorem A.2). Absoluteness of A follows from Lemma B.3 because each conjunct is absolute and

(3) The prefix (QJ: a-+R(w,)) is equivalent to:

where Q € { V , 3 } . Let $ ( f , a ) be the formula u C w , ... ~ . By (2) and ’Lemma B.2, $(f,a ) is R (w,)-absolute; by (1, (iii)),

i.e., $ ( f , a ) has property (*) of Lemma B.3; therefore, both

are R (a,)-absolute. The last assertion of (3) follows from the first two by induction on n. 11. Construction of the formula

Below we shall denote by 2, the set of all quantifier-free formulas of LUJ{ E}). For n 2 1, a formula ‘p is Z, if it has the form

(3Xi)(VXz)...(Qxn>@, where X , , . . . , X , are disjoint countable sets of variables 8 is a Z, formula, and Q is V or 3 depending on whether n is even or odd, respectively. To present the formula without prenex form we have to code the language and we do this so that symbols and formulae are coded as maps with domain a natural number and values in R(p2). The codes for the

412

AN

Lw,aIFORMULA

WITH NO PRENEX FORM

basic symbols are as follows: Symbol

Notation

1

1 -

-

A V 3

A V -

v

i7-

3

(

(

-

1

)

-

E

E

x ua -

‘a

X

X,

r

t

‘

for each a < w ,

when X c (0,Jtr<wI}=Var, and a = ( a l u , ~ X } for any set t E R ( y )

If a, b are finite maps with, say, Dom(a)= n E w and Dom(b) = m E w , then let ab denote the map such that Dom(ab) = n m,and ab(x) =

+

i

ifx
b(x-n)

The code for atomic formulas is obtained by juxtaposition of the codes for their constituents; for example, the code for “ ( u a ~ : o B ) ”is the map f such that Dom(f) = 5 , and f(0) = (0,5>,

f (1) = (1, a>, f(2) = (0,

e>,

f(3) = (1

P>,

7

f(4) = (036)

If c is a countable set of codes for formulae, then is the code for the conjunction of those formulas; note that r\F is the map f such that

APP. BI

413

CONSTRUCTION OF THE FORMULA

Dom(f)= (0,l } and

f( 1) = (3,

u {Range( 8 ) lg E4).

Thus, it is easy to see how codes for L,l,I-formulas can be obtained just as for first order languages. Using Lemma B.4 it is easy to prove by induction on the structure of formulas that the code of any formula belongs to R(w,) (indeed, it belongs to H(w,)). We denote the code of a formula cp by ‘cp’. We must next make the following series of definitions. Each of the concepts defined below is R(w,)-absolute; in each case, inspection of the formula defined shows that its absoluteness follows at once from Lemmas B. 1-B.4. (1)

At(g)

f-,

(3a,P Ew,)(g=

<E?t)pvg=

E

5);

so cp is atomic e.At(‘cp’)

and At(rcp’) w R(w2)F At[‘cp’]. (11)

~ ( b ) = { f ( f = Ao ~ r f = V i i for some a c b , Z < w , ) u { f l f = 7h for some h E b } u { f l At(f)} u 6;

so if b is a set of codes for formulas, then C ( b ) is the set of codes for formulas obtained by one application of A, V or 1 to members of b. Note that b E R (a2)+ C ( b )E R (w,). (111)

z E So c-f (3f)(Fnt(f) A Dom(f) = 0, AWP E w i ) ( f ( p ) = C ( U Range(frP)) A z E U Range(f)));

so for any formula cp ‘cp’ E So w cp is

Z,.

Note that S,E R(w,). [Proof: z E So + (3!f)(Fnt(n A ...); hence there is f :w , + R ( w , ) such that S 0 c U Range(n; since Range(f)€R(o,) (Lemma B.4(1,(iii))), it follows that S,ER(W,)]. Also note that by Lemma B.4

414

AN

Lu,u,FORMULA

WITH NO PRENEX FORM

P P P . €3

so K is a transitive, well-founded ordering of So and { x I x K a } E R(w,). By Theorem A.2 we can use K to make definitions by induction. (V) q ( z , x ) = a t , z E S , A F n t ( x ) A [@a, Ew &(z = ‘tiaE vcl V z = c,= up’) A ff = { Ly, P }) v ( 3 y E Dom(x))(z = 7 Y A ~ Y =)a ) v (3t Dom(x))((z = r\tv z = A a = Ux [ t ] ) ]

vi)

so, if x E R ( o , ) , then

R (w,)

I= q (z, x) = a w

q (z, x) = a ;

now define Fv(z)= a

z E So A ( I d ) [Fnt(d) A Dom(p) = So A(vx E &,)(d(x)= q(x,dr { Y IYKx}) A a = d(Z))];

t)

so if rp is a 2 , formula, then Fv( ‘rp’) also note

=a

w a = { (Y I v, is a free variable of rp} ;

APP. BI

415

CONSTRUCTION OF THE FORMULA

Studying this definition we see that if cp is Z, then PO('cp'9f)

moreover,

* Nw,)I-cp,[fl;

R (a21I- Po[sJ-1* Po( s,f), (VII) We now give a definition schema for S,: z E S,

so

( g a l . ..anc o 1 ) ( 3 gE S,)(a,. ..a, are countable, pairwise disjoint

t)

A Z = ( 3 X,,)(V

rcpl~Snej cp is Z,

and R(o,)t=gES,

Thus,

and

* gE'S,.

X O 2 ) . * * ( Q

Xon)g).

416

f-,

AN

L,,,,

FORMULA WITH NO PRENEX FORM

“4PP. B

@ a , . ..a,, cwl)(3g E &,)[a,. ..a, are countable, pairwise disjoint Az=(3

. X , ) . - . ( QX u n k

so

T,(‘rp’,y) e rp is Z, with one free variable and R(w,)t-rp[y].

Now let T ( x , y )be the formula v , , , T , ( x , y ) . If T were equivalent to a Z,, formula for some nEo, then clearly it would exist an r E w and a Z, formula T ( X ) such that

which is absurd. Thus, we conclude that T is not equivalent to any Z,, formula and hence it is the formula we required.

REMARK. It is a routine matter to extend this counterexample to the case of regular uncountable K , instead of 0,. Details are left as an exercise for the reader.

APPENDIX C

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

Throughout this book we have taken a purely model-theoretic approach to infinitary languages. Since it is sometimes possible to deal with infinitary languages using a Hilbert-type axiomatic approach, we shall describe in this Appendix the principal results which have been obtained in this direction. * We begin by recalling that for a given language L, Val(L) is defined to be the set of all sentences of L which are true in all structures of the appropriate similarity type. These are the so called valid sentences of L. Since this definition involves the totality of all structures one might rightly expect Val(L) to be of cyclopean proportions. However, in certain cases it turns out that Val(L) can be alternatively described by very simple combinatorial procedures. For example, in the case of first order logic, Val(L,,) can be generated from certain finite sets of valid formulas by one or two simple rules, which when applied to valid formulas give valid formulas. The theorem which asserts that all valid formulas of L,, can be so obtained is called the completeness theorem (for L,,), and was first proved by Godel. Unfortunately, completeness of this type is only possible for very few of the languages LKA. As well as providing a manageable tool for dealing with Val(L), completeness results also show that Val(L) lies low in an appropriate hierarchy of definable sets. For example, in the case of first order logic, Val(L,,) is recursively enumerable. This suggests that a possible method of showing incompleteness is to analyze the definability of Val(L) that a satisfactory completeness result would establish, and then proceed to show that Val(L) is not definable in such a way. We shall use an argument of this kind to show that La,,, and a number of other infinitary languages cannot be complete in a satisfactory way. This situation will resemble to a large extent Tarski’s proof that 417

418

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

[App. c

Th((w, +, .,O, 1)) is not first-order definable over ( w , +, .,O, 1). Furthermore, the method of proof of the undefinability of Val(L,,,I) is an adaptation of the method of proof of Tarski’s result. Note that above we have been a little vague with statements of the form “Val(L,,) is recursively enumerable”. Of course what one means is that if the formulas of L,, are coded by integers in some natural way, then the set of codes of members of Val(L,,) is a recursively enumerable set of integers. Hence, to deal with an arbitrary language L in such a way, we shall need a structure 6, (called the coding structure for L) and a particular one-one map cL of Form(L) into (0,I (called the coding map for L). The choice of the coding structure and map is not unique and is motivated by intuitive considerations. In first order logic one usually requires that the coding map be “effective” in some way (this means, in essence, that one can recognize which elements of the coding structure are actually codes, and that any formula could be reconstructed from its code in a finite number of “computable” steps). For example, for L, Godel used originally the coding structure (0, + , .,O, 1) and the well known coding map obtained by using the sequence of prime numbers a n i the Chinese remainder theorem. One could as well use the structure (H(w), E rH(w)) of all hereditarily finite sets. In the case L = LLA, where K and A are infinite cardinals, K 2 A, and K is regular, it is most natural to use the structure (H(K),E r H ( K ) ) of all sets hereditarily of power less than K . This corresponds to the idea of conceiving of L,,-formulas as set-theoretical objects.’ The choice of the coding map largely depends on the particular way L,,-formulas are defined. The coding map we have defined in Appendix B is quite suitable for our purposes here.

