Classification Theory Revised Edition

CLASSIFICATION THEORY AND THE NUMBER OF NON-ISOMORPHIC MODELS

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 9 2

Editors

J. BARWISE, Stanford H. J. KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam

NORTH-HOLLAND AMSTERDAM. NEW YORK * OXFORD .TOKYO

CLASSIFICATION THEORY AND THE NUMBER OF NON-ISOMORPHIC MODELS R E V I S E D EDITION

S. S H E L A H The Hebrew University, Jerusalem, Israel

1990

NORTH-HOLLAND AMSTERDAM'NEW YORK.OXFORD.TOKY0

ELSEVIER SCIENCE PUBLISHERS B.V Sara Burgerhartstraat 25 P.O. Box 211 lo00 AE Amsterdam, The Netherlands Dietributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y.10010, U.S.A.

Library of Congress Cataloging-in-PublicationData Shelah, Saharon. Classification theory and the number of non-isomorphic models/S. Shelah. -- 2nd ed. p. cm. -- (Studies in logic and the foundations of mathematics; v. 92) Includes bibliographical references. ISBN M - 7 0 2 W 1 1. Model theory. I. Title. 11. Series. QA9.7.S53 1990 51 1’.8--dc20

8S29756 CIP

First edition : 1978 Second edition : 1990

ISBN : 0 444 70280 1

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1990

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the Publisher, Elsevier Science Publishera B.V., P.O. Box 211, lo00 AE Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the Publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or idem contained in the material herein.

Printed in Great Britain

CONTENTS

Contents. . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . ix Introduction. . . . . . . . . . . . . . . . . . . . . . . xi Introduction to the revised edition . . . . . . . . . . xv Open problems . . . . . . . . . . . . . . . . . . xvii Added in proof . . . . . . . . . . . . . . . . . . xxiii Notation . . . . . . . . . . . . . . . . . . . . . . . . . xxxl

CHAPTER I. PRELIB~INARZES ....

............ Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q 1. Preliminaries and saturation . . . . . . . . . . . . . . 8 2. Order. stability and indiscernibles . . . . . . . . . . . . CHAPTERI1. RANKSAND INCOMPLETE TYPES.

1 1 1 9

........ Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q 1. Ranks oftypes . . . . . . . . . . . . . . . . . . . . Q 2. Stability. ranks and definability . . . . . . . . . . . . Q 3. Ranks. degrees and superstability . . . . . . . . . . . . Q P. The f.c.p., the independence property and the strict order property . . . . . . . . . . . . . . . . . . . . . . .

18 21 29 41

CHAPTER I11. GLOBAL THEORY .

82

.............. Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q1.Forking . . . . . . . . . . . . . . . . . . . . . . . Q 2 . The finite equivalence relation theorem . . . . . . . . . Q 3. The instability spectrum . . . . . . . . . . . . . . . . Q 4 . Further properties of forking . . . . . . . . . . . . . . V

18

62

82 84 94

101

108

vi

CONTENTS

f 6. The fist stability cardinal f 6. Imaginary elements

............... .................. $ 7. Instability . . . . . . . . . . . . . . . . . . . . . .

122 130 137

CHAPTERIV. ~

150

........... f 0 . Introduction . . . . . . . . . . . . . . . . . . . . .

f f f f f

E

M

1. The set of axioms 2 Examples of F’s

O . .D. .~ .

...................

................... . 3. General properties of F-primary models . . . . . . . . . 4. Prime models for stable theories . . . . . . . . . . . . 6. Various results . . . . . . . . . . . . . . . . . . . .

CHAPTER V . MOBE ON TYPESAND SATURATED MODELS.

... f 0 . Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Orthogonality. regularity and mhimelity of types . . . . . f 2 . Dimensions and orders between indiscernible sets . . . . . f 3. Weighted dimensions and superstability . . . . . . . . . f 4. Semi-regular and semi-minimaltypes . . . . . . . . . . f 6. Multi-dimensional theories . . . . . . . . . . . . . . . $ 6. Cardinality-quantifiers and two-cardinal theorems f 7 Ranksrevisited

.

. . . . .

............ .......

.. CHAPTERVI . SATURATION OF ULTBLPRODUCTS

...... $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Reduced products and regular filters . . . . . . . . . . f 2 . Good filters and compactness of reduced products . . . . . f 3. Constructing ultrafilters . . . . . . . . . . . . . . . . f 4 . Keisler’s order . . . . . . . . . . . . . . . . . . . . f 6. Saturation of ultrapowers and wtagoricity of yuuudoelementary claases . . . . . . . . . . . . . . . . . . f 6. Saturation of ultralimits . . . . . . . . . . . . . . . . CHAPTER VII . CONSTRUCTIONOF MODELS .

......... $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Skolem functions and generalizations of eaturativity . . . .

150 152 157 174 183 204 223 223 230 240 249 267 284 289 305 321 321 324 333 345 370 379 390 397 397 400

vii

UONTENTS

Q 2. Generalized Ehrenfeucht-Mostowski models . . . . Q 3. On the f.c.p., uniform trees and ID(T)I > IT/= KO . Q 4 . Semi-definability . . . . . . . . . . . . . . . .

.

$ 6 Hanf numbers of omitting types

...

...

... ............

411 419 426 432

.

CHAPTER VIII THE NUMBEROF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CUSSES . . . . . . . . . . . . . . 440 $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . $ 1 . Independence of types . . . . . . . . . . . . . . . . . $ 2 . Unsupmtable theories . . . . . . . . . . . . . . . . Q 3. Saturated models and the w e h = lTll . . . . . . . . . $ 4. Categoricity. etlturation and homogeneity up to a cardinality

440 444 455 464 47 1

.

CHAPTER I X CATEUOBJCITYAND THB: NUMBER OB MODELS IN ELEMENTARY CLASSES. . . . . . . . . . . . . . . . . . 479

..................... Q 1. Supratable theories and categoricity . . . . . . . . . . Q 2. On the lower parts of the spectrum . . . . . . . . . . .

479 481 497

CHAPTER X . CLASSIFICATIONFOR F&-SATURATED MODELS

508

Q O . Introduction

$ 0. Introduction . . . . . . . . . . . . . . $ 1. Preliminaries . . . . . . . . . . . . . . $ 2 . The dimensional order property . . . . . . . $ 3. The decomposition lemma . . . . . . . . . $ 4. Deepness . . . . . . . . . . . . . . . . $ 5 . Deep theories have many non-isomorphic models $ 6. Infinite depth . . . . . . . . . . . . . . $ 7 . Trivial types . . . . . . . . . . . . . . CHAPTERX I . THEDECOMPOSITION THEOREM .

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . .

$0. Introduction . . . . . . . . . . . . . . . . . $ 1. Stationarization . . . . . . . . . . . . . . . . $ 2. The axiomatic treatment . . . . . . . . . . . . $3. Specifying the axiomatic treatment . . . . . . . .

508 509 512 520 527 533 548 550

557 557 557 561 572

viii

CONTENTS

XI1. THE MAIN GAPFOR

THEORIES .

590

$0. Introduction . . . . . . . . . . . . . . . . . $1. On FA"and F{ . . . . . . . . . . . . . . . . . $2. Stable systems . . . . . . . . . . . . . . . . $3. On good sets . . . . . . . . . . . . . . . . . $4. The otop/existence dichotomy . . . . . . . . . . $5. From the (Xo.2)-existence property to the (A. 2)-existence property . . . . . . . . . . . . . . . . . . . $6. The book's main theorem . . . . . . . . . . . .

590 591 598 603 608

XI11. FORTHOMAS THE DOUBTER . . . . . . .

622

$0. Introduction . . . . . . . . . . . . . . . . . $ 1 . Can the models be characterized by invariants ? . . . . $2. On having many models. no one elementarily embeddable into another . . . . . . . . . . . . . . . . . $3. On the Morley conjecture . . . . . . . . . . . . $4. ](Ha. T)for a large enough . . . . . . . . . . . .

622 623

CHAPTER

CHAPTER

COUNTABLE

616 620

627 634 643

........................ 653 $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . 653 5 1. Filters. stationary sets and families of sets . . . . . . . . 653 5 2. Partition theorems . . . . . . . . . . . . . . . . . . 659 5 3. VarioUereeults . . . . . . . . . . . . . . . . . . . . 666 Historicalremazka . . . . . . . . . . . . . . . . . . . . 673 Referen....................... 684 Index of definitions and abbreviations. . . . . . . . . . . . 691 Index of symbols . . . . . . . . . . . . . . . . . . . . 703

~P~PNDIX

ACKNOWLEDGEMENTS

My reseamh in this area started with my thesis, and I thank M. 0. Rabin for his kind guidance. I thank the Israel Academy of Science for partially supporting my research, aa well aa the United StatesIsrael Binational Science Foundation (grant 11 10)and the NSF (grants 14PH747 and MCS-08479). Much of the proofreading and checking waa done industriously by M.Abramsky who waa my research assistant from 1973 to 1976,and for this I am grateful to him, aa well as to L. Marcus who did a similar work previously. They are responsible for the fey paragraphs which are in good English. I owe a special debt of gratitude to D. Ehrman, J. Alon and mainly D. Sharon for typing, correcting and recorrecting the manuscript. Lastly, I thank J. Bddwin, G. Cherlin, S. Koppleberg, D. Monk, M. Rubin and P. Schmidt for detecting various errors and inaccuracies.

ix

This Page Intentionally Left Blank

INTRODUCTION

The aim of this book is to represent works of the author on claesi6ication and related topics. The author, in a moment of insanity, believed this would be the easiest way to represent his work. There is no point in trying to convince you that I think the book is important; since otherwise I would not have spent my time writing it. Anyhow, I have said it in the introduction to [Sh 75~1,which is supposed to be a propaganda for this book. There is also no point in explaining how the subject evolved because I say almost everything in the introduction to [Sh 741. Let us note only that the founding stone is [Mo651 (there are historical notes at the end of the book). So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting ycuueelf to the reading, and solving the exercises till you know it by he&. Unfortunately, I suspect the reader is looking for advice on how n& to read, i.e. what to skip, and even better, how to read only some isolated highlights. If you are generally not interested in model-theory but are interested in ultrafilters and ultraproducts, you can read Chapter IV, Sections 1, 2 and 3. There, many claesical results are represented but also some new ones. We prove that there is a A-good not A+-goodultrafilter iff A is regular (improving Keialer, cf. [Ke 651); show that p < 2” A pwo = p iff there is an ultrafilter D over A and ni < w such that n,/D = p, thus anawering a question of Keisler (cf. [Ke 0781). We also answer one of the questions from [CK 731 (due to Keisler): there is a regular ultrafilter D over A > No, D not good, but n,/D > No* n,/D = 2”. Also an HI-incomplete D over I is A-good iff for every order J, JI/D is A-saturated. Also Chapter VII is quite isolated. The first section deals with a generalization of A-saturated, so that such models exist in more cardinalities. The second section presents a generalization of Ehren-

n

n

xi

n

xii

MTaODUCTION

feuchtMoetowski models, and w u m e only the knowledge of Skolem functions. The third section uses the second, and some definitions from previous chapters, and pretends to show the generalization of Section 2 is useful, so e.g., we have a model generated by a tree of indiscernibles and this is important for unsuperstable theories. The fourth section again requires no prerequisite, and there we prove that for any theory, and p 2 2ITl, it has p-universal, p-stable models of arbitrarily high cardinalities. In Section 5 we present the theorems on Hanf numbers of omitting types. Chapter VIII is devoted to the number of non-isomorphic models, but for this we use only some definitions (or rather equivalent conditions explicitly stated and used as alternative definitions).Also Chapter VII, Section 3 is used in Chapter VIII, Sections 2 and 3. Most of Chapter VIII, Section 1 is quite easy (and has real results) and there is not much dependence. In Chapter VIII, Section 2 we have somewhat heavier material; you can start by reading Theorem 2.2(1): if T is unsuperstable, h > IT1 is regular, then T has 2”non-isomorphic models of power A. Chapter VIII, Sections 2 and 3 are aimed at proving the same for any h 2 I TI + 24,; but Chapter VIII, Section 3 and the later parts of Section 2 me, in their present form, suitable for your too arrogant students (but no guarantee). Section 7 of Chapter I11 is independent, and contains results on the possible K(h) = sup(lS(A)I: IAI s A} and related problems. Also the combinatonal appendix needs no prerequisite, but it is not so interesting. If you arrive here I hope you understand that you should really start to read from Chapter I. Now Chapter I, Section 1 is a restatement of some classical theorems (compactnese and LowenheimSkolem which are proved later, elementary chains, saturated models, etc.). Now the reader should know what are firat order theories, formulas and satisfaction; and no more; but I would be very interested to meet a successful reader who has not read before a significant part of [CK731,or [Sa 721.We here a h make some conventions. Chapter I, Section 2 contains central definitions (stability, indiscernibility) and some theorems on existence of indiscernibles and the connection between unstability and order. Chapter 11, Sections 1 and 2 are easy and central. Here ranks are introduced and stability of formulas and theories are really introduced, i.e., we give many equivalent conditions. Chapter 11, Sections 3 and 4 are quite peripheral and independent. In Chapter 11, Seotion 3 we investigate more deeply ranks, and connect

INTRODUOTION

...

Xlll

them with superstability. Only Claim 3.12 of Chapter I1 is used in Chapter 11, Section 4. In Chapter 11, Section 4 we investigate some more properties of formulm and theories: the finite cover property, the strict order property and the independence property. Almost the only use later is in the investigation of Keisler orders and s&~rationof ultrapowers, in Chapter VI, Sections 4 and 6. In Chapter 111,Sections 1, 2, 3, 4 and 6 are very necessary for what follows; Sections 1-4 can be read only successively, Section 6 depends mainly on Chapter 11, Sections 1and 2. Unlike Chapter I1we concentrate here on complete types and investigate forking, and in Sections 1-6 deal with stable T only. Forking is introduced in Section 1, investigated in Section 2, used in Section 3 to (almost) give the stability spectrum theorem, and in Section 6 really to give it. In Section 4 we me the connection between ranks and forking, use them to prove results on both sides. I n Section 6 we extend our models so aa to have names for equivalence classes, this is important when we look for canonical forms. In Section 7 we investigate various properties, like in Chapter 11, Section 4, mostly for unstable T. Chapter IV deals with prime models. As we have to deal with five kinds of prime models, we give an axiomatic setting (in Section 1) fmd which examples satisfy which axioms (in Section 2, the results sum up in a table at the end). In Section 3 we prove theorems in this axiomatic setting (prime models exists, they realize only isolated types, etc.). Section 3 uses only Chapter IV, Section 1, but in Chapter IV, Section 2 extensive use of Chapter 111, Section 4 is made. The fourth section is somewhat heavier; we here concentrate on stable theories, and give characterization of prime models (implying uniqueness). Only a few lemmas from it are used later. The fifth section contains scattered results, most of them do not require much knowledge. The main theorem says, e.g., that for countable stable T, if a primary model (i.e. a prime model constructed step by step) exists, the prime model is unique. We also prove a theorem on the existence of a model of T omitting < 2ITI complete types. Chapter V contains some of the deepest results of the book. It uses Chapter I V but usually not in a deep sense, the reader would better go to its detailed introduction. Now in Chapter VI we turn to ultraproducts, Sections 1 , 2 and 3 do not refer to anything. I n Sections 4, 6 and 6 we use Chapter 11. The firat half waa already reviewed. In the second half we investigate

xiv

I"F&ODUOTION

Keisler order on theories ([Ke 671) and get quite a complete picture (to complete the picture we should know more on unstable theories without the strict order property), and find, quite accurately, how saturated am ultraproducts and ultrdimits. Chapters VII and VIII were already reviewed. Chapter VIII, Section 4 is to a large extent a summary of results, and has some not so hard theorems. In Chapter IX,Section 1we prove the categorioity theorems, relying on various previous resulta (including Chapters V and VIII). We also prove some results on the number of models in elementary claases (in Chapter VIII we concentrate on pseudo-elementary claeeee), and when a model can be represented as a union of strictly increasing chain of length 6. In Chapter IX, Section 2 we deal with the number of countable models of a superstable T ;and what can be the number of F&,-models of T in He (it is 1, IaI' (p s 21Tl), or 2 2la1),and just models when T is totally-transcendental. The Appendix contains the combinatorid results we need.

Note. (1) Each chapter hae its introduction, it may be wise to look at it again during the reading. (2) Exercises am scattered randomly among the sections. Some are the result of the public pressure due to the prejudice that examples clarify notions, others me variants of theorems I want to mention, or remnant of obsolete proof. And some were the result of my preferring to give an exercise with a generous hint rather than a theorem with a thin proof. (3) The change in the name of the book is not incidental, but a change in point of view during the yeam in which the book waa written, explained in [Sh 76~1. The reader may ctlso want to know which of my papers becomes obsolete by this book. So this is the fate of [Sh 69a], [Sh 70a], [Sh 7Ob] [Sh 711, [Sh 71b], [Sh 71d], [Sh 72a], [Sh 741 and [Sh 7481. Also [Sh72] except Section 2 which deals with the uncountable caw, [Sh 72b] except the purely combinatorid part and the theorem on L,,,,. But in [Sh 71b], the proof of the last theorem is not covered. We did not deal here with the results on non-elementary claases ([Sh 691, Section 6; [Sh 69~1,[Sh 701, [Sh 76a] and [Sh 7601).

INTRODUCTION TO THE REVISED EDITION

In this edition four new chapters (X, XI, XI1 and XIII) have been added. (In addition, many corrections have been made, including many which previously were in the “Added in Proof’’ part of the First Edition.) The additional chapters present the solution to countable first order T of what the author sees as the main test of the theory (XII, 6.1).

In Chapter X we introduce the dop (dimensional order property) and show that it is a very meaningful dividing line for superstable theories: if it holds, T has many &-saturated ( = Fgo-saturated) models; if it fails, we have a structure theory: every &-saturated model M of T is Fio-prime over a non-forking tree of models (M,,:qeI c u’A) (each M,,F;fo-primeover 9). If the tree I were to have an infinite branch, the theory is deep and still has many models. Otherwise, we can compute the number of #,-saturated models. This chapter is an improved version of [Sh 83a], [Sh 84bl. Mainly the IE case is explicated and the facts on trivial (regular) types are gathered in one section. In Chapter X I we prove the needed decomposition theorems assuming (*)

there are prime atomic models over MI U 4 when &, 4M,, A& {illl,&&} is independent over 4.

