A Guide to Classical and Modern Model Theory (Trends in Logic)

A GUIDE TO CLASSICAL AND MODERN MODEL THEORY

TRENDS IN LOGIC Studia Logica Library VOLUME 19 Manag ing Editor Ryszard W6jcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences , Warsaw, Poland Editors Daniele Mund ici, Department of Math ematics "Ulisse Dini ", University of Florence, Italy Ewa Orlowska, National Institute of Telecommunications, Warsaw, Poland Graham Priest, Department of Philosophy, Unive rsity of Queensland, Brisbane, Aus tralia Krister Segerberg, Department of Philosophy, Uppsa la University, Sweden Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany

SCOPE OF THE SERIES Trends in Logic is a bookseri es covering essentially the same area as the j ourn al Studia Logica - that is, co ntemporary formal logic and its applications and relations to other disciplines. These include artifi cial intelli gence, inform atics, cog nitive science, philosoph y of science , and the ph ilosoph y of language. However, thi s list is not ex haustive, moreo ver, the range of applications, co mparison s and sources of inspiration is open and evolves over time.

Volume Edito r Ryszard Woj:kki

The titles published in this series are listed at the end of this volume.

A GUIDE TO CLASSICAL AND MODERN MODEL THEORY by

ANNALISA MARCJA University of Florence, Italy and

CARLO TOFFALORI University of Camerino , Italy

KLUWER ACADEMIC PUBLISHERS DORDRECHTI BOSTON I LONDON

A C.LP. Catalogue record for this book is available from the Library of Congress.

ISBN 1-4020 -1330-2 (HB) ISBN 1-4020-1331-0 (PB)

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Preface This book deals with Mod el Theory. So the first que stion t hat a possible, recalcitrant reader might ask is ju st: What is Mod el Theory? Which are its intents a nd applications? Wh y should one try to learn it? Another, mor e particular que stion migh t be th e following on e. Let us ass ume, if you like, t hat Mo del Theor y deser ves some a ttent ion. Wh y should one use this book as a guide t o it ? T he answer t o t he former qu esti on may sound problema tic, bu t it is quite simple, at leas t in our opinion. For , Model Theor y has been developin g, since its bir th , a number of methods a nd concepts t hat do have t heir int rinsic relevan ce, but also provide fruitful and not abl e a pplications in various fields of Mathematics. We could mention her e its role in Algebra and Algebraic Geom etry, for instance the analysis of differentially closed fields (and th e resul ts on t he differential closure of a differential field) , or p-adic fields (and th e asy mpt otic solut ion of Art in's Conjecture), as well as the recent Hru shovski 's mod el t heoret ic a pproac h t o classical problems, like Mo rdellLang's Co njecture or Manin-M umfo rd's Conjecture. So Mod el Theory is to day a lively, spright ly and fertil e research area, which sure ly deser ves t he attent ion of t he mathem atical world a nd, consequent ly, its own references. This recalls t he latter qu estion above. Act ually there do exist some excellent t extbooks explaining Mod el Th eory, suc h as [56] a nd [57]. Also Poizat 's book [131] should be mentioned; it was writt en more th an ten yea rs ago, bu t it is st ill up-to-d ate, and it has been recentl y t ranslate d in En glish. In add it ion more specialist ic references t reat ad equ ately some particular fields in Mo del Theory, such as stability t heory, simplicity t heory, o-minimality, classification theory and so on. Nevert heless, we believe t hat t his book has its own role and its own originality in t his setting . Ind eed we wish t o address t his work not only to t he experts of th e area, bu t also , and mainly, to youn g people having a basic knowledge of mod el theory and wishing to proceed towards a deeper a na lysis , as well as to mathematicians which are not directly involved in Mod el v

VI

PREFACE

Theor y but work in related and overlapping fields, such as Algebra and Geometry. Accordingly we will emphasize t he frequ ent and fruitful connections between Model Theor y a nd t hese branches of Math em a tics (differe nt ially closed fields, Artin 's Conj ecture, Mordell-Lang's Conjecture and so on ). In each case, we aim at giving a detailed report or , at leas t , at sket ching th e main ideas and techniques of th e model t heoretic approach. Our book wishes also to follow a historical perspective in introducing Model Theory. Of course, t his do es not mean to provide a full history of Model Theory (although such a project could be interesting and wor thy of some attention), but ju st to inser t any basic concept in th e historical fram ework where it was born , and so to better clarify the reason s why it was introduced. Hence, after shortly recalling in Chapter 1 basic Mod el Theory (structures and theories, compactness and definability), we deal in Ch apter 2 with quantifier elimination, in particular with the work of Alfred Tarski on algebraically closed fields and real closed fields. We will discuss the role of quantifier eliminat ion in Mod el Theory, bu t we will t reat briefly also its int riguing role in the P = N P problem within the new mod els of computation (such as t he Blum-Shub-Smale approach , and so on). C ha pter 3 will be concerne d with Abrah am Robinson 's ideas: mod el completeness, model companions, existent ially closed structures . We will consider again algebraically closed fields and real closed fields, but we will illustrate also other crucial classes, like differentially closed fields, sep arabl y closed fi elds, p-adically closed fields and, finall y, existent ially closed difference fi elds (a rather recent mat ter , with som e rem ark abl e applications t o Algebraic Geometry). Cha pter 4 deal s with imaginary elements. They a re esse nt ially classes of definable equivalence relations in a structure A , so elements in some quo tient structure. We describe Shelah 's const ruct ion of A eq , englobing these classes as new elements in th e whole st ruct ure, and we show that th ese imagin ary elements can be somet imes eliminated , because the corresponding quotients ca n be simulated by som e suit a ble definable subsets of A . Chapters 5 and 6 are devoted to Morley 's Theorem on uncountable categoricity. Actually its proof will be given only in C ha pter 7, but here we describe Morley's ideas -algebraic closure, totally transcend ental theories, prim e models, an so on- and we illustrate t heir richness and th eir applicat ions. We will be led in this way t o one of t he main topics in Mod el Theory, nam ely t he Class ificat ion Problem. We will ex plain in C ha pter 7 t he more relevan t ideas in t he formid abl e work of Shelah on t his mat ter (sim plicity, stability, superstability, mod ula rity) , a nd we will discuss t heir significa nce in some

PREFACE

vu

important algebraic classes, like differential fields, difference fields, and so on. We wish also to deal with the Zilber program of classifying structures up to biinterpretability, in particular with Zilber's Conjecture on strongly minimal sets, and its brilliant solution due to Hrushovski. Also Chapter 8 largely owes to Hrushovski. In fact, after illustrating in more detail the natural connection between Model Theory and Algebraic Geometry, we will describe the Hrushovski proof of Mordell-Lang conjecture; we will refer very quickly also to the Hrushovski solution of the related Manin-Mumford conjecture. In particular we will realize how deeply Model Theory, actually both pure Model Theory and Model Theory applied to algebra are involved in these proofs. The final Chapter is devoted to a (comparatively) recent and fertile area in Model Theory: o-minimality. We will expound the basic results on 0minimal theories, and we will discuss some intriguing developments, including Wilkie's solution of a classical problem of Tarski on the exponentiation in the real field. We assume some familiarity with the basic notions of Algebra, Set Theory and Recursion Theory. [65], [66] or [78], and [121] respectively are good references for these areas. Incidentally, let us point out that we are working within the usual Zermelo - Fraenkel axiomatic system, including the Axiom of Choice. We also assume some acquaintance with basic Model Theory, such as it is usually proposed in any introductory course. However, Chapter 1 is devoted, as already said, to a short and somewhat informal sketch of these matters. As its title states, this book aims at being only a guide. We do not claim to provide an exhaustive treatment of Model Theory; indeed our omissions are likely to be much more numerous and larger than the topics we deal with. But we have aimed at giving an almost complete report of at least two crucial subjects (w-stability and o-minimality), and at providing the basic hints towards som e conspicuous generalizations (such as superstability, stability, and so on). In a similar way, we have treated in detail some key algebraic examples (algebraically closed fields, real closed fields, differentially closed fields in characteristic 0), but we have provided at least some basic information on other relevant structures (like p-adic fields, existentially closed fields with an automorphism, differentially and separably closed fields in a prime characteristic). In conclusion, we do hope that the outcome of our work is a sufficiently clear and terse picture of what Model Theory is, and provides a report as homogeneous and general as possible. Incidentally, let 11S say

viii

PREFACE

t hat t his book is not a lit er al t ranslation of t he form er it alian version [108]; all t he material was revised and rewrit ten ; ou r t reat ment of some t opics, like qu an tifier elimination and model com pleteness , are entirely new ; an d we have adde d some relevan t matters , such as prime models and Morley 's T heo rem on uncount able categorica l t heories .

Contents 1

2

3

Structures 1.1 St ru ctures 1.2 Sentences 1.3 Embeddings 1.4 The Compactness Theorem 1.5 Elementary classes a nd t heories 1.6 Complete theories 1.7 Definable sets 1.8 References . . . . .

1

1 5 9

18 20 30 35 42

Quantifier Elimination 2.1 Elimin ation sets . . . 2.2 Discrete linea r orders . 2.3 Den se linea r orders . . 2.4 Algebraically closed fields (and Tarski) 2.5 Tarski aga in: Real closed fields . . . . 2.6 pp-elimination of quantifiers and modules 2.7 Strongly mini mal th eorie s . 2.8 o-minimal theories . 2.9 Computational aspects of q. e. 2.10 References .

43

Model Completeness 3.1 An introduction . . 3.2 Abraha m Robinson 's t est 3.3 Mo del complet eness and Algebra 3.4 p-adic fields and Artin 's Conj ecture . 3.5 Existentially closed st ruct ures . . 3.6 DCFa

85

IX

43

47 52 54 61 68 76 78

79 82

85

88 91 96 103

109

x

CON TENTS 3.7 3.8 3.9

4

5

6

112 115 119

E lim inat io n of im ag inarie s 4.1 Interpretability . 4.2 Imaginary elements . . . . 4.3 Algebraically closed fields 4.4 Rea l closed fields . . . . . 4.5 The elimin ation of imagina ries sometimes fails . 4.6 References . . . . . . . . . . . . . . . . . . .. .

1 21

Morley ra nk 5.1 A tale of two cha pters 5.2 Definable sets .. 5.3 Types . 5.4 Saturated mod els . . . 5.5 A parenthesis: pure injecti ve modules 5.6 Omi t ting ty pes . . . . . . . 5.7 The Morley rank , at last 5.8 Strongly minimal sets . 5.9 Algebraic closure and definable closure 5.10 References .

133

w -stabili t y

181

6.1 6.2 6.3 6.4

181 184 192 196 209 217 220

6.5

6.6 6.7 7

SCFp and D CFp ACFA . . . References.. ..

Tot ally tran scend ent al t heories w-stable groups w-stab le fields . Prime models . D C Fo revisited Ryll-Nardz ewski 's Theorem , and other things References .

C lassify ing 7.1 Shelah 's Class ification Theor y. 7.2 Simple t heories . . 7.3 St a ble theories 7.4 Superstabl e t heories 7.5 w-stable th eories .. 7.6 Classifiabl e th eories .

121 123 126 129 131 132

133 133 136 143 150 156 158 168 172 180

221

221 227 235 239 242 261

CONTENTS

7.7 7.8 7.9 7.10 7.11

8

9

xi

Shela h's Uniqueness Th eor em . . Mo rIey's Theorem Biinterpretability a nd Zilber Conjecture Two algebraic examples References .

270 273 279 286 289

Model Theory and Algebraic Geometry 8.1 Int rodu ction . 8.2 Algebraic varieties , ideals , types . . 8.3 Dimension and MorIey rank . . . . 8.4 Mor phisms and definable functions . 8.5 Manifolds 8.6 Algebraic gro ups . 8.7 The MordelI-Lang Conjecture 8.8 References .

291 291 292 294 297 299

O-minimality 9.1 Int roduct ion . 9.2 The Monotonicit y Theor em 9.3 Cells . 9.4 Cell decomposition a nd other theorems . 9.5 Their proofs . 9.6 Defina ble groups in o-rninimal struct ures. 9.7 O-minimalit y and Real Analysis . 9.8 Variants on th e o-minimal t heme 9.9 No rose wit hout thorns. 9.10 References .