I. A review of (positive) completeness and definability results for infinitary logic We now review without proofs some positive completeness and definability results for infinitary languages. This subject was first investigated by Karp, who obtained most of the known results in this area. For the proofs the reader may consult KARP[l], where a complete exposition can be found.

’

KARP [ l ] uses instead ordinal structures of the form ( a , E r a ) (for appropriately chosen a). This choice-which is based on the idea of conceiving of infinitely long formulas as transfinite sequences of symbols-makes matters unnecessarily complicated (for example, the choice of a coding map).

APP. CI

419

REVIEW OF COMPLETENESS A N D DEFINABILITY RESULTS

First and foremost among infinitary logics stands LUI,, which enjoys the simplest and richest completeness and definability properties. Indeed, Karp discovered that LWI,has a complete axiomatization (consisting of the basic system of axiom schemes and rules of inference described below) which is a simple and natural generalization of one of the well-known axiomatic systems for first-order logic. This system is adequate for proofs in the sense that its theorems are exactly the valid sentences of Lu,, (note: proofs in this system have countable length). The proof of this result can be found in KEISLER[3], lecture 4, where Henkin-type techniques are used. Using this result it can also be shown that Val(L,,,) is 8,-definable over the coding structure 6= (H(o,), E r H(w,)) (see KEISLER[3], lecture 9). However, there is no system of axioms and inference rules for LWI,which is adequate for deductions, in the sense that for any set Z of L,,,-sentences and any sentence rp: Zkcp e Zkrp.

[This follows from Exercises 2.3.1 and 2.3.2, Chapter 2, $3.1 This last remark also applies to any language LKA. In order to review results for other infinitary logics LKA,we introduce the following sets of axiom-schemes and rules of inference.

Quantificational axiom-schemes

provided that no variable of X occurs free in rp. (VX)cp+Suby(::rp,'

where X

Var,


The right-hand side denotes the formula obtained by substituting each variable term f(o) in 'p.

DEX

by the

420

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

[App. c

provided that 'p contains no free occurrence of the variable u E X within the scope of a quantifier binding some variable of the term f ( u ) . Equality axiom-schemes t=t

for any term t,

for 0 < 6 < K and (uCI [ < 6 ) c Fv('p), provided that no variable in U C < 8 ( F ~ ( tFv(t;)) J~ is bound in 'p. Propositional rules of inference

Modus ponens:

Conjunction:

Quantificational rule of inference

Generalization :

The deduction system consisting of the axiom-schemes and rules displayed above will be called the basic system for LKA.These axioms and rules belong to any of the systems of proofs for infinitary logic considered below. Karp's completeness theorem says that the basic system is a complete axiomatization for La+,. For other infinitary logics no such neat result is possible. We have to enrich the basic system with further axioms and rules which express set-theoretical laws. This is so because the very meanings of the formulas of LK,, when X 2 wl, presupposes set-theoretical facts. We shall consider the following:

APP. Cl

REVIEW OF COMPLETENESS AND DEFINABILITY RESULTS

42 1

Distributive axiom-schemes

For a given infinite cardinal y, (D,) denotes the following axiom-scheme

where, for a gven collection (qtI[< y) of formulas, each qa@ is either qr or qq6 for some 5 > y , and for every g : y + y the set ( ( ~ a , ~ ( ~ ) I a < y } contains both qI and i q Efor some [< y.’ Set-theoretical rules of inference

For a given infinite cardinal y, (IC,) denotes the following rule of inference y-independent choices:

7


where the sets Xo, Xt, YIGVar have cardinality
’

These distributive axioms were first considered by Chang in the study of the representation problem for Boolean algebras. Thus, they are sometimes referred to as “Chang’s distributive laws”.

422

AXIOMATIZABILITY, COMPLETENESS A N D DEFINABILITY RESULTS

[App. c

REMARK. In order to make these rules less mysterious, let us write them as axiom-schemes. Thus the axiom-scheme corresponding to (IC,) takes the form:

(*I

Ac<,(3XE)y+

uc<,~E)Al
For example, if X e = Fv(@ for each [< y , this axiom holds in a structure 8 iff the collection of sets { $15 < y } , where 9: = { f:Xe+18zI( 18t= q+[f]} and cpF#0, has a choice function. There is a similar interpretation for (DC,) related to a generalized form of the axiom of dependent choices. Note that (*) is a formula of L,, only when y
APP. CI

SCOTT’S UNDEFINABILITY THEOREM

423

It can be shown that the axioms and rules added to the basic system in the last mentioned results cannot be removed without making the resulting system incomplete. Likewise, if the rules (DC,) are replaced by the corresponding rules (IC,) and X > w, the resulting axiom system will not be sufficient for proving all the valid sentences of LKA;for example, KARP[ 11, 9 12.4, proves (4) If the similarity type p contains a binary relation symbol, then there is a valid formula of LKA(p),K > X > w , stating an axiom of countable dependent choices, which is not provable in the basic system plus the axioms (D,) and the rules (IC,) for all y < K . The completeness results (1)-(3) listed above imply that the corresponding sets Val(L,,) are first-order definable over the coding structures (H(K),E ~H(K)). Indeed, by arguments similar to those used in KEISLER [3], lectures 8 and 9, it is possible to establish that: (5) For K and h satisfying the requirements of (1) and (2a) above, E ~H(K)) (i.e., definable by a firstVal(L,,) is &-definable over (H(K), order formula of type { € } of the form 3xVyO(x,y), where all quantifiers occarring in O are bounded quantifiers). ( 6 ) In the cases (2b) an d (3), Val(L,,) is XI-definable over (H(K),

E

r

H (K )).

The reason why quantifier prefixes of the form 3V are needed in the cases (1) and (2a) (whereas existential quantifiers suffice in the case Lu,J is the presence of the distribution axiom-schemes (D,). In the cases (2b) and (3) K is strongly inaccessible and only distributive axioms (D,) with y < K are used. This implies that the set yY of all maps g : y + y which is mentioned in (D,) is an element of H(K),and a corresponding universal quantifier becomes bounded, thus allowing Z ,-definability. In contrast to the preceding results, we shall next see that when K is a successor cardinal, the set Val(L,,) is so complicated that it is not even E 1H(K)).This result proves conclusively LKK( €)-definable over (H(K), that the logics L,+,+ do not admit a satisfactory complete axiomatization. Indeed, one may expect that a reasonable axiomatization for a logical system should always produce a set of theorems which is set theoretically definable (i.e., first-order definable over the coding structure), but Val(L,+,+) does not satisfy any definability requirement, even if we go as far as to allow definability by L,+,+-formulas. 11. Scott’s undefinability theorem

This result was discovered by Scott around 1960, but it was published for the first time in Karp’s book [I], Chapter 14. The proof given below is

424

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

[App. c

essentially the same as that of L~PEZ-ESCOBAR [6]. We shall carry out the details of Scott’s theorem only for the language L,l,,l. The proof for the general case of L,+,.+is a straightforward generalization of the proof given below. As it was said before, we shall use @ = (H(w,), E rH(w,)) as coding structure for LylUl.We shall only consider formulas of similarity type { E } , where E is a binary relation symbol. This restriction can be considerably weakened. The coding map will be the one used in Appendix B. In that appendix we used (R(o,), E R(w,)) as coding structure, but an examination of the coding rules of pp. 41 1 4 1 3 shows that those rules do actually code Lulul( E)-formulas by hereditarily countable sets. This is easily proved by induction on L,I,I-formulas, by observing first that finite maps into H(wJ belong to H(o,), and then observing that (i) if rp is A‘k, where ‘k is a countable set of L,I,I-formulas, then ‘rp’ (the code of rp) is the map such that Dom(j) = (0, l } and

f(0) = (091>, f(1) = (3, U {Range(‘#’)

since

< No and

I$ E q } ) ;

by induction hypothesis Range(‘$’) E H(w,) for each

$ E ‘k, it is clear that Range(f) E H ( w , ) ; (ii) if rp is the formula (3X)rc/, where X c V a r = {ua ( a
+

f(0) = (0,5), f ( l ) = (0,3),

f(2) = (2, a),

f(3) = (0,6),

f( n) = $’( n - 4) for 4 < n < Dom( ‘I)’); since by induction hypothesis Range(‘$’) EH(w,), then it is clear that Range(j) EH(w,). We also said before that Scott’s argument is an adaptation of Tarski’s theorem on the undefinability over ( w , +, .,O, 1) of the set of sentences valid in arithmetic. However, there is one important difference: while (a,+, .,O, 1) is not characterizable up to isomorphism by any set of first

APP. CI

SCOTT’S WDEFINABILITY THEOREM

425

order sentences, the coding structure @ is characterizable up to isomorphism by a single LwIwI(E)-sentence. This observation allowed Scott to obtain an even stronger theorem; namely:

SCOTT’S UNDEFINABILITY THEOREM (for LulWl). The set Val(L,I,I(E)) of all valid sentences of L,l,l(E) is not definable over Q by an L,I,I(E)-formula. Before proving this theorem we shall need some preliminary lemmas. LEMMA C.l. ( a ) @ is characterizable in L,,ul (E), i.e., there is a sentence y of this language such that for every structure % of type { E}, %I-y

H

%=@,

(b) @ is pointwise definable in Lul,(E), i.e., for every h EH(w,) there is a formula cp, E Form,(L,I,I(E))such that for all h’ EH(wl), @I-cp,,[h’]w h=h’.