Chapter XI1 is the crux of the matter. We prove that the negation of (*) above implies that in models of T we can define a relation which order a large subset of ;the definition being aR6 iff a type p ( z ,a,6) is omitted. Chapter XI11 is intended to exemplify that Theorem XII, 6.1 fulfills its aim. For this we consider several questions which are

‘“w(

xv

xvi

INTRODUCTION TO THE REVISED EDITION

solved using its partition to cases. In Section 1 we deal with “when La,,-equivalent models of cardinality A are isomorphic ”. In Section 2 we deal with computing IE(A,T).Considering what was done in Chapter X, the main problem is t o show IE(h,T)= 2A when T has otop. Usually, the reader thinks this is immediate, forgetting that elementary embedding does not necessarily preserve the omission of types. In Chapter XII, $3,we prove the Morley conjecture : for countable first order T, No< A < y * I ( h , T)< I @ , T) except when T is categorical in N, (so (VA > N,)I(A, T)= 1) but I(No,T)= No. In Chapter XIII, $4, we compute I ( h , T ) for A large enough (though we do not know if a relevant invariant can be >No but

< 2x0). For a discussion on the significance of the new chapters, see [Sh 85bl. We thank Leo Harrington for hearing the proofs in Chapter XI1 while they were generated, and the reader may thank him for persuading me to prove the otop there using finite sequences rather than some considerably more complicated and infinite versions. We thank Saffe and Hrushovski for pointing out the need for corrections, Grossberg for proofreading VIII, and the Israel Journal ofMuthemutics for allowing us to use portions of [Sh 831 and [Sh 83a] which appear there. Last, but not least, we thank Damit Sharon and Alice Leonhardt for typing the new material and corrections. S. SHELAH November 1987

OPEN PROBLEMS

A. Completions Here we deal with problems which just complete theorems in the book. We feel existing methods will suffice. (1) Complete the computation of the possible function I(A,T) ( A 2 No,T countable complete first order theory). See XIII, $93 and 4. Though some think this was “the problem”, I could not make myself excited about it. Still it would be nice to know. The most appealing part is : ( l a ) What can SND(T)be? If it is a limit cardinal > K O ,what can k(T), Z( T)be ? Hopefully SND(T)E (w 1)U {&, (2’~)’} : even so, we shall need another cardinal invariant to compute I(A,T)for N, 6 A < If we are interested in I(&, T), too, we should ask ( l b ) Does the assumption I ( N o ,T)= No,T superstable, bound the depth of T ?[ ( w + 1) is possible.] (2) What can A be from Theorem V, 5.81 (3) Is DP(T)> lTl+possible ? Remember DP(T)< S(lTI), all successor < [TI+are possible; however, cf. IT1 > Noimply S(lT1)> /TI+. (4)Suppose A 2 K,(T)(or even A = K,(T)), M is F,”-atomic over A , and for every I E M indiscernible over A , dim(1,A , M ) < A. Then, is M necessarily F,”-prime over A ? Even various weakenings are open. You can assume A > IT1 [is regular] or it can be to prove for Ceq(this strengthens the hypothesis - we have more I ’ s ) , or try the parallel for Fi. Note that this is an attempt to improve the characterization theorem IV, 4.14 ; note also that for T superstable this is true (see IV, 4.18). ( 5 ) Suppose T is stable (D(T)( > IT(‘“,(Vx < ,u)[x‘~< JD(T)I]K

+

xvii

xviii

OPEN PROBLEMS

regular ~ K ( T Prove ). that for every A = Aat,,@eslk d I v ( ~ ~ “a,m z , @m,)I, d I v ( ~ , ~ ~ 2*0 (using cardinality quantifiers). (4) Prove the main gap for strongly A-saturated models (i.e. for Fi-saturated models strongly A-homogeneous, i.e. if (ai:i < a < A ) r M ( b , : i < a < A ) s M realizes the same type, then some automorphism of the model takes one to the other). The following problem should be easier to solve. (5) Prove the main gap (etc.), for the class of Nosaturated models (Tcountable). (6) Classify when we replace “isomorphic” by “L,,,uequivalent ”. (6a) Classify the possible Scott heights of models of T. See the works of Nadel and [Sh71a]. This does, of course, have some connection to the Vaught conjecture. (7) We know that { A :T has a rigid model of power A} may be “bad” (see [Sh 76~1).But what for IT(+-saturated models? What about the number of non-isomorphic ones when a t least one exists? See [Sh 83b] on very partial positive results. (8) Vaught conjecture, i.e. (for complete countable T) I ( K o ,T)> No=I ( N o ,T)= 2*0. We can look also at variants of it: the number of minimal, models or rigid models. Morley [Mo] proves I ( N o ,T)> K, *I(No, T)= 2’0, and the proof applies to many other cases (but this really belongs to description set theory, where stronger theorems have since been proven, e.g. that of Burgess; see [Sh 84bl). Some people think this is the most important question in model theory as its solution will give us an understanding of countable models which is the most important kind of models. We disagree with all those three statements.

xxii

OPEN PROBLEMS

D. Further reaching problem (1) Define " T has a structure theory over a class K" and classify accordingly. It seems reasonable to look a t specific K's. A reasonable interpretation is: for some K , A for every model M of T there is a model I E K ,and functionsf:M-t"I, g : M - t A such that if a : ~ M ( = t 1,2,i < n),g(a2)= g(a,2) and ( f ( a 2 ) :i < n), (f(a,"): i < n) realize the same atomic type in I , then (a: : i < n), (a,":i < n ) realizes the same type in M ; moreover, if F,BeKI,h < h and c,J realizes the same atomic type in I , then

(3a c M ) [f(iz)= F

A g(a) = &] iff

(3a EM)[f(a) = Z A g ( a ) = a].

So the main gap says which T has a structure theory over K = { I :I a tree with d o levels and depth ITI+2Ko” by “ A > ITI},“(2No)+”by “K1”. Let us turn to the list of open problems. 6. Problem A7 The problem was: “IS every unidimensional, stable T,superstable ? ” Hrushovski answers positively, using the interpretation of groups. 7. Problem C4

The problem was: “Prove the main gap from a Fi-saturated strongly A-homogeneous model”. For A = KO, we compute the possible spectrum functions [Sh 88aI. Also, the related problem of a A-resplendent model, A > IT1,was solved.

8. Problem 0 2 (Classification over a predicate) This is essentially solved (but using independence results for nonstructure, and much is not yet written, see [PSh 851, [Sh 85d] and [Sh 86bl). 9. Problem 0 3

The classification of a universal class (i.e. MEK if and only if every finitely generated substructure of M belongs to K) is essentially solved see [Sh 85e] and [Sh 891, but not completely written. see a work by Grossberg and the On the categoricity of author (on the categoricity of w successive cardinals) in preparation ; and a work by Makkai and the author (and more in preparation).

xxvi

ADDED IN PROOF

10. Chapter III, Section 5 (on instability and D ( T ) )

Newelski had shown that we cannot in general improve the results there. 11. SND

E. Hrushovski has proved that for countable first order T , SND(T) satisfies CH. We report below on other relevant works of the author. 12. Better invariants

We prove that if T is superstable without dop, for characterizing up to isomorphism He-saturated models we need only finitary invariants; more specifically we can use the theory of the model in the logic with quantification over sets which are the algebraic closure of finite sets [in a""]and has quantifiers on dimension. 13. Multidimensionality

As we say previously the results of V, $ 5 can be improved. I n fact we have now a typescript [Sh 891 proving the following. Assume T is stable (first order complete) and it is multidimensional (see V, $ 5 ) , then there is a (stationary) type orthogonal to the empty set [hence : if T is stable in N, and a < p then T has 2 21,-al pairwise non isomorphic F;t -saturated models of cardinality N,]. 14. Isomorphic ultrapowers

We proved that it is consistent with ZFC, that there are countable models M , N which are elementary equivalent, but for every ultrafilter D on w , M"/D and N"/D are not isomorphic, moreover Th(M) = Th(N) does not have the strict order property and even for different ultrafilters we do not get isomorphic ultrapower. This will probably appear in the proceedings of the MSRI 10/89 symposium in set theory. It also seems that, e.g., in some models of set theory, some ultrafilter will not give isomorphic ultrapower in the use of Ax Kochen.

ADDED IN PROOF

xxvii

15. Universal models

In a work in preparation by Kojman and the author more is done on characterizing the possible classes { A : T has a universal model of power A } for T first order, e.g., T is countable with the strict order property or unsuperstable and K, < 2'0, then T has no universal model in K,. 16. Resplendency

We note that the following are equivalent for a first order T : (a) T is stable, (b) if M is a ITI+-saturated model of T satisfying the following then it is saturated: for every c i € M for i < ITI, if q is a consistent theory extending Th(M,ci)i(A, a) is defined inductively by >,(A) = A, and for a > 0, >(A, a) = 23(A*B); aa(No)is written aa. A+ is the cardinal successor of A; "'A = &s,, (L(M)I,then M is A-saturated ifl M is A - m p a c t . (3) Every A-saturated model is A-honzogeneoua and A-universal. (4) Every ( c No)-univer8al, A-homogeneous model is A-saturated. ( 6 ) If p I A, then every A-saturated model is p-saturated, every hcumpact nzodel is p - m p a c t , and every A-universal model is p-universal.

(6) Let A > Nobe a limit cardinal. If M is p-saturated (p-homogeneous) for every p c A, then M is ~-8aturated(A-homogeneous).(Thisdoes not hold for universality but holds for compactness.) (7) If M is A-saturated ( A - m p a c t ) and LoE L(M), then M 1 Lo is

A-saturated (A-compact). ( 8 ) Every strongly A-homogeneous model is A-homogeneow. Proof. (1) Immediate. (2) Immediate. (3) For this it suffices to prove:

CLAIM 1.10: Let A E B E IMI, IAI < A, IBI I A, f an ( M , N ) elementary mapping, Dom f = A. If N is A-saturated OT N = M is A-homogeneoue, t h n we can extend f to an ( M , N)-elementary mapping 9, Domg = B.

A u {bt: i < a} (so lBal c A for a < IBI) and we shall define by induction on a 5 IBI ( M , N ) elementary mappings f a ; fo = f, fd = Ut h 2 IAI then there is I' II'l > h such that I' is an indiscernible sequence Over A . Proof. First we shall show: (**)

There exist B, C, A E B E C E 1611, ICl I A, and p ES"(C, M ) such that (1) for all C',C G C' s IN[, IC'l I X p has an extension p' ES"(CI)realized in I - "C' which does not split over B (in particular p does not split over B), and (2) for all E' E /MI there is E C such that tp(E, B) = tp@', B).

E

I,

OH. I, 9 21

ORDER, STABILITY AND INDISUERNIBLES

13

By way of contradiction assume -,(**). We define by induotion t m hmwsing aequence {B,},,, suoh that B, s ]MI,lBfl 5 A as follows: Bo = A , Bd = uf A L IAI, then there is J IJI > A, which is an indiscernible sequence over A.

c I,

Proof. Immediate by the two preceding lemmaa. Remark 1.J is in fact an indiscernible set over A , see 2.9 and 11,2.13(1) and (9). Remark 2. If K is regular, and for all IAI < K and m < w , ISm(A)I< K, and there is A < K for which 2.6(*) holds, then for all A and I with. IAI < K s IIl, there is J c I, IJI 2 K, J an indiscernible sequence over A. The proof is similar. LEMMA 2.9: If I is a A-indiscernible sequence over A but not a Aindiscernible set over A then there are n < w , cp(Zo, . . .,P - l ; 8) E A, and 8 E A such that cp(fo,. . , P - l ;8) is connected and antisymmetric over I.

.

Proof. Immediate.

THEOREM 2.10: Suppose >

xOsjb p, then there is an indiscernible set J G I of length A+.

Proof.Use the ideas of the proof of 2.6 and 2.10, and use 11, 2.16. EXERCISE 2.1: Suppose A. < A, < < A,, I E IMI, IIl > hl: and ill is stable in A, for i s n, then there is an (n + 2)-indiscernible sequence J c I, IJI = &+. EXERCISE 2.2: (1) Find a model of power N,, which is stable in No, but there is no indiscernible sequence of length N, in it. (2) Moreover, for some symmetric tp = p(z, y), there is no p-2indiscernible sequence of power N, in it. (Hint: Use a dense Specker order for ( l ) , and add a well-ordering to it for (2) (see e.g. [GS 73]).) (3) In (1) and (2), show that we can choose a model with'a countable, N,-categorical theory. EXERCISE 2.3: Suppose A c B c C and for every 5 E C there is a 8 E B such that tp(E, A) = tp(6, A). If p ES"'(B)does not split over A, then p has a unique extension in Sm(C)which does not split over A. EXERCISE 2.4: Generalize 2.12 to (A, A)-stability. EXERCISE 2.5: Suppose A is regular, M a model, and for every A c /MI, ]A1 < A implies JS"'(A,M)1 < A, and in M there is no ordered set

OH. I, 8 21

ORDER, STABILITY AND INDISCERNIBLES

17

of sequences of length p for some p < A. Then for every I c 1M1, III 2 A there is an indiscernible set J c I, IJI = A.

EXERCISE 2.6: In 2.12 (cf. Exercise 2.6) add an A E IMI, [ A ]< of p( IA I < A) and demand that J is indiscernible over A. EXERCISE 2.7: (1) SupposeS E S;(A), 9 = cp(3; g), 181 > IAI. Prove that there are li, E A, pk € 8 (n, k < o),and t E (0, l}, such that cp(Z; B,$ €pic iff n < k (iff +Z; $pk,of course). (Hint: Generalize 2.10(1) and 2.11.) (2) Generalize 2.10(2) similarly. EXERCISE 2.8: Let cp = q(Z;g),

h be regular, 5 E A E J M1 , (A1 < A. Show that if b = 5-En. ^E, I{p E Smn(A,M ) : cpn(F;b)e p)I 2 A, then for some B G (MI, ( B J5 m(n - l), I(p E P ( A u B): cp(Z; E) E ~ } 1 J A.

EXERCISE 2.9: (1) Suppose T is stable, A regular, IAI < A =. IS(A)I < A, and p < A =. p" < A. For any lAol < A and elements ah (i < K , a < A) there is a set S c A, IS1 = A, such that for any i(0) < . < i(n) < K , {(af,O),. . .,ap)):a ~S)isindiscernibleover A,. (Hint: You can msume T is stable in some p < A by 11, 2.13 or see 111, 4.23.) (2) Generalize (1) when we replace the condition on T by a condition on M . EXERCISE 2.10: Let M be an infinite K+-saturatedmodel, and define the model M" m follows:

pf"I

=

Ff(a)=

p 1U"pfIY a

a[.]

ifaEM, if a E "IMI,

PM"= IMI and M

E

ME.

Show that if M is A-stable, A = A", then M" is A-stable; and if Th(M) is A-stable, A = A", then T h ( P ) is A-stable; but if A < A" or M is not A-stable, then M" is not A-stable; and if A < A", Th(M") is not A-stable. Also show that Th(M") depends only on Th(M) and K .

CEAPTmR 11

RANKS AND INCOMPLETE TYPES

II.0. Introduotion Section 2 (and Section 1 on which it relies) are the cornemtones of the book, but they a,re not difficult. This chapter has esaentidy two interlocked aims; the first is to investigate the family of formulas cp(Z; a)(aE Q)('p = 'p(Z; g) fixed); or other words the family of the seta 'p(Q, E Q). We try to classify them by complexity, in Sections 2 and 4. Clearly this hae various implications for the theory T.The second aim is to investigate various notions of ranks (e.g., possible values, equality between distinct ranks), we deal with it in Section 1; somewhat in Section 2, and mainly in Section 3. The properties we shall deal with me the order property ( =instability), the f.c.p. (the hite cover property) the independence property and the strict order property. We say T haa such a property if some 'p(z; g ) has the property (or, equivdently, by our theorems, some 'p(Z;g) has the property). Essentially we prove that the order property implies the f.c.p. and the order property is equivalent to the disjunction of the independence property and the strict order property. Other properties of this sort are suggested in Chapter 111, Section 7, (K&T) = a, K ~ J T=) 00, essentially). The most important of these properties is stability (=not the order property) with which we shall deal in Section 2. The main theorem (2.2) lists many propertiea equivalent to unstability of a formula, some of them are helpful in proving assertions about stable formulas, and some are helpful in proving assertions about unstable formulas. For unstable formulaa the important properties are: the order property (i.e. for some a,,, &,(n < W ) t: 'pp,, 6,J if€ 2 > k) (this shows the similarity to the "classical " cam-the theory of dense linear order); and for every X there is an A, IAI I; X < lQ(A)l.Among the properties important for stable 'p m:l&(A)I I; IAI + KO (=stability 18

OH. 11, 8 01

INTRODUOTION

19

in every A); P ( p , v, a)s P ( Z = Z,9 , 2 ) < w ; and every cp-m-type over A is definable by a formula over A, of a fixed form. For theories we get a similar theorem (2.13): T is stable in every A = iff T is stable in some A iff every v(x;8 ) is stable iff every ~ ( 38);is stable iff T does not have the order property (i.e. no formula orders an infinite set of sequences), ifF every infinite indiscernible sequence is an indiscernible set. We also prove that if IIl 2 A > IAl, A regular and A is finite, then some J E I is A-indiscernible over A, IJI = A. In Section 4 we shall deal with the other properties. We say v(Z; 0 ) does not have the f.0.p. if for some k, < KO we can strengthen the compactness properties of EE to: if p is a set of cp-m-formulas,and every subset of p of cardinality sk, is realized, then p is realized. The “classical~yexample of a theory with the f.c.p. is the theory of one equivalence relation such that for every n < w there are (insnitely many) E-equivalence claases of n elements. It appears that for stable T, the possession of the f.c.p. is equivalent to several natural properties (see the f.0.p. Theorem 4.4). In particular if T has the f.c.p., then there are a formula E ( z , y; 2) and sequences Z,, such that E ( z , y, En) is an equivalence relation with k, equivalence classes and n 5 k,, < No(this shows the a 5 i t y of any stable T with the f.c.p. to the classical example, and is helpful in proving assertions about such theories). On the other hand if T does not have the f.c.p., then Rm[8(3;a), rp, A] = k is an elementary property of a; and there is k;,@ < KO such that for each a, for all A 2 k;,, the value of P [ B ( Z ; a), 9,A] is fixed; and the value of Mlt[8(Z;a),q] is bounded by some ki,, < KO; and for each finite A, there is k, < KO such that any set of (A, m)-formulas p is realized iff each q E p , IqI < k, is realized; and every (A, m)-type has a finite subtype of bounded cardinality which has the same rank Rm(- ,A, A). Let =(A) be the first regular cardinality p such that IAl 5 A =]l?t(A)l< p. It appears that the number of possible functions is small: for stable v = rp(Z; 0): n ( 1 < n < KO),and A + . For unstable v = v(Z; 8):Ded, A and (2”+ ;when they are distinct, rp satisfies the second case iff v(Z; ti) has the independence property. The strict order property of T is equivalent to the existence of a formula ~ ( 2g); defining a quasiorder (i.e.,a transitive, reflexive, anti-symmetric relation) with infinite chains. In Section 4 we also st& to investigate dimensions. Ranks were invented and investigated for their use in stability. Several kinds of ranks were used, and most of them are particular caaes of P ( p , A , A), on which we concentrate. We investigate them

20

RANKS A N D INCOMPLETE TYPES

[CH. 11,8 0

alao when there is no apparent application; more information is obtained in Chapter 111, Section 4, and Chapter 5, Section 7. Rm(p,A, A) is interesting mainly for A = 2, No,00 and A = L or A

finite. What is the meaning of the rank Rm(p,A , A)? For finite p , we can say that it measures the complexity of the family of sets {a: a realizes p u { c p ( ~ ; 6))) for v E A , 6 E 6. The basic properties are proved in Section 1. The rank is monotonic in the parameters (decreasing for p and A, increasing in A), and every type has a finite subtype of the same rank. When A 2 No we can extend p while preserving the rank, i.e., if p is over A, there is a complete q 2 p, Rm(p,A , A) = Rm(q,A, A); when A = 2, v E A there is no si such that

Rm[pu {cp(?C;

A , A]

= Rm(p,A ,

A) for t E (0, 1);

so for A = L, p over A , p has at most one extension to a complete type over A of the same rank. For every A, the maximal number of such complete extensions is, essentially denoted by Mlta(p,A, A). In Section 1 we also prove that for limit A, Rm(p,A, A) = Rm(p,A,A+). In Section 2 we prove that for finite A, Rm(p,A, 2) < w or Rm(p,A , 00) = a;and Rm[e(Z;a), A, k] = 1 is an elementary property of a. In Section 3 we complete the investigation on the stability spectrum for countable theories. If IT1 < 2H0, then T is stable in IT1 ifF T is stable in some A < 2H0iff T is stable in every A 2 ITI iff R(x = x,L, 2) < 00. Also T is superstableiff T is stable in every A 2 2ITI iff T is stable in some A < AHo iff R(z = x,L, 00) < 00. But the aim of Section 3 is the investigation of R and D. We prove that Rm(p,A, A) is < 141 + Noor 00 usually (i-e.,A = a, or A 5 No). Also ah = R m (p,A A) , (except for being decreasing) is essentially arbitrary for KO I A I I TI +,has restricted changes for A s No,is fixed for A > (2lT1)+.In fact it is fixed for A > lAl+ + No except when for some p , ! A ] + s p 5 (2A)+, lAl+ < A < p implies a, = CO, and A > p + implies aA = a, < 00; and a,,= 00 except possibly when p is a limit cardinal (rememberthat for limit p , a,,= a,+) so ah (A > Id 1 No)has at most three values (see Theorem 3.13 and Exercise 3.16). Moreover for A = L, A s KO we can compute Rm(p,A, A) from R m ( p , A 2) , (Exercise 3.21); and R m ( p , A ,a),for finite p , is usually directly characterized as the maximal a for which TCl +(Ap, h) is consistent for some (A, a)-functionh. We deal much with Dm(p,A, A), +

+

+

OH.