313

301 304 310 313 318 320

324 329 339 341

346 347 348

Bibliography

351

Index

363

Chapter 1

Structures 1.1

Structures

The aim of t his chapter is to sket ch out basic model t heory. We wish to summarize some key facts for people already acquainted with them , but also , at the sa me t ime, t o introduce t hem to people unfamili ar to logic, and perh ap s disliking too man y logical details. Accordingly we will use a rather colloquial tone. The fundam ent al qu estion t o be a nswered is: what is Model Theor y? As we will see in mor e detail in Section 1.2, Mod el Theory is -or, mor e precisely, was at its beginning- the study of t he relationship between math em atical formul as and st ruct ures sat isfying or rejecting th em. Bu t , in ord er t o fully appreciate t his matter , it is advisa ble for us pr eliminarily to recall what a st ruct ure is, and which kind of formulas we ar e dealing with. This section is devoted to th e form er concept . Structures are an algebraic notion . Actually, since Galois, Algebra is not only th e solving of equations, or literal calcu lus , but becomes the science of st ruct ures (gro ups, rings , fields, and so on). This new direction gets clearer at the beginning of t he last cent ury, with Steinitz's work on fields and, later , t he publication of th e Van der Waerden book . What is a st ruct ur e? Basically, it is a non-empty set A , with a collect ion of distinguished elements, op er ations, and relations. For instance, t he set Z of int egers with th e usual op erations of addition + and multiplication . is a st ruct ure, as well as the same set Z with th e ord er relation ~ . Note that , in th ese examples, th e underlying set is t he same (th e integers) , but , of course, t he st ructure changes: in t he form er case we have t he ring of integers, in t he lat ter t he int egers as a n ord ered set . To make t his kind of difference among struct ures clearer, we have t o choose a lang uage, in oth er word s t o specify how many distinguished 1

CHAPTER 1. STRUCTURES

2

elements, how many n-ary operations and relations (for every natural n i= 0) we want to involve in building our structure. So, when we discuss the integral domain of integers, our language needs two binary operations (for addition and multiplication), while, in the latter case, a binary relation (for the order) is enough. Notice that the language of the ring case works as well for all the structures admitting two binary operations, and hence possibly for structures which are not rings; for instance, the reals with the functions f(x, y)

= sin(x -

y),

g(x , y)

=e

X

'

Y

for all x and y in R provide a new structure for our language, but, of course, the algebraic features of this structure are very far from the basic properties of integral domains. Accordingly it is advisable, from a general point of view , to distinguish the constant, operation and relation symbols of a language L and the elements, operations and relations embodying these symbols in a given structure for L. Symbols are something like the characters in a tragedy (like Hamlet) , while their interpretations in a structure are the actors playing on the stage (Laurence Olivier, or Kenneth Branagh, or your favourite "Hamlet"). In this framework, we can at last provide a sharp definition of structure. We fix a language L. For simplicity, we assume that L is countable, hence either finit e or denumerable (but most of what we shall say can be extended without problems to uncountable languages). Definition 1.1.1 A structure A for L is a pair consisting of a non empty set A, called the universe of A, and a function mapping

(i) every constant c of L into an element cA of A, and, for any positive integer n,

(ii) every n-ary operation symbol f of L into an n-ary operation fA of A (hence a function from An into A),

(iii) every n-ary relation symbol R of L into an n-ary relation RA of A (hence a subset of An).

The structure A is usually denoted as follows

Let us propose some examples, which will be useful later in this book.

1.1. STRUCT URES

3

E x a m ples 1.1.2 1. A gr aph is a non empt y set A with a binary rela tion P both irreflexive and sy mmet ric. Hence a gra ph can be viewed as a struct ure A in the lan gu age L consisting of a uniqu e binary relation sy mbol R , with RA = P. Also a non em pty set A par ti ally ordered by some relation ~ ca n be regarded as a struct ure in t he same lan gu age L ; this time, RA =~ '

2. A (mul tiplicative) groupy is a structureof th elan gu ageL = {I , ., -I} , where 1 is a constant , . and - I are op er ation sy mbols of arit y 2 and 1 resp ectively. 19 represents the iden ti ty element in y, while .9 and 9 -1 denote the product and th e inverse op eration in y . Actually one might enrich L with some addit ional symbols; for instance, one might introduce a new binary operation symbol [ ] corresponding to th e commutat or operation in y . Bu t, for a and bin G, [a , b] is ju st a . b· a -I . «:' , so [ ] is not really new , and is impli citly defined by L . Actually we will prefer L later ; bu t it is not ewor th y t hat L can capture a nd express some fur th er operation s (and relations a nd constants) of y besides t hose liter ally interpreting its sy mbols. 3. A field K is a structure of t he lan guage L = {O, 1, +, - , .} where 0 an d 1 a re const an t , and +, - and · are op er ation sy mbols (eac h having a n o bvious interpretation in K ). Alte rnatively, K ca n be viewed as a struct ure in th e lan gu age L' = L U {- I } with a new operation symbol - I ; obviously, - I has to be int erpreted wit hin K in t he inver se op eration for non zero eleme nts of K . However , accor ding to t he gener al definition of struct ure given before, - I JC should denote a 1-ary op er at ion wit h dom ain K . So we run into t he problem of defining 0- 1 ; thi s ca n be over come by agreeing, for instance, 0- 1 = 0, bu t t his solut ion may sound slight ly artificial. So we will pr efer t o adopt below t he language L when dealing wit h fields. Ind eed , when a and b are two elements in a field K, then a = b- I can be equivalent ly expressed by saying a . b = 1. 4. An ordered field is a st ruct ure in the language L = {+ , - , " 0, 1, ~ } obtained by adding a new binary relation sy mbol ~ ' It s interpret a tion in a given orde red field is clear: the ord er relation in t he fi eld . 5. Let N denote the set of natural numbers . 0 is a n eleme nt of N ; t he successor s (mapping eac h natural n into n + 1) is a 1-ary fun ction from N t o N. Giu seppe Peano poin ted ou t t hat t he Induction Principle (t ogether with t he a uxiliary condit ions that s is 1 - 1 but 0 is not in

CHAPTER 1. STRUCTURES

4

its image) fu lly characterizes (N , 0, s). A suitable language to discuss this st ructure should include a constant symbol and a I-ary operations symbol.

°

6. Let IC be a (countable) field . A vectorspace V over IC ca n be regarded as a st ructu re in the language LK = {O , +, - , r (r E K)}, where is a con stant , + and - are operation symbols with arity 2 and 1 resp ect ively, and , for eve ry r E K , r denotes in LK a l-ary op eration symbol , to be interpret ed inside V in th e scala r multiplication by r . The other symbols in LK are interpreted in the obvious way. The assum pt io n on the cardinality of K has the only role of ensuring LK count a ble . Moreover, what we have said so far easily ext ends to right or left modules over a (countable) ring R with identi ty ; the corresponding language is obviously denoted by L R . As already said, we should distinguish symbols and interpretations, for inst a nce, a binary relation symbol R and t he relation RA embodying it in a st ructu re A (sometimes an order rela tion in a partially ordered set, bu t elsewh er e possibly the adjacency relation in a graph). But , to avoid too many complications, we will often confuse (and actually we already confused ) t he la nguage symbols and their" mo st natural" interpretations. For instance, in Example 1.1.2,6, we denoted in the same way the addition symbol + of LR and its obvious interpretation in a given R-module M , namely the addition in M . We will be interested in several alg ebraic notions concerning st ruct ures. In part icul ar embed dings play a crucial role in Model Theory. So let 's recall their definition .

Definitio n 1.1.3 Let A and B two struct ures in a language L . A h omomorphis m of A into B is a function f from A into B su ch that

(i) for every constant c of L, f(c A) = cB; (ii) for every positive integer n , for eve ry n-ary operation symbol P in L and for eve ry sequen ce ii = (aI , ... , an ) in A n , f(pA(ii)) = pB(J(ii)) (hereaf te r f( ii) abridges (J( al) , .. " f( an)) ); (iii) for eve ry positive integer n , for eve ry n-ary relation sy mbol R of L and for every seque n ce ii in A n, if ii E RA , th en f (ii) E R B .

.r is called an e m bed din g of A into B if f

is injective and, in (iii), f(ii) E R B im plies ii E RA for every ii in An. When there is some embedding of A

1.2. SENTENCES

5

into E, we write A ~ E. An isomorphism of A onto E is a surjective embedding. When there is some isomorphism of A onto E, we say that A a nd E a re isomorphic and we write A ~ E. Finally, an endom orphism (automorphism) of A is a homomorphism (isomorphism) of A onto A. Definition 1.1.4 Let A and E be two structures of L such that A ~ B. If the inclusion of A into B defines an embedding of A into E, A is called a substructure of E , and E an extension of A .

Now let E be a st ruct ure of L , and A be a non- empty s ubset of B . We wonder if A is the dom ain of a subst ruct ure of E. On e promptly reali zes t hat t his may be false. Ind eed (i) if c is a constant of L , it may happen that cB is not in A;

(ii) if F is an n-ary op eration sym bol in L , it may happen t hat t he restriction of F B to An is not a n n- ary op eration in A, in other words that A is not closed under F B ;

(iii) on th e cont rary, if R is an n- ary relation symbol in L, then R B n A n is an n-a ry relat ion in A. So A is not necessa rily t he domain of a substr ucture of E. However the closure of A U {cB : c constant in L} with res pect t o t he ope rations pB , when F ranges over the operation symbols in L , does form the dom ain of a substructure of L , usually denoted (A) , and called the substructure generated by A : in this case A is said to be a set of gener ators of (A) . Notice that these notions ca n be introduced even in the case A = 0, provided that L contains at least a const ant symbol. E is called finitely ge nerated if there exists a finite subset A of B such that E = (A). F ina lly, let L ~ L ' be t wo langu ages, A be an L-stru cture, A' be an L ' st ruct ure such that A = A' a nd the int erpretations of the sy mbols in L are the same in A and in A'. In this case, we say that A' expands A , or also that A' is a n expansion of A to L ' ; A is called a restriction of A'to L.

1.2

Sentences

Given a language L , after forming the st ruct ures of L, one builds, in a complementary way, the formul as of L, in particular th e sentences of L , a nd one defines when a formula (a sente nce) is tru e in a given structure. T his is t he realm of Logic rather th an of Algebra.

CHAPTER 1. STR UCTURES

6

Act ually t here ar e several possible ways of introducing formul as and truth , acc ording to our tastes or our math ematical purposes. We will limit ourselves in t his book to the first order framework. Let us sket ch briefly how formul as and tru th a re usu ally introduced in t he first order logic. For simplicity let us work in t he par t icular sett ing of na tural numbers (full genera l det ails and sha rp definitions ca n be found in a ny hand book of basic Math ema tic al Logic , such as [153]). Conside r the natural numbers a nd t he corresponding st r uct ure (N , 0, s), where s denotes the success or function . The corres po nding language L includ es a con stant (for 0) and a 1-ary operation sy mbol (to be int erpret ed in s). As an nounced at t he end of the pr eviou s section, we denote these symbols by still using 0 and s : this is not completely correct , but simplifi es our life. In the first ord er set t ing, formulas ca n be built by using additional sy mbols • count a bly many eleme nt vari abl es VD, V i, ... , V n , ... (just to resp ect our count a ble fram ework ; oth er wise we can use as man y variables as we need) , • t he basic connectives /\ (a nd) , V (or) , --, (not) (a nd even ---+ (if ..., t he n) , H (if and only if) if you like) , • t he quantifiers V (for all) a nd :J (t h ere exists ) , • parenth eses (, ) a nd a sy mbol = to be int erp reted everywhere by t he equality relation. At this point one form s the te rms of L . Essent ially t hey are polynomials; in our case t hey are built st a rting from t he con stan t 0 and t he variables V n (n na tural) and using t he op er ation symbols (so s in our sett ing) . The second step is to construct the atomic formu las of L: basically they are eq uations between t er ms, but , when th e language includ es a k-ary relation symbol R , we have to include every state me nt saying that a k-uple of terms satisfies R . At this poin t t he formul as of L are built fro m t he atomic on es inductively in th e followin g way: 1. one can negate, or conj unct, or disjunct some given formulas and get new formulas --'0: , 0: /\ (3, 0: V (3 ;

2. one ca n t ake a formul a Vvno: , :Jvno:;

0:

3. nothing else is a formul a.