PROOF.(a) Take y to be the conjunction of the sentences Q,.( i= 1,. ..,4) of Examples 1.2.16 and 1.2.18. (b) cp,,(vo) is defined by induction on the rank of h, as follows: if p(h)=O (i.e., h = B ) , let cp,,(vo) be 13u,(u,Evo); if p(h)>O and cpx has been defined for all xEH(w,) such that p(x)
Since each h EH(o,) is countable, cph belongs to Lul,(E). By induction on max{p(h),p(h’)} it is easily verified that @kcp,,[h’] w h=h’..

This lemma shows, in particular, that the code ‘cp’ of each formula cp of LulWl is LUI,-definable over Q.Next we show that various sets of formulas and other related sets are first-order definable over Q. LEMMA C.2. There exist L,,(E)-formulas Form, Form’, Sent, with one free variable and an LUu(E)-formula Df with two free variables, such that: Form@=

{ h E H(w,) I there exists a formula h =‘ip’},

ip

E Form(L,I,I( E)) and

426

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

[App. c

(Form”)@= { h E H(w,) I fhere exists a formula rp of L,+,,(E) having uo as its onb free variable, and h = ‘rp’}, (Sent)@ = { h €H(w,) Ifhere exists rp ESent(LUIaI(E))and h = ‘rp’}, (Df)@ = { ( h , h ’ ) l h , h ’ E H ( w , ) and h’=‘rph’}.

PROOF.The definition of these formulas is very much a routine matter. As illustration we outline the construction of Form, leaving the rest as exercises for the reader. Recall that the following set-theoretical notions and operations are first-order definable: ‘‘u is an ordinal”, “ u is a function”, “ u is a countable set” (denoted by Ord(u), Fnt(u), 6
At(g)-(3a,p)[Ord(a)AOrd( P ) A ( g = Y ~ ; ~ ~ v g = < E i j ~ ) ] , C’(.) is the following set-theoretical operation: C ’ ( b ) = b u {flAt(f)} u { f l f = I h for some h E b } u { f l f = XZ or f = VZ for some a 6 , Z < No} u { f l for some a c o r d , Z < No,and some h E b,

Note that C ’ ( b ) is a definable operation, and that if b EH(wl) is a set of codes for LWI,,-formulas,then C ’ ( b ) E H ( w , and ) it is the set of codes for formulas which are either atomic, or in b, or obtained by one application of -1, V, A, 3,or V to members of b. The details of this assertion are left to the reader. (111)

Form(u)-(3f)[Fnt(

f ) A Vx(x E Dom(f)-Ord(x))

A (v’P)[Ord( P)-+f( A 0E Range(f)l.

u

=

c(u Range(f 1P)>1

The verification that Form actually defines the set displayed in the statement of the lemma is again left as an exercise for the reader. The explicit definition of the formulas Form’ and Sent uses slight modifications of (IV), (V) and the definition of Fv, in Appendix B. The formula Df is easily constructed using the preceding steps and the inductive definition of rph given in Lemma C.l(b).

APP. CI

SCOTT’S UNDEFINABILITY THEOREM

427

Next we want to define the notion of satisfaction in @, i.e., a formula which for every h E H ( w l ) and EForml(L,I,,(E)) expresses the fact that @ k + [ h ] .Since every member of H(wJ is LUI,-definable over 6 (Lemma C.l(b)) and the notion of formula with one free variable is first-order definable over @ (Lemma C.2), we can do this by using (the code of) the LUIUl( E)-formula

+

3U(g?h(U)A+(U))

which, by Lemma C.l(b), has the same meaning as @ k + [ h ] .

LEMMAC.3. There is a (first-order)formula Sat with three free variables, such that for all h,, h , , h 2 E H ( w , ) , @ + Sat[h,, h , , h2] e there exists a formula

+

E Form,(L,I,I(E)) such that h,= ‘4‘ and h, = r(3Uo)(fp),(Uo) A+(u,))’.

PROOF.According to our definition of Appendix B, the code of the formula ( 3 u , ) ( f p , , ( u , ) ~ + ( ~ , ) )is the map f such that Dom(f)= 8 and

For j = O , 1, 2, 3, 4, 5 and 7, let $(w) be the set-theoretical formula with the only free variable w which defines f(j). For example, O,(w) is the formula “ w x ( 1,0)”, i.e., vz[z E

V Z = u 2 ) A 13 u ( u E u3) A v w l ( w1 E u4*W= u3)A v w2(w2 E u l + + w 2 w u4) A v w 3 ( w 3E U2++w3= u4V w3=u3))].

428

[App. c

AXIOMATIZABILITY, COMPLETENESS AND DEPINABILITY RESULTS

Instead of the formula defining f(6), we consider the following: 86(w,uo,u3)= w43,Range(u3) u Range(u,)). Let now B(u,, u,, u3) be

Loosely speaking, 8 defines the “function” obtained by replacing ‘(ph,’ and by the variables u3 and uo respectively. Now let Sat(uo,ul, u,) be

r+’

Formo(uo)~~~,(Df(u,,~3)~e(~01~2,~3)). The following chain of equivalences, which can easily be checked by the reader using Lemma C.2, completes the proof of Lemma C.3.

@ + Sat[ho,hl,h21 @ @I- Formo[h,] and there is h3EH(wl) such that @+Df[h,,h,] and 8+8Ih,,h,,h,l -3 @ + Formo[h,], there is h, such that h3=r9)hl(u0)1and h, is a function, Dom(h,) = 8, h 2 ( j )=f ( j ) for j = 0,. . .,5,7 and h,(6) = (3, Range( h3)u Range( h,)) H there is a formula )I of Lw,wl(E)with u, as its only free variable such that h, = ‘+’ and h, is a function Dom(l.3 = 8, h,(j) f o r j =O,. ..,5,7, and hz(6) = (3, RangeCcp,,(u,Y) u Ranger+’))) @ there is a formula of Lwlwl(E) with uo as its only free variable such that h,=‘+’ and h2=r(3u,)(cp,l(u,)~+(u,))’.

+

The next theorem is known as the fixedpoint theorem.

THEOREM C.4. For every 8 EForm(L,,,,(E)) having uariable, there exists 1c, E Sent(LwIwI(E)) such that

uo us its

onh

free

@++~3u,((pr~.(u0)~8(u0))

(or, equiualently, @++ iff@+@[‘+’]).

PROOF.Let cp(v,) be the formula ( 3 ~ ~ ) ( S a t ( u , , ~ , ,Au8(ul)). ~) The required sentence 4 is (3uo)(qr4,(uJA$J(u,)).

APP. CI

429

SCOTT'S UNDEFINABILITY THEOREM

-

Indeed, by Lemmas C.2 and C.3, the following are easily seen to be equivalent:

@+J,

-

there is h E H(w,) such that @ k (cp., A +)[h] there is h eH(w,) such that h =rep' and @ + +[h] e @ +"I * there is h , EH(w,) such that @ + Sat['#', '#', h,] and 6 F 8 [h,] e there exists h , €H(w,) such that h, ='(3vo)(cp.+.(uo)AC$(U~))'and 6r-@[h,l there exists h, EH(w,) such that h , = ' I ) ' and @ k B [ h , ]e @ + (~uo)(P,v(Vo) A @(uo))-

-

We now prove the analog of Tarski's theorem on the undefinability of truth. THEOREM C.5. There does not exist a formula 8 of Lulul(E)with vo as its on& free variable such that for all aESent(LUIuI(E)),

(*)

@+a@

-

g

~

q

~

~

~

~

.

PROOF.Assume that such a formula 8 exists. Applying Theorem C.4 to 1 0 we get a sentence J, such that

(**I

Q+J,

Case 1. @P+. Then, by (**), @+ (*) when (I = J,. Case 2. @ + -14. Then, by (**), dict (*) when a = i$.

@+

+ q r q ] .

is['+'].

But these two facts contradict

@+e['J,'].Again, these two facts contra-

Thus, the assumption that 8 exists leads to a contradiction.

Finally, we prove our main result. THEOREM C.6 (Scott's undefinability theorem). The set of all (codes o f ) valid Lu1J E)-sentences is not LUlJ E)-definable over 6. PROOF.Suppose, by way of contradiction, that there exists an LulJE) formula Val(uo) with vo as its only free variable such that for all h EH(w,): (*) @ + Val[h] e there exists J , E S ~ ~ ~ ( L , ~ , ~such ( E ) )that k # and h=rJ,'.