11,5 13

BANKS OF TYPES

21

which is a version of Rm(p,A , A) (essentially we require that the many contradictory extensions are finite) and we use it to prove assertions about R for A > lAl+ + No and prove that for A > IAl + Nothey am usually equal. We also give a method of constructing counterexamples (Exercise 3.20) and many exercises. +

IK.1. Ranks oftspea DEFINITION 1.1: Let p be an m-type, A a set of m-formulas, A a cardinal (possibly finite),or A = 00. We define the rank Rm(p,A, A) or Rm[p,A , A] by defining inductively when P ( p ,A, A) 2 a, a an ordinal. (1) Rm(p,A,A) 2 0 when p is a (consistent) type. (When p is inconsistent we stipulate P ( p ,A, A) = -1.) (2) Rm(p,A , A) 2 6 for 6 a limit ordinal if Rm(p,d, A) 2 a for all a < 6. (3) P ( p , d , A) 2 a + 1 iffor a l l p < handall finiteq E p thereare types {qi}isrrwhich are d-m-types (i.e., rn-types whose formulas m e all of the form ~ ( 38); or +E; a) where ~ ( 35);is in A) such that: E q, (or vice (i) for i # j there is a formula such that q~ E qi, l~ versa). In this cme we say that qi and q, are explicitly contradictory. (ii) P ( q u q i,A, A) 2 a for all i Ip. (4) Now if Rm(p,A,A) 2 a but not Rm(p,A,A) 2 a + 1 we say Rm(p,A, A) = a. (It is easy to see by induction on a that Rm(p,A, A) 2 a implies P ( p ,A, A) 2 p for all /3 5 a.)If Rm(p,A, A) 2 a for all a we define P ( p , A, A) = 00. DEFINITION1.2: If Rm(p,A, A) = a # 00, define the mu2tipZicity: (1) Mltl(p, A, A) is the first power po such that there is a finite q E p such that q has no more than po A-m-types qi satisfying 3(i), (ii)from Definition 1.1 (po may be finite). (2) Mlta(p,A, A) is the first power po such that there are no more than po A-m-types qi satisfying 3(i), (ii) from Definition 1.1 for p = q. If Rm(p,A,A) = 00, define Mltl(p,A, A) = Mlta(p,d, A) = a. If Mltl(p, A, A) = Mlta(p,A, A) we shall just write Mlt(p, A, A). If a statement is true for both Mltl(p, A, A) and Mlt2(p,A, A) we shall write Mltl(p,A, A). If A = KO we may omit it.

Remark. Clearly if Rm@,A, A) # 00, then 0 < Mlta(p,A, A) 5 Mltl(p, d, A) < A and if p is finite, then Mlta(p,A, A) = Mltl(p, A, A). In fact Mltl(p, A, A)+ < A. See Lemma 1.9.

22

RANKS AND INCOMPLETE TYPES

[m.II,§ 1

Ndation. If p = {8(z; a)} we may write P [ 8 ( z a), ; A, A] instead of P[{B(z; a)},A, A]. The same convention applies to Mlt and D (see Definition 3.2). 8pcial Cases. P ( p ,A , A) 2 1 if for all p < A and all finite q E p there are Am-types {q,},s,, each of which is consistent with q, but every two are explicitly contradictory. P ( p ,A, A) 2 2 if for every p < A and every hite Q E Z, there Am-types {q,},s,, which pairwise explicitly contradictory and P ( q u q,, A , A) 2 1 for all i. Example. Let M be a model with equivalence relations {Ea: a < a,} such that a < /3 =+ E, E En, E, divides every equivalence class of En into infinitely many classes, every equivalence dam of 1, is infinite, and if a, = y + 1, then E, has h h i t e l y many equivalence classes. Let Ta*= Th(iK) and let 6 = 6(Ta')be the saturated model of Ta' of cardinality I?. Then for every a E 1 6 1: (1) Rl[{z = a}, L,a]= 0, (2) R1[{zEoa}, L,001 = 1, (3) R'[{zE@}, L,001 = 1 + a. Remark. Tawadmits elimination of quantifiers and is totally transcendental (see Definition 3.1).

Proof. (1) ClearlyRl [{z = a}, L,a]2 0, and there are no q,, q1 such that qo u {z = a} and q1 u {z = a} are consistent while q,, q1 are explicitly contradictory, otherwise a would realize qo and ql. Thus R1[{z= a}, L,A] 2 1 for A 2 2, in particular for h = ao. (2) First we show R1[{zE,a},L,ao]2 1. Let p be any cardinal. Choose elements a,, i s p, in the 1, equivalence class of a and let r, = {z = a,} u {z # a,: i # j s p}. Clearly they are pairwise explicitly contradictory, but a, realizes {zE,a} u r,. In order that R1[{zEoa},L, a] 2 2 there must be two explicitly contradictory types p,, pl such that for i = 0 , l R1[{zE,a}upi, L,a]2 1. In particular {zE,a} upi = q, is consistent, i = 0, 1. By Theorem 1.6, extend q,, ql, to complete types q:, q: over the same sets and of the same rank. But E, E E, for all a < a, so q:, q: can be explicitly contradictory only if there is a, such that [z = a,] E q:, [z # a,] E q: or vice versa. But then from the proof of (1)it is clear that R1(q,*,L, ao) 2 1. So R1[{zEoa},L,ao] I2. (3) We leave this &B an exercise for the reader.

aII, 8 11

RANKS OF TYPES

23

DEFINITION 1.3: p t- q when every sequence realizing p realizes q. If p t- q and q t-p we write p = q: p is equivalent to q. Remark. For finite types, p I- q is equivalent to (VZ)[A p A q], and for infinite types to the condition that for every finite q‘ E q there is a finite p’ E p such that p’ t- q’. --f

THEOREM 1.1: (1) If p1 t-p2, then Rm(p,,A,A) s Rm(p2,A, A). I n p ~ t k t this h ~ie the m e w h Pa E p,. (2) If e q d i t y hQZd.8 in (l),t h Mlt’(p,,A, A) s Mlt’(p,,A, A). Proqf. (1) We prove by induction on a that Rm(pl,A,A) 2 a =Rm(p2,A , A) 2 a. For a = 0 or limit there is no problem. Assume for a and prove for a + 1. Let R”’(p,,A, A) 2 a + 1 and let qa E pa be h i t e , p < A. We shall h d typea {q,}, . p% are pairwise explicitly contradictory, p,, c py, and P ( p V , 9,A) 2 a,by the minimality of a. So R"(pv,9,A) 2 a + 1, a contradiction. Hence P ( p v ,9,A) = a, so R"(3 = 8,9,A) = 00. As A was arbitrary, by 1.11 P ( 3 = 3, 9,a)= 00. LEMMA 2.8: (6)

(7).

Proof. Exercise. Notation. From now on in this section, R ( p )or P ( p )denotes P ( p ,9,2).

LEMMA 2.0: (1) Let p be an arbitrary m-type. R ( p ) 2 n iff q 9 ,n) = {#(Zn; q :#(Z;a) Ep, q E n2}

u ( ~ ( 3g n~l r );n [ k l : q E "2, k < n } is &tent. (Note t7& &'I n) = I'(9, n ) = I'{;+(q, n).) (2) For any rn-type p , for all k, n < o,Rm(p,9,k ) 2 n + 1 iff {#(Z,;

a): v E " + l k ,#(Z; a) ~ p }

u {v(Z,,: 2t.f) 3

-,'p(Ep; 2:'):

q, p E n+lk, Y E nzrk,Z(v) = rn < n

+ 1,

v=qtm=prm,i=q[m],

j = p b l , i # j} i8 consistent. P ( p , 9,k) = n, Mlt(p,9,k) 2 k, < k iff the set of fomnulas given above is inconsiStent, but if thefurther wnditbna q[O] < k,, p[O] < ko are added, the set ie d e t e n t . (3) For any m-type p for all n < o,Rm(p,9,No) 2 n + 1 ifl

{$(a,; a):Y €*+I u {p)(%,,;

*j)

w , #(Z;a) El)}

= +zp;

* f ) : q, p E n+lo, vE

Z(v) = m

2 such that for every 7 E "'2, 1, U p,, [p,, = {Q,,l,(Z; a,,ln)R[nl: 72 < $(q)}]i8 COWi8t&.

Remark. From 3.1 it follows that (1) and ( 2 ) are equivalent also for infinite p (by 1.2). If in ( 2 ) we replace A by No,the equivalence is still correct, but the claim is stronger with A. If we replace A by A1 > A, the implication (1) * ( 2 ) is not correct.

DEFINITION 3.1. If Rm(Z= 3, L, 2 ) < 00, T is called totally tranecendental.

Proof of 3.1. ( 2 ) ==

(1) is trivial by 1.3(1).

(3) + ( 2 ) We shdl prove by induction on a that Rm(pup,,, A , A) 2 a for every q E * > 2 . As p o = p , this suffices. For a = 0 it follows by the

consistency of p,, u p ; and for a a limit ordinal from the induction hypothesis. So let a = p + 1 , q E " > 2 ; then { p v :q Q Y E "2) is a family 0f2~0d-m-types which are pairwise explicitly contradictory. By 1.1(1) and the induction hypothesis Rrn(pv up , A , A) 2 hence Rm(p,,up ,A , A) 2

p+

1 = a.

(1) =- (3) We shall define by induction on n < w subsets

U, of lAl+ + No of cardinality lAl+ + KO, formulas Q , , E A for q ~ " ' 2 and sequences for q E ">2, i E U,,such that R'"(p!,,A, 2 ) 2 i for i E U,, 7 ~ where ~ 2p i = p U {pTlr(Z;@$)'['I: k < Z(7)). For n = 0 let U, = {a: a < 141 + No}and p'o = p, so by (1) the induction hypothesis holds. For n + 1, for every i < IA I + Nochoose j = j(i), i < j E U,, so Rrn(rJ,, A , 2 ) > i for q E "2, hence for some Q!, E A and a!,, Rm[& u {tp!,(Z; a!,)'},A , 21 2 i, t E (0, l}. Hence for some Un+lc {a: a < 141' + No}, lUn+lI = l A l + Ho,foreverya,pE Un+l, q E "2, v; = Q:. Let for q E "2, Q,,= Q; (for any a E Un+L) and for when q E , > 2 and $ when q E ,2. Hence is i E Un+l, +

+

+

is consistent so some assignment Z, show that (3) holds.

H

6,,,gv H iiv realizes it. These av

42

RANKS A N D INCOMPLETE T Y P E S

[OH.

11,s 3

Remark. For A = L them conditions are equivalent for difFerent m's by I, 2.1, i.e., (1)1 o (2), o (3), for d 1, myn. Proof. (3)==(2)trivially, by the same A. (2)*( 1) Suppose not ( 1). For every p E @ ( A ) there ie a finite qp G p suoh that P ( p , A , 2) = P ( q p y A 2) y < co (by not (1)). By 1.6,p(1) # ~ ( 2* ) Q P ( ~ )# qfis hence l8?(A)I s I{qp:p

E 8?(A)}l s

+

S

I@:

q a finite A-m-type over A}\

+ KO,

a contradiotion.

(1) * (3) By 3.1 we know that (1)implies condition (3)from 3.1. Letting A = U {a,,: r) E @>2}, and q,, be a type in &(A) exfending p,, [we use the notation of 3.1(3)] we see

I&WI because if r) #

Y

2

E "2, let n =

V q l k ( Z ; $Ik)nrkJ

EI)n~

k,,:r ) E'21

=P o

min{k: r)[k] # v[k]}. Then

l Q , , l k ( % q,k)"rkJ

= q J v l k ( R a"Ik)v[k' E q v .

On the other hand, lAl I; I">21.N0 = No.

CONCLUSION 3.3: (1) If IT1 < 2n0, T ie stable in IT1 (or in some A < 2b), then T ie shbk in every cardinality p 2 ITI. (2) A totally tranrrcencEental T i s stable in every A 2 I TI. DEFINITION 3.2: We define the

degree Dm(p,A, A) ( p ,A , A are as in Definition 1.1) by defining inductively when Dm(p,A, A) 2 a (a an ordinal): (1) D"(p,A, A) 2 0 when p is a (consistent) type (when p is an inconsistent set of m-formulas, we let the degree be -1). (2) D"(p,A, A) 2 8 (8 a limit ordinal) if D"(p,A, A) 2 a for all a < 8.

OH.

11, 8 31

RANKS, DEQREES AND SUPER8TABIIJTY

43

(3) Dm(p,A,A) 2 a + l i f f o r d l p < Aandallfhitarspthereare, a finite q z r , n < w, $(Z;0 ) E A, and sequences a{,i 5 p such that: (i) Dm(qu {$@;at}, 4 A) 2 a, (ii) {$(z;a,):i 5 p} is n-contradictory (or n-inconsistent) over q, i.e., for every w E p, IwI = n, t=-,(3Z)(AiEw $(3;a{)A A q). (4) Exactly as (4) of Definition 1.1.

Remark. We ignore the degree for A < X,, and will be mainly interested in the c&88 A > ITI+.We can always replace q by A q.

U

a,)},A, A)

=

y:

(4) If p is an m-type over A, A 2 No,A is given, then there i s q , p E q E &?m(A) m h thud

W p ,4 4 = D"(q, 4 A). Proof. The proof of (2) is like l . l ( l ) , 1.3(1); the proof of (1) is like 1.2; the proof of (3) is like 1.7 and the proof of (4) like 1.6.

LEMMA 3.5: (1) If X E {R, D},p a$nite type and 00 > Xm(p,A, A) 2 a, then for some A-m-type q, Xm(pU q, A, A) = a. (2) If X E{R, D} a = (21Tl)+,Xrn(p,A, A) 2 a, then Xm(p,A, A) = 00. If X = R we need only a = (21*l+No)+.

Proof.(1) We suppose there is no such q, and prove by induction on

82

a that q = p vq,,

q1 a A-m-type, Xm(q,A,A) 2 a implies

Xrn(q,4 A) 2 8. For 8 = a this is the aasumption, for 8 a limit ordinal it follows by induction. For = y + 1 > a, and e.g., X = D, let rl c q be finite and p < A. Then there are a finite r z rl and formulaa $(Z;a,) (i 5 p, $ E A ) n-contradictory over p v r such that

Dm[pu r u {$(Z;at)},A, A] 2 a

44

[CH.

RANKS AND INCOMPLETE TYPES

11,f 3

(they exist as Dm(pu q , A , A) 2 a implies Dm(pU q,d, A) 2 a So by the induction hypothesis

+ 1).

D"[p u er u {#(% q},4A1 2 y As this holds for any finite erl, Dm(q,A , A) 2 p. Thus Dm(p,A, A) = 00, a contradiction. The proof for

X

=

R is similar.

(2) By 1.2 and 3.4(1) we can assume p is finite; if X = R we can also assume IT1 = lAl 24,. If (2) d m not hold, then by part (1) there are finite p , (i < a) such that X m ( p , , A ,A) = i. By 1.1(1) and 3.4(2) we can assume p , = {Ol(3;q)}.By the definition of a, for some i < j < a, 8, = 8, and tp(q) = tp(z,); but Xm(pr,A,A) = i < j = Xm(p,,A, A), a contradiction.

+

DEFINITION3.3: Let &(a) be the set of strictly descending nonempty sequences of ordinals < a. A (A, a)-functim is a function

A : &(a) --+ ((~(3,z), #(E; g), n ) : Q E L,# E A , n < o,Z(Z) = m}; say h(q) = (v,, #, n,) (for some fixed m). Let 8 = 8(z; a) be an m-formula, p a cardinal, A a (A, a)-functionand U G &(a). Define T,"(8,h)to be the following set of formdm containing free variables of the form &,,

rfce,h) =

{(33)[w ;a)

A

= ( ) or

r)

r)

u {+z)[/\ r)

E

A

0 c ISIt,)

E

~ ~ ~~ , I i~ . v l i )( A 3vqlI(~; ; r,lI,vl~I-l~~~]:

U ;and v E " p where k

#,@; iir,v-dA

=

Z(r))}

'~~(3; %,v)]:

lEW

U,w

= p , IwI

= n,, v E "p where k=Z(q)- 1

If U = &(a), we write F,*(8, h). THEOREM 3.6: Let IT1 + 1. c cfp. Then Dm[8(3;@ , A , p + ] 2 a if there M a ( A , .)-function h euch that I',*(8(3; ti), h) is wn&tent.