and a va ria ble

Vn

0:,

(3, . . .

a nd form new formul as

1.2. SENTENCES

7

For a and /3 formul as , a --+ /3, a H /3 ju st abr idge -,a V /3, (a --+ /3)1\ (/3 --+ a ) respectively. Let us propose some examples in ou r framework of natural numbers. The injectivity of s ca n be expressed by t he following formul a in our language L while the formula

Vvo-,(O = s(vo))

says that 0 is not in t he image of s. Actually these formulas a re sentences (each occurring variabl e is un de r the influence of a correspond ing quantifier). In general , an occurrence of a vari abl e v in a form ula a is bounded if it is und er th e influence of a quantifier Vv, ::lv, and free otherwise; a is called a sen tence if, as already said, each occurrence of a variable in a is bounded. Wh en writing a( v), we want to emphas ize th at th e va riabl es freely occu rring in th e formula a are in the tuple v. 2. and 3. are very restrictive conditions, a nd are the distinctive peculiarity of first ord er logic. Actually, in Mathematics, one sometimes uses V and ::l on s ubsets (rather t ha n on elements ) of a st ruct ure. This is ju st what happens in ou r set t ing concerning (N , 0, s) with resp ect to t he Indu cti on Principle. In fact, Induction says for every s ubset X of N , if X cont ains 0 and is dosed under s, then X=N. This statement uses V on su bsets, and t his is not allowed in fi rst order logic. Accordingl y, the Induction P rinciple ca nnot be writ ten (at least liter ally in the form proposed som e lines ago) in t he first ord er fra mework. T his might look very disappointing: consequent ly, one may search more powerful and expressive ways of constructing form ulas , for instance by allowing qu an t ifica t ion on set va riables (t his it the so-called second order logic). But actua lly first order logic enjoys several important a nd reasonable technical theorems, th a t get lost a nd do not hold any more in these alternative world s. We will discuss these res ults later , but it may be useful to quote already now a theorem of Lindstrom say ing (very roughly speaking) that "first ord er logic is the best possible one" (see [11] for a det ailed exposition of Lindstrom theor em). However , formul as and sentences ar e not sufficient to form a logic. Wh at we need now to accomplish a com plete descrip tion of our setting is a not ion a truth. We want to define when a sentence of a lan guage L is true in a struct ure A of L , and , more generallly, when a sequence ii in A makes

8

CHAPTERl . STR UCTURES

a formul a a(v) tru e in A. This ca n be done in a very nat ur al way, saying exactly what one expects to hear. For instance t he sentence 3v (v 2+1 = 0) is t rue in t he complex field ju st becau se C contains some elements ± i satisfying t he equation v 2 + 1 = 0; a nd in t he ord ered field of reals yI2 makes t he formul a v 2 = 2 A v 2: 0 t rue because satisfies both its condit ions, while -yI2, or 1, or other elements ca nnot satisfy t he same formula. See again [153], or any handbook of Mathematical Logic for mor e details on t he definition of first ord er t rut h. We omit t hem here. Incid ent ally we note t hat, according t his not ion of t rut h, a V {3 ju st mean s -{.a A ....,{3) , and Vvna says t he same thing as ....,3vn( ....,a) . So we could avoid t he con nect ive V a nd t he qu an tifier V in our alph ab et an d, consequent ly, in ou r inductive definition of formula, and to introduce a V {3 and Vvna as abbreviat ions, ju st as we did for a ---+ {3 and a H {3. In this perspective, formul as are obtained from t he atomic ones by using A, ...." 3 a nd nothing else. Mo reover one can see t hat, acco rding t o t his definition of t rut h, up t o suita ble manipulations, eac h formula ep( w) ca n be writ ten as

where QI , . . . , Qn are q uantifiers , v = (VI, .. . , v n ) a nd a(v, w) is a qu an tifier free form ula, and even a disj unction of conj unctions of atomic formul as and negations. (*) is called t he normal form of a formula . When ep( w) is in its normal form and every qu antifier Q i (1 ~ i ~ n) is uni versal V (existential 3), we say t hat ep(w) is universal (e x ist e nt ia l, resp ecti vely). Before concluding t his section, we would like t o emphas ize t hat t he st udy of t his t ruth relation bet ween struct ures and sentences is ju st Model Theor y, at least according t o the feeling in t he fifties. In fact , one says t hat a struct ure A is a model of a sentence a , or of a set T of sentences in the lan gu age L of A , and one writes A 1= a, A 1= T resp ectively, whenever a , or every sente nce in T , is true in A . Mo del Theor y is ju st t he st udy of this relati onship betwe en st ruct ures a nd (sets of) sentences. Tarski provides an authoritative corroboration of t his claim , when he writes in 1954 [158J

W hit hin t he last years, a new branch of metamathem atics has been developing. It is called theory of m odels and can be regarded as a part of the semantics of formali zed theories. T he problems st udied in the theory of mo dels concern m ut ual relations bet ween sentences offormalized theories and mathematical systems in which these sentences hold.

1.3. EMBEDDINGS

9

It is notable that this Tarski quotation is likely to propose officially for the first time the expression theory of models. Accordingly, one might fix 1954 as the birthyear -or perhaps the baptism year- of Model Theory (if one likes this kind of matters). Actually, several themes related to the theory of models predate the fifties; but one can reasonably agree that just in that period Model Theory took its first steps as an autonomous subject in Mathematical Logic and in general mathematics.

1.3

Embeddings

We already defined in 1.1 embeddings and isomorphisms a mong structures of the same langu age L. We followed the usual algebraic approach. However there are alternative and equivalent ways, of more logical flavour , to introduce these notions. Let us recall them. First we consider embeddings. Theorem 1.3.1 Let A and B be structures of L , f be a function from A into B. Th en the following propositions are equivalent :

(i) f is an emb edding of A into B; (ii) for every quantifier free formula If'(if) in L and for every sequence ii in A, A F If'(ii) if and only if B F 1f'(J(ii)); (iii) for eve ry atomic formula If'(if) in L and for every sequence ii in A , A F If'(ii) if and only if B F 1f'(J(ii)) . The proof is just a straightfoward check using the definitions of embedding, term and (atomic or quantifier free) formula. Referring to definitions is a winning a nd straightforward strategy also in showing the following characterizations of th e notion of isomorphism. Theorem 1.3.2 Let A and B be structures of L, f be a surjective function from A onto B. Th en the following propositions are equivalent:

(i) f is an isomorphism of A onto B; (ii) for every quantifier free formula If'(if) in L and for every sequence ii in A , A F If'(ii) if and only if B F 1f'(J(ii)); (iii) for every atomic formula If'(if) in L and for every sequence ii in A , A F If'(ii) if and only ifB F 1f'(J(ii)); (iv) for every formula If'(if) in L and for every sequence ii in A , A if and only if B F 1f'(J(ii)).

F If'(ii)

10

CHA PTER 1. STRUCTURES

It can be ob ser ved t hat , when f is any em bedding of A int o B, for every qu an ti fier free formula a( 5, w) in L and every sequence ii in A ,

if A

F= 3wa(ii, w),

t hen B

F= 3wa(J (ii), w)

or also, equivalently, if B

F= Vwa (J (ii ), w) , t hen A F= Vwa (ii , w) .

Definition 1.3.3 T wo struc tures A and B of L are e le m e nt a r ily equivalent (A == B) if they satisfy th e sam e senten ces of L . As a n eas y co rolla ry of T heor em 1.3 .2, we have: Theorem 1.3.4 Isom orphic str uctures are ele m entarily equivalent. Co nversely, it may happen t hat eleme ntarily equivalent st ructures A a nd B a re not isomorphic. We will see count erexam ples below. However it is an easy exercise t o show t hat , for finit e structures, eleme ntary equivalence a nd isomorphism ar e j ust t he sa me t hing. Now let us int rod uce a related notion: parti al isom orphism . Definition 1.3.5 Let A an d B be structures of L. A partial isomorphism between A and B is an isomorphis m bet ween a substruct ure of A and a su bstructure of B. A and B are said to be partially ieomorphic A ~p B if there is a non em pty set J of partial isomorphisms bet ween A and B satisf ying the back-and-forth prop ert y: for all f E I ,

(i) fo r eve ry a (: A, there is som e gE l such that f

~

9 an d a is in th e

~

9 an d b is in th e

domain of g ,.

(ii) f or eve ry b E B , there is some g El such th at f im age of g.

E xample 1.3.6 Two dense linea r ord erin gs wit hout end points A = (A, s;) a nd B = (B , S;) a re parti ally isomorphic. In fact, let I include all t he possib le isomor phis ms between a finit e substruct ure of A and a finit e subst ruct ure of B. I is not empty, becau se, for every a E A and b E B , a t-+ b defines a par ti al isomorphism in I. Now take a ny f E I ; let ao < a l < . . . < an list t he elem en t s in t he dom ain of f a nd bo < bl < .. . < bn t hose in t he image of f ; so f (ai) = b, for eve ry i S; n .

1.3. EMBEDDINGS

11

Pick a E A, and notice that th ere exists some b E B such that, for every i ~ n, a; ~ a ~ b, ~ b. This is trivial when a is in th e dom ain of f . Otherwise, one uses the facts that B has no minimum when a < aa, th at B has no maximum when a > an, and , finally, th at t he order of B is dense in the remaining cases. Define gEl by putting Domg

=

Domf U {a},

g ~ I,

Img

=

Imf U {b} ,

g(a) = b.

Clearly g satisfies (i). (ii) is proved in th e same way. Remark 1.3.7 • If A ~ B, then A ~p B. In fact , let f be an isomorphism of A onto B. I

= {f}

does satisfy (i) and

(ii) . • Conversely, partially isomorphic structures may not be isomorphic. Indeed one can find two struct ures that admit a different cardinalit y, and yet a re pa rtially isomorphic. For instan ce, t his is t he case of two dense linear ord erings without end points. We have j ust seen that th ey are always partially isomorphic, indipendently of their cardinalities; in particular (R,~) ~p

(Q, ~ ).

But one ca n also find partially isom or ph ic non isomorphic struct ures with the same cardinality. For instance, st ill conside r den se linear orderings without end point s, a nd notice t hat (R, ~) ~ (R + Q , ~) ((R + Q, ~) denotes here the disjoint union of a copy of (R, ~) and a copy of (Q, ~) , where (R, ~) precedes (Q, ~)). Both (R, ~) a nd (R + Q , ~) hav e t he continuum power. But they ca nnot be isomorphic, becau se (R + Q , ~ ) , unlik e (R, ~ ), cont ains some countable intervals, a nd any order isom or phism maps countabl e int ervals onto countabl e intervals. However, with in countable mod els, parti ally isomorphic st ruct ures are also isomorphic.

Theorem 1.3.8 Let A and B be countable partially isomorphic structures. T he n A ~ B.

12

CHAPTER 1. STRUCTURES

T he pr oof is obtained as follows. F irst one list A and B in some way A = {an: n E N },

B = {b n : n E N} .

Let J be a set of parti al isom orphism s bet ween A and B ensur ing A c::=.p B. Du e to (i) and (ii) one enla rges a given 10 E J by defining, for every nat ur al n, a function In E I such t hat , for a ny n,

2. an is in the do main of [z « , 3. bn is in the image of

fz n+l.

Pu t 1= UnENln. Owin g t o 1., I is a function; 2. impli es t hat its dom ain is A, a nd 3. ensures that its image is B. In ord er to conclude that I is a n isom orphism , we have t o check t hat , for every atomic formula cp (v) of L and every sequence ii in A, A 1= cp (ii) if a nd only if B 1= cp (J (ii)). Bu t t his is easily don e, as t here is some n such t hat ii is in t he domain of In' and In rest rict s I and is a n isom orphi sm between its dom ain and its image. A noteworthy consequence of t he t heorem is Corollary 1.3.9 (Cantor) Tw o countable dense linear orderings without

endpoints are isomo rphic . Hence lineari t y, den sit y and lack of end poin ts cha racterize t he ord er of rat ional s up t o isomorphism . It should be und erlin ed t hat Cant or 's origina l proof used a different a rgume nt; but a subsequ ent a pproac h of I-Iausdorff a nd Huntington inaugu rated t he back-and-forth method. In fact, wh at th ey did was just firstly to obser ve that two dense linea r ord ers without end points are partially isomorphic (according to our mod ern terminology) , a nd consequently to deduce that , if one adds the countability ass umpt ion, th en isomorphism follows; t he lat t er poin t can be easily generalized t o arbitra ry structures (and actua lly t his is what Theorem 1.3. 8 says) . Now let us compa re

c::=. p

and

=.