430

AXIOMATIZABILITY, COMPLETENESS AND DEFINABILITY RESULTS

[App. c

Let y be the sentence which characterizes @ up to isomorphism (Lemma C.l(a)); then, for every # ESent(LUI,,(E)):

(**I

@t-#

by (*) we infer,

(***)

*

ky+#;

w @~==Val[‘y+#’].

@t-#

Recall that y+# is i y v # , so that ry-++’ is the map

f(0)= (0,2),

f( 1) = ( 3 , Range(‘ i y ’ ) u Range(‘#’)). As in Lemma C.3, let O,(w) be the formula w 4 0 , 2 ) and let Ol(w,uo,ul) be the formula w e ( 3 , Range(u,)u Range(u,)). Let 02(uo,ul,u2) be

Fnt(u,) ~Dom(u2)st!2/\Vw((0,w)Eu,t,~,(w)) A Vw(< 1,w>E ~ ~ - ( 8 w,,uo,u,)).

Let Quo) be

We show that for every +ESent(LUIUI(E)) we have

@++

(****)

* @+Or#’].

Indeed, by the definition of 0, Lemma C.l and (***), the following equivalences hold:

@t-O[‘#’I * there are h,, h , ~ H ( w , such ) that ,!ij~cp,,~,[h,], @t-O,[r#’,h,,h,] and @ + Val[h,] .a there are h , , h2EH(wl) such that h , = ‘ i y ’ , h, is a function, Dom(h,) = 2, h,(O) = ( 0 , 2 ) , h2(1) = ( 3 , Range(h,) u Range(‘+’)) and @ 1- Val[h,] w there is h, EH(w,) such that h, = i y v #’ and @I- Val[h,] H Q k Val[‘y+#’] ts

@+#.

Thus (****) holds. But this contradicts Theorem C.5, and this contradiction shows that a formula Val with the property (*) does not exist. W

APPENDIX D

RESULTS ON THE CONSTRUCTIBLE UNIVERSE

In this appendix we present a brief survey of the most outstanding results in the metamathematics of set theory obtained in the sixties by application of model-theoretic ideas and methods. These results throw light on the structure of the constructible universe for set theory (i.e., the model (L, E )) under various large cardinal hypotheses. The first result in this direcfion is Scott’s theorem that the existence of measurable cardinal implies the negation of Godel’s axiom of constructibility (Theorem 0.4.29); this result uses the ultraproduct construction. The subject was taken up and further developed in Rowbottom’s dissertation [ 11. But it was Silver, in his thesis [ 11, who obtained far reaching results by combining the techniques of partition calculus (Chapter 0, 5 5 ) with the ideas of construction of models from indiscernibles developed in Chapter 2, $2. Some of Silver’s results were further refined by KUNEN[ 11, [2], using as a tool the method of iterated ultrapowers, discovered by Gaifman. We also mention the known results concerning (relative) consistency of partition cardinals of the forms K + ( w ) ~ < and ~ K+(w,),<~. The basic tool for the application of the partition calculus and model theory to the study of the constructible universe is the following set of conditions: (1) K , AECNAW
E Form(L,,( E)), x is a sequence of elements and for all AECN- { w } such that Range(x) G L(A), (L(A), E ) t- tp,[Xl.

(3) There is a closed class C, CN - { w } c C c ON, such that (a) “C is homogeneous for L”: 43 1

432

"4PP. D

RESULTS ON THE CONSTRUCTIBLE UNIVERSE

For every QJ E Form(L,,( E)) and every x, y , increasing, denumerable sequences with range included in C, (cp,,x)~Satw ((~,y)ESat. (b) "C generates L": If z E L, then there is QJ E Form(L,,( every y E L,

E)) and x E C" such

that for

(q,y*x) €Sat w y = z . (i.e., every element of L is definable in terms of some elements of C). (c) If AECN-{w}, then A is the Ath element of C. Condition (2) gives meaning to the notion of satisfaction in (L, E) (in spite of the fact that L is a proper class); (L, E ) ~ - Q J [just ~ ] means (QJ,x) E Sat. Under conditions (1) and (3) the following becomes a well-defined set of natural numbers: the set of all Godel numbers (in a suitable arithmetization of the language) of €-formulas satisfied in (L, E) by increasing sequence of uncountable cardinals. This set is called O#, and can also be defined as the set of all Godel numbers of formulas QJ such that

The following theorem lists some of the important consequences of properties (1H3):

THEOREM D.l. Under the assumptions (1)-(3), the following are true: (i) (Silver-Solovay) O# $ L; (ii) (Silver-Solovay) O# is a A; subset of w ; (iii) L contains a weakly compact cardinal (L,

€)+

"up# is weakly compact";'

(iv) ( L , E)+ wp*+(w)>" Using the results of Chapter 0, $ 5 1,5, condition (iv) is easily seen to imply (i). Indeed, if O# EL, then Lot = L (cf. property (4), p. 1 I), whence

(L, E ) F w,+(w),'".

' Recall that XLais the denotation of the class X in the model La.

APP. Dl

RESULTS ON THE CONSTRUCTIBLE UNfVERSE

Since i ( w ) denotes the first

K

433

such that K+(w)Z<~, we get

(L,E)ki(Ld) w (Corollary 0.5.17), and this property relativizes to (L, E), we conclude

(L,

E)+

-

3 a ( a EInAw<

(Y

< wl),

which evidently is false since (L, E) is a model of ZFC. Likewise, one can prove directly that (iii) (i). The relevance of (ii) above lies in the fact that on the basis of ZFC all 2: subsets of w are constructible (Shoenfield). In 1967 Solovay proved, using a different method, that the existence of a non-constructible A: subset of w is consistent with ZFC. We also remark that:

LEMMAD.2. (I) implies 9 ( w ) n L

< No.

PROOF(In sketch). From GODEL[l], p. 54, we borrow the fact that every x E 9 ( w ) n L equals some F( p), with p < wl, where F is the function that enumerates all constructible sets; this implies that 9 ( w ) n L G L(w,) and, furthermore, (L(w2), E)+ “9(a) can be put in one-one correspondence with some ordinal”. By (l), this also holds in (L(w,), E), whence 9 ( w ) n L ( w , )

9( w ) n L c L(w,), the result is proved.

< No; since

Once these consequences of (1)+3) are established, the next step consists in searching for conditions under which these three properties hold. The following theorem lists some such conditions.

THEOREM D.3. Each of the following conditions (A)-(E) implies (1)-(3): (A) (SILVER [I], [2]) For some uncountable cardinal K there is a set X C K of power K such that ( X , E Y X ) is a set of order-indiscernibles for (L(K), E). In particular, (B) There is a cardinal K such that K + ( C O , ) , < ~ . (C) (KUNEN[1],[2]) There exists an efementaly map from (L, E) into (L, E) which moves some ordinal. (D) (KUNEN[ 1],[2]) There is some infinite cardinal K such fhat (L(K), E) has a proper elementary substructure of power K .

434

RESULTS ON THE CONSTRUCTIBLE UNIVERSE

"4PP. D

(E) (KUNEN[1],[2]) There exists a cardinal K such that for any binary relation R C K X K and every h < K , the structure ( ~ , hR, has an elementary substructure % of power K where An 131 is countable.

>

A cardinal as in (E) is called a Rowbottom cardinal; the reader can easily restate (E) in terms of partition properties; thus, he can prove that K

is a Ramsey cardinal

* K is a Rowbottom cardinal.

(A)-(E) do not exhaust the list of known properties that imply (1)-(3). A remarkable one, not listed above, is a model-theoretic statement known as Chang's conjecture. Combining (B) and (E) of the preceding theorem with Lemma D.2, we obtain: D.4 (Rowbottom). If either there exists a Rowbottom cardinal COROLLARY or a cardinal K such that K + ( w ~ ) ~ < " ,then there are on!y denumerably many constructible subsets of w.

To close this Appendix we mention some of the known (relative) consistency results concerning partition cardinals; all the results, except the last, are due to Silver. THEOREM D.5. If K+(u),<", then there is a cardinal X < K which is I f : indescribable for each m, n E w . In particular, any such K is larger than the first weakly compact, strongly inaccessible cardinal > w . THEOREM D.6. If K+(w)~<" and ilh' is a transitive model of ZF (or even of a weaker set theory) containing K, then ilh'I= K+(w)2<". COROLLARY D.7. If ZF+(~K)(K-+(W),<") is consistent.

is consistent, then ZF+

(v=L)+

(~K)(K+(W)>")

In particular, V = L is (relatively) consistent with the existence of weakly compact, strongly inaccessible cardinals > w and II",indescribable cardinals for each n, m E w. This result should be compared with

D.8. (ROWBOTTOM [ 11). ZFC -t(v= L) THEOREM sistent.

+(~K)(K+(w,)~<")

is incon-

Thus, K+(w)~<" is the strongest property (in terms of size) known to hi. (relatively) consistent with V = L.