Remark. For the if part no restriction on p is needed. Proof. (e) Suppose r,*(O(Z;a), h) is consistent, and let it be realized For r ) E &(a) u {< )} and by the assignment #,,v H 6,,,, Z,,v H v E "p, where k = Z(r)), let pn,v =

a)} u {#qli(z; h 1 1 , v t i ) : tJ

0
0 and D m ( p < > , < > , A , p2+ a. ) As p 0 , < ) = {O@; a)},this proves one direction. (+) We prove by induction on a, for all O(Z; 6)at once. a = 0 h is the empty function. a = 6 (a limit ordinal). So by the induction hypothesis, for every /3 < a there is a (A, /3)-functionhBfor which I',*(O(Z; a), hB)is consistent. For Y E &(a) define h(v) = h,ml+l(~). It is easy to check that a combination of the assignments realizing the I',*(O(Z; a), hB) realizes

r,*(eg; a),a). a = /3 + 1 By

the definition there are p(Z; a), n < w and Amformulas $@;6,) (i 5 p ) n-contradictory over cp(Z;E) such that Dm[p(Z;a) A $(Z; &),A, p'] 2 /3 and p(E; a) k O(Z; 6). Hence, by the induction hypothesis for every i 5 p there is a ( A , /3)-function h, such that I'y*[p(Z; a) A $(Z; 6,),h,] is consistent. Define ( A , a)-functions hf (for i 5 p ) as follows: If r] E de(/3), hf(r])= h,(r]),if r] = (p), k'(r]) = (p, $, n> and if r] = (/3)^u, u E &( then /I) hf(r]) , = h&) (clearly this exhausts all possibilitiee). If for some i 5 p, Fy*(O(Z;a),h*) is consistent, we me through. Otherwise, for each i some finite subset of it is inconsistent, hence for some finite u(i) c &(a) I';tf)(e(Z; a),hf)is inconsistent. The number of possible u(i)is 5 1.1 + No,and for each finite u the number of possible hf r u i s j I T I . A s c f p > IT1 + IaI,thereisVcp,IVI = p s u c h t h a t for all i E V,u(i) = u,h' 1 u = h* 1u.W.1.o.g. u is closed undertaking non-empty initial segments. Now using the $(Z;6J (i E V ) and p(Z; 8) we can ewily show that I';(O(E; a), h*) is realized, a contradiction.

46

R A N K S A N D INCOMPLETE T Y P E S

[CH.

11, 8 3

Proof. (1) It suilices to prove (1) for finite p, so by the remark to 3.6 it suffices to prove (2).Notice that 3.6 holds when ITI < cf p = a. (2), (3) By 3.6, for some (A, p)-function h, F,*(Ap, h) is consistent. NOWwe define for i < p, 0 < k < W ; v k , #k, n k , 7; such that 7: E &(/A), E(qi) = k, q;[k - 11 > i, h(qi 1 I ) = (vl, n,) for 1 5 E s k. For k = 1 there = (~1,#1,ni, such that .{I < p : h((a)) = < ~ i ,$1, nJ}1 = p, and for i < p choose q: = ( a ) such that a > i, and h((a)) = (ql,#1, nl). If we have defined for k , there am ( P k + l , # k + l , nk+l such that = (P)k+l, # k + l , nk+l>}l = p ; < p: and for i < p choose q i + l = qg-(a) where a > i, h(q:-(a)) = these v k s #k, nk Prove (2), (3)* ( ' P k + l , #k+l, n k + l > *

THEOREM 3.8: If A >

l!Pl+,

then Dm(p,A,A) =

Dm(p,A,ao).

Remark. If you wonder why we do not require only A > (IdI then see Exercises 3.10 and 3.14.

+ KO) , +

Proof. Let p = IT1 +,and w.1.o.g. p is {O(Z; a)}. If Dm(p,A,p + ) 2 a, then by 3.6 (when a < p) and 3.7 (when a 2 p ) there is a (A, a)-function h such that F,*(O(Z;a), h) is consistent. Hence for every A, Ff(O(3;a), h) is consistent; hence P [ O ( Z ; a ) , A , A] 2 a. By the monotonicity of Dm(O(Z;a ) , A , A) in A (by 3.4(2)) it follows that for every A > p, D"[O(Z;a),A , A] = D"[O(Z;a),A, p + ] . NOWwe can e d y finish the proof by proving by induction on a that D"[O(Z;a),A, p + ] 2 u implies Dm[O(Z;a), A, a]2 a using what we have just proved (for every p).

Proof. We shall define, by induction on k c w, formulas tpi and natural numbers rnk such that (1) for every k, there are a,,, q E " > p , such that (i) for every q E "p, p,, = p u {v:(Z; 0 < n < w} is consistent, (ii) for everyq E "p,n < w, w c p, IwI 2 mi+,, {&+l(Z; Gv(i)): i E w} is inconeietent.

CH. 11,s 31

RANKS, DEGIREESAND SUPERSTABILITY

47

Letf(k) = min{n: m: > 2) u {a). (2) For every k, if f ( k ) < w , then f ( k + 1) > f ( k ) or Z = f ( k ) = f(k + l), and mf > m:+l, (3) if n < f ( k ) , then tpl = &+l. If we succeed in defining them, and define tpk as yz for large enough k ( f ( k ) > n), then clearly 3.9 is satisfied by g~, = Alsis, ?@; Yj) A A p. It is also clear that for k = 0 there me such &, mz (by 3.7(3)). So it suffices to prove, that if q!,,m!, are defined for Z I k, 0 < n < w , then we can define &+l, m:+l, 0 < n < w . Now we can assume there are a,, r ) E O>p, such that (1)is satisfied by v;, m:, (0 < n < a),a,; and (iii) ifr)E@>p,Z< w , i l I ia s . - - sil < p,jl s j, I . . - s j, < p (wherei, = i,+l oj, = j,+,fora = 1,. . .,Z - l)andvl,. . . , v , E ” > ~ then the two sequences

h

Tin- 2 } ,so 1B1 5 No, B c A , IS,(A)I 2 IS,(B)I 2 2 b becaum the types {Q~I&; avIk)v[kl:k < w } are consistent and pairwise contradictory for v

E "2.

EXERCISE 3.1: Prove that i f p 2 Id1 + No,p+ 2 A, and Rm(p,A , A) 2 S(p), then R*(p, d, A) = co. [On S(p) see Definition VII, 6.1(3) and Theorem VII, 5.6.1

CH.

11, 8 31

RANKS, DEGREES AND SUPERSTABILITY

55

EXERCISE 3.2: Prove that if p + 2 A, X = R or X = D and T is stable, X m ( pA, , A) L [(No+ I A l ) Y ] + , then X*(p, A, A) = 00 (forX = D use Exercise 3.10).

EXERCISE 3.3: For

a formula B = B(Z;a) define the model Qo.

universe is (6 E Q : 6 C Orb; a]}, and its relations a m

.

Re = {(Zit . . 4): Q C v[Zly y

Its

.. . 4, a]} y

for every q~E L(T). Assume T is stable or at least that for every q ~ , P(e(z;a), v, 00) < a. Prove that P [ B ( Z , a), L, A] computed in Q is equal to R(x = z, Lo,A) computed in Go. Moreover, for each A c L there is do c Lo (and vice versa) such that = Ido)and P [ B ( Z ; a), A, A] computed in Q is equal to Rm(E= Z,Ao, A) computed in Qo. Moreover we can assume that for every v(Z;g) E A there is p o ( f ;go)E do,and vice versa, such that (i) for every 6 E Q there is go E Q0 such that for every 3 E Go, 5 = @OY

* *

.>,

(E

.

k c p [ ~ ~ , .. ; 61

A

A B[Z,; a]

iff Cro C qo[z;6O],

1

(ii) for every 6 O E CoyCro C (W)

6O)

or there is 6 as in (i).

EXERCISE 3.4: Give an example of a theory T suchthat D(x = x , L, 00) = 1, R(x = x , L, 00) = 00. (Hint: See 4.8(2).) EXERCISE 3.5: Give an example of a theory T,, ITaI = 1.1 such that D(x = x , L,00) = a, R ( z = z,L, 00) = 00.

+ KO

EXERCISE 3.6: Give an example of a theory T,1111 = A, such that R(x = z,L, p ) < 00 iff p > (2"+. (Hint: It has A one-place predicates

Onls. 1 EXERCIBE 3.7: Prove that if A is finite, and for every k < o, P ( p ,A , k) 2 n, then P ( p ,A, KO) 2 n.

DEFINITION 3.4: (1) Define CR(p, A) as an ordinal or

as follows: CR(p, A) 2 0 iff p is a complete type, and CR(p, A) 2 a + 1 8 p is a complete tspe and for every p < A and finite B c Domp, there are pairwise contradictory complete types qt (i 5 p), p t B E qi, and CR(qi, A) 2 a(2) T is transcendental if CR(p, 2) < 00 for every type p . 00,

56

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 3

EXERCISE 3.8: Prove for CR(p, A) theorems parallel to l . l ( l ) , 1.2, 1.3(1), 3.1 and 3.2. EXERCISE 3.9: Investigate CR(p, A). (Hint: See V, Section 7.) EXERCIBE 3.10: (1) If T is stable, show that in (3) of the definition of D”(I,,A, A) we can restrict q to the union of r and a finite clg(A)-rn-type. (2) If in addition A = clg(A) we can replace “{y5(f;a,); i s p} is n-contradictory over q” by “ { y 5 ( f ; a,);i s p} is n-contradictory” (i.e., n-contradictory over the empty set) and thus omit q completely and use just r. (3) Hence show that if T is stable and A = clg(A) it is sufficient to define a (A, a)-function as a function

A: &(a) -, {(#(f; g), n ) : y5 E A , n < w , Z ( f )

= m}

and we a n then replace IT1 by (Id1 + No)in 3.6, 3.7, 3.8 and 3.11.

EXERCISE 3.11: Show that in the previous exercise the stability is indeed necessary. EXERCISE 3,12: Let p be a finite rn-type, A c L; prove that the following conditions are equivalent when < 2n0, and always (2) * (1) * (3). (1) R”(p,A , 2) = a, (2) for some A, I{q € @ ( A ) :I, = q}l > IAI lAl No, (3) for some A, IAI s KO, [{q E S ~ ( A )p: c q}l 2 P o .

+

+

EXERCISE 3.13: Relativize 3.3(1) and 3.14 to 6(f;8) (as.in Exercise 2.5) and check the other theorems to see if they can be relativized similarly. EXERCISE 3.14: (1) Show it is possible that Dm(f= Z,$, N,) = 00 but Dm(f= f,y5, A) = 2 for every A 2 8,. (Compare with 3.8.) (2) Show that in (1) we can replace two by any a. (3) Show that in any example of (l), T is necesmrily unstable. [Hint: Let T, be the theory saying: P , Q form a partition of the universe; P,(O < n < w ) a m disjoint subsets of P ; xRy -+ P ( z ) A Q ( y ) and S(z) = { y : z R y } (i.e., we look at P as a family of subsets of Q); the sets S(z) for z E PI are pairwise disjoint; P,(cO), P,(c,) for 1 E “w, and

CH. 11, f 31

RANKS, DEQREES AND SUPERSTABILITY

57

the c,’s are distinct; and let It be a one-to-one function from O’w into w , for every r ) E n < w and distinct

n

21 E Pn+1,

S(cnlk)

kSI(n)

n

S ( z I )= 0. I < htn)

Now T1has a model completion with elimination of quantifiers satisfying (1) for 51, = zRy.1

EXERCISE 3.15: Show that in Theorem 3.11, we can replace “ p is regular ” by ‘‘cf p > I TI ” and if T is stable we can replace “p 2 I TI , pf < p (i < [All) * ~ l < l A l l if A = K,,, or No+a+l, ifA < No+,.

For each u < Nu+,, or u = 00 we can expand M to M, by one-place predicates such that 2 is replaced by u, ID(T,)I = Ha+,, lTll = N, where T, = Th(M,).] We can use Exercise 3.20 for more general examples.

EXERCISE 3.17: Prove that in Claim 3.12, when 0 is finite, we can replace “ K = log, log,I@I ” by “ ~ ~ “I 0 I@[ ” (see 1.7 of the Appendix). EXERCISE 3.18: (1) Prove that the following conditions on A 5 x are equivalent: (A),,, For some T, IT1 I A, ID,(T)I = x. (B) For some Boolean algebra B, llBll IA and on B there are x ultrafilters. (2) If A, x satisfy (B) for a 1-homogeneous B, show that in Exercise 3.6 we can replace (2*))+by x+. EXERCISE 3.19: Generalize 3.9 to Dm(p,A, p + ) : (1) for stable T, (2) in general.

58

RANKS AND INCOMPLETE TYPES

[CH.

11, $ 3

EXERCISE 3.20: Let R(A, T ) = R1(z = X, L, A), Mlt(A, T) = Mlt(z = z,L,A). Suppose T,is a complete theory in L, (i < K ) , L, pairwise disjoint, (Vz)P,(z)E T,. Let M , be a model of T,, M = Mi (i.e., IM/= Uip, such that for every r ] E *p,

P U {vnlr(%anlr) A

l(pV(8;

&): 1
p iff there are p,, (n < and v;(z; 4) for

7 = (, A, k ] = 8

+ n:Sk,

Mlt(pn u {$},A, k ) = 1 : s k

then

QUESTION 3.31: Is Exercise 3.30(4) best possible? (Thisis a problem in finite combinatorics.) EXERCISE 3.32: Suppose Z(Z) = Z(g) = m and O(Z,g; C) k cp(Z; 7i) A $(g; 6), q ( Z ; G) k (3g)O(Z, 5; C), and for every 2, O(Z; 2; C) is algebraic. Prove that for every infinite A, Rm[v(Z;Ti), L,A] 5 R"'[$(Z;6),L,A].

CH.

11, 8 31

61

RANKS, DEGREES AND SUPERSTABILITY

DISCUSSION 3.16 : The following generalizes Exercise 3.3, and shows how the work of Poizat can be put in our context. Let p be a type over a set A. Define 6, = (a, a)asA;Q, is a model with universe {a E C: a realizes p } , and relation R, = {E E Q,: E satiafies tp}, for any tp = tp(3, a), si E A. In 11, Exercise 3.2 we deal with the case p a finite type, then Q,, is a saturated model, and so we can translate results on the stable case to the non-stable case. Here Q,, is not necessarily sctturated, but it is like the Q when we deal with kind I1 in [Sh 76~1.As said there, all results on ranks, degrees and also forking goes through, and some caaes are discussed in detail. Let me elaborate a little more: CLAIM 1: Q, is a i - h o g e n w w r ; moreover, i f (ar:i < u), (br:i < u) realizes the same quantifier-free formulas in 6, (equivalently, the same form* with parametersfrom A in a), then there is an autommphisnz f of Q, which taka (ar:i < u> to (b,: i < a) (and, in fact, is a ratridion of an automorlphh of Q which is the identity on A).

Proof. Find the required automorphism of Remark. This would hold also for Easily:

IQ’I

Q.

r E &A).

= {a: tp(a, A ) E F},

CLAIM 2: (1) Q, is i-compact for quantifier free form&, ( 2 ) Q, satisfies the assumption of kind 11, hence all recrults on forking and ranke generalized.

in

But we may still be interested in the connection between what occurs Q and in 6 ,.

ASSERTION 3: (1) A set I E IQ,I is discernible in Qp iff it is indiscernible in Q over A. (2) Two such injinite sets are equivalent in 6, iff they are equivalent in Q. ( 3 ) For B E C E IQ,l, and IQpl, tp(si,C, Q,) fork over B iff tp(Si, C u A, Q ) fork over B u A. (4) Similar equivalence holds for stability in a power. Procf. E.g. (3) p = tp(8, C,6,) does not fork over B iff some infinite indiscernible set I over C of sequences realizing p has, under auto-

62

RANKS AND INCOMPLETE TYPES

[CH.

11, 5 4

morphism of Q, which are the identity over B, a bound number of images up to equivalence, so we finish. For usual rank there is no nice translation, but if in the definition we do not take a h i t e subtype but deal with the same type, or deal with C P (and A = 0 for simplicity) the situation will be like 11, Exercise 3.3.

II.4. The f.c.p., the independence property and the strict order property DEBTNITION 4.1: (1) ~ ( 3g ; ) has the finite cover property (f.c.p.) if for arbitrarily large natural numbers n there are go,. . ., such that CT(~Z)

A c~(z;a') k-zn

but for every 1 < n, C(~Z)

A

cp(z;ak).

k 0 (n(2) 2 0 for we can choose k > n(1) and n(2) # 0 by the choice of Z(0)). Clearly JpfZo)+ll tends to infinity with k; and for proving (5), it suffices to prove that Mlf[B(Z; z k ) , A] = Mlt[p$O),A] tends to infinity with k. We prove that, more specifically, Mlt(p$O),d)2 Ip$O)+lI IpjZ0)I.Let A be such that all p k are over A. For each Q E pfZ0)+l- pfco), by the choice of pZo)+l , Rm[p$O)+ - { Q } , A , No] 2 n(0) - Z(O), so there is q,,, ES"(A), pZ0)+l - {tp} c q,,, and Rm(q,,A,No) 2 n(2). But Rm(pfc'o)+l, A, No) I n(0) - Z(0) - 1 so 1~ E q. Clearly the qVys show that Mlt(pfc",A) 2 lp$o)+ll- Ip$O'I. (5), + (4)m Suppose A, k, 8 exemplify (6),,,;so there are a,(Z < w) such that Rm[8(Z;a,),A, KO] = k and Mlt[O(z; a,),d] 2 1. So for some A, I{q E S ~ ( A , )P: [ { 8 ( z a,)} ; u q, A , KO] = k}l 2 1, SO if I 2 24n,by 1.6 of the Appendix there are tp;, 8; (i < n ) such that R"(pf, A, No) = k, p ; = ttp;(z;b y ( * = j ) : j5 i} u p(z; a,)} andyfEcll(A).SowecanfindZ(n+ 1) < wandtp, suchthatfori < n < w = (p,, and let for i < n, A pi(,,, = 8,(Z; EF). If (4), is not true, there are k,bt, such that: for every E, P [ d t ( Z ; a), A , No] = k 4 C@] and similarly for 8 , $ . By a compactness argument we can find a, 8, (j< w) such that: @)

P [ 8 ( Z ;a),A,

KO] = k,

Rm[{8(Z;a)} U {Cp,(Z; 6j)u(j=f):j I i}, A,

KO] = k

contradicting the definition of the rank. (a),,, => (8),,, We choose A, k, 8 exemplifying (4),,,, so that k is minimal. Clearlv for everv 8,. the property P [ 8 , ( S ; Zl),A, KO] 2 k is elemen-

66

[CH.