Theorem 1.3.10 Pa rtially isom orphic structu res are eleme ntarily equiva-

lent. In fact let A a nd B be parti ally isomorphic st ruct ures in a la nguage L , and let I be a set of parti al isomo rp hisms between A and B witnessing A c::=. p B.

13

1.3. EMBEDDINGS

Then one ca n show th at, for every choice of a formu la

A

l=
<=>

B

l=
Note that , when

(a) for eve r y a E A , th ere is 9 E In su ch that 9 extends f and th e domain of 9 in clude s a ,

(b) for ev er y bEE , th ere is 9 E In su ch that 9 ex ten ds f and th e im age of 9 in cludes b.

Now let us deal again with arbitrary embeddings. The charact erizatio n of isomorphism given in Theorem 1.3 .2 (iv) su ggests the following notion . Definition 1.3.12 Let A and B be structures of L. An em bedding f of A into B is called elementary if, f or every f orm ul a
F

CHAPTER 1. STRUCTURES

14

We say that A is elementarily embeddable in E, and we write A < E, when th ere is some elementary embedding of A in E. A is called an elementa r y s u b stru ct ure of E if A ~ B a nd the inclusion of A in B defines an elementary embedding of A in E (hence, for every formula
1. Every isomorphism is, of course, an elementary em-

2. If A is element arily embeddable in E, then A and E are elementarily equivalent (just restrict the definition of element a ry embedding to sentences

(a) In the la nguage L = { <}, consider the following

A

= (N -

{O} , <) ,

E = (N , <).

The inclu sion of N - {O} in N defines a n embedding of A in E which is not elementary because 1 is a mini mal element in A , but not in E (in other words, satisfies 3w(w < v) in E, but not in A) . On th e cont rar y, t he successor function from N in N - {O} is an isomorphism between E an d A. (b) The real field R is a subfield of the complex field C , and hence is a substructure in th e language L = {O, 1, +, " - }. However R is not an eleme ntary substructure of C . In fact, for every positive real a, a satisfies t he formula 3w(w2 + V = 0) in C , but not in R . On t he other hand, we will see later t hat every embedding of algebraically closed fields (or real closed fields) is elementary. Another rem arkable class of embeddings concerns existe nt ial formulas. Definition 1.3.15 Let A and E be structures in a language L. An em bedding f of A in to E is called ex istential if and only if, for every existe ntial formu la
A

F
{::} E

F
In mo re detail, we require that, for every qua ntifier free formula a( a, w) in L a nd for every sequence a in A ,

(*) A

F 3wa (a, w) if and

only if E

F 3wa (J (a), w).

1.3. EMBEDDIN GS

15

When A ~ Band f is the inclusion of A into B, we say that A is an e xist e ntial s ubst ruct ure of B. Wh en t here is som e existe nt ial embedding of A in B, we say t hat A is existentially e mbedde d in B and we write

A <1 B. Let us discuss bri efly existent ial embeddings f . First of all, it is clear t hat element a ry embeddings ar e existent ial, t oo. Furthermore, notice that (*) ca n be weakened to require if B

F 3wo:(J(a), w), then

A

F 3wo:(a, w)

becau se t he inverse implic ation if A

F 3wo:(a, w), then B F 3wo:(J(a) , w)

is satisfied by every embedding. Now, a quantifier free formula 0:( V, w) ca n be equivalent ly written (by sta nda rd proposition al tec hniques) as a disjunct ion

v O:j (v, w)

j5,s

where s is a natural number and, for every j ::; s, O:j (v, w) is a conj unct ion of atomic formulas (eq uations) and negations. On e eas ily dedu ce t hat 3wo:(v, w) ca n be equivalent ly written as

V 3wO:j( v, w). j 5,s

It follows t hat

f is exist enti al if a nd only if B

F 3wo:(J(a) , w) implie s

A

F 3wo:(a, w)

for every a in A and for every finite conj unct ion 0: (V, w) of equations and negations. Now let us deal again with ar bit ra ry elementary embeddings. Let L be a lan guage, A be a structure of L. In ord er to examine the st ruct ures element a rily equivalent to A , L is ju st the language we need. But, within t hese models , one meets t hose where A is elementarily embeddable; and t hese struct ur es actu ally require a richer lan guage, em phas izing th e fact t hat A embeds (element ari ly) in eac h of them . This lar ger language is built by adding to L a constant sy mbol for every a E A, a nd will be called L (A) ; t he new constant corresponding t o a could be written Ca, in ord er to remind a bu t t o avoid an y confu sion with a. But we will often denote it ambiguously by a (for simplicit y's sake) .

CHAPTER 1. STRUCTURES

16

A itself becomes a structure of L (A) provided we int erpret quite naturally, for every a E A , t he consta nt cor res ponding t o a in a. T he result ing structure will be deno ted A A. Wh ich a re t he st ruct ures elemen tarily equivalent t o A A in L (A )? They are ob t ain ed as follows. Take an L-struct ure B where A embeds elementarily, say by f ; for every a E A , let f (a ) int erpret t he constant of a . On e gets in t his way a struct ure BJ(A) of L (A), and it is easy to check t hat BJ(A) == AA· Conversely, every st ructure elementarily equivalent t o A A ca n be ob t ain ed in t his way. Mo re generally, for every struct ure A of L and for every s ubset X of A , one ca n introduce a new langu age L(X) by adding to L a new const ant for every element x in X. Ax is the L(X)-structure expanding A and interpreting, for every x EX , the constant symbol corresponding t o x in x it self. Of course, t here do exist ot her structures elementaril y eq uivalent t o Ax. Let us see how to construct t hem. Let B be a struct ure of L.

Definition 1.3.16 A f un ct ion f fro m X in to B is called elementary if , f or eve ry formula cp( if) in L an d for eve ry sequence i in X , A

F cp(i )

{:}

B

F cp (J (i )).

Not ice t hat , when X = A, an eleme nt ary fu ncti on from X in B is j ust an eleme ntary em bedding of A in B. Moreover , for any X , an elementary fu ncti on f from X in B enjoys t he following propert ies. (i)

f is 1 - 1 (use CP (V I' V2) :

VI

= V2) .

(ii) f- I itself is a n elementary fu nction (from f (X) in A).

(iii) A an d B are elementarily equivalent (apply t he definition of elementary fun ction to the sente nces cp of L). Now take an L-structure B. Let f be an element ary fun ction from X into B , a nd, for every x E X , let f( x) int erpret the constant of x in L(X). On e gets in thi s way a st ruct ure BJ (X ) of L(X) , and even a mod el eleme ntarily equivalent to Ax. Conversely, every st ruct ure == A x ca n be obtained in t his way. Remark 1.3.17 Let cp( V, w) be a formula of L , d be a sequence in A , i be a seq uence in X. Fussy people will like t o distin guish (a) A F cp(ii, i) (in t he sense t hat A satisfies t he L-formula cp (v, w) if V, ware embodied by ii, i resp ect ively) ;

17

1.3. EMBEDDINGS

F i.p( ii , x) (in the sense that Ax sat isfies the L(X)-formula i.p( if, x) if if is embodied by il);

(b) Ax (c) AA

F

i.p(il, x) (in th e sense that AA satisfies th e L(A)-sentence

i.p(il, x)).

However one easily shows that (a), (b) and (c) are equivalent. So we will use t he sim plest notation (that of (a)) to mean any of these conditions. We conclude t his section by mentioning withou t proof two theorems on elementary emb edd ings. We will state them for sim plicity in t he case when the involved embeddings a re ju st inclusions but they extend to arbitrary (elementary) embeddings. T he for mer theorem provides a crit erion to check whether a given subset of a struct ure B is the domain of an elementary substructure of B (remember the discu ssion at th e end of 1.1) . Theorem 1.3.18 (Tar ski-Vau ght) Let B be a structure of L , A be a subset of B. Th en th e follow ing proposition s are equiv alent:

(i) A is the domain of an eleme ntary substructure A of B; (ii) fo r every fo rmula a( if, w) of L and for every sequence ii in A , if B F :3wa(ii, w), then there exists an element b E A such that B F a(ii, b) . Now let us introduce the latter result. Take a set I to t ally ordered by a relation ~. For every i E I , let A i be a struct ure of L. Suppose t hat , for every choice of i ~ j in I , A i is a substruct ur e of A j: this means th at A-~ C - A J and • for every constant c of L ,

cA ;

= cA j ,

• for every n-ary operation symbol F of L , F A; is t he restriction of FAj to Ai , • for every n- ary relation symbol R of L,

RA ;

= RAJ n Ai .

Therefore we can build a new structure A in L , having domain A = U iEI Ai (and hence including Ai for all i E 1), and inter pret ing t he symbols of L as follows: • for every constant c of L,

cA

= cA ; where i is any elemen t in J;

CHAPTER l . STRUCT URES

18

• for every n-ary op er ation sy mbol F of L an d for every choice of aI , ... , an in A,

where i E I sat isfies aI, . . . , an E A i; • for every n-ary relation sy mbol R of L and for every choice of aI , ... , an in A, where i E I sat isfies aI, ... , an E A i. It is clear that

A is well defined a nd ext ends Ai for every i

E I. Straight-

fowa rd techniques show: T heorem 1.3.19 (Elem ent ary C hain Th eor em) S uppose that , fo r eve ry choice of i ::; j in I , A i is an elemen tar y substr ucture of A j. Th en, fo r every i E I , Ai is an elemen tar y substruct ure of A .

1.4

The Compactne ss T heore m

T he Compactness Theorem is t he most powerful to ol -a nd inde ed a key feature- in classical Model T heory. It states Theorem 1.4.1 Let S be an infinite set of sentences in a languag e L . S up-

pose that eve ry fin it e subset of S has a model. T hen S has a model.

Notice t hat the converse is obvious, because a mod el of S is a mod el of every (finite or infinit e) s ubset of S . But t he t heorem ensures t ha t , if every finit e subset of S has its own model (hence different subset s may admit different mod els), then th ere is a global model sat isfying all the sentences in S. In fact , there a re severa l sit uat ions where the Compactness Theorem a pplies and guarantee s satisfiability for sets S of sente nces for which it is very difficult to imagin e a genera l model directl y, bu t it is quite simple t o equip every finite subset wit h a suit a ble private mod el: we will see some of them later in 1.5. In fact t his section and (impli citl y) t he next one will be devot ed t o discu ssin g t his fu nd am ental t heorem and , in det ail: • its proof; • its name;

19

1.4. THE COMPACTNESS THEOREM

• its role in producing " nonstandard" and , in some sense , unexp ect ed models, and , as th e reverse side of t he same medal , in bounding t he expressiveness of first order logic and in excluding t hat some fam ilia r principles, like induction on naturals, may be writ ten in any way in t he first order fram ework ; • in spite of t his, its plau sibility, supported by metamathematical conside rations on t he nature of math ematical pro ofs; • finally , some word s a bo ut t he already mentioned t heorem of Lindstrom saying t hat the Compactness Theorem , to gether with a related resu lt (th e downward Lowenheirn Skolem Th eor em) fully cha racterizes first ord er logic. As said, let us begin by discussing the proof. There are several possible ways t o show t he Compactn ess Theorem . For instance, t here is an approach based on t he algebraic noti on of ultraproduct a nd du e t o Keisler (see [39]) . Another classical pr oof was propo sed by Henkin . Let us outline very qui ckly its idea. Wh at we have at t he beginning is a( n infinite) set of sent ences S such t hat every fi nite subset has a model. Some tec hnica l -and non t rivial- pr elimin ary wor k shows t hat t here is no loss of gene ra lity in ass um ing t hat S satisfies t wo fur th er cond itions: 1. S is com plete, in oth er words , for every se nte nce rp in L , eit her rp or -' rp is in S;