APPENDIX E

REAL-VALUED MEASURABLE CARDINALS AND THE TREE PROPERTY

In this appendix we present a simple and elegant argument due to SILVER [ 11, [2], which shows that all real-valued measurable cardinals' enjoy the tree property considered in Chapter 3, 13.C (property (10)). There we proved that, for strongly inaccessible cardinals, this property is equivalent to weak compactness. However. Theorem E.l below implies that if the exisience of real-valued measurable cardinals is consistent with ZFC, then the tree property does not imply strong inaccessibility.

THEOREM E. 1. Zf K E RM, then every tree of power branch of power K .

K

K

has the tree property:

in which all levels have power


has a

PROOF.Let ( A , T ) be such a tree; without loss of generality we may assume that A = K . Let La be the athlevel of ( K , T ) ; we can also assume that L, =,0 for a 2 K , since otherwise the tree has a branch of power K . Notice that L,#,0 for a < K , for otherwise the tree has power < K . For X E Klet: R(X)=

{ y EKIXTy}

u {X}.

Let p be a additive measure on define for a < K :

K

and set t ( x ) = p ( R ( x ) ) . Finally

w (a)= sup{ t (x) I x E La}.

< K and p is < K-additive. Hence We remark first that p(L,) = 0 since we have p( U,,,L,) = 0 for each y < K . Clearly for a < K : UXEL.R(X) = K - U s < J , ;

' See Definition 0.4.7 (b). 435

436

[APP. E

REAL-VALUED MEASURABLE CARDINALS

hence p ( U X E L , R ( x ) ) =1, and since < K there is at least one x EL, for which t ( x ) > 0. Hence w ( a ) > 0 for each a < K . Observe next that xTy implies that R ( y ) c R ( x ) ; hence t ( y ) < t ( x ) . If a < P, for each y ELB there is XEL, such that x T y ; therefore w( P )

< w(a).

Thus, w is a decreasing map from K into (0, I]. But cf(K) > No (Corollary 0.4.8), and since there is no well-ordered or conversely well-ordered sequence in [0,1] of cardinality >No’, it follows that w is eventually constant: there are Po€ K and ro€(O, 11 such that w ( a ) = ro for every a 2 Po. Let

S=

-

{x€K(t(x)>

jr0 and x € L , for some a 2 Po}.

By the preceding conclusion, S S = K. We also have:

n L, #&3

for each

Po< a < K ; therefore

for if y E S we show that there is x E S n LBOsuch that y E R ( x ) ; clearly there is x E L B osuch that x T y , i.e. y E R ( x ) , and since xTy implies t ( y ) < t ( x ) , it follows that x E S. Since < K and K is regular, we conclude

SnR(I)

= K. that there exists some z E S n LBOfor which Suppose that S n R ( z ) has two T-incomparable elements, u, u say. Then R ( u ) n R ( u ) = f J and since R ( u ) u R ( u ) c R ( z ) , we have: t ( z ) > t ( u ) + t ( u ) > ro, contradicting w( Po)= ro. Therefore any two members of S n R ( z ) are T-comparable, i.e., S n R ( z ) is a chain. It is easy to verify that the set ( S n R ( z ) ) U { x E K I x T z } is a branch; it has power K .

The preceding theorem is important in view of Solovay’s consistency results concerning real-valued measurable cardinals, which prove that the following theories are equiconsistent :

ZFC + “there is an 2M cardinal” ZFC + “there is an RM cardinal ZFC+“2’oECRM”.

< 2*0”

I One way of proving this is by recalling that (0, I] has the “Souslin property”: every family of disjoint open intervals of [0, I] is countable.

APP. El

REAL-VALUED MEASURABLE CARDINALS

431

Therefore if “ R M Z 0 ” is consistent with ZFC, the statement ‘‘2’0 has the tree property” is also consistent with ZFC. The GCH fails in this situation. We mention in passing that Kunen has recently announced a considerable improvement of Theorem E.l but, seemingly, at the sacrifice of the simplicity of the above argument (see SILVER[2], pp. 4&47). Finally, we mention the following results of Jensen which relate weak compactness with another property of trees; these concepts grew out of work on a different problem in set theory.

THEOREM E.2 (JENSEN[2]). The following statement holds in (L, E), the constructible universe: ‘tfor every tree”.

K

E CN-{ u},K

is weak& compact iff there is no K - Souslin

(cf. Definition 0.3.3(c)). Since in Z F + V = L it is provable that every weakly compact cardinal is strongly inaccessible, we get:

E.3 (JENSEN [l]). V = L COROLLARY

“there is an N,-Souslin tree”.

The negation of the last assertion is known under the name “Souslin’s hypothesis”; it is equivalent to a classical conjecture about the structure of the order of the real numbers.

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w. R E I N H A R I ) . l [I]

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[I]

SUBJECT INDEX

absolute formula 407 absolutely complete field of sets 29 accessible, -cardinal 4 -ordinal 271 addition, ordinal 8 additive measure, - completely 34 - countably 33 -finitely 32 -A34 admissible, -conjunction 264 - disjunction 264 admits a union 95-96 alephs 3 4 No-categorical 137, 37 1

atom 28 atomic, -formula 64 -measure 38 -model 371 atomless Boolean algebra 37 1 automorphism, -group 80, 123, 128 -theorem 123 axiom of constructibility 11 axioms and rules of inference for infinitary logic 419421 axiom-schemes, -distributive 421 - equality 420 -propositional 419 - quantificational 419420

N1-1

-free group 383-384 - Souslin tree 437 a-extendable cardinal 204 a-quantifier closed 333 athapproximation 357 algebra, - atomic Boolean 219, 222 - atomless Boolean 371 - complete Boolean 219, 222 - finitely generated 74 -a- 33 - topological universal 2 18 ancestral relation 404 antichain 32 Archinedian, -field 82 -group 136 arithmetic, - saturated models 107-108 - LUlw-axioms 72-73 - number of countable models 105 arity 60'

back and fGrth, - characterization of elementary equivalence 321 - method 308-310 back and forth property, - general form 329-330 - less than X at a time 314 -one at a time 311, 319 Baer-Specker theorem 38638 1 Barwise-Kunen-Morley theorem 278-28 1 basic deduction system for L,, 419-420 Benda's theorem 387-391 beth, - exponentiation laws 6 -number 5 --a 5 Beth definability theorem 133 Boolean, - combination 89, 296 - o-algebra 33 Boolean algebra, - atomic 2 19, 222

445

446

SURJk('7 1NI)I:X

- atomless 37 I - complete 219, 222

categorical. ~- No- 137, 371 bound occurrence 64 -K127, 128 bounded quantifier formula, 263, 408 chain 32, 153 - absoluteness of 408 Chang. bountiful class of groups 215 - distributive laws 421 isomorphism theorem 135. 345 346 canonical, c theorem o n Hanf and Morley numbers - isomorphism 354 230-233 -map 30. 175. 176 Chang-hi-Suszko theorem. IS3 - monomorphism 179 - counterexamples to 153- 155 cardinal, 3 characteristic subgroup 80 - accessible 4 characteristically simple group 80. 216 - beth 5 choices, - cofinality of 4 dependent 42 1 -compact 159 independent 42 I - countably real-valued measurable 35, chosen along 390 436 class. - countably two-valued measurable 37 - elementary 91 - exponentiation laws 6. 7 ~elementary in the wider sense 91 - extendable 204 - languages 65 -- y-incompact I59 n y m a l 326 - inaccessible 5 -of ZF I - incompact 159 - pseudo-elementary 91 - limit 4 - pseudo-elementary in the wider sense 91 - measurable 37. 42. 180 -- quantifier-closed 333 -number o f a set 3 transitive 2 - nrhorder characterizable 290 closed. - II;-describable 183 - expression 64 - II",indescribable 183 formula 64 - Ramsey 56. 204 relatively 15 - real-valued measurable 35. 435 ~ 4 3 7 set o f sentences 266 - regular 4 term 62 -~ Rowbottom 434 under subformulas 87 - 2;-describable 183 code 411 413 - 2;-indescribable 183 coding. 411 412 - singular 4 -map 418 -strong limit 5 structure 41 I. 413. 418. 424 - strongly accessible 4 cofinality 4 -strongly compact 159, 199-202 collapsing isomorphism 3, 406 - strongly inaccessible 5 compact. - strongly incompact 159 - cardinal 159 -successor 4 topological space 2 18. 2 19-22 I - two-valued measurable 37. 42 compatible 102 -weakly compact 159. 184-185 complete. -weakly inaccessible 5 -- absolutely 29 cardinal sum. 70 Boolean algebra 219. 222 -extended full 71. 301 - field of sets 28. 29 -full 71 filter 28 -simple 70 < K - 28 ~