RANKS AND INCOMPLETE TYPES

11, 8 4

tary (as it is equivalent to -,V14kR”[e,(Z;a,), A , KO] = I, and by the minimality of k). Moreover also the property “Rm[8(Z;a),A, KO] > k or Rm[B(Z;@ , A , No] 2 k and Mlt[O(Z; a ) , A ] 2 1” is elementary (it is equivalent to the existence of Qt,, E A (i # j c 1) and 6‘, such that

A R”[{e(Z;a)} u {Ti,,@; &,,):

i < E},d, KO] 2 k

j A ; q,p E 'A;

Z(U), v = 7 1 n = p n, i = q[n],j = p[nl}. For notational simplicity let +t = f ( t ( Z * ; g), = +(Z*; Z). By Lemma 2.9(2) it is clear that for every cp-m-type q Rm(q,cp,A) 2 k iff {+(Z*; Z)} u {Q(Z*; a): cp(Z; a)t E q} is consistent. Taking in Theorem 4.4(3) A = {+, +O, +l}, m' = Z(Z*^Z) the result follows. (B) Since for every A 2 2, and p : (0 Rrn(p,cp, 4 I no -def - Rm(- = 3,cp, 2) < w , (ii) Rm(p,cp, n ) = R m ( p cp, , a) ( n IA) implies Rm(p,cp, n ) = R Y P , A); it suffices to prove: for every n there are a(n),p(n) < w such that for any set (*I of cp-m-formulas, p , if for every q c p , IqI I a@), RYq, cp, R n ) l 2 n , then R"(p, q,00) 2 n. (Take n2 = max{p(n): n s n , } , so for (A) k = m&x{a(n):n Ino}.) We prove it by induction on n. For n = 0, Rm(p,cp, A ) 2 0 iff p is consistent; hence by "not (3),,," our assertion holds. Suppose we have proved for n, and we shall prove for n + 1. By Theorem 2.12(2) for every q E O'2, k, Z < w there is a formula +[J such that =

+

cp9

Rm[{cp@;Si,)R['': i < Z(q)},cp, k ] 2 Z iff kgJISi.o .,.,82(q)-1]. Let m, = Z(1)where cp = cp(Z; g); and A, = {I#")*":q E a(n)r2}. Let k(n) < w be such that

68

[OH.

RANKS AND INCOMPLETE TYPES

11, 8 4

(i) every inconsistent set of A,-m,-formulas F has an inconsistent subset of power < k(n),and (ii) there are no a,, Sl such that C&; a,] = 7p[6,; at] for i # j < k(n) [k(n)exists by "not (3)," and Theorem 2.2 and 4.31. Now let P(n + 1) = 24k(n)+ /3(n),and a(n + 1) be such that if r i a a of power 0.

t<jSn

It is not hard to prove that Tinais consistent, by building a model for it. It is also easy and standard to prove it has elimination of quantifiers, and is complete. By this it can be shown that no formula 'p = 'p(Z; 9) has the strict order property. On the other hand, clearly y(x; y) = zEy has the independence property (hence the order ProPY)*

a)

THEOREM 4.9: If 'p@; i8 urntable, A < K < Ded A, then there is an A such that IAI 5 A, ISg(A)I 2 K . (See Definition 1.4 of the Appendix.)

Proof. By Exercise 1.4 of the Appendix, there is an ordered setJ, IJI 2 K , with a dense subset I, IZl = A. As 'p(Z; 9 ) is unstable, by Theorem 2.2 'p has the order property. Hence by the compactness theorem there are BE, 8 E J such that: for every t E J, {'p(Z; B8)if(tsE):8 E J)is consistent. Let A = (J{aE:8 E I}. Clearly IAI s I Z l - Ho = A. For every t E J let p t = {'p(Z; aE)if(t Ded, A. So by Definition 4.4,there is A, IAI s A, 15;(A)I r: Ded, A. As Ded, A is regular, by Theorem 4.10(2)' p ( f ; #) has the independence property. So by Theorem 4.10(1)for every A there is an A, IAI s A, ISz(A)I 2 2A,

[OH. 11,8 4

RANKS AND INOOMPLETE TYPES

76

hence Kr;(A) > 2A. But always IS;(A)I 5 2 1 4 + N o , hence Kc(A) I; (2"+. So if p is unstable, Kr;(A) = Ded, A (for every A) or Kt$!(A)= ( 2A) (for every A). So suppose p(5; 8) is stable. For every n, if for one A, Kc(A) > n, then there is A, IAl I; A, ISg(A)I 2 n. It is easy to find B E A, IBI 5 KO such that ISg(B)J2 n; hence for every p, aa p 2 K, 2 IBI, Kr;(p) > n. Hence if for some A, Kr$(A) = n, then for every p, Kr;(p) = n. As for every A, and a, ti realizes a type in 8; (A), clearly IS;(A)I 2 1, son > 1. So suppose Kr;(A) 2 No for some A. Then for any n there is A,, l@(An)l 2 2". We can define by induction on k < n, sik, ~ [ ksuch ] that +

1{p ES;(A,):p(5;sil)n[ll~p for every 1 < k}l 2 2n-k and {p@;al)n[lI: 1 < k - 1) u {p(z;a k - 1 ) 1 - n t k - 1 1

1

is consistent. For simplicity suppose every $11 is zero. Hence {p(?; jjl)if(l n* and r # q implies pa $ r . Let d(2; E) (a E IMI) say: {p(Z; go): $(go;Z)} is consistent, has rank (Rm(-, 9,KO)) = n*, and extends pa (0, E exist by Theorem 4.4). We could have chosen p1 such that n* is minimal. Then

E(Zl, Za; Z) = 8(2,; Z)

A 0(Za; E ) A

(Vg)[$(g; 21)

$(g; Z,)]

is as required.]

QUEATION 4.8: Suppose the condition of Exercise 4.6 holds. Is there an equivalence relation over M with exactly A equivalence classes ?

CHAPTERI11

GLOBAL THEORY

III.0.Introduction The main interest and motivation in this chapter is in uncountable theories. The results of Chapter I1 are sufficient to find the stability spectrum of T (i.e., the class of cardinals in which T is stable) when T is countable. We shall prove: THE STABILITY SPECTRUM THEOREM: For any stable T,there is a cardinal K ( T ) I I TI such that T is stable in A iff A = A Ax for every h < DedlAI. (Thus we get a complete characterization of Kr(h) for countable T,and a better picture in general). We also find when there are > h pairwise contradictory types of cardinality < K over a set A, IAI = A. Those proofs indicate an alternative proof to the stability spectrum theorem for h 2 2ITI without using forking or ranks (see Exercise 7 . 8 ) . In Sections 1-5, T will be stable and A finite in all unstarred theorems and l e m m . No unstated assumptions are made on T or A in starred theorems and lemmm.

III.1. Forking Notation.For an index set U let 2, = {xu:u E U}and for a sequence Ti = ( u ~ ) ~let + ,Pa = A type in 2 , over B is a consistent set of formulas using only variables from P, and parameters from B. The

type is complete if it is maximal among the types in Z, over B. For an m-type p and a set I of sequences of length m let p ( I ) = { Z E I : a realizes p}. Similarly we define cp(I;6) for a formula ~ ( 26). ; Insted of %Awe sometimeswrite just A.

DEFINITION 1.1: tp,(B, A) = {cp(Z6; a): 6 E B,a E A, Ccp[6; a]},tp*(6, A) = {cp(Ze; 8):si E A , E E 6, C cp[E; a]}.

DEFINITION 1.2: p split8 strongly over A if there is an indiscernible sequence I = {Z:n < w } over A and a formula cp such that cp(Z; ao), -,cp(Z; al) ~ p (Compare . Definition I, 2.6.) Remark. Clearly ifp splits strongly over A then p splits over A.

DEFINITION 1.3: The formula cp(Z; ti) &ideaover A if there we n < and sequenoes a’, 1 < w, such that (1) @(Ti, A = tp(8, A), (2) {cp(Z; #); 1 < w} is n-inconsistent (see Definition 11,3.2(3)(E)).

w

DEFINITION 1.4: The type p in I , forb over A if there are formdaa cpo(Zo;So), . .., cpn-l (3”- l; Z-l),with Z c 3, for k < n, such that ( l ) p V k < n vk(*;a k ) , (2) cpk(P;ak)divides over A for dl k < n. Instead of writing “{cpO(z; a)} forks” we write “cp(2; a) forks”. Note. p is not in general a type over A ; see Corollary 1.3.

LEMMA 1.1*: (1) If

g, k$(g; E) --+ cp@; 13)and cp(1; E) divides over A , then #(g; E) d i v k h Over A. (Inparticular this holds when $ is obtained by adding dummy variublea to cp.) ( 2 ) Ifp i s a type in Za, @finite) then p forks over A iff there are formula ZE

.

. .,cp’-’(Za; 3 - l ) smh thUtp k V k < n cpk(Za; ak),andcpk(Za; 7ik) divides over A for k c n. (3) cp(Z; 8) d i v k h Over A iff there are n c w and a sequence I = {a‘: 1 c w } indiscernible Over A such that ti = iio and {cp(Z; 8): 1 < w} is (po(Za; go),

n-ietent. (4) If cp(E; a) d i v k h over A then cp(I;a)forbover A. (6)p f o r b over A iff some finite &type q of p forb Over A type in Iafor some finite a). ( 6 ) tp*(C, B) fwks ovw A ifffor tp(8, B))f o r b Over A.

8Ome

(so q is a

E E C, tp*@,B) (OT qUiVdently

QLOBAL THEORY

86

[CH. 111, 8 1

(7) I f p , c p , f o r i < j < Sa~nop,forksoverA,tlaenU,,,p,doea not fork over A. (8) If A c B, q and p are types in the same variables, q t- p [ p E q] and q doecr not fork [aplit ~trongly]over A then p doea not fork [aplit strongly] over B. (9) I,fpu{$'(Z; a')}forksOverAforl < n < w,thenp~{V,,,,#(Z;ti~)} forks over A.

Proof. ( 1 ) is immediate and (2) follows from the parenthetical remark in (1).

(3) If the stated condition holds then clearly cp(Z; a) divides over A. Now suppose cp(Z; a) divides over A. Then there are n < w and sequences a', 1 < w such that tp(si, A ) = tp(2, A), and {cp@; 8 ) :1 < w} is ninconsistent. By Ramsey's theorem and compactness there are E l , 1 < w such that (6': 1 < w} is an indiscernible sequence over A , tp(6', A) = tp(a, A) and {cp(Z; 6l):1 < w } is n-inconsistent. So there is an automorphism P of Q such that F 1 A is the identity and F(6O) = a. So (P(6'):1 < w ) is the required sequence. (4) Immediate. (A converse is Exercise 4.15.) (5) Since i f p t- V,,,, qi then for some finite q c p , q t- V, a, contradiation. (2) Since q forks over A, let q k V k < n q k ( Z ; ak)as in Lemma 1.1(2) where ~ ~ (Z k3) divides ; over A. Take A , = {vk(Z; gk): k < n} and let d 2 dobe finite. clearly q k p U {Vkl < n(q) or I{n < w : Clp[8;7in]}I < n(q). Define 6" = aannSian+land $(Z; 6") = Q(Z; aan)A l ~ ( Z ; Zan+l). Clearly (6": n < w } is indiscernible over A . If w E w , IwI = n(tp), then {#(Z;6"): n E w} is inconsistent; otherwise, letting E realize this set, I{n < w , Ccp[E; Z]}l 2 1{2n:n E w}l = n(rp) and I{n < w : Cl~[8; a"]}l 2 1{2n + 1: n E w}1 = n(tp),Contradiction. Thus by Lemma 1.1(3)#(Z; 6O) divides over A. In addition p I- $(Z; 6O) since tp(Z; ao),+%; a,) ~ pHence . p forks over A. (2) * By definition, p k V k < n rpk(Z; ak)where tpk(8;Z k ) divides over A. So by Lemma 1.1(3)there are sequences (or sets; it is the same by stability) {a:: i < w } indiscernible over A, = iZk, and there are n(k) such that { q k ( Z ; a:): i < w } is n(k)-inconsistent. Let B = Dom p v {@:k < n , i < n(k)}andletqE8m(B)extendp.SincepI-Vk p. By Corollary 3.3 K ( T )s ITI' and 80 IBI 5 ]TI.Let B E ]MI,11iK11 = ITI. Then p < I{~EB(A):Bp = B}I iI { ~ E B ( Ap) :does not fork over B}I s l{p E 8(1M(U A ) :p does not fork over B}I = I{p E 8(lM1): p does not fork over B}I (this by Lemma 2.16(2)) s ISr(lNl)l. But clearly IS(1MI)l s 2ITl, contradicting p 2 2ITl.And if T is stable in po and 11M11 = IT1 s popthen IS(lMI)I 5 po contradiofing po s p.

THEOREM 3.7: If p < p X d n then T is n d 8tabk in p.

Proof. Since p < p e M nthere is K < K(T)such that p" > p. Let K be the minimal one. Thus p e E = p. Now since K < K( T), for some m K < ."( 2') 80 by Theorem 3.1, (3), we have an inoreeeing sequence A,, i s K , a type p E Sm(An),sets I' = {q:j < p} indiscernible over A,,I' c A,, md cpi(Z; $) EZ) such that j > 0 =+ yv@; 4)EZ). Now for r ) E " p define an elementary mapping F,,suoh that (1) Dom F,,= U {$: j < p, i < Z(r))}, (2) 7 = p i Fnextends F,,, (3) if Z(q) = i then for every j:

(4&-(a = &r(al), (B)q-($) = FV(ad),

(C) for everg a # 0,j = &-(o>(G). The definition prom& by induction on Z(r)). For Z(7) = 0, F,,is the

CH.

111,s 31

THE INSTABILITY SPECTRUM

105

uy N,,, and T is not stable in Ha+, so T is not superstable. (3) Clearly K"(T)5 K"+'(T)(by extending the m-type i n Theorem 3.1 to an (m + 1)-type). For every regular K there is a strong limit cardinal A > 2ITI of cofinality K , so A'" = A, A" > A. If K < K ( Tthen )

106

GLOBAL THEORY

T is not stable in h by Theorem 3.7, and if K h by Lemma 3.6. SOfor regular K ,

K

[OH.

2 ~~(2') then

III,f)3

T is stable in

< K ( T iff ) K < K,(T),SO we finish.

DEFINITION 3.2: I is a maximal indiscernible set over A in M if I is indiscernible over A , I E ]MI,and there is no J 3 I, J E IMI,J indiscernible over A. DEFINITION 3.3: dim(I, A, M) = min{l JI: J is equivalent to I , and J is a maximal indiscernible set over A in M}.If A = 0 we just write dim(I, M).If for all J as above dim(I, A, M )= IJI, we say that the dimension is true.

Remark. This definition differs technically from Definition 11, 4.5, but no confusion will arise.

LEMMA 3.9: I f I is a d m a l indhcernible set over A in M ,then 111 K ( T )= dim(1, A , M ) ~ ( 5 "and ) ifdim(I, A , M ) 2 ~ ( 5 " )then ~ the dimension is true.

+

+

Proof.Let J be a maximal indiscernible set over A in M ,equivalent to I, and for which IJI = dim(I, A, M). Clearly IIl 2 IJI, so that IIl + K ( T )2 IJI + K ( T ) .Thus it is sufficient to prove that III + K ( T )I I JI + K ( T or ) in fact IIl I IJI + K ( T ) . By way of contradiction assume IIl > IJI + ~(2'). From Corollary 3.6 we have I, E I, !IlI 5 IJI + ~ ( 2 "such ) that I - I, is indiscernible over A u U I , u UJ. From the equivalence of I and J we can find an infinite I* such that (I- 11)u I*is indiscernible over A u UJ and J u I*is indiscernible over A (take I*= (8,: n < w } where the type 8, realizes over A u I U J u U {al:1 < n} is the average of I over the same set). Now with the aid of I*we shall show that (I - I,)u J is indiscernible over A. Let a,,. . .,8, be distinct sequences in (I - 11)u J , E E A , r p ( f , , . . .,3,; E) a formula. Let 6,, .. ., 6, be any distinct sequencesin I*. Define 8; aa follows: for 8, E J, 6, for 8, E I - I , .

u u

Now Crp[zi1, . . ., 8,; EJ o Crp[8;, . . .,8:; El since (I - I,)u I* is indiscernible over A U UJ, and hp[8;, . . .,8:; El o C&, . . ., 6,; a] since J u I*is indiscernible over A. But then the truth of rp(8,, . . .,8,; a) is

CH.

111, Q 31

THE INSTABILITY SPECTRUM

107

not dependent on the particular a,. Thus (I - I,) u J is indiscernible over A, in contradiction to J’s maximality in M. LEMMA 3.10: (1) Let M be an ( K , + K ( 2’))-saturatedmodel. Then M is h-saturated iff for every infinite indiscernible I c 1M1,dim(I, M ) 2 A. (2) We can replace the msumption “ (N, + K ( T))-saturated” by “every type which is almst Over some A c IMI,IA I < K ( T ) ,is realized in M ” (i.e.,by “F~tn-satu*ated”. Bee Definition%I V Y1.1 and IV, 2.1). Proof. (1) If M is h-saturated then clearly there is no indiscernible set of power < h which is maximal in M.So one direction is clear. Now assume M is not h-saturated, h > K, + K ( T), and let p E #“(A), A c IMI,IAI < A, be a type omitted in M. Extend p to p ~ P ( l M l ) and choose B c “1, IBI < K ( T )so , that q does not fork over B (by Corollary 3.2). By 1.12 and 2.13(2) since M is (N, + K(T))-saturated there is I c ldll which is an infinite indiscernible set such that p = Av(I, IMI).Let J z I be an indiscernible set maximal in M. If dim(I, M ) < A, we are through. Otherwise IJI 2 &(I, M ) 2 A. By Corollary 3.6 there is J’ c J , IJ‘I s lAl + K ( T )such that J - J’ is indiscernible over A ( K ( T< ) A, IAI < h so IJ’I < hand thus J - J ‘ # 0). But then since Av(J, A) = Av(I, A) = p 1 A = p we have that all the sequences in J - J’ realize p, contradiction. (2) Similar proof. Of THEOREM 3.11: If { d f ( } { < h iS an ~nCreaeing8modele, and K ( T ) Cf 6, then = ui cf(S), and in partioular h > K(T).First assume cf S > No so we can use 3.10(1). Let I c IiKl be an infinite indiscernible set. Let B c lH), IBI < K(T),be such that Av(I, ]MI) does not fork over B. Since K( T ) s of S and cf 6 is regular, there isj < 6 such that B E 1M,1. By Corollary 2.13(2) there is J E lM,1 equivalent to I and since M,is h-saturated there is J’ z J , J ‘ c IM,I, indiscernible and IJ’I 2 A. Thus by Lemma 3.9 dim(I,M) = dim(J’,M) 2 h and by Lemma 3.10(1) M is h-saturated. If cf S = 8, we can use 3.10(2) instead of 3.10(1) (by 2.13(1)) and easily reprove a suitable version of 2.13(2) (by Exercise 2.4). THEOREM 3.12: If T is A-stable then T h a h-saturated nzodel of 7A.