2. S is rich: if S contains a sentence of t he form :Jva:( v), t hen t here is a constant symbol c in L such t hat a:(c) is in S . At t his point a qui te art ificia l const ruct ion produces t he model we are looking for. Ba sically, th e domain is ju st the set of t erm s without variables in L (the so-called Herbrand universe of L) ; this is non-empty owing to 2. The L-stru cture arises in a rather reasona ble way. 1 a nd 2 play a key role in showing t hat what we build is a model of S . It is worth emphasizing t hat t he model we get in our pro of is count able (for a cou ntable L ; when t he lan guage has a larger ca rdina lity -X , it is easy t o check t hat our argument st ill work s and produces a mod el of power -X). So , as a byproduct of t he Henkin proof, we have t hat , when S has a mod el, t hen S has a countable model: t his is t he so ca lled Downward Lowenli eim Skolem T heorem, and is a notable resul t . Vve sha ll discuss it s relevance in 1.5 .

s:

CHA PTER 1. STRUCTURES

20

Now let us t reat t he reason s of t he t heorem nam e. Act ually compactness recalls topology. In fact , we will see later in Chapter 5 that th e th eor em has a topological content a nd implicitly says that a cert ain topological s pace is compact . We shall see within a few lines in 1.5 t hat t he Com pactness Theorem produ ces some strong a nd severe expressiveness rest rictions in first order logic. For inst a nce, we will show t hat , ju st owing to Com pactness, condit ions like finit eness, or pop ula r statements such as t he Minim um Principle, cannot be expressed in a first order way. On t he other hand, one sho uld agr ee t hat what t he Compact ness T heorem says is a qui t e reas ona ble statement, especially if one conside rs t he following corolla ry. Let 5 be a set of sente nces of L and a be a sentence of L ; we say that a is a logical consequ enc e of 5 , a nd we wri te 5 F a , when a is true in all the mod els of 5 . Corollary 1.4.2 If 5

F a , then there is a fi nit e su bset So of 5 such that

So Fa. Proof. Clearly 5 F a if and only if 5 U {-,a} has no models. But , owing to Compactness, t his is equ ivalent t o say t hat t here is a finit e subset of 5 U {-,a} without a ny model s. Wi th no loss of generality we can assum e that this finite set is of the form So U {-,a} where So ~ 5 . But, aga in, stating t hat So U {-,a} has no mod els is equivalent t o say th at So F !.p· ...

Now, anot her fun dament al result in first order logic, deeply related t o compactness -the Com pleteness Theorem- says that one can ex plicit ly prov ide a notion of provabili ty accompanying and support ing this concept of consequence in such a way t hat t he logical consequ ences of a given 5 a re ju st what is proved by 5 at the end of a sequence of rigorous dedu ctions. So what compac t ness in conclusion emphas izes is t he finit ary nature of math emat ical pr oofs; t his feature can be regard ed as a n aut horitat ive witness in its favour and, t ro ugh it , as a support to first orde r logic.

1.5

Elementary classes and theories

W hen conside ring, for a given lan gu age L , structures, formul as and t ruth, tw o problems a rise qu ite naturally: (a) given a set T of L-sentences, "classify" th e models of T (their class will be denoted M od(T));

21

1.5. ELEMENTARY CLASSES AND THEORIES

(b) given a class K of Le-structure, "characterize" t he set of t he L-sentences true in all the structures of K (this set will be called t he theory of K and denoted T h( K )). According to its declar ed intent s, Model T heory should be mainly concerned with P roblem (a) . However , also (b) arises quite naturally in the model t heoretic framework. For inst ance, consider a class K formed by a single structure, like t he complex field , or t he real field. We will see lat er that K cannot be represented as M od(T ) for a ny T. Bu t it may be qui t e interesting to realize in an explicit way which sentences are true in t he only structure in K. However we have to admit t hat the previous statements of (a) an d (b) a re somewhat vague and unprecise . F irst of all, what do classifying or characte rizing mean? This is not a minor question ; on the contrary, it is a very delicate and central matter. For inst ance, t he classification problem for a class of structures touches and overlaps several basic ope n qu estions in Algebra . So we should be more detailed about t his crucial point . Of course, one ca n reasonably agree that a classification should ident ify isomorphic structures. Bu t t his is still a partial and indefinite answer; we shou ld fix more precisely which criteria, tools and invariants we want to use in our classification problem . We shall try to clarify these fundamental questions in the next chapters. Her e we limit ourse lves to discuss other points, mainly concern ing (a). T is a set of sentences in a langu age L . 1. Let a be a sentence of L, and suppose that every model of T is also a model of a (so a a logical consequence of TT 1= a). Hence M od(T) = Mod (T U{a }). Conseq uent ly we can assume wit h no loss of generality for our purposes t hat

for every sentence a of L , if T

1= a , t hen a E T.

A set of sentences in L with this property is called a theory of L. It is a simple exercise to show that, given a set T of sentences of L, T is a t heory if and only if t here is a class K of struct ures of L such that T = Th (K ) (hint : ({:::) is clear; t o show (=?) use K = M od(T) ). 2. A theory T is called consistent if and only if T satisfies one of the following (equivalent ) conditions: (i) for every sentence a of L , either a

rf. T

or -,a

(ii) there is some sentence of L which is not in T;

rf. T;

CHAPTERl. STRUCTURES

22 (iii) M od(T)

-# 0.

To show t he equivalence among (i) , (ii) and (iii) is a n easy exercise. (i) says t hat t he consist ent theories are just t hose excluding any con tradiction. (iii) ensures t hat t hese th eori es are exactly t hose ad mitting at leas t one model. It is clear t hat, within Problem (a), we are exclusively interested in consist ent th eori es. So we can assume in (a) th at T is a consi stent th eor y; accordingly her eafter theory will always abbreviat e consisten t th eory. A rigid model theoretic perspective might limit th e classification analysis to t he classes of models of (consistent) theories . Bu t op en minds could prefer a more gener al study, providing an abstract t reat ment of th e class ification problem for arbitrary classes of structures. Hence it is worth underlining that there do exist clas ses K of L-struct ures which are not of th e form K = M od(T) for any theory T of L. We propose here some examples; t he Compactness Th eorem is a fundamental tool in t his set t ing.

Definition 1.5.1 A (non-empty) class K of struct ures of L is said to be elementary (or also axio mat izab le) if there is a set T of se nte nces of L (without loss of gen erality, a theory T of L ) such that K = M od(T ). Now let us propose a ser ies of exa mples, as promised. P ar t of t hem aim at poin ting ou t t hat seve ra l classes of st ruct ures a re explicit ly eleme nt ary becau se t heir definitions ca n be naturally wri t t en in a first order way. Bu t oth er cases are not element ar y: it is here that the Com pactness Theorem plays its role and first ord er logic shows it s expressiveness bounds.

Examples 1.5.2 1. Let L = 0 (so the str uct ures of L a re th e non -empty sets), K be the class of infinit e sets . "Infinite" mean s " admitting a t le a st n + 1 elements for every natural n" . Given n, the property "there are at least n + 1 elements" can be expressed in a first ord er way by t he following sentence of L o;

1\

:Jvo .. . :Jvn

-'( Vi = Vj ) .

i<j~n

Hence K

= M od(T) where T = {(Yn :

n E N }, and so K is elementary.

2. Let again L = 0, bu t now let K be th e class of finit e (non-e mpty) sets. " Finite" means " having a t most n + 1 elements for some natural n" . Given n , the pro posit ion" there are at most n + 1 elem ents" can

1.5. ELEMENTARY CLASSES AND THEORIES

23

be expressed in a first order way by the sentence -,an +l . But now M od( {-,a n +! : n E N}) is not K, indeed it equals the class of the sets having only one element. So the approach in 1 does not work any longer. However, assume that K is elementary, hence K = M od(T) for a suitable set T of sentences of L. Put

T' = Tu {an: nE N}. Let T6 be a finite subset of T'. For some natural N, T~ ~ T U {an: n EN , n

< N}.

Notice that TU {an: nE N, n::; N} (and hence T6) has a model: it suffices to take a finite set with at least N + 1 elements. At this point , owing to the Compactness Theorem, we deduce that T' itself has a model. This is a set both finite (as a model of T') and infinite (as a model of an for every natural n). We get in this way a contradiction. Hence K is not elementary. Notice that this a rgument works as well for every class K of finite arbitrarily large structures (in the sense that, for every positive integer n, there is a structure in K whose size is larger than n). A class K of this kind cannot be elementary; in other words, the theory of K does admit infinite models, too; notice that this applies, for instance, to the class of finite groups, as well as to the class of finite fields . So one can wonder which are the infinite models of the theory of these finite structures. We will consider the particular case of fields later in Example 6. Now let us deal with orders. 3. Let L = {::;} where j, is a binary relation symbol (which we confuse, for simplicity, with its interpretation below -an order relation-). In L we consider the class K of linear orders. It is easily seen that K is elementary, because the properties defining linear orders are first order sentences (and indeed universal first order sentences) of L. For instance, linearity can be expressed by

The set of the logical consequences of these sentences is the theory of linear orders ; it is formed by the sentences true in every linear order.

CHAPTER 1. STR UCTURES

24

In the sa me way the class of dense linear orders without endpoints is elem ent a ry ; in fact, density is st at ed by

Vvo VVI :J V2 (vo <

VI

-+ Vo < V2 1\ V2 < vd ,

and lackness of end points by V VO:J VI ( VI V VO:J VI

(vo <

VI

< Vo),

(vo <

VI )

abbrevi ates here Vo:::; VII\ --, (VO= vd ).

The set of the logical cons eq ue nces of the se nt enc es quoted so far is the theory of dense linear orders without endpoints; we will denote it by DLO-. It is form ed by t he se nt ences tru e in every dense linear order with no endpoint s. Recall t hat (Q, :::;) , (R , :::;) are dense linear orders without endoints , and consequently their theories include DLO- (and one may wond er if they actually equal DLO-). The reader can ch eck directly t hat t he followin g clas ses of L-structures are eleme nt a ry: • dense linear orders with a least but no last elem ent, or a last but no least element, or both a least and a last element, • infinite discrete linear orders with or without end points (an order is discrete when ever y eleme nt, bu t t he leas t on e -if any- , has a pr edecessor a nd eve ry eleme nt, but t he last on e -if any- , has a suc cess o r) . 4 . We still work in L = {:::;} (wh er e j, is a binary relation sy m bol), bu t this t ime we deal with t he clas s K of well ordered sets (so ordered set s w here eve ry non-empty subset has a least element ) . Hen ce (N , :::;) E K owing to t he Minimum Principle, while any dense linea r order (A, :::;), even with a minimum, does not lie in K (in fact, given b > a in A , which is t he least elem ent > a in A?) . So the situation is , in so me sen se , opposite t o the last (elem entary) ex a m ple 3. Suppose that K is element ary, so K = M od(T) for a s uitable set T of L-sen t ences. Pu t L' = L U {cn : n E N} wh ere, for every natural n , Cn is a co nst a nt sy m bol, a nd in L' look at the following se t of sentences

T ' = TU

{Cn+1

<

Cn

:

nE N }.

25

1.5. ELEMEN TA RY CLASSES AND THEORIES

Let

Tb be a finite subset of 1", t hen t here is some natural N Tb

~ T U { Cn+ l

< Cn

: n

such th at

EN, n ~ N}.

Then Tb has a model because T U {Cn+ l < Cn : n EN , n ~ N} has : it suffices to t a ke the well ordered set (N , ~) , to interp ret Co, Cl , . . . , CN +! in N + 1, N, . .. , res pectively, and any fu rth er constant C n (with n > N ) ar bit ra rily. By the Compactness Theorem, 1" does admit a model A' = (A, ~ , (C;;') nEN) .

°

Let A = (A , ~) , th en A is a mod el of T, a nd hence is a well ordered set; however it contains t he non- em pt y subset

X={C;;': nEN} admit t ing no minimum , becau se, for every natural n, C;;~ l < c;;'. So we get a contradiction. Conseq uent ly K is not eleme ntary. In oth er words th ere a re linearl y ord ered sets which are not well ordered but sat isfy the same first order sentence as well ord ered sets. 5. Let now L = {O , 1, +, - , .} be our la nguage for fields. We cons ider in L t he class K of fields. K is elementary. In fact t he definition itself of field can be writ ten as a series of first order sentences (in most cases , of universal first order sentences) in L . For instance

says that a ny non zero element has an inver se. Also t he class of algebraically closed fields is elementary, although the corres ponding check is a lit tle subt ler. In fact what we have to say now is that , for every natural n , any (monic) polynomial of degree n + 1 has at least one root. So the point is how to quantify over polynomials of deg ree n + 1. However recall that such a polyn omi al is ju st an ordered sequence of length n + 2 of element s in th e field : the first is t he coefficient of degree 0, t he last is t he coefficient of degree n + 1 (and equals 1 if we deal with mon ic polynomials) ; so what we have to write is j ust , for every n ,

(where vi has the obvious meaning, for ever y i ~ n

+ 1).