~

~~

~

~

~

~

~

~

~

SUBJM'I INDEX

- LXA-diagram 98 - lattice 222, 223 - linear ordering 222, 223 -over v 409 - partial ordering 221. 222 - relative to a frame 265 -sentence 256 - set of sentences 68 -structure over X 178 - structure over a set. 225 - type 103 - ultrafilter 28 - uniform space 218 completely additive measure 34 completeness results. 158, 417 -for L,, -for L,,, 128 - for infinitary logic 4 2 2 4 2 3 componen 1. -connected 352 , o f type 352 - principal 357 conjunctive-universal-existential formula 90 connected, -component 352 -elements 352 -relation 2 -set 352 connectives 59 consistency property 266 consistent. - relative to a frame 265 - set of sentences 68 - type 102 constant, - individual 60 -term 62 constructibility, II -axiom constructible, - results on 4 3 1 4 3 4 11 -set -universe II correlated classes of formulas 394 countable embedding theorem 346 countably, -additive non-trivial measure 33 - real-valued measurable cardinal 35, 436 - real-valued measure space 35 - two-valued measurable cardinal 37

447

-- two-valued measure space 37 criterion, - Tarski's 95

definability, - counterexamples to 143-144 - property relative to a language 132 - results for valid sentences of infinitary logic 423 - theorems 132-1 34 definable, - explicitly 132 - implicitly 132 -L145 - LKA- 179 -La,372 defining axiom 113 denotation 66 dense linear ordering. -elimination of quantifiers I I2 -- saturated 109-1 12 dependent choices 421 describable cardinal. -IIL183 - 2;- 183 diagonal, -- of a sequence of classes of ordinals 20 - of an operation 21 diagram. '97 - complete LKA-98 direct, -- product 7&71, 373 -sum 70-71. 373-374 directed, - < p - 96 -p96 -set 96 discrete topological space 217 distributive, - axiom-schemes 42 1 -lattice 222 divisible abelian group 378 domain 2, 3 downward Lowenheim-Skolem theorem, 205-2 12 -counterexamples to 212-215 Ehrenfeucht-Mostowski theorem 119-121 element, - index of 357

448

SUBJECT INDEX

- order of 352 -terminal 352 -type of 352 elementary, -class 91 - class in the wider sense 91 -map 95 -substructure 95 elimination of quantifiers 112, 322-324 embeddable, - LK),- 94 embedding 94 endowed with Skolem functions 115 endpoint 138 enumerating function, 13 -properties of 13-14 equality, - axiom-schemes 420 -symbol 59 equivalence, -'P265 equivalent, - LK,- 69 - +- 297 Erdos-Hajnal theorem 55-56 Erdos-Rado theorem 44-45, 46-48 q,-set, 109-1 10, 212-213 - isomorphism of 3 I 1-3 12 existential, -formula 89 - -universal formula 90 expansion, 69 - inessential 69 explicitly definable relative to a sentence 132 exponentiation, -laws 6, 7 -ordinal 8 -semi-weak 5 -weak 5 expression, 63 -closed 64 extendable cardinal 204 extended full cardinal sum 71, 301 extension, -of maps 310 - of partial isomorphisms 3 11, 3 13 -property 177, 199 extensional, -relation 2

extensionality axiom 302 extremally disconnected space 219 false 68 field, -Archimedian 82 - multiplicative group of 93 -of complex numbers 128 -of a relation 2 field of sets, 28 - absolutely complete 29 - < r-complete 28 - < r-complete generated by a set 29 field of subsets 28 filter, 28 -A-regular 384 - < K-complete 28 - non-principal 28 - w-incomplete 384385 -on a set 28 -prime 28 7principal 28 -proper 28 - properties of 385-386 finite sets 73-74 finitely generated algebra 74 fixed point theorem 428 formula, 64 -absolute 407 -atomic 64 - Boolean combination 89 -bounded quantifier 263, 408 -closed 64 - conjunctive-universal-existential 90 -existential 89 - existential-universal 90 - induction 64 - M-complete over u 409 - nth order 72 -of L,, 64 -Of Lmx 65 65 -of L, -II: 72 -positive 88 -prenex 90 - prenex-universal 284286, 289 64 -property - quantifier-free 89 - reduced 89, 326327 -Z: 72

449

SUBJECT INDEX

-tree 138 -universal 8 - universal-existential 90 Fra‘i sse-Ehrenfeucht theorem 321-322 frame, 264 - for a formula 264 -Skolem 265 -tidy 265 free, - abelian group 378 64 -occurrence full cardinal sum 71 function, -domain 3 -enumerating 13 -image 3 -image set 3 - inverse image set 3 -range 3 - scattering 205 fupction symbol, 60 - arity 60 - defining axiom 113 fundamental theorem 109, 316317

P .

y-compact language 159

y -incompact,

- cardinal 159 - language 159

r-,

-map 93 - morphism 93, 329 -substructure 94, 329 generalized continuum hypothesis 7 generated, -finitely 74 -subgroup 78 - substructure 3 10, 373 Gregory’s counterexample 363-370 group, - rc,-free 383-384 - characteristically simple 80, 216 - divisible abelian 378 - free abelian 378 - ti-free 379 -of automorphisms 80, 123 - similarity type of 78 - simple 79, 216 - torsion 79, 216 - torsion-free abelian 378

-transitively

normal 86, 216

Hanf,

- incompactness

theorem 166170 -lemma 227 -number 227 - -Scott theorem 184 Hausdorff space 217 height, - of a well-founded structure 274 - of an element of a tree 31 Helling’s theorem 251-255 hereditarily of power < K 82, 83, 213-214, 263 holds 68 homogeneous, -model 109 -set, subset 43, 51 - set for a structure 119 -structure 109 -weakly 109 -weakly& 109 homomorphic image 94 homomorphism 94 hull, -Skolem 116 hyperinaccessible, -of type 0 15, 19 -of type 1 15 -of type TJ 21 hyper-hyperinaccessible of type TJ 26 hypothesis, - generalized continuum 7 - limit cardinal 7-8 ideal 29 image, 3 -#et 3 immediate successor 32, 84 implicitly defined 132, 136137 inaccessible, -cardinal 5 -of type p 12, 203 -weakly 5 incompact, - cardinal 159 -language 159 -strongly 159 incompactness theorem 166170 incomparable 32

450

SUBJECI INDEX

inconsistent 68 independent choices 421 indescribable cardinal, --rIz- 183 -E:183 index 357 indiscernibles, 119 - < - 119 - automorphism theorem for 123 -models generated by 118-128 -n-\k- 269 -order119 269 -stretching theorem 121-122 -subset property 121 individual constant symbol 60 induction, - definition by 403405 -on formulas 64 - on reduced formulas 327 - on well-founded relations 3, 403 inessential expansion 69 initial, -ordinal 3 - segment 77, 271 interpolants 129 interpolating 129 interpolation, - counterexamples to 142-143, 262-263 -property relative to a language 131 -theorems 129-132 interval, -af401 -open 112 inverse image set 3 isomorphism, 2, 94 - canonical 354 - collapsing 3, 406 - A-partial 3 10 -partial 310 iteration, 21 - normalized 2 1

--*-

ti-categorical 127, 128 ti-free group 379 K +-accessible ordinal 271 K semi-accessible ordinal 274-275 < K-complete, - field of sets 28 -filter 28 . +

Karp’s theorems 3 17-3 19, 3 19-320 Kueker, - lemma 338-339 -theorem 383 Kueker-Keisler theorem 382 Kunen’s theorems 205, 273-274. 287-288 L-definable 145

LA,

--definable

179 of 59-60 - -embeddable 94 --equivalent 69 - expressions of 63 - formulas of 64 --map 94 --substructure 94 --theory 68 I-,+,-accessible ordinal 271 L,,-definable 372 A-, -additive measure 34 - extension property 3 1 1, 3 13 - homogeneous 109 - partial isomorphism 3 10 - regular filter 384 105 -saturated - universal 108 language. 130 - y-compact I59 - y-incompact 159 - incompact I59 - singuIar cardinality 85 - strongly incompact 159 lattice, - complete 222, 223 -distributive 222 lemma, - Hanf 227 - Kueker 338-339 -pivotal 34-35 - Rowbottom 51 - Sierpinski 49 length of a tree 31 less than A at a time back and forth property 3 14 limit, -cardinal 4 -ordinal 4 -points 14 - description

SUBJECT INDEX

limit cardinal hypothesis 7-8 linear map 190 linear ordering, - complete 222, 223 logical consequence 68 logical symbols 59 logically valid 68 Lopez-Escobar’s theorem 256-260 Lo; theorem for , L 174-175 Lowenheim-Skolem theorem, - downward 205-2 12 - “poor man’s’’ 401 - upward 117-1 18, 224225 M-, -absolute formula 407 ~- complete over c’ 409 Magidor’s theorem 202 Mahlo’s theorem 19 makes A-regular 384 Makkai’s theorem 93 map. canonical 30, 175. 176 - coding 418 -- elementary 95 -extension of 310 r- 93 - LKA- 94 -linear 390 -partial 93 maximal type 103 measure, -atomic 37 completely additive 34 - countably additive 33 -- A-additive 34 - normal 42. 57-58 measurable cardinal, 37, 42, 180 - countably real-valued 35 - countably two-valued 37 - real-valued 35, 435437 -- two-valued 37 measure space, 33 - countably real-valued 35 - countably two-valued 37 - real-valued 35 - two-valued 37 metric space 100, 218, 224 metrizable topological space 217, 218 model, 68 ~