108

GLOBAL THEORY

[OH. 111,8 4

Proof. We define an increasing sequence {&f,};s), such that IIM,II = A, M,,= U,< M,and every p E B( IM,I) is realized in M,+ It is easy to see that M,,is cf(b)-saturated. Thus if A is regular then MAis Asaturated.

If A is &gulax then A = x; A, SO M,+,:is A,-eaturated. Now if K ( T ) cf(A) then by Theorem 3.11 we have that MAis A,-saturated, for all i. Thus M Ais A-&urated. If ~(2’)> cf(A) then A A. So by Theorem 3.7 T is not stable in A, contradiction.

EXERCIBE 3.1: In Lemma 3.4 prove that we have 111 < ~ ~ ( 2 ” ) . EXERCISE 3.2: Give an example in which K ( T )is singular. EXERCISE 3.3: Prove ~ ~ ( 2= ’ )~(2’).[Hint: Suppose A, (i < K ) is increasing, tp(an6, A,+l) forks over A, for i < K . By 4.16 tp(iZ, A, +1) forks over A, or tp(6, A, + u a) forks over A, u a.3 EXERCIBE 3.4: Prove that in Lemma 3.4, when J is infbite, there is a minimal I = I , among the possible I’s, i.e., I , is contained in every possible I . EXERCISE 3.5: Prove the paxallel of 3.11 for A-aomp&ctnesa.

III.4. Further properties of forking THEOREM 4.1 : Let p E Bm(B), A E By then p doe0 not fork Over A iff for allJinite A , Rm(p,A , No)= Rm(pt A , A , KO). The analogous theorem for stationary types is: THEOREM 4.2: Let p E Bm(B),A E Bythen (1) p M etatimry Over A i+@for all Jinite A , P ( p , A , KO) = P ( p r A , A, KO)and Mlt(p, A, No) = 1. If p M etationary wer A and the rn-type q 2 p , then q doe% not fork Over A iff for all Jinite A and all A, 2 s A s KO,R”(p,A , A) = Rm(q,A , A). (2) p it3 e t a t i m r y over A iff p is etation4zry Over B and &oes not fork over A . ( 3 )In (1) we can restrict ourselves to all large enough A .

OH.

m, 8 41

FURTHER PROPERTIES OF FORKING

109

Proof of Theorem 4.1. The direction e follows from 1.2(2). Now w u m e p does not fork over A. Let 8 realizep. Let q = stp(8, A), r = stp(8, B). By 2.6(1) (and since q t - p t A, r t-p),for every finite d, R"(r, d, No) = Rm(pu r , d , No) = Rm(p,d,No); Rm(q,d,80)= W q U P t A , d , NO) = R"(p A , d, No).So it is sufficient to prove Lemma 4.3.

LEMMA 4.3: Let q = stp(8, A), p a type realized by 8 which does not fork Over A , and q E p . Then for everyfinite A , 2 5 A s No, (1) Rrn(q,d,4 = Rrn(p,d, 4. (2) Mlt(q, A, A) = Mt(p,4 A). Proof. First we prove (1) for A < KO. Let p be over B, where A G B. By 11, 1.1 Rm(q,d,A) 2 R m ( p , d A). , By 11, 2.13 and 11, 2.2 Rm(q,d, A) = n < w for some n. By 11, 2.9(2) there are cpf;j(E A ) , iZ$j for q E ">A, v E "A, i < A, j < A) by 2.6(3) such that (i) stp(b,, A) = stp(7i,,A), q E "A atp(6f;j,_A) = stp(7it;', A), q E ,A'" i < A, j < A. (ii) t p * ( u {b,,: q E "A} u U {&j: q E n > A , i < A, j < A}, B) does not fork over A, and extend the corresponding type of the By 2.9(1) 6,,(7E "A) realizes p . Clearly hpnll[6n; 6;i,lat when 1 < n, 9 E "A, i # j < A; and i = q[l],t = 0 or j = q[Z],t = 1. So by 11, 2.9(2) (now using the other direction) Rm(p,d,A) 2 n = Rm(q,d,A). So we prove the equality. The proof of (1) for A = KO,and of (2) is similar using suitable parts of 11, 2.9. Now we return to

Proof of Themem 4.2. (1) Suppose p is stationary over A. So it does not fork over A, hence by 4.1 for every finite d, Rm(p,A , No) = R"(p A , 4 No). Let M be a (I BI + ITI +)-saturatedmodel, B c "1. Let r E rSm( 1M1) p E T, T does not fork over A , and let 8 realize r . By 2.14 p k q =deI stp(si,A), hence by 4.3 Mlt(p,d, A) = Mlt(q,d, A) = Mlt(r,d, A) for

r

+

110

QLOBAL THEORY

[CH. 111, 9 4

every finite A, and 2 IA I8,. But by 11, 1.10(1) Mlt(r, A, A) = 1 so Mlt(p,A, A) = 1 and we finish. Now suppose p is not stationary over A. If it forks over A, then by 4.1 for some finite A, Rm(p,A,8,) # Rm(p 1 A , A, 8,). So assume p does not fork over A. By 2.9(1), if p = tp(8,, B), q = stp(8,, A) then p If q. So for some E E FEm(A)p u {-,E(Z; a,)} is consistent; let si, rertlizeit.By2.6(1)Rm[p u {E(Z;aI)},A,KO] = Rm(p,A,N,)forZ = 1, 2 (where A = {A')).But E(Z;8,) k -,E(Z; a,). Hence Mlt(p, A, KO) 2: 2. The second sentence is emy, so we have finished the proof of 4.2(1). (2) The proof is immediate by part (1). From 4.1 we see COROLLARY 4.4: Let A E B E C and p E Sm(C).p does not fork over A i$p does not fork over B and p 1 B does not fork over A . THEOREM 4.6: Letp be an m-type over A , q a A-m-type,A not necessarily finite, swh t?ut p U q (bconeistent and) f o r b over A . Then for A 1 8,: * ( p , A , A) < 00 * Rm(pu q, A , A) < Rm(p,A , A).

Remark. (1) This improves 1.2. Note that in 1.2(4) we have some unspecified A,, and in 1.2(3) we have a formula which divides. (2) In the c o m e of proving this theorem we shall show: (*)

If p forks over A, A c B, then there is an elementary mapping F, F 1 A = the identity and P ( p ) forks over B.

Proof. Let M be a p+-saturated model ( p = IAI + A + I TI)such that A c [MIand q is a type over [MI. Assume that a = Rm(p,A,A) = Rm(pu q, A , A) < 00 and we will get a contradiction. Since A 2 No,by 11, 1.6 and 11, 1.1 there is q' E S T ( ~ M such ( ) that Rm(pu q', A , A) = a. By the definition of rank there axe .c A such q'. Let them be {q,: j < j , < A} and let 8,realize p u q,. Choose A, c 1M1, IA,I 5 IT1 such that tp(a,, [MI) does not fork over A, and let B = U, ECN(Eo,@ u Y).

CH.

111, 5 61

THE FIRST STABILITY CARDINAL

125

DEFINITION 6.4: Let l7,@, Y E FEm(B). l7 is

independent over Qi (mod Y )iffor all E E 27, E does not depend on @ u (27 - {E))(mod Y ) .

LEMMA 5.6*: (1) For every 0,E there is a flnite Y E Qi such tluct ECN(E, @) = ECN(E, Y). (2) If E depend%on @ (mod Y) then for someflnite l7 E @, E depends on 27 (mod Y ) . Proof. (1) Let n = ECN(E, @). Then F(@,8, n + 1) is inconsistent. Let F' be a finite inconsistent subset and let Y be the set of E E FEm(B ) appearing in I", I" E F(Y,8, n + 1 ) so ECN(E, Y ) I n. But by Lemma 6.4(2) ECN(E, u') 2 n, so we have equality. (2) Immediate by part ( 1).

LEIKMA 6.6*: If El depends on @ u {Ea} (mod Y ) ,but El doe% not depend on @ (mod Y ) ,then Ea depencls on Qi u { E l } (mod Y) and even on { E l } (mod Y u @). Proof. Let a realize p and denote pg = p u {E(Z;a): E E l7}. Let a:, . . ., ail (a!, . . . , a:,) be representatives of the equivalence classes of

El ( E a )which contain sequences realizing p. Let S f = { k : 1 I k Inl, p& u { E l @ ;a;)} is consistent}. Since El depends on @ u {Ea} (mod Y ) but not on @ (mod Y ) there is k E S1 such that p&y u {Ea(Z;a), El(%;ah)} is inconsistent. Let a* realize u { E ~ ( za;)}. ; Thus for E E @ u Y , I. E[a*;a] A E1[a*;ail. Also for some m 5 na k Ea[a;a%]so pgUyu {Ea(Z;a;)} is consistent (it is realized by 3).But p&,u~Bi) U {Ea(Z;a:)} is inconsistent as every sequence which realizes it also realizes p&, u {Ea(Z;a), El(%;a;)}. Thus ECN(Ea,Qi u Y U { E l } ) < ECN(Ea,Qi U Y ) IECN(Ea, Y ) , and the conclusion follows. LEMMA 6.7*: If for all i < a E' does not depend on Qi u {Ej: j < i) (mod Y ) ,then { E L i: < a} is independent over Qi (mod Y ) . Proof. By Lemma 5.6 it suffices to prove this for finite a. For a = 0 or a = 1 it is trivial. Now suppose the claim is true for a but fails for a 1. For some i < a E* depends on l7 = @ u { E f :j < a 1 , j # i}

+

+

(mod Y ) ,but ELdoes not depend on l7 - {E"} (mod Y ) .So by Lemma 6.6, with E' for E l , E" for Ea, l7 - {E"} for @, and Y for Y , we get a contradiction.

126

QLOBAL THEORY

[CH. 111, f 6

LEMMA 6.8*: Let l7 be independent over @ (mod Y). Then: (1) for every E there is a Jinite II* c l7u @ a c h that l7 - 17* is independent over @ u {E}(mod l7* u Y). (2) For every Q1 there is l7, E nu@, < 1a11++ No, such that n - n1is independent over @ U (mod U n1). Proof. (1) By Lemma 6.6(1) there is a finite l7* G l7 u @ such that ECN(E, Y U @ u n) = ECN(E, Y u n*). Clearly by 6.4(2) l7 - l7* is independent over @ (mod Y u n*). Suppose that - 17* is not independent over @ u {-E> (mod Y u n*). Then for some E* E l7 - l7*, E* depends on @ u { E } u (l7 - 17* - (E*}) (mod Y u n*). But E* does not depend on @ u (l7 - l7* - {E*}) (mod Y u l7*) (by 5.4(2): the monotonicity properties of dependency). So by Lemma 6.6 E depends on @ u (I7- I!*) (mod Y u n*), hence ECN(E, Y u l7*) > ECN(Z, 17* U Y U @ u (l7 - l7*)),contradiction to the definition of l7*. Now the claim follows by Lemma 6.4(2). (2) Similar.

LEMMA 6.9*: If No < A I A* (see Lemma 5.1)and A is regular, then there isaJiniteYc{E::i< A}andSsA,lSl =A,suchthatl7={E::i~S} is independent over the empty set (mod Y) and ECN(Er, Y u (l7 - {E:})) 2 Bfori€S.

Proof.Let l7, = {E:: i < j};by Lemma 6.1 ECN(E,*,nj)2 2. If 6 is a limit ordinal less than A then by Lemma 6.6(1) there is f ( 6 ) < 6 such that ECN(E,*,n d ) = ECN(E,*,nf(d)). By the regularity of A and 1.3( 1)of the Appendix there is i, < A such that So = (6 < A : f (6) = i,} is of cardinality A. For each 6 E Sothere is afinite ITdE l7,,such that ECN(E,*,n d ) = ECN(E,*,n,,)= ECN(E,*,nd). As I17i,l < A, IS,l = A, A is regular, and l7*Jhas c A finite subsets, for some finite l7,S = (6: f (6) = i,, n6= l7} is of cardinality A. The result now follows by 6.7. LEMMA 6.10: Let A, I A* be singular. Then there is Y G { E r : i c A,}, IYI = cf A,, and S E A,, IS1 = A,, such that {E:: i ES}is independent over the empty set (mod Y )and ECN(E:, Y)2 2 fbr i ES. Proof. Let A, = ~ o < c r < c f h Aa o where Aa is regular and for 0 < u < < cf A,, AB > A, > cf Ao. Let l7, be defined aa in the previous lemma.

CH. 111, 6 61

THE FfR8T STABILITY CARDINAL

127

Now define by induction finite Y, E nAa+, and S, c lSal = such that {Ef : i ELI,} is independent over UB A, a contradiction). Thus 5.13(2) holds, i.e., 2No 2 A + . I n other words A < 2Ho; so ID(T)I < 2No and thus IT1 = ID(T)I by Lemma 11, 3.15. Of course there is a countable A for which IS(A)I 2 2% (by Theorem 11, 3.2).

Thus T is not stable in any power < 2% But by our hypothesis if A 2 2n0, A = ACE(*) then T is stable in A. Thus (2) holds.

130

OLOBALTHEORY

[CH.

111, Q 6

COROLLARY 6.16: If T is superstable then ID(T)I or ID(T)( + 2 b is thefivst power in which T i s stable and it i a I8(A)Jfor some countable A.

LEMMA 6.17: If T is countable and superstable, but not stable in So,then there is m such that IDm(T)I 2 No.(Thus T is not N,-categorical by I X , 1.6.)

Proof. Similar to the proof of Lemma 6.13.

III.6. Imaginary elements

DEFINITION 6.1 : (1) The sequence 6 is defined by the formula;@I a) if ~(6 a);= {6}. It is defined by the type p if 6 is the unique sequence which realizes p . It is definable over A if tp(6, A) defines it. (2) The formula v(Z;a) is algebraic if v(Q; 3)is finite. The type p is algebraic if it is realized by finitely many sequences only. The sequence 6 is algebraic over A if tp(6, A ) is algebraic. (3) The definable closure of A, dcl A is {b: b definable over A}. (4) The algebraic closure of A, rtcl A is {b: b algebraic over A}. If A = acl A we say A is algebraically closed. (6) When dcl A = acl A, cl A will be their common value.

LEMMA 6.1 : (1) The type p defines a sequence iff for some finite q c p , A q defines that sequence. (2) The type p is algebraic i;fsfor some Jinite q c_ p , A q is algebraic (moreover, we can choose a q such that p and q are realized by exactly the same sequences). Proof. Immediate.

LEMMA 6.2: (1) A c dclA c acl A. (2) When A G By dcl A c dcl B and acl A c acl B. (3) If B = dclA tiLen B = dcl B. (4) If B = acl A then B = acl B. (6) 6 &a definable [azgebraic]over A if 6 E dcl A [6E acl A]. Proof. Easy, e.g., (4) If c E acl B then for some b,, . . .,bk E By n < w and formula 9, b(3 '"z)~(z,b,, . . .,bk) A ~ [ cb,,, . . .,b,]. AS b, E B =

OH.

111, 8 61

acl A there are #,[b,; a,]. Define

O(z;

*

.

131

IMdGlINARY ELEMENT8 EA,

n(i)
214, h regular, for some w, E ldl, n < w , S , = { p E S ~ ( A )P,(p) : = w,foreveryn < w}hascardinality Th.Letd, = { t p , : i E ~ , < , w , } , so by assumption lA*I < ( A1, hence by (ii) lST*(A)l < A. As h is regular, for some p o E ST.(A),

Inn qV- 0. i<jsn

(2) For two distinct B-equivalence classes x,/E, x2/E, the family of

sets

{z: z E q , zRy} for y E x,/E forms a partition of x l / E

(VX?.l)(ZBY -BY), +

( V q ) ( i ~ E-+ y ~ Z ( Z EA Z~ R z ) , (vZyZ)(Z&/

A ZR2 A

2 E y j - y = 2).

(3) These partitions are independent, moreover, for m (vzi,. *

*

n < w,

(i\T z ~ y i i\ l z ~ z i

,Xn)(vyl,- - yn)(vz)[ 8

s

A

1-1

111

A -@IEYk A 1x1 = x k )

isi ) ~ ( 2 ah
em

EXERCISE 7.3: Suppose T does not have the independence property; and there are K cp,, $a, i < K , a < w such that (where Z(Z) = m) for any r) E =w, {cp,(Z; G)K(a*"[fl): i < K , a < w } is consistent. Prove K < K&,(T). EXERCISE 7.4: Prove that in Defhition 7.3 we can w.1.o.g. take n, = 2.

QUESTION 7.5: Prove that q,,(T) = KE,(T)= KLJT). QUESTION 7.6: ISKm(A)= K1(A)? EXERCISE 7.7: Suppose A E By p €Sm(A). T does not have the independence property. Then there are q E Sm(B),q 2 p and C G By ICl < K&,(T)such that: if {a,,:n < w } is an indiscernible sequence over A u C,and a,, E Bythen cp(3;a,) E q e cp(3;a1)E q. Remurk. Instead of "T does not have the independence property" we can add "where {cp(Z; a,) = +Z; n < w } is k-contradictory for some k < w y y .

EXERCISE 7.8: Prove Theorem 3.8 by the,method of the proof of 7.3 (changing the definition of Pa). QUESTION 7.9: Investigate K F ~ ( TIf) . T does not have the independence property is K ~ ~ = ( TK )~ ~ ( at T )least , when both are regular cardinals? PROBLEM 7.10: Characterize the possible functions R(A), and K(A, K ) = sup{ISI + : S is a set of 1-types, contradictory in pairs; p E S == 1p1 < K ; and the types in S are over some A, IAI 5 A}. (Possibility: Maybe the function A(A,@) should be used, where @ = (w,: i < K ) , w, a subset of {j:j < i} and k(A, a) = sup{IZ n +: Z E rc2A; Z closed under initial segments; and for i < K , I{7 w,: r) E Z } ~IA} (remember r) is a function with domain {j:j< i}).}

CH. 111, f 71

149

INSTABILITY

EXERCISE 7.11: Suppose in P,in acl(A v a) there is I independent over A, assuming T is stable (see Definition 4.4). Prove K ~ ~ > ~ (IIl.T ) EXERCISE 7.12:There is a theory T without the strict order property, Kcdt(T)= 00,K ~ ~ ( I TI,also by Fi,and if T is stable, by Fftoo. Procf. We should prove that if B , C E A, (ql,C)EF; where q1 = tp(6, A U a), and (p, B) E F : where p = tp(8, A ) then (p,, B) E F;, wherep, = tp(zi, A u 6). Let q = q1 r A .