26

CHAPTER 1. STRUCTURES T he logical consequences of t he sent ences liste d so far form the theory of algebraically closed fields, usually denoted ACF . Now let p be a pri me, or p = o. Also the class of (algebraically closed) fields of characteristic p is elementary; for, it suffices to add to the previous sentences the one saying that the sum of p t imes 1 is 0 when p is a prime, or, when p = 0, the negations of all t hese sentences. In conclusion, for every p prime or equal to 0, we can introduce the theory ACFp of algebraically closed fi elds of charact eristic p. Among the algebraically closed fields in characteris tic 0 recall the complex field C , as well as the (countable) field C o of complex algebraic numbers; th eir th eories contain ACF o, and one may wonder if they eq ual ACF o. Recall also that ever y field K has a (minimal) algebraically closed extension K; in particular, for p prime, Z / pZ is an example of algebraically closed field in characteristic p . Since we are t reating fields , let us consider agai n finite fields , a nd, mor e exactly, the infinite models of their t heory we met in exam ple 2: th e so ca lled pseudofinite fields. As observed before, one can ask which is the structure of these fields. J. Ax equipped them with a very elegant axiomatizatio n, explaining the essential nature of finite fields in t he first order setting: in fact , pseudofinite fields are just the fields K suc h that:

* K is perfect, * K has exactly one algebraic extension of every deg ree, * every absolutely irreducible variet y over K has a point in K. All these conditions ca n be written in a fi rst ord er way, although this is not imm ediat e to check. 6. A first order language for the class K of ordered fields is L = {O, 1, +, - , ., ::;}. K is elementary in L because it equals M od(T) where T is the set of the following sentences in L: (i) the field axioms (see Exa mple 5) ; (ii) those characterizing the linear orders (see Exam ple 3);

(iii) the sentence saying that sum s a nd prod ucts of non negative elements are non negative

1.5. ELEMENTA RY CLASSES AND THEORIES

27

Also t he class of real closed orde red fields (t hose satisfying t he Int ermediat e Value Property for polynomials of degr ee ~ 1) is eleme ntary, it suffices to add t he new sentences : (iv) for every natural n ,

+ v I . W + ... + Vn . ui" + w n +1 A u < w -+ u < v A v < w A Vo + vI . V + ... + Vn . v n + v n +1 = 0). A

-+

0 < Vo

T he logical consequences of (i), (ii), (iii) , (iv) form t he theory of real closed ordered fields, usu ally denoted R C F . Examples of real closed ord ered fields a re t he ord ered field of t he real numbers R , as well as t he (coun t abl e) ord ered field R o of real algebra ic numbers. T heir t heories includ e RC F , and one may wond er if act ua lly t hey equa l RC F . 7. Let R be a (countable) rin g wit h identi ty. Conside r t he lan gu age L R = {O, +, - , r (r E Rn of (left ) R -m odules. T he class of left R modules is elementary becau se it equ als t he class of mod els of t he following sentences in L R: (i) t hose ax iomatizing t he a belian groups in t he la nguage wit h 0, a nd - ;

+

(ii) for every r , s E R , if r +s and r ·s deno te t he sum and t he product (respecti vely) of rand s in R,

\lvo( (r

+ s) vo =

rvo + sVo) ,

\lvo((r · s )vo = r( svo)) , \lvO\lvl (r( Vo

+ VI) =

rvo + rvt} ,

(iii) finally, if 1 denotes th e identity element in R , \I Vo( 1Vo = vo).

T he logical consequences of t he previous sentences form t he t heory n T of left R-modules. Of course , t here is no reason to pr efer the left to t he right , at least in t his case ; indeed, one ca n check t hat even t he class of right R -modul es is element ary, and consequent ly one can introduce t he theory T n of right R-modules.

28

CHA P TER 1. STRUCTURES

Let us come back to our classification probl em for element ary, or also noneleme nt ary classes. The following fundam ental t heorem can suggest t hat , even in the elementary case, t his problem is not simple, as the class of models of a theory T ca n includ e ma ny pairwise non-isomorphic structures. Theorem 1.5.3 (Lowenheim-Skolem] Let T be a theory in a (countable) language L . Suppo se that T has some infinite model. T hen, for every infinite cardinal A, T admits some model of power A.

T he proof just uses Co mpactness in the extend ed fram ework of languages of a rbitrary cardinali ties. In fact one enlarges L by A many new constant sy mbols c; (i El , III = A) and one gets in t his way a n extended language L' . In L' one considers the following set of sentences

T'

= TU {--'(Ci = Cj) :

i, j E I, i # j }.

Any finite portion T~ of T' has a mod el; in fact it turns out that , for some finite subset 10 of I ,

so, in order t o obtain a mod el of T~ , it is sufficient to refer to a n infinite model A di T , as ensured by t he hypothesis, and t o interpret the finitely many cons t ants c; (i E 10 ) in pairwise different eleme nts of A. At this point , Com pact ness a pplies and gives a mod el of T ' (hence of T) of power :S: A. Bu t t his mod el has t o include the A many distint interp ret at ions of the c/s, a nd so its power is exactl y A. T herefore, if a theory T of L has at least an infinite model then T has a mod el in each infinite power (and two mod els with different cardinalit ies ca nnot be isomorp hic). Of cou rse, one may wond er how strong is the assu mp tion th at T has some infinit e mod el. Not so much , if one recalls that a th eory T ad mit t ing finite models of a rbitrarily large size must admit also some infinite mod els. Another reasonabl e question may concern how many mod els T adm it s in any fixed infinite cardinal A. One can check t hat t heir number cannot exceed 2\ bu t this upper bound can be reached, for every A, by some suit able T's. T he opposite case, when T has just one mod el in power A (up to isomorphism) , will be of some interest in the next chapters; we fix it in t he following definition. Definition 1.5.4 Let T be a theory with som e infinite model, A be an infinite cardinal. T is said to be A-categorical if and only it any two models of T of power A are isomorphic.

1.5. ELEMENTARY CLASSES AND THEORIES

29

We wish to devote some more lines t o t he Lowenh cim-Skol em th eorem . Among other things , it confirms th at elementary equivalence is a weaker relation than isomorphism. In fact, t ake a n infinite struct ure A , an d use the Lowenh eim-Skolem to build a model A' satisfying the same first order sentences as A but having a differen t cardinality, It is easily checked that A , A' are elementary equivalent; bu t , of course, t hey ca nnot be isomorphic. Now recall what we pointed out in 1.2 : the Induction P rinciple (in its usu al form) cannot be written in the fi rst order style in t he la nguage for (N , 0, s) because first order logic forbids quantification on set vari abl es. However, as far as we know , one might find an equivalent statement that ca n be expressed in t he first order set t ing; in this sense, Induction migh t becom e a first orde r statement. Well, the Lowenheim-Skolem theorem excludes this extreme possibility. For, t he Induction P rincip le cha racterizes (N , 0, s) up to isomorphism , while the Lowenh eim-Skol em theorem ens ures us that any tentat ive first order equivalent translation (even involving infinitely many se ntences) has some uncountable models. So this translation cannot exist . The Lowenh eim-Skol em t heorem emphasizes other similar expressiveness restrictions in first ord er logic. For instan ce, it is well known t hat t he ord ered field of teals is, up to isomorphism, t he only complete ord ered field (here completeness means t hat ever y non-empty upperly, lowerly bounded set of reals has a least upper bound , a greatest lower bound resp ectively). So completeness ca nnot be expressed in a first order way, because any tentative first order tran slation sh ould be true in some real closed field wit h a noncont inuum power. On t he ot her side, we will see that the Lowenh eim-Skolcm t heore m is a very useful and powerful technical tool in first ord er mod el theory (just as the Com pactness Theorem ). And actually the expressiveness restrictions rema rked before are only the other side of the picture of these technical advantages. This is j ust th e content of the Lindst rom theorem quo ted before in 1.2. Indeed , what Lind stro rn shows is t hat, if you have a logic (nam ely a reasona ble syst em offormulas a nd truth) and you demand t hat your logic satisfies the Compact ness Theorem and the weak er form of the Lowenh eim-Skolem Theorem , ca lled Downward Lowenh eim-Skolem Theorem , introduced in 1.4 and requiring -for countable lan gu ages- th at any set of sentences adm itting a mod el does hav e a count able model , then your logic is t he first ord er logic. In t his sense the first order framework is (Leibni zianl y) the best possible one .

30

1.6

CHAPTER 1. STRUCTURES

Complete theories

Let us deal again now with one of the main themes in Model Theory, Le. class ifying struct ures in a given class K . Due to our first order setting, we limit our analysis to elementary classes K = Mod(T), where T is a first order t heory. T his choice is not so partial and na rrow as it may ap pear. In fact, it certainly includes th e cases when T is exp licit ly given and eq uips K with an effect ive list of first order ax ioms , as in the positive examples of the last section; but it is also conce rned wit h oth er , and worse sit uations. For instance, think of the t heory T of finite sets, or groups, or fields, or, in general, of a class of finite arbitrarily la rge structures, so t hat T has also infinite models. Alternatively, think of the th eory T of a single infinite structure A: due to the Lowenheim-Skol em Th eorem, T has some models non-isomor phic to A. In t hese cases, T is introduced by sp ecifying some crucial models, but this does not determine in an exp licit way a priori which first order sentences belong to T , and which are excluded; inde ed we could just be interested in finding an effective ax iomatization as in the previous examples, and we could aim both at descri bing T and also -as a relat ed matter- at classifying its mod els. These are th e settings we wish to consider. Actually we should also admit that we have not clearly explained up to now which kind of classification we pu rsue; however we have ag reed t hat this classification should identify isomor phic models but disting uish non isomorphic structures. Also, we have seen that isomorphic mod els sat isfy t he same order orde r sentences. So a pr eliminary classification is j ust up to elementary equivalence, an d aims at distin guishing non elementarily equivalent structures. Once this is don e, we could restrict our analysis to structures satisfying t he same first ord er conditions; Le. fix a structure A and classify up to isomorphism t he models of its theory T = Th({ A}) (by t he way, let us a bbre viate for sim plicity 1'h( { A} ) by Th( A)). Which is an intrinsic syntactical characterization of such a theory 1'? Basica lly it is " complete" according to the following definit ion. Definition 1.6.1 A (consistent) theory T of L is said to be complete if, for every sentence

In fact, it is easi ly observed on the ground of the definition of truth in first order logic that , given a structure A in a la nguage L , for every L-sen tence ip, either

31

1.6. COMPLE TE THEORIES

On t he other hand , every complete t heory T ca n be represented in t his way. In fact , fix any mod el A of T . Clea rly T ~ T h(A) . Conversely, let ep be a sentence of Th (A ), t hen -' ep (j. T h(A) and so -'ep (j. T; as T is complete, ep E T . Notice t hat t he same arg ume nt shows t hat, if T ~ T ' are consistent t heories a nd T is cornplete, t hen T = 'I", Now not ice what follows. Rem a r k 1.6.2 Every (consistent) t heory T of L ca n be enla rged in at least one way to a complete t heory in L. In fact , it suffices t o conside r T h(A ) where A is any mod el of T . A complet e t heory exte nding T is called a com plet ion of T .

So our classification project ca n be organized as follows. • First , determin e struct ures up to elementary equivalence, words find all t he complet ions of a given t heory T;

III

other

• t hen, class ify up t o isom orphism t he mod els of a complete T . We deal in t his section wit h t he form er problem , hence wit h complet ions an d , definitively, wit h com plete t heor ies. Incomplete t heor ies are easy to meet. Example 1.6 .3 For inst an ce, t he t heory of groups it is not complete (as t here a re both ab elian a nd nonabelia n groups, a nd commutativity ca n be written in a universal first ord er sent ence) . In th e same way, the t heory of fields is not complete (why?) . Also the th eor y of linear orders is not com plete (as t here are both dense and non dense t otal orde rs, as well as ord ers with or without a minimum or a maximu m).