45 1

-atomic 371 - existence theorem 266 - generated by indiscernibles 118-128 modus ponens 420 monomorphism, 30, 94 - canonical 179 Morley numbers 228-229 Morley’s theorem on omitting types 243250 Morley-Lopez Escobar-Helling theorem 255 morphism, -r- 93,329 -partial 93 Mostowski’s theorem 292 multiplicative group of fields 93 n-ary 60 natural structure 10 (n,P)-good pair 252 Nebres’ theorem 156 negative occurrence 88 n-C-indiscernibles 269 n-good pair 247 non-logical symbols 60 non-principal filter 22 normal, - class of formulas 326 - measure 42, 57-58 - ultrafilter 42, 203-204 normal subgroup, 78 - generated 78 normalized iteration 2 1 v-wide 347 number, - Hanf 227 - Morley 228-229 nth order, - characterizable cardinal 290 -formulas 72 number of places 60 occurrence, -bound 64 -free 64 -negative 88 -positive 88 a-,

-chain 153 -family 153 - incomplete filter 384385

452

SUBJECT INDEX

w b t e r v a l 401 omegas 3 4 omit 102 one at a time back and forth property 311. 319 open interval 112 order, - -indiscernibles 119 - infinitary properties of 398 -of an element 352 - scattered 100 -type 2 ordinal, 2 -addition 8 - arithmetical operations 8 - cofinality of 4 - exponentiation 8 -initial 3 - K+-accessible 271 - K +-semi-accessible 274-275 -limit 4 - L,+, -accessible 271 -product 8 -successor 4 pair, ---good 247 - (n,P)-good 252 partial isomorphism, 310 - Cartesian product of 374 -reduced product of 389 partial ordering, - complete 221, 222 .rr,-numbers 12-13 IIf-indescribable cardinals 185

n:,

- -describable cardinal 183 -formula 72 - -indescribable cardinal 183 pins down 271 pivotal lemma 34-35 Polish space 224 positive, -formula 88 - occurrence 88 prenex form, - for higher order formulas 72 - in infinitary languages 91 prenex form theorem, -counterexamples to 151-152, 41 1 4 1 6

-weak 157 prenex formula 90 preservation theorems, 134, 153, 156 -counterexamples to 153-155 preserved, - under unions of chains 153 - under unions of w-families 153 prime filter 28 principal, -component 357 -filter 28 - type 371 product, -direct 7&71, 373 - of partial maps 374, 389 -ordinal 8 -reduced 30 projection 373 proper filter 28 property, -back and forth 311, 314, 319, 329-330 - censistency 266 - definability 132 -extension 177, 199 -formula 64 -interpolation 131 -Souslin 57 -subset 121 -tree 185, 435 propositional, - axiom-schemes 419 - rules of inference 420 pseudo-elemen tary, -class 91 - class in the wider sense 91 *-9

- equivalence

265 - indiscernibles 269 quantificational,

-axiom-schemes

419420 - rule of inference 420 quantifier-closed 333 quantifier-free formula 89 quantifier-length 96 quantifier-rank (=depth), 90 - restricted notion 325 -wide notion 325 quantifiers, 59 -elimination of 112, 322-324

SUBJECT INDEX

Ramsey, - cardinal 56, 204 -theorem 45 rank 10 real-valued, - measurable cardinal 35, 435-437 - measure space 35 realize 102 reduced, - formula 89, 326327 -power 30 -product 30 reduct 69 regular filter 384 relation, - ancestral 404 -connected 2 -domain 2 -extensional 2 -field 2 :range 2 -well-founded 2 relation symbol, 60 - arity 60 relational, -structure 65 -system 65 relative, -closure 16 -- definability property 132 -interpolation property 131 - PC-class 92 - PC,-class 92 relatively closed set 15 relativization 407 restricted, 70 - notion of quantifier-rank 325 rigid 145, 361, 363 restriction 70 p-directed 96
453

-propositional

420 420 - set-theoretical 421 - quantificational

satisfaction, 66-67 of 427 - properties of 67 satisfies 66-67 saturated structure, 105 -existence theorem 1 6 1 0 7 scattered order 100 scattering function 205 Scott, - generalized isomorphism theorem 3 4 C 343 - isomorphism theorem 135, 345-346 -sentence 135, 137, 343, 346 -theorem on normal measures 42 - undefinability theorem 425, 429-430 second order, -implicit definition 137 -Lmm 130, 131 segment, - initial 77, 271 semantically consistent 68 semi-accessible ordinaI 274-275 semi-weak exponentiation 5 sentence, 64 - complete 256 - interpolant 129 - interpolating 129 - Scott 135, 137, 343, 346 separable space 224 set, - finite 73-74 -hereditarily of power < K 82, 83, 213214, 263 -homogeneous 43, 51 -homogeneous for a structure 119 -rank of 10 -transitive 2, 82 set-theoretical rules of inference 421 Shelah’s theorem 239-240 Shepherdson-Mostowski theorem 3, 406 Sierpinski lemma 49 2: subsets of o 433 - coding

z,;

- -describable cardinal 183 -formula 72 - -indescribable cardinal 183

454

SUBJECT INDEX

Silver’s theorem 435436 Silver-Solovay theorem 432 similarity type, 60 -endowed with Skolem functions 115 -operations on 61-62 - Skolem 115 simp1e, - cardinal sum 70 - group 79, 216 singular, -cardinal 4 - cardinality languages 85 Skolem, -frame 265 -hull 116 -structure I15 -type 115 Souslin, -property 57 -tree 32, 437 space, -compact 218, 219-221 - complete uniform 2 18 -discrete 217 - extremally disconnected 2 19 - Hausdorff 2 I7 -metric 100, 218, 224 - metrizable 217, 218 -Polish 224 - separable 224 -Stone 218-219 - T,(i=O, ...,4) 217 - topological 2 16 spectrum 140 Stone space 218-219 stretching theorem 121-122 strong limit cardinal 5 strongly, - accessible cardinal 4 - compact cardinal 159, 199-202 - inaccessible cardinal 5 - incompact cardinal 159 - incompact language 159 structure, 65 -coding 41 1, 413,418,424 -complete over X 178 -complete over a set 225 - denotation in 66 -natural 10 - relational 65

-Skolem 115 -tidy 115 - universe of 65 -well-founded 274 subformula, 87 - closed under 87 subgroup, - characteristic 80 -generated 78 -normal 78 -thin 379 subset, -homogeneous 43, 51 -property of indiscernibles substructure, 69 - elementary 95 -r- 94,329 - generated 3 10, 373 - LKA- 94 successor, -cardinal 4 - immediate 32, 84 -ordinal 4 sum, -cardinal 70 -direct 70, 373-374 -disjoint 70 - extended full 71, 301 -full 71 symbol, -equality 59 - functional 60 - individual constant 60 -logical 59 - non-logical 60 - relational 60

121

Tarski, -criterion 95 - union theorem 96-97 term, 62 -closed 62 -constant 62 terminal element 352 theorem, - automorphism 123 - Baer-Specker 38C38 1 - Barwise-Kunen-Morley 278-28 I - Benda 387-391 - Beth definability 133

SUBJECT lNDEX

--hang

337 isomorphism 135, 345-346 on Hanf and Morley numbers

455

- upward Lowenheim-Skolem 117-1 18, 224225 --hang’s - Voptnka-Hrbatek 202 230-233 theory, 68 - Chang-Kok-Suszko 153 -, L 68 - completeness for Lo,, 133 thin subgroup 379 - countable embedding 346 tidy, - definability for L,,, 133 -frame 265 - downward Lowenheim-Skolem 205-212 -set of sentences I15 - Ehrenfeucht-Mostowski 119-121 - structure 115 -- Erdos-Hajnal 55-56 7,-space ( i = O , ...,4) 217 - Erdos-Rado 4445, 4&48 topological, - existence of saturated structures 106107 - group 218 - fixed point 428 - module 218 - Fraisse-Ehrenfeucht 321-322 -ring 218 -fundamental 109 -space 216, 217 - generalized Scott isomorphism 34Ck343 torsion group 79, 216 - Hanf‘s incompactness 166-170 torsion-free abelian group 378 - Hanf-Scott 184 transitive, - Helling 25 1-255 - class, set 2, 82 -interpolation for Lo,, 129-130 -closure 82 ‘Karp 3 17-3 19, 3 19-320 transitively normal group 86, 216 -Kueker 383 translation 353, 401 - Kueker-Keisler 382 tree, 31, 84 - Kunen 205, 273-274, 287-288, 433-434 - antichain of 32 - L6pez Escobar 256-260 -binary 32 -KO; for L,, 174-175 -branch 32 - Magidor 202 - chain of 32 -Mahlo 19 -height of element 31 -Makkai 93 - infinitary axioms 84 - model existence 266 - K-Souslin 32 - Morley’s on omitting types 243-250 - length of 3 1 - Morley-Lopez Escobar-Helling 255 - length (=height) of element 3 1 - Mostowski 292 - level of 31 -Nebres 156 - path (=branch) of 32 - “poor man’s” Lowenheim-Skolem 401 - property 185, 435 -preservation for L,,, 134 - root of 32 -Ramsey 45 -well-founded 137-138 - Rowbottom 434 true 68 -Scott 42 two-valued, - Scott’s isomorphism 135, 345-346 - measurable cardinal 37 - Scott’s undefinability 425, 429430 -measure 37 - Shelah 239-240 type, - Shepherdson-Mostowski 3, 406 - endowed with Skolem functions 1 15 - Silver 435-436 - of an element 352 - Silver-Solovay 432 - similarity 60 - stretching 121-122 -Skolem 115 - Tarski’s union 9 6 9 7 type of formulas, 102 - Ulam 37, 38 - compatible 102 - Ulam-Tarski 3 9 4 0 - complete 103 - Chang’s