162

[OH. Ivy8 2

PRIME MODBL8

For x = f this is 111,4.13. For x = s, t , we first claim that if Ti, realizes p , 6, realizes q, then 8,^6, realizes tp(an6, A). Because tp(Zl, A ) = tp(8, A ) for some elementary mapping f,f 1 A = the identity and f (a,) = a; so w.1.o.g. Ti = 6,.AS C E A , (q, C) E F;, tp(6, A ) I- tp(6, A u a),SO tp(iZ-6, A ) = tp(Tin6,, A), so the claim is proved. Hence if 8, realizes p necesaarily tp(aln6, A ) = tp(a-6, A), SO tp(iZl, A u 6)= tp(& A u 6) = p,. SO p I- p,. But for some F;-type p' over B, p' E p , p' I- p , hence p' I- p1 hence ( p , , B) E q. For 1: = a the proof is similar. We first show that if a,, 6, realize stp@, B), stp(6, C) respectively then tp(Ti-6, A ) = tp(Siln6,, A). Hence stp(Z, B) I- tp(Ti, A U 6) so we finish by 2.1. We are left with x = 1. Let ~ ( 5 3 Z)~ ; be a formula. As in the proof of 2.8 we can find n < w and, for 1 < n, 'pI = cpI(Za; ?&,a,) such that for every E E A u 6, there are I < n and E' E 8, 8' E A such that hp(Za; 8) = (pI(Za,6,8'). As (ql,C)E F there ~ are types p i c q1 YC, p i = qi(zh), where $1 = #I(%; zap 21) = FI(Za; 536, zI)*Let (I\ = p1 1' $1, [dl < {0&3, $): i < (p:(} be closed under conjunctions (by 2.2(2)) where i$E C. For every 1 < n, i < lp:l let 0&(Za,

z ITI,wehaveIp*l < h ; a n d o f c o u r s e p * s p r B . Now it is easy to check p* I- p1 v. +

r'

THEOREM 2.10: A d m (VII) is satisfied by Fg when a,f, T is stable, or x = t , I , h is regular.

2 = s

or x =

Prmf. Let B c A , p = tp(si, A U C), p = tp*(C, A), r = tp& u C, A) and (p, B u C) E FRY(p, B) E F;. We must prove B) E F; for 8 E C where = tp(a-8, A). CLAIM 2.11: For sets A , C, suppose for every 8 E C pi I- qe = tp(8, A), and p' I- p = tp(& A u C), p' is over A u C, (p', pi are closed under conjunctions).Define ?-&=

{(3%d)[#(3a, E l

E y

5,%d; 6')

a E cy #(za;

Then ri,c I- ra,e = tp(ii-E, A ) .

4,E2)]: a, 6,) Ep', 0@CyEd; )'6

A t)(?t&, z y

E pied}.

OH. IVY8 21

EXdlldPLJES OB

163

F

Proof of 2.11. Let 'p = 'p(ZayZ, 6,) E T,,,, and we shall prove r;,E I- (p, and tGs will suffice. By the hypothesis p' I-p and t'p[7i, 8,6,] hence p' I- 'p(Zdd,a, 6,) ~ p Hence . (as p' is closed under conjunctions) there exists # = 2, a, 6,) ~ p (we ' add dummy variables if necessary to make room for 2) such that a E C,6, E A and #(ZayE, a, 6,) I- (p(ZaYE, 60). Hence b * [ Z , a, 6,] where 'p*(Za, 4, 6,) = (E6)[#(Za, Z, Zd,6,) + (p(Zad, $, 6011. I- 'p*(ga, Za;6,) hence for some Hence (p*(Zay%a; 6,) E qa-d, SO @ a ~ %; 6 8 ) E d - d , F(Ed)(vZd)[O(Z&Z d ; 63)+(p*(Ze~Z ~6 Y a)Ia

It is easy to check that (33d)[$@tw

%Y 4, 61)A

@ a ~ Zdp

6311 I- CP(%F,, %Y

60)

and that the antecedent of this implication belongs to &. So we finish. Continuation of the poqf of 2.10. Clearly if p', qi are over B u C, B respectively (and w.1.o.g. C n A E A), then T $ , is ~ over B ; so the caae x = s is immediate. If x = t , then I C 1 < A as (p,B U C) E Fi and there are p', qi satisfying 2.11 such that 1 p' 1, I < A. As h is regular in that case it follows that IrL,,l < h so we finish. The C= 2 = f follows by 111, 4.16. So now we deal with the m e 2 = I ; again we assume h is regular. Let 2 E C, and 'p(Za, Za; E ) be a formula. By the hypothesis there is p' c p closed under conjunctions, lp'1 < A, p' is over B u C and p' I- p 'p. Let p' = {#l(Z,,, E, a(i),6 :): i < a < A}, and 'pr(ZayZd(:);6*,6:) = (VZ,J[#1(Z4yZ,,Zao, b:) -+ 'p(Z4, ZEy6*)], for a(;)E C, 6 : E B, 6* E A . By the hypothesis (q,-dcf),B) E Fi so for some &acn c qpdct, 1 B y IQ&l < A d-aco I- Qa-aco t' 9:. Let +

q;-&l) =

@*,I: j < .(i) < A}

{4.,(5,%(1);

(again w u m e it is closed under conjunctions), 6;,,E B. Let e&(Z8~

za; gtj)

= (3Zd(l))[#l(% %Y

ZrZ(f)>

6:)

A

ei,j(3t~Zd(f); 6 : , j ) 1 .

Let r& = {O&(z8,Z,; 6f.J :j c a(i),i c a}.It is easy to check r;,, E r,,a is over B and of cardinality < A . It suffices to show T L , ~k r5, I+ Q. For let 'p(?i?,,, Za; 6*) E r,,,, (8* E A ) so k'p[iZ, E, 6*]. Hence 'p(Z,,, 2, b*) ~ p , hence for some i < a, #&,, E, a(i),6:) I- 'p(Zay8, 6*)hence b(fia)[${(Z8Y

SO

t'p:[~, a(i),6*,671, hence

q)

'p@aY

EY

6*)].

'pr(~~, it&:);6*,6:)

E q ~ - d ( { ) hence for

164

[CH. Iv,6 2

PXME MODELS

some j < a(i) of d

6:,,)

~f,j(% gd(f); @,j)

A

I- &Zd, Zd(,); 6*, 6:). So by the definition

(3zd(f>)t#f(Zd,

hence

% zd(f;, 6:)

%Y

zd(O;

ef,j(Zdt

%d(f),

#f(%

hence A

'

qP(zdY

6f,j)1 I- d z d Y

%; 6*)

6*)1

d&(za,zd;67,J I- d z d , zd;g*)

so we finish also the case x = 1. We are left with the case x = a. First we prove

LEMMA 2.12: Let T be atable (1) If B E A, tp(Ti, A) ~ q ( B ) , E ~ a c l U l ( Ti)thtp(Ti^c', B A)EF$(B). (2) Let B c A, tp(Ti, A) E Fa@) iff tp(Ti, ad A) E F"(clc1 B).

Proof.(1) Let O(z; 8, 6)be algebraic, 6 E B and C@, 8,6]. By 2.4( 1) for ,$I we can assume s(z; a, 6) I- tp@, B u a). Let e ( l q a, 6) =

. .,B,,},

Z0 = 5. By 111, 2.6(3), (1) there are Ti', 5; such that stp(TinZon ^Z,, B) = stp(Si'^ELn. .^$, B) and stp(Ti'nZLn- ^-' cn, A ) does not fork over B. So stp(Ti', A) = stp(8, A), (by 111, 2.6 and 111, 2.9) hence we can msume Ti' = Ti. So (EL,. ., 5;) is a permutation of (EOy ,En), hence we ctln w u m e Ei = $. So tp(Si^Z,, A) does not fork {Eo,.

- ..

--

.

...

.

over B. By 2.1 it suffices to prove stp(8"Zo, B) I- tp(Ti-Zo, A). If this is not true there is Ti*^8*, stp(Si*^Z*, B) = stp(Ti^ZoyB), tp(Ti*^Z*, A) # tp(Ti^E,, A). So necessarily tp(Si*^E*, A) forks over B. Again we can assume Ti* = Ti, hence for some i E* = Ef, contradiction. (2) Immediate, by 2.1 part (4), and 111, 6.3(4). Conclusion of thproof of 2.10. We have to prove it for x = a. By 2.2(11), it suffices to prove it in P. Let C' = acl(B u C), so by 2.12(1) tp*(C', A) EF$(B)and by 2.12(2) tp(Si, A u C') E F"(B U C') = F"(C'). SO tp@, cl) I- tp(Ti, A u C'), stp*(C', B) I- tp(C', A). Now use 2.11 with p' = tp*(Ti, B u C'), qi a type almost over B, which is over A, and such that qi I- tp(E, A) (qi exists by 2.1). This proves the assertion.

LEMMA 2.13: A d m (VIII) i8 8athfied by

for x = t, 8, a, f.

Proof of 2.13. Let A, (i < 6) be increasing, p E ~ ~ ( UAf), , , ,pf = p 1 A, E F$(B).We must prove ( p , B) E F;. For x = f,it follows by 111, 1.l(7).

OH.

Iv, 9 21

EXAMPLES OF

F

165

For x = 8, t let q, be an Fg-type over B yq, E p,, q, t p,. I f x = s , p rBisanFi-type,p t B t q , t p , h e n c e p t B t U , , , p , = p . If x = t , qo is an Fi-type, qo t p o ;and clearly q, G p o ,hencep, t q, t p,, hence !lo I- Po Uf < d Pr = P. For x = a, by 2.1, if li realizes p, stp(8, B) t p , hence stp(li, B) I- p hence (p, B) E F:.

LEMMA 2.14: Adom (IX) is satisfied by F : for x 6 < Cf A, it8 ConclUsiOn hkh.

=

t , 8, U , f; for 1, if

Proof. Let 6 < cf A,. A,, B, are increasing with i, p ES"'(U, I T I , T ~ z e m T G e t a b k i n 8 0 m e p< 2A. (2) Adoma (X.l), (XI.l), (X.2) and (XI.2) are satiejkd by if A r K(T). (3) A d m (X.l), (XI.I), (X.2) and (XI.2) are eatiefied by F{ if T io etable. (4) Ax(X.1) ie eatisjied by Fi if T ie stable, A r (TI (e.g., by every countable stable T). (6) A d m (X.l), (XI. l), (X.2) an& (XI.2) are eatiefid by FW, if T k totally tranecendmtal. Proof. Part (1) follows by 2.15(2)(3), 2.16(1) and 2.17(1), (2). Part (2) follows by 2.16(2)(iii);part (3) by 111, 1.4. Part (6) follows by 2.15(4), 2.16 and 2.17(1). So we are left with (4). Let p be an m-type over A, and {cp,(P; g,): i < ITI) a list of the formulas in L. Define by induotion on i < I TI formulaa $,(1;a,) (a, E A)

PRIME MODELS

168

[OH. I v ,

52

such that p , = p u {$,(3; Z,): j < i} is consistent. If we have defined $,@,a,) for i < a < ITI,clearly p , is consistent also when a is a limit ordinal. Choose $,@; a,) so that Bm(p,+,,tp,, 2) is minimal. It is easy to check that for some q a ~ S ; ( A )P , ~ + ~Hence I-~~ plT, . has a unique extension in Sm(A):U,< I TIq,. So Ax(X.1) (and a little more) holds.

LEMMA 2.19: When x = t , 8, I , f,F = Fsf,F 8ati8&? Ax(XI1). Also for x = U, T 8table t h b holds. Proof. For x = t , 8, for some Fg-type q E p C,, q I-p 1 B (as (P t B, Ci)E Fg) hence q I- P t Ca; but as ( p ,Ca) E R,p 1Ca 1p , hence q b p , SO (p,Ca) ~ F s fFor . x = u the proof is similar using 2.1(4). For x = f Clearly p does not fork over B (asC, E B, (p,0 2 ) E Pi) and p t B does not fork over C,, hence by the transitivity of forking (111, 4.4) p does not fork over C,. We are left with x = I , so for every 'p, for some q E p 1 Ca, 191 < A, Q I-p 1' tp, it is also clew that p t C, kp IC,, hence p 1C, 1 q, so for every $ E q there is a finite re E p 1C,,re 1 $. Now r = {rs: 41, E q} E p C,, and r I- q I- p 1 'p, so (p,C,) E Fi.

u

+

DEEWITION 2.8: Ff:= {(p,A): p E#"(B),where m < h and p does not split over A}.

w, A

E B, IAI
IT1 it holds by 2.2(8), 2.8(2).) [Hint: Let I be indismrnible, IIl = A, ~ E IC, = I - {a}, and B = A = 0. Remember h ( q ) = A+ + IT1 by 2.2(3).] +

QUESTION 2.5: Show that Ax(V1) may fail for Fi,when A is singulax CfA 5

pi.

EXERCISE 2.6: Show that for A singular Ax(VI1) may fail for FA, (notioethat by 2.2(8), h 5 12’1). [Hint:As in Exercise 2.3 but P,(a) = c,, B = 0 , A = {at: a < &, i < K } , afrE Pi(&)and Efr(c,,a;) iff a = 8.1 QUESTION 2.7’: Show that for A singulax Ax(VI1) may fail for F:, with IT1 = K (=of A),

OH.

IVY8 21

EXAXPLES OF F

171

EXERCIBE 2.8: Show that Ax(VI1) may fail for Fa when T is unstable. [Hint: As in Exercise 2.1, and let b E R(Q),Q C - , E ( b ,c, a). Let B = 0, A = {b}, c = {a},B = (c}.] EXERCILYE 2.9: Show that Ax(1X) may fail for Fi,when cf A 5 I TI, and we can choose I TI = cf A. [Hint: Let L(T)consist of equality only; af (i 5 A) be distinct elements. B, = A, = {a,: j < i} for i < h and

P

= tP(aA,

Ul A; in fact we can .choose a superstable T).[Hint: (1) Tin*,(2) Let T be T A +but the counterexample is for Teq(for TA*see Exercise 11, 2.3, for "eq" see 111, Section 6). Clearly ~ ( 2 1 ~ 9 .= ) A++, ITe'] = A+. Let I, (a < A + ) be the set of one-to-one functions from u into A, Z = U,Z,, and choose a, E Q for S E Z such that a$,at iff

PRIME MODELS

172 8

1i

=t

1i; then A

=

[OH. I v ,

42

{a8/Ei:8 E I , i < A+} c P, 'p = P = ( z )form a

counterexample, (3) Let A be regular

T

= Th(N),

M

=

(AA x

0,.

. ., P,,, . . .,. ..,Ev,. . . ) , , s ~ > h , v s ~ A

where P, = {(p, n ) E 1611: r) Q p} and ( p , n)E,(r), m ) if€ for some a p t a = 7 1 a = v 1 a, p(a) = r)(a)# v(a) or p = r). The counterexample is for TW (assuming N < a): A = {(p, n)/Ev:v E AA, ( p , n ) E 1N1,v # p}, and 'p = P=WI

EXERCISE 2.14: Show that Ax(X.2)may fail for Fi, T stable, ITI = A. [Hint: Let T = TA,A regular (see Exercise 11, 2.3), choose u,E(L: for 7E + 1) such that a,,Eiaviff r ) i = v i. The counterexample is for Teq:A, = 0, A , = (a,,/#,:i < A, r ) E "}, A , = A , u (a,,:r) E A(A + l), (A + 1)E Range(r))}and 'p = PI(z).]

r

EXERCISE 2.15: Show that Ax(XI.l) (hence (XI.2)) may fail for FL, T stable, A = ITI.[Hint: Let P,(i < A) be pairwise disjoint, infinite one-place predicates c E 1C1 - U, Pi(a), ci E Pi( a), A = 0, C = {q: i < A}. SO tp(c,A) is F&-isolated, it has a unique extension in S(C):tp(c, C), which is not Fi-isolated.] EXERCISE 2.16: Show that Ax(XI1) may fail for (Tunstable). [Hint: Use T,a, b, c from Exercise 2.8: we let C1 = 0, B = C, = {a}, A = {a,c}, P = tp(b, A).] EXERCISE 2.17: Show that (VI) may fail for Fr. [Hint: Use the theory of the rationals as an ordered set, B = C = 9, A = {l}, 6 = (3), a = (2).] EXERCISE 2.18: Show that Ax(XI.1) (and XI.2)) may fail for FX. mint: Let T be as in Exercise 2.11, hint 2. We let A = 0, B = 0, C = Pi(C), choose a EQ(&)and let p = tp(a, A). (We use the notation of Ax(XI.l).)

u

EXERCISE 2.19: Show that Ax(XI1) may fail for Fhp. [Hint: Use the previous example. C, = 0, A = Cp = {P,(a)},B = Po@),p = tp(a, B).] EXERCISE 2.20: Prove 2.12 for F( instead @.

CH.

Iv, 8 21

EXAMPLES OW F

173

EXERCISE 2.21: Assume Ax(III.1)and (VII).If Ax(X.1)[or Ax(X.2)] holds, then it holds for v = p(P; a) too. EXERCISE 2.22: Show that A3(T)= KO iff A3(T)= KO. [Hint: Suppose A3(T)> No,then T is not totally transcendental (by 2.15(4)); let M'be a IT1 +-saturatedmodel of T . Then A = [MI,q = {7v(z; 8): R(cp(P;a), L, 2) < 00) exemplifies A3(T)> KO.Now A3(T)> Noimplies A3(T)> Noby 2.16(2), so as A3(T),A3(T) 2 24, we finish.]

QUESTIONZ.Z3:Showthatbetween~~(T), IT[,X(T)A,(T)(1 = 1 , 2 , 3 ) there are no connections except those from 2.15(1), (6), 2.16(1) Exercise 2.22. [Hint: (1) T , : Let E, (i < A,) beindependent equivalencerelations, each with two equivalence clwes. Then K J T , ) = KO, ITl/ = A,, A,(T,) = A1(T1)= AS(T1) = Aa(T1)= KO. A3(T1)= A3(T1) = A:. (2) T,: Let AI = (A(a)3,. . .,P,,. . .) where, for r) E A(2)>2Pn -{v E *ca)3:7 Q v}, and A, = 2

({a}appears only when @ = 2a + 1, a < a,). For y odd, replace 2 by 1, and {a} appears only when @ = 2a, a < a,. Clearly f8+, = U y efSyA (A = h(F)) satisfies our requirements. Now fa. is the required mapping. (2) The proof is similar.

Remurk. In fact the regularity of h(F) is needed for 3.3, 3.6 but not in this proof. THEOREM 3.10 [Ax(I)]: (1) If B is F-constructible over A, then B is F-primitive over A. (2) If B i s Fqrimury over A, then B is F-prime over A.