On the other hand , the previou s remark poin ti ng out t hat a compl et e T is the t heory of any mod el of T seems to provide a great deal of complete t heories; bu t these exa mples are not satisfactory. In fact , as already said, wh at we reason abl y ex pect is t o have complet e t heories T equipped with a n explicit list of basic axioms, ensuring that t he sente nces in T are ju st t he consequences of t hese axioms . Now, when we look at T h(A ) for some struct ure A (t he field of com plex numbers , or t he ordered field of reals, an d so on ) , t his list of axioms is lacking; indeed we could wish t o obt ain such a bas ic axiomatization in t he mentioned cases . A possible strategy t o solve t hese problems might be t he following. Given A , prepa re a t entative ex plicit axiornatization an d t he corres ponding t heory T. Of course , A should

32

CHA PTER 1. STRUCTURES

be a mod el of T . At t his point , check if T is complete, by some suitable procedures. If yes, T = T h (A ). Unfortunately, checkin g completeness for a theor y T as before is not simple. We mention here a celebrate d sufficient (but non-necessary) condit ion, found ed on t he notion of A-ca tegor icity. Theorem 1.6.4 (Vaught) Let T be a theory of L . Su ppose that every model of T is infinite and T is A-categorical fo r som e infinite cardina l A. T hen T is complete.

Proof. Suppose to wa rds a con tradi ction t hat T is not complete; let

0. Th en the theory l oo of infinite sets is complete.

Clea rly T has no finite mod els. Mo reover t wo (infinit e) sets in Proof. t he same power a re isomorphic; in oth er words l oo is A-cat egor ical in a ny A ~ ~o. Co nsequently l oo is complete . • Corollary 1.6.6 Th e theory DLO- of dense linear orders without endpoints is complete .

Proof. DLO- has no finite mod els; this is easy to check, by using density or th e a bsence of end points. Moreover th e fa mou s theorem of Cantor on den se linear orde rs recalled in 1.3 ensures t hat DLO- is ~o- cat egori c al: every countable linear ord er wit hout end poin ts is isomorphic to t he ord er of ration als. Hence DLO- is com plete . • Recall t hat both (Q, :::;) and (R , :::;) are models of DLO - ; it follows t hat DLO - = T h( Q,:::;) = T h (R, :::;) . Corollary 1.6.7 For every p = 0 or prim e, the theory ACFp of algebraically closed fields of characteristic p is complete.

1.6. COMPLETE THEORIES

33

Proof. An algebraically closed field lC is always infinite; in fact , if ao, .. . , an are distinct elements of lC , the polynomial (x - ao) .... . (x - an) + 1 in K [x] has a root cv in lC ; cv cannot equal ao, ... , an, a nd so is a new eleme nt . At t his poin t , in order to apply Vaught 's Theorem , we have t o prove '\-categoricity for some infinite '\. But t his is ju st a consequence of St einitz 's a na lysis of algebraically closed fields . For, this analysis esse nt ially impli es (in our t erminology) that , fixed p = 0 or prime, t he t heory AC Fp of algebraically closed fields of characteristic p is '\-categorical for every uncount abl e cardina l ,\ (so Vaugh t 's Theorem applies and yields cornplet eness) . Let us recall briefly why (we will provide an alte rnat ive, detailed proof of t he complete ness of AC Fp in Chapter 2). Any algebraically closed field in characteristic p can be obtain ed as lC = lCo(S) where lCo is the prime subfield of lC (hence lCo is isomorphic to the rational field if p = 0, or to the field with p elements if p is prime) , S is a t ra nscendence basis of lC (namely a maximal algebra ically inde pende nt subset ), and - denotes the algebraic closure in lC. Fur t hermore t he isomorphism ty pe of lC is fully determined by t he cardina lity of S (the tra nscendence degree of lC). Accordingly, one ca n realize that AC Fp has • ~o pairwise non isomorphic count a ble models (correspondingly to the

t ranscendence degrees 0, 1, ...,

~o) ,

• for every uncountabl e cardinal '\ , exactly one isomorphism class of mod els of power '\ , becau se all t hese models sha re the same t ran scendence degr ee '\. Hence AC Fp is '\-categorical in ever y cardinal ,\ > com plete. ..

~o ,

and consequent ly

In particular, two algebraically closed fields lC l and lC z ha ving the same characteristic p but different transcendence degrees d l -I d z a re elementarily eq uivalent, but cannot be isomorphic. Hence, when lC l a nd lC z are countable, t hey are not even partially isomorphic. Notice also that t he field of complex nu mbers is a model of ACFo, and so AC 1"0 eq uals it s t heory: we find in this way an explicit list of axiom s (that of AC 1"0) for the theory of the complex field. Let us propose a further application of Vaught 's T heorem t o deduce completeness. Perhaps at t his poin t some one may expect t o meet R C 1" and the theory of the real field in ou r list of exam ples. But we have to delay this a ppoint ment . Ind eed , RC F is complete (and hen ce equals the theory

34

CHAPTER 1. STRUCTURES

of t he ordered field of reals) , bu t RCF is not A-cat egori cal for any infini t e ca rdina l A. So a different approach is necessary: we sha ll follow this new strategy in t he nex t cha pter. On t he contrary, Vau ght 's Theorem applies t o vectors paces over a countable field . Let us see why. C o ro lla ry 1.6 .8 Let JC be a countable fi eld. Th en the theory KT ' of infinite vectors paces over JC is complete.

Proof. Clea rly KT ' has no finite models. Moreover we know that two (infinit e) vectorsp aces with the same dim ension over JC a re isomorphic. Consequ ently, for every cardinal A bigger than ~o, there is a unique isomorphism class for all the JC-vectorspace of power A (in fact , eac h of them has dimension A). In other word s, KT ' is A-cat egorical for every ca rdinal A > ~o. By Vaught 's Theor em , KT ' is complete . • On t he cont rary, KT' may not be categorical in ~o . In fact , when JC is infinite, JC , JC2, .. . , JC{ No) are count abl e JC-vectorsp aces wit h distin ct dimensions, and so ca nnot be isomorphic, hen ce t hey are not even par tially isomorphic. So eleme ntary equivalence ca nn ot impl y pa rti al iso morphis m (a nd isomorphism ). T he read er may check directly what happens when JC is finite . We conclude t his section by introducing a nother noti on relat ed to completeness. It will be used in Chapter 3 t o show th at RC F is complete. Recall t hat a complete th eor y T equa ls T h(A) for every model A , and hence a t heory T is complet e if a nd onl y if a ny t wo mod els of T a re eleme nt arily eq uivalent . Definition 1.6 .9 A theory T is model complet e if every em bedding of models of T is elem entary.

It is eas y to exhibit t heories which ar e not mod el com plete . For instance, t he pr eviou s examples 1.3.1 4 ensure t hat t he t heory of (N , <) , as well as th e t heory of fields , are not mod el com plete . On th e cont rary, it is not sim pie t o give explicit exam ples of mod el complete t heories. Cha pter 3 will be devoted t o t his poin t , and to discussing t he relevan ce of t his notion within Mod el T heo ry.

1.7. D EFINABLE SETS

1.7

35

Definable sets

Formulas include equa t ions , and Algebra aim s at finding solut ions of equations. Mo re gener ally, given a lan gu age L , a formula
Definition 1.7.1 Let A be a structu re of L , n be a positive in teger. A subset D of An is called definable in A if there is a formula
T he eleme nts of A occ urring as constants in t he formul a
~

Y ~ A a nd D is X-defina ble, t hen D is V-definable, too;

(iii) D is defina ble if a nd only if t here exists a finite subset X of A such t hat D is X -definabl e (( <=) follows from (ii) ; in orde r to show (=}) , ju st let X be t he set of t he parameters in a defini ng L(A)-formula

36

CHAPTER 1. STR UCTUR ES

An eleme nt a E A is X -definabl e if it s singleton is. Of course, every a is A-definabl e (by v = a). Bu t , when a rJ. X , things are not so t rivial. Remark 1.7.2 Fi x a struct ure A of L a nd a positive integer n. 1. Let X be a subset of A . T he X-definable subset of A n form a subalgebra of the Boolean algebra of all t he subset of A n.

In other words, both A n a nd 0 are X-definable (by t he formulas VI = VI and '(VI = vd respectively) , and , if Do and D I are two X-definable subsets of A n, t hen even th eir union DoUDI , t heir intersection DonD I and the compleme nt A n - Do are X-definabl e (if
2. Definable sets are closed also und er projections, in t he following sense. Let D be a subset of A n definable in A. Let Jr be t he pro jection of A ont o some fixed i ::; n coordinates. T hen Jr(D) (a subset of Ai) is st ill defina ble. For instan ce, if t he L(A)-formula
In fact let D be a finit e subset of A n, a nd let d~ , . . ., d~ be its elements . For every j ::; t, put dj = (djl ' . . . , djn)' T hen

V 1\

j9 l:::;i:::;n

(Vi = dji)

defines D in A . In particular, in a finit e struct ure A every subset of A n is definable; moreover , owing to 1, in any st ructur e A even the cofinite subsets of A n a re definable. 4. A is infinite, then there exist som e subset s of A n which are not definable in A.

T his follows from a simple cardinal counting argument . In fact , we know that there are 21A1 dist inct subsets of An, while the subset s of An definabl e in A cannot exceed the L( A)-formulas defining t hem. Con sequ entl y t he definable subsets of A n ar e at most IAI (for a countable L , of course) .

1.7. DEFINABLE SETS

37

An ex plicit exam ple of an infinite struct ure wit h a non-definable subset is t he following. Let L = 0, so t he struct ures of L a re ju st t he nonem pty sets A . Take a n infinite set A. We have see n t hat every finite or cofinite subset of A is definable. We claim t hat no other subset of A is definable. In fact, let D be a subset of A such t hat both D and its com plement A - D are infinite. Suppose t owa rds a cont radict ion t ha t D is definabl e, a nd so D =

Hence algebraic curves are exa mples of algebraic varieties. Moreover every algebraic vari et y as before is definabl e in J( , for instance by the (quantifier free) formul a

1\ qj(v) =

O.

j9

So one may wond er how close and deep t his connection bet ween definable sets in J( a nd algebraic varieti es in J( is. Of course, we ca nnot expect t hat a ny definabl e set is a variety (alt hough t his is cert ainl y t rue when J( is finite). In fact , owing t o Remark 1.7.2 ,1 before, every finite Boolean com bination of definabl e sets is also definable. Bu t , wit h respect t o t his point , algebraic varieties behave in a different way.

38

CHAPTER 1. STRUCTURES

• The union of t wo (a nd consequently of finit ely many) algebraic vari eties in an algebraic varie ty. This is a simple exercise of Algebra, essenti ally using the fact that , in a field K , t he product of two nonz ero elements is different from O. • T he intersection of two, or finitel y man y, or even infinitel y many algebraic vari eti es is st ill an algebraic variety. This is a trivial exercise in the finite case, and a deep t heorem in Algebra -known as Hilbert 's Basis Theor em- otherwise. Notice that these properti es (together with the easy observation t hat K " and 0 ar e algebra ic variet ies -for, they are th e zero sets of t he zero polynomial, and of a ny non zero constant polynomial in K[x'J resp ectively-) show t hat th e algebraic var ieti es of K" ar e the closed sets in a suitable topology of K" (the Zariski topology) . However • t he complement of a n algebraic vari ety of K" is not necessarily an algebraic vari ety of K" : So t here are definabl e sets of K" which are not algebraic varieti es. Ind eed Algebraic Geometry introduces the noti on of constructible set t o define a finite Boolean combination of algebraic variet ies of K": Remark 1.7.2 ,1 before ensures t hat every constructible set is definable. In certain fields K the converse is also true, and hence definable j ust means constru ctible. For instance, this is what happens when K is an algebra ically closed field (and so, in particular , when K is t he complex field). This is not a t rivial result , bu t a deep t heorem of Tarski a nd C hevalley, and will be discussed in t he next C ha pter .