456 -consistent -maximal -omitted - principal -realized

SUBJECT INDEX

102 103 102 37 1 102

Ulam’s theorems 37, 38 Ulam-Tarski theorem 39-40 ultrafilter, 28 - < K-complete 28 - normal 42, 203-204 -on a set 28 ultrapower 30 ultraproduct 30, 31 undefinability theorem 425, 429430 union, 96 - admits 95-96 - of chains 153 -of w-families 153 -theorem 96-97 universal, - -existential formula 90 -formula 89 - structure 108 109 -weakly universe 65 upper directed 96 upward Lowenheim-Skolem theorem, 1 17118, 224-225 - “poor man’s’’ 401

valid, 68 -sentences of L 417 value of a term 66 variables, 59 -bound 64 64 -free Vopenka-HrbaEek theorem 202 weak, - exponentiation 5 - prenex form theorem 157 weakly, -compact 159, 184-185 -homogeneous 109 - inaccessible 5 - incompact 159 -A-homogeneous 109 -A-universal 109 - universal 109 well-founded, - f a m u la tree 138 - induction principle 3, 403 -relation 2 - structure 274 -tree 137-138 well-ordering, 74-78 - non-definability of 256-260 wide, - notion of quantifier-rank 325 -Y347

LIST OF SYMBOLS

ZF ZFC VBG MK

Zermelo-Fraenkel axiomatic system for set theory ZF plus the axiom of choice Von Neumann-Bernays-Godel axiomatic system for set theory Morse-Kelley axiomatic system for set theory the membership relation; the ordering of the ordinal numbers the class of all ordinal numbers inclusion, proper inclusion the restriction of a relation R X " to a set Y cX the isomorphism relation

c

(x,

<j

f:A+B Dom( R ); Range( R )

-

f [XI X

CN CN' w: N

the order-type of an ordered set ( x , < ) a mapfof A into B domain of the binary relation R ; range of R the image of a set X under the map f the cardinal number of the set x the class of all infinite cardinal numbers the class of all cardinal numbers the map enumerating all infinite cardinal numbers set-theoretic union limit (=supremum) of a set of ordinal numbers the first infinite cardinal 457

1 1

2 2 2

3-4 4

4 4

458

LIST OF SYMBOLS

Cartesian product; Cartesian power sum and product of cardinals exponentiation of cardinals the power-set of the set x the set of all subsets of x of power < A the cofinality of the (limit) ordinal a the successor. predecessor, of the ordinal

P

K+;K-

Sing Reg AC SAC In

the cardinal successor, predecessor, of a cardinal K the class of all singular cardinals the class of all regular cardinals the class of all accessible cardinals the class of all strongly accessible cardinals the class of (strongly) inaccessible cardinals the class of weakly inaccessible cardinals the athbeth-cardinal starting from K the athbeth-cardinal weak exponentiation of cardinals semi-weak exponentiation of cardinals the function "aleph-index" of beth- cardinals the generalized continuum hypothesis the limit cardinal hypothesis (ordinal) sum of ordinal numbers (ordinal) product of ordinal numbers (ordinal) exponentiation of ordinal numbers the collection of all sets of rank < cr the rank of a set x the class of all sets a model for ZF, ZFC the a t hnatural structure the class of all constructible sets, the class of all sets constructible from a

4 4 4 4

4-5 4-5

5 7 7-8 8 8 8 9 10 10 10 10 11

LIST OF SYMBOLS

the axiom of constructibility the class of .$'h-order fixed points relative to X the enumerating function of & ( X ) the class of all limit points of X CON Mahlo operation applied to X LON the operation dual to M(X)

x* F, F W , fi, FD

*F M(R-q)(X)

diagonal of a sequence of classes of ordinals the normalization of a sequence of classes of ordinals iterations and various classes associated to the operation F normalized iterations and various classes associated to an operation F the dual of an operation F the qth generaiized iteration of M relative to a well-ordering R

MRiX1 M (AC) M E (...( ME(AC))...) Dij

rl,E,A,/:T

the < K-complete field of subsets of a set X generated by a family 5 C 9( X ) equivalence modulo a filter '3 ; the equivalence class off modulo 9 the reduced product (ultraproduct) modulo 5 of a family of sets { A , 1 i E I } the canonical map of A into A '/5 height on an element of a tree length of a tree athlevel of a tree measure space countably real-valued measure space (measurable cardinal)

459

11 12-13 12-13 13 14 15 19 19 20 20 21 21 2 1-22 22

25 25 25-26 27 29 30 30 30 31 31 31 33 35

460

LIST OF SYMBOLS

real-valued measure space (measurable cardinal) countably two-valued measure space (measurable cardinal) two-valued measure space (measurable cardinal) the set of r-element subsets of X

RM 2CM 2M

the least cardinal rn such that rn+(n)t the class of all cardinals rn such that rn-t(rn): ( = t h e class of all weakly compact, strongly inaccessible cardinals) the set of all finite subsets of M the smallest cardinal rn such that

rn+(A)>" K-+(a>:,

K+(a);",

; ( a ) (for ordinal a ) L A 1-

,A, 3,=,v,

(infinitary) logical symbols

Rel, Fnt, Ind P.,Y

(metalinguistic) symbols for similarity types arity functions

35 37 37 43 43 44 46,48 50 51 4, 51

52

58 59-60 59 60 60-6 1 60 6 1-62

the set of all terms of type p ; (metalinguistic) symbols for terms the set of all formulas, of all formulas with Y free variables, of all sentences, of LKA(

P)

the set of all atomic formulas of similarity type p, the set of all free variables of a formula cp

62

64

64

LIST OF SYMBOLS

infinitary (class) languages the universe of a structure 8 the denotation of a symbol S in a structure 8 f*g

tYfl

the value of a term t in a structure % for the assignment f

the L,,-theory of 8 LK,-equivalence

the class of all isomorphs to substructures (of power
Aut( G), Aut(8) H(K)

similarity type for groups normal subgroup subgroup (normal subgroup) of G generated by A the group of automorphisms of G the set of all sets hereditarily of power < K

46 1

65 66 66 66 66 66-67 68 68 68 69 69 69

69 69-70 70 70-7 1 71. 71 72 74,256 75 76 78,382 78 78 80, 123 82

462

LIST OF SYMBOLS

the transitive closure of x the set of all subformulas of cp the set of all positive formulas of LKh(p) the set of all reduced formulas of LK,( p) the set of all quantifier-free formulas of LKh(

the set of all Boolean combinations of length < K of formulas in A (for A C Form(L,,)) the set of all existential formulas of LK,+ the set of all universal formulas of L,, the set of all universal-existential (conjunctive universal-existential) formulas of L, the set of all prenex formulas of L,, the quantifier-rank of cp

82 87 87 88 89 89

89,296 89 89

90 90 90 91 91 92 92 92 92 94 94 94 94

the union of a set K of structures the quantifier length of r the diagram of M the complete L,, (L,,)-diagram of M

96 96 97 98 110 113 114

LIST OF SYMBOLS

the Skolem hull of X in 91; the Skolem hull of 3 Form,(X, <) X(Z; ( X , <>;a)

rn(Z,C) I= LO Interp(L, L') Def(L, L')

SP(W I, SI, c

the spectrum of K the classes of all incompact, of all strongly incompact, and of all compact cardinals the complete structure over h

- (2%- ) describable, indescribable cardinal; T; EP the extension property the Hanf number of a language L (set of h ( L ) ,h ( X ) sentences X ) the class of well-ordering of cofinality at least 2<' Morley numbers for omitting (LWu-) in,, n, types the Hanf number of L,,, ( = h(L,+,))

the set of admissible conjunctions of 9

the set of all accessible ordinals the first ordinal not in AC,, the height of ( X , R ) "K

cp-

'4

Q,

is +-equivalent to 4

463

114-1 15 115

116 119 119 123 131 131 132 140 159 178 183 177 227 224 228,229 230 26 1 264 265 266 267 27 I 273 274 28 1 297

464

0

LIST OF SYMBOLS

the partial map between substructures generated by the empty set families of partial maps

L,,-equivalence

up to quantifier-rank p

sequences of families of partial isomorphisms

the order of a the direct sum of the structures 9li (i E I )

310 311 313 3 14 319 3 19-320 3 19-320 324 326 326 328-329 329-330 330 352 353 373 390 40 1 404 432