Proof. (1) Let d = (A, {ai: i < a}) be an F-construction of B over A and let B* be an F-saturated set, A c B*. We should find an elementary mapping f from B into B*, f t A = the identity. Define by induction on /3 I CL elementary mappings f8 from .at@) into B* so that f, extends fE for /3 c y. Let fo be the identity over A, and for limit 8, then by fd = U f l < a f s * I f f 8 is defined, ( ~ E B4) Y E F (PE = tp(aE, B*, so f8(236) is Axiom (I), (fj(p8)~ f6(B8)) E F, and DomLfs(pfl)] realized by some i;B E B* (as B* is F-saturated). Extend f8 to f8 + by defining f8 + 1(7iB) = 68. Clearly f a is the required mapping. (2) Immediate.

,

180

PRIME MODELS

[CH.

Iv, 0 3

LEMMA 3.11: (1) If A G B c C, C is F-primitive over A then B is F primitive over A. (2) If A c B E C, C is F-prime over A, B F-saturated then B is Fprime over A. (3) The same for (F,p) instead of F.

Proof. (1) Let B' be an F-saturated set, A G B'. Then there is an elementary mapping f from C into B', f 1 A = the identity. Let g = f 1 B, so g is an elementary mapping from B into B', g 1 A = the identity. Hence B is F-primitive over A. (2) Immediate from (1) and the definition. (3) Proved similarly. CONCLUSION 3.12: Let p(F) < a0 then: (1) [Ax(I), (X.l) and (XI.l)]Over every A there is an F-prime model. (2) [Ax(II.l), (III.l),(111.2), (VII) and (XI.1);h(F) regular] I n every F-prime set B ouer A , every sequence realizes over A an F-isolated type. (3) [Ax(I), (ILI), (III.l),(111.21, (V.2), (VII) and (XI.1);A(F)regular] If B is F-prim over A, C E B, ICI < cf h(F) then B is F-prime over A U C . The Same hoZds.for F-atomic.

Proof. (1) Immediate, by 3.1(2), 3.10 and Exercise 3.1. (2) By 3. I( 2) there is an F-primary set B* over A. So by the definition of F-primeness there is an elementary mapping f from B into B*. f 1 A = the identity. By 3.2 for every 6 E B* tp(6, A ) is F-isolated. Hence for every 6 E B, tp(f (6),A) is F-isolated. But as f 1 A = the identity, 6, f (6) realize the same type over A, so we finish. (3) By 3.1(2) over A there is an F-primary set B*; and by the definition of F-primeness there is an elementary mapping f from B into B*, f F A = the identity. By 3.6 B* is F-primary over A uf ( C ) , and by 3.10 B* is F-prime over A uf (C). As A uf(C) c f (B) G B* by 3.11(2) f (B) is F-prime over A u f (C). (By Ax(1) the property of being Fsaturated hence F-prime over a set is preserved by elementary mapping.) Hence B is F-prime over A u C. The proof for F-atomic is similar.

LEMMA 3.13 [Ax(III.I), (VI) and (VIII)]: Suppose A,C,, (i < a) are given sets, and for every 5 EC,, for some BE E A, tp(E, A u U j < C , j )E F(Bi). Then for every i < a, 5 E C,, tp(E, A u U j + C,) , E F(BE).

Proof.By 1.2(2) (and Ax(III.1) and (VIII))it suffices to prove that for

CH. I v ,

8 31

181

PROPERTIES OF F-PRIMARY MODELS

u,*,

i < a, E EC,,a E C,, tp(E, A LJ 2) E F(@). Let a = where 2, E C j ( l ) 0 s 1 s k, andj(0) < . . . < j ( n ) < i < j ( n + 1) < < j (k).We prove by induction on 1 I k + 1that tp(E,A u a,-. .-a, EF(&). For 1 s n + 1 this holds by Axiom (111.1).If it is true for 1, 1 s k,it is true for 1 + 1 by Axiom (VI). For 1 = k + 1 we get what we want. Z o n e .

-

---

THEOREM 3.16 [Ax(I), (III.l), (111.2), (V.2), (VIII) and (XI.1); h(F) regular] : (1) 8uppose A c [MI, M is F-saturated, and F-atomic over A. Letp, ~ S ( A ) f oi r< a,and A* = A u {c: c E IMI, c realizes a p f for an i < a}. Then hf is F-atomic over A*. (2) For F = q,we can assume thcct pf's are types almost over A , when T is etable. Proof. (1) Let 7i E 1M1, by Ax(XI.1) a s p = tp(Z, A ) is F-isolated, p has an extension q E ~ ' " ( A which *) is also F-isolated. So for some C G A*, ( Q , C ) E F(so ICl < X(F)) and some 6~ ]MI realizes q. Let C = {c,: i < ICl}. By the hypothesis for every E E C u 6, tp(E, A ) is Fisolated, say tp(E, A ) E F(B,). By Axiom (111.2) and the regularity of X(F), tp,(C u 6, A ) E F(B), where B = {Be: E E C u 6}. Hence by Ax(V.2), for every a < ICl,

u

p a = tp(c,, A u 6 u {cf:i < a})E F ( Bu 6 u {cf:i < a}). Now we define, by induction on i, c; E 1M( such that the mapping

fa(a s ' ICI), f,(a) = a for a E A, f,(6) = 7i and f&) = c;, for i < a, is

elementary. It is trivial to definef, and fa (6 a limit ordinal). For note that by Ax(I), f,(p,) is F-isolated, (wa < h(F)) hence some ci E 1M1 realizes it. Let C' = {c;: i < ICl}; as tp(c;, A) = tp(c,, A), c, E A* clearly C' c A*, and IC'I < X(F).We shall provep* = tp(B, A*) E F(C'), and thus finish. By Ax(VII1)it sufficesto prove tp(& A u C' u E) E F(C') for any E E A*. As before, we can extend ficl to an elementary mappingf', whose domain is c [MI,and whose range is C u Rangef,,,. By the choice of C, and Ax(III.I), tp(6, Domf') E F(C), hence by Ax(1) tp(7i, Rangef') E F(C') so tp(Z, A u C' u E) E F(C'). (2) Work in Ceq.By 2.12(2)Meqis F-saturated and also F-atomic over A , hence it is F-atomic over acl A (see 2.1). Now w.1.o.g. each pt is over acl A ; hence, by 3.14(1) Meqis F-atomic over (acl A ) U A*, and by 2.1 Meq is F-atomic over A * so M is F-atomic over A*. LEMMA 3.16 [Ax(III.2) and (V.2); X(F) regular or [Ax(III.2), (V.1) and

PRIME MODELS

182

[OH.

Iv,

3

(IX)]]: If C is F-atomic over A , and IC - A1 s h(F) [or IC - A1 I cf h(F) when A(F) is Singular], then C is F-constructible over A.

Proof. Let C - A = {c,: i < a = ICl}, and let A, = A u {c,: i < j}. It suffioes to prove tp(c,, A,) is F-isolated. As h(F) is regular [or as IC - A1 5 cf h(F)] and Ax(III.2) holds, tp,({c,: j Ii},A ) E F(B) for some B c A, IBI c h(F). Hence, by Ax(V.2) [or by l.l(4)] tp(c,, A,) E F(B u {c,: j c i}).

THEOREM 3.17: (1) [Ax(XI.2) or Ax(XI.l), a c cf h(F),and Ax(IX)] If A, E A, f o r i c j Ia then there are sets B, (i s a)Buch t7t.d A, E B,, i < j =- B, c B,, B, U A , is F-constructible over A , U U, KO then Fk-saturation is equivalent to Fi-saturation (by 2.2(6), (7)). Similarly for other concepta (primeness, etc.).

EXERCISE 4.2: Show that in 4.3(2) we cannot omit the assumption that h 2 K ~ ( T ) . DEFINITION 4.1: For x = t, 8, a, h 2

K,

let

F;,= = {(p,B): for some A, m; p ES”(A), B c A , IBI < fork over B and p is an Fg-isolated type}.

K,P

does not

Note that (p, B) E Ff,= does not imply (p, B) E Ff.

LEMMA 4.4: (1) For x p

i8

=

t , 8 , a, K 2 K(IT),p

E @ ( A ) ,p

is e-i8olaterl i#

F,,-GO&ed.

(2) I n tlce following conceptsfor x = t , 8 , a; it does not matter w&kr

F = V, or F = Ff,%:F-construction A, or (B) p = h is regular ( > No). Then Ff (z= t, s) does not satisfy Ax(XI.1). (3) The same a8 (1) replacing tp by stp and the conclzcsion is p < K ( T ) . (4) If in (3) p 2 h and in the hypothesis we add to: case (i): tp(6, A) is Ff-isolated, case (ii): for every finite S E p, tp(6, A,) is B'g-isolated; and (A) p > h or (B) p = h is regular > KO. Then Ff does not satisfy Ax(XI.1). Proof. (1) Suppose case (i)holds but p L ha(T).Then there is q E &'"'(A,), p = tp(6, A ) E q and a < p such that q 1 A, k q (this can be done by the definition of P ( T ) and by renaming the ail's which is possible by 3.13 applied to F&). By renaming, we can assume 6 realizes q, without changing the hypothesis. By Ax(V1) for Fm, w tp(6,A) Vtp(6, A u a , ) a h tp(iZ,, A) Y tp(iZ,, A u 6). So there are E E A and 'p such that b[6; a,, i5J but tp(iZ,, A) u {-,'p(6, f,E)} is coneistent, and let Z i i realize it. By assumption, there is an elementary mappingf,f 1A, = the identity,f (a;) = a,. We can extend it to an elementary mapping g whose domain includes 6. Then g(6) realizes q A, but not q 1A,+1, a contradiction. So we are left with case (ii),and we reduce it to case (i) as follows: Let So = {a < p: tp(6, A,) If tp(6, A,+l)}. Now w.1.o.g. So = p. [ w e define by induction on a 5 p sets &a) E p, S(a) increasing and continuous, lS(a + 1) - S(a)l < KO.If S ( a )is defined tp(6, As(,))Y tp(6, A,) 80 for someS(a + I), #(a) E S(a + I), IS(a + 1) - S(a)l < No,a €#(a + I), tp(6, A,(,)) br tp(6, As(,+1,), and S(a + 1) is minimal under those conditions. Let S(a + 1) - &a) = {~,(,.o), - . ,%&k(a))}, a,* = %a,o)na,(=.l) -. .nnow the az's satisfies all the hypotheses on the iZ,'s, and for them So = p.] So for every a there a m n(a) < o,i(Z,a) < a (for I < n(a)),formulas 'pa,and 5, E A such that C'p,[6, a,; 7il(o,,), ;E,] but tp(6, A,) u {-,'pa(%; a,, ...,EJ} is consistent. If KO < x 5 p, x regular, by 1.3(1) of the Appendix there is a stationary Sx E x, such that a € S X* n(a) = n, i(l,a ) = i(Z) (for I < n). Then let A' =

.

.

...

192

PRIME MODELS

[OH.

Iv,0 4

A u U1 8, is regular, or (B)z = t , 8; cf h 2 ha(T)+ HI, OT (C) z = a, cf h z K ( T )+ N,. If for every i < 6, E E C,,for .9ome @ E A tp(E,A,) E FS(&), where A, = A U c,,and for i < 8, tp(6, A,) i8 F-i80latfd,then tp(6, A,) i8 F-?kO&e&.

Prmf.W e prove it by induction on cf 8. If cf 8 < cf A, thia is immediate by Axiom (IX). Suppose the conclusion fails. Then we can eaaily define by induction on i < cf 8, a(i) < 6, and a, E C,,, such that (1) j < i 3 a ( j ) < a(i), (2) tP(6, A,{)) LJ tP(6,AM:) u 4 ) for 5 = t, 8, (3) stp(6, A,,)) Y stp(6,A,,, u a,) for x = a. W.1.o.g. 6 = cf6 (by renaming), so we already know 6 2 cfh. If A is regular, z = t , s we get by 4.12(2) case (ii) (for p s f6)that Ax(XI.l) fails for Ff contradicting an assumption. If h is regular z = a, we get a similar contradiction by 4.12(2) case (ii). So we have proved case (A) of 4.13. In case (B) of 4.13 apply 4.12(1) case (ii) and in case (C) of 4.13 apply 4.12(3) case (ii).

Remark. I n 4.13 for x = t , s we need not assume T is stable.

DE~MTION 4.2: Let cfh 2 K ( T )+ N,, z = t , 8 , ( t . Then M is d e d Ff-admissible over A if: (1) bd is q-saturated. (2) A c IM(, and M is F$-atodc over A.

OH.

Iv, 8 41

PRJME MODEL9 FOR STABLE THEORIES

(3) For any infinite indiscernible set I (of elements) over A ,I G

Av(I, 1iK1)is F,"+-isolated.

193

]MI,

Note. If there is no I aa in (3) then of c o m e (3) holds vacuously. THE FIRST CHARACTERIZATION THEOREM 4.14: Suppose cf h 2 K ( T )+ K,, x = t,8,a, F = FZ, F 8Udi8fies A X ~ (x.1) M and (xI.1). If h i8 8i747UhrY2 = t, U%8U??M 8Udi8$&9 Ax(XI.1)fc" arbitrarily large regular p < h. For x = t, h 2 h3(t)or at leaet Pi-saturated implie8 h - v t . Then: (1) i8 Fprime over A M i8 F-admissible over A . (2) The Fprime model over A is unique up to i s m p h i m over A . ( 3 ) If M I is F-adrnissible over A,, 1 = 1,2; f an elementary mapping fr.m A , onto A , then we extend f to an i 8 0 t ~ ~ p hfrom ~ s n M1 ~ onto Ma. Remark. In the proof, if x = a, you should replace t p by stp in many cmes. Proqf. (1) If M is F-prime over A , then M is F-saturated, by definition, iK is F-atomic over A by 3.12(2) (using 4.7 instead of 3.2), and if I c ldll is an indiscernible set of elements over A then Av(I, !MI) is Ff+isolated by 4.9(3). Hence M is F-admissible over A . As there is an F-prime model over A by 3.12(1), the other direction follows by (3). (2) Follows by ( l ) , (3) and the existence of an F-prime model over A by 3.12(1). ( 3 ) Assume x = t or x = a (the result for x = 8 follows by x = a as h > KO (see Exercise 2.30, 2.2(6), (7))). Let h = & A, be such that if A is regular, = K,(T) K,,A, = A ; and if A ,is singular then A, is regular, = cf A ;and if x = t , A singular then Fi(,) satisfies Ax(XI.l) ;and h(i) = A, > lil, A, 2 A, + K ( T )+ K,. For every C E lMIl 1 = 1, 2, there is A, = h(E) such that tp(E, A,) is F&-isolated (we can assume 1M,1 n IiKaI = 0 ) . We shall define elementary mappings fa (a < x ) such that (i) ,!?< EL implies fa extends fs, (ii) A, c Domf, = A: c lM1l, A , E Rangef, = A: c IMaI, (iii) M,, M , are F-admissible over Dom fa, Range fa respectively, (iv) if 2 = 1, 2 E E lMI1, then tp(E, A f ) is FE-isolated, where p = A, h@). For the checking of F-admissibility in (iii) notice that by (3)conditions (1) and (3) &om Definition 4.2, are immediate; hence only (2) (on being F-atomic) needs proof, and it follows from (iv).

x

+

x

+

z, No;or (3) z = 8, t, T totaliy tramcendentul. Then (1)M is F-prime over A iff M $8 F-~admis~ble over A. (2) The F w m e &l Over A is unique up to ~eornorlph~sm, Over A. (3)If M , ie F-*admissible over A,, 1 = 1, 2, and f is an elementary mapping from A, onto A, then we can extend f to an i8omOr.ph~87?~ from Mi onto Ma. Proof. As in 4.14,it suffices to prove (3).For simplicity restrict ourselves to thecase 5 = a. Clearly it suffices to prove Claim 4.19(2)below, by applying it to qo = {z = z},y = 1.

CH.

m,6 41

PRJME MODELS FOR STABLE THEORIES

199

CLAIM 4.19:8uppoee M , , A , (1 = 1, 2) and f are as in 4.18(3). (1)If a: E lMll, tp(a:, A,) = f[tp(ar, A , )] then we can extend f to an elementary m y p i n g from A , u {a E lMll: a realize0 stp(a:, A,)} onto A , u {a E 1M,1: realize0 stp(af, A,)}. (2) If qf (i < y ) are l-typeaOver A,, then we can extend f to an elentent a y mapping from A , u {a E lMll: a realize0 8ome qi} onto A , u {a E 1M,1:a realiz~f(qf)}.

Proof of 4.19. We prove by induction on a that: 4.19(2) holds, when D(q,, L, 00) < a and 4.19(1)holds, when D[stp(a:, A,), L, 003 s a. Caae I. a = 0. Then (2) is empty, and in case (1) by definition of degree stp(ar, A,) is algebraic, hence is realized by a* only, so extend f by lettingf(a:) = a:. Case I I . a > 0, we have proved (1)and (2)for all fi < a, and we now prove (2) for a.

O.Wehaveproved(1)foreveryp < a,and(2)forfi s a, and we shall prove (1)for a.

200

PRIME MODEL8

[CH.

m,8 4

Let I, E lM,l be a maximal independent set over A, whose elements realize stp(a:, lM1l). Let I, = { a ; : j < a,}and a; = a: so if a, 2 w, I,is indiscernible over A, (by 111, 1.10 and 111, 2.9(1)). By Definition 4.3, 4.9 and 4.10, we can Mume that (A) w1 I a, so, for some regular p s A, for every i < A tp(ai, A, u {a;:j < i}) is FUO+lil +-isolatedhence w.1.o.g. a, = A, or (B) a, I w. By F-homogeneity of (Y,,a)croAl it is easy to see that the same case occurs for I = 1, 2. If (B) holds, define by induction elementary mappings f,,,n < w, such that: (i) fo = f, f,,+l extends f,,, M,[M,] is F-admissible over Domf,, [Rangef n l . (ii) ut, i < n belong to the domain 0ff3,,+,. (iii) at,i < n, belong to the range 0ff3,,+~. (iv) Domf,,,+, = Domf3,,+, u {a E lYll:a realizes stp(af, A,), and tp(a, Dom fsn + forks over A l}. This is easily done. Iff3,, is defined, tp(a:, Domf,,,) is F-isolated (by Definition 4.3, part (2)), hence f3,,[tp(4, Domf,,)] is F-isolated, so by by some bi, and letf3,,+,(u;) Definition 4.3, part (2), it is realized in Y,, = bi, D0mf3,+1 = Domf,,, u {a:}. If n = 0 we can let bi = a% = a:. The definition of f3,, + 2 is similar by Definition 4.3( 2) and that of f3,, + is by the induction hypothesis on 3n + 2, by (1) [notice that if a realizes stp(a:, A,) then tp(a, Domfl) k stp(a:, Al); and by 111, 1.2(4), forking implies decreasing in degree]. Now U,,