2. (Defi na b le set s a n d Real Algebraic G eom e t r y ) Let L = {O, 1, +, " - ::;} be our language for ord ered fields. Fix a n ordered field K, and a positive integer n . Algebraic Geometry studies the sets of the elements of K " satisfying disequations like q(x) ~ 0 where q(x) E I<[x'J, and calls semialgebraic set any finite Boolean combinat ion of t hem. It is clear t hat every semialgebraic set is definabl e in K , and even by a qu antifier free formula (a Boolean combina t ion of atomic formu las q(v) ~ 0) . A t heo rem of Tarski and Seidenb erg ensures t hat, when K is a real closed ord ered field (in particular when K is t he ordered field of reals) , t hen t he definable sets of K" are exactly t hose semialgebraic . So a close conn ection a rises bet ween Model Theory and Algebra ic Geom etry also in t his fram ework . The Tar ski and Seidenb erg t heorem will be t reated in detail in t he next chap ter.

39

1.7. DE FINABLE SETS

Notice also t hat the order relation ~ is definable in the real field R even within t he la nguage of fields {O, 1, +, " - }: in fact, it suffices to recall that the non negative reals are exactly the squares, a nd hence to define by t he for mul a

:3W(VI - V2 = w 2).

Conseq uently every semialgebraic set D in R is definabl e in the real field even wit hin the language of fields, just by rep lacing any formula

q(iJ)

~ 0

(with q(x) E R[X]) by t he equivalent for mula

:3w(q(iJ) = w2 ) . However , notice that the lat t er formu la requires a quantifier. 3. (Definable sets and recursive sets) Now we consi der t he langu age L = {+ , .} and, in L , t he structure (N , + , .). F irst notice t hat every natural n is 0-definabl e in (N , +, .). T his can be eas ily show n by using an ind uction arg ument on n; if n = 0 or n = 1, ju st take t he form ulas "VI

= 0" :

VI

+ VI

=

VI,

"VI

respectively, while, for n ~ 1, n "VI

= n

+ I"

= 1" :

VI . VI

=

+ 1 is 0-definable

VI

A --, (" VI = 0" )

by

: :3z 0:3 z 1(" Zo = I " A" Z l = n" A VI = Zo

+ zt).

+, .) defina ble sets just equal 0-definable sets. (N , +, .) is deeply related to recursion theory. Let us

Consequently in (N ,

Definability in see why. A basic aim in recursion theory is to provide a sharp definition of the notion of elgorithm . According to the Church and Turing thesis,

algorithm mean s Turing machine, in t he sense t hat t he problems with a solving algorit hm are ju st t hose handled by a Turing machin e. Actua lly t he Churc h-Tu ring mo del of computation dat es back to t he thirties, a nd, from the practical point of view, is undoubt edly sur passed by the new advances in computer science. However, as an abstract and theoretic proposal, it is still valid

40

CHAPTERl . STRUCTURES and commonly ag reed , at least in any discrete setting. This is clearly our framework when we are dealing with natural numbers. Incidentally, recall that every discrete context ca n be eas ily tran slat ed int o the world of natu ral numbers, owing to the G6del coding procedures , equipping effectively eac h element with it s own natu ral label. So let 's work wit h the str uct ure (N, +, .). On the ground of the Church a nd Tur ing thesis, one can define in a sha rp way which are the subsets D <;;; N " (with n a positive integer) ad mitting a decision algorithm , nam ely a pro cedure running in finitely many steps and establishing, for every if E N " ; if if is in D or not. These sets D ar e called recursive. A crucial result in Recursion Theory -indeed a key step within t he proof of G6del First Incompleteness Theorem- ensures t hat every recursive set is definable in (N , +, .). More generally, any recursively enumerable D <;;; N " is definabl e in (N , +, .). Rec all t hat rec urs ive enumerability is a weaker notion t han rec urs iveness, a nd ju st req uires th at there is some algorit hm effectively listi ng t he element s of the involved set D . On the cont rary, there do exist some subset of N " which a re not recursively en umerable (and hence are not even recursive) , but are definable. These remarks witness that now definable sets ar e a very complicated class, becau se t hey inherit t he complexity of t he class of recursively enum erabl e sets wit h t he cor res pondin g int rinsic hierarchies, and possibl y more. So, when dealin g with (N , +, -} , definable sets are not so clean as in the complex, or in the real field . 4. (Definable sets and decidable theories) We again work in the language L = {+, .}, but t his time we examine the structure (Z , +, .). Ident ify nat ural numbers a nd non negative integers. T hen N beco mes an 0-definabl e subset of Z ; in fact , a celebrat ed theorem of Lagran ge ensures t hat, among the integers, t he non negative element s a re just the sums of 4 squares. Hence the form ula

defin es N inside (Z , +, .). Also the sum and product op erations in N are 0-definable in (Z, +, .), because they just restrict to N the add it ion and multiplication in Z. So we can conclude that the whole structure (N , +, .) is " 0-definable" in (Z, +, .). Now it is well known that the t heory of (N, +, .) is not decidable, in t he sense t hat the set of the natural codes of t he se ntences t rue in (N , +, .) is not recursive. In other words , no gener al algorit hm can

41

1.7. DEFINABLE SETS

decid e, for every se ntence

(N , +, .) F
{:}

Consequently eve n t he t heo ry of (Z ,

(Z,

+, .) F
+, .) is

undec idable.

T he method sketched here is oft en used t o show und ecidabili ty for t heories int erpreting as described other t heo ries whose undecid ability is known , or also, spec ula rly, t o deduce decid abilit y for t he t heories that ca n be interpret ed in the previous way in some other de cidable theory. Hence defin ability plays a crucial role also within the deci sion problem for theories. 5. (M o d u le s and pp-definable subgroups) Let R a (countable) ring wit h identit y. Conside r the langu ag e LR = {a, +, - , r (r E Rn of R -modules, a nd in LR t he class R - M od of (left ) R -m odules. A formula
:J :JW

•

t V- = B . t W- )

where A a nd B a re ma trices wit h coefficients in R an d suitable sizes, . denot es t he usu al row-by-column mul tiplica ti on for matrices, and t is t he t rans pose op er ation. Equivalentl y, if one put s v = (VI , ... , v n ) , W = (WI , ... , wm), A = ( r i j)i,j a nd B = (S ih) i ,h , where j ra nges fro m 1 t o n , h from 1 t o m, a nd i fro m 1 t o some s uitable positive int eger t , then
1\ ( L

:JWI ... :Jwm

r ij Vj

l ~ i 9 l ~ j~ n

=

L

Si h Wh ) .

l ~h~m

Hen ce, in every R-module M ,
has some solution in M m. Let us examine some part icul a r cases.

CHAPTER 1. STR UCT URES

42

• Let r E R,


is a subgroup of M , and even a submodule at lea st when R is commutative, or, sim plerly, when r is in t he centre of R . In general it is easy to check that , for ever y pp-formula
A t heorem of Baur and Monk ensures that , in every R-modu le M , a ny defi nable sets is a finite Boolean combination of eosets of pp-defina ble subgroups (see Ch ap t er 2).

1.8

References

There exist severa l excellent handbooks providing t he backgrounds of Mathemat ical Logic necessary in this chapter. Among them , let us mention once again Sho enfield [153], but also Malitz [103] or Ebbingha us-Flum-T homas [37]. The key reference for basic Model Theory is [18]. We also refer t o Devlin [31] for Set Theory, t o Od ifredd i [121] for Recursion Theory and to Jacobson [65] for Algebra. [173] is the historical sour ce qu oted at t he end of 1.2 . Ax's analysis of pseudofinite fields is given in [3]

Chapter 2

Quantifier Elimination 2 .1

Elimination sets

Let L be a la nguage. It may happen that two different L-formu las ip (v) a nd ip'(v) admit t he same meaning in a structure A of L, or in a class of L-structures, for inst an ce among t he models of a given L-theory T . For example, in t he ordered field of reals (and even in every real closed field), t he form ula ip(v) : v ~ 0 (being nonnegati ve) is t he same t hing as ip'(v) : :Jw(v = w2 ) (being a square) . Simila rly, in the orde red dom ain of int egers, ip (v) : v ~ 0 (being positive) has t he same interpret at ion as ip' (v) : :Jwt3w2:J w3:J w4 (v = L 1
\fv(ip(v)

+7

ip'(v)) E T ,

equivalently whe n for all mod els A of T. The notion of elimination set arises qui t e naturally a t t his point. An elimination set for T is a set F of L-formulas suc h th at every L-for mula ip(v) is T -eq uivalent to a suitable Boolean combination of formulas of F . Clearly the set of all the L-formul as is a n elimination set for T . But, of course, t his is not an interesting case, a nd we reasonably expect simpler sets

43

44

CHAPTER 2. Q UA NTIFIER ELIMINATION

F. In particular, when the set of atomic formulas in L is an eliminat ion set for T, we say t hat T has t he quantifier elim ination in L. In detail

Definition 2.1.1 Let T be a theory in a language L. T has the elimination of quantifiers (q.e.} in L if and only if every formu la
• an effecti ve procedure translating an y L-sentence into a T-equ ivalent Boo lean combinat ion of sente nces in F (or even an effective

45

2.1. ELIM INA T ION SETS

redu ction of a ny L-formula into a T -equivalent Boolean combination of formulas in F ); • an algorithm to decid e, for ever y Boolean combination tences of F, whet her 0' is or not in T.

0'

ofsen-

Then , clearly, T is decidable, and actually we have got a decision algorit hm (by successively applying t he pr eviou s t wo pro cedures). 2. Another noteworthy application of qu a ntifier elimi nation conce rns definabilit y. In fact , if F is an elimination set for T , t hen the definabl e sets of a mod el A of T reduce to

where

where

x in A.

3. A third application regards the classification of com pletions of T . Recall th at T is possibly incomplete; but we know t hat T ha s some (non-uniqu e!) complete ext ension in L. So we are led to conside r th e pro blem offi ndin g a ll t he com plete extensions ofT in L , in other word s classifyin g t he isomorphism classes of model s of T up to elementary equivalence. Now, if A and B are two models and A is not element arily equivalent to B, t hen there is some se nte nce

1, and - (- m) if m < -1 ; simila rly for n). T his implies that the complet e extensions of ACF are fully determined by t he characterist ic of t heir models , a nd hence coincide wit h t he theories AC Fp where p is 0, or a pri me.

46

CHAPTER 2. QUANTIFIER ELIMINATION 4. Finally, let us deal with model completeness. Assume that T has quantifier elimination in L. We claim that, in this case, every embedding between models of T is elementary, in other words T is model complete. In fact, let A and B be models of T, f be an embedding of A into B. Given a formula
A

1=
{:}

B

1=
{:}

B

1=
As VV(
A Hence

f

1=
is elementary.

This chapter is devoted to illustrating several key examples of quantifier elimination, starting from the earliest (Langford's results on discrete or dense linear orders) to include those perhaps most classical and celebrated (Tarski's elimination procedures for the real and the complex fields). We shall treat other eliminations sets as well (most notably, Baur-Monk's ppelimination theorem for modules over a given ring). These examples will lead us to introduce two basic notions in Model Theory, strong minimality and o-rninirnality respectively. We shall discuss them at the end of the chapter. The final section will be devoted to some computational aspects of the quantifier elimination procedures. It should be underlined that the interest in quantifier elimination arose several years before the official birth of Model Theory. In fact it was at the beginning of the twentieth century that Lowenheim and, later, Skolem provided some procedures translating formulas into a simpler form avoiding quantifiers (they are, more or less, the artificial method we sketched at the beginning of this section). Moreover, the earliest explicit examples of quantifier elimination in some specific algebraic structures treat discrete and dense linear orders and date back to the twenties (they were obtained by Langford in 1927). In these results, as well as in Tarski's theorems, the major emphasis seems to be on decidability: the elimination of quantifiers is a step towards decidability, just as described before. But over the years this emphasis on decidability reduced and was replaced by an increasing interest in definability. Actually, definability is the main theme where Model Theory and quantifier elimination meet.

2.2. DISCRETE LINEAR ORD ERS

2.2

47

Discrete linear orders

We begin here our ana lysis of qu a ntifier eliminable t heories. F irst we t reat infini te linear ord ers. Acco rdingly our basic lan gu age is L = {:s;}. More pr ecisely we deal wit h: • t heories of discrete linear ord ers (in t his section) , • t heories of dense linear ord ers (in t he next one). As already said , the quantifier elimina t ion results in t hese cases were firstl y shown by Langford in 1927; Tarski pursued t he an alysis t o get decidability a nd to classify th e com plete theories of infinite discret e a nd dense total orders. Recall t hat a (n infinit e) linear ord er A = (A , :S; ) is discrete if a nd only if

(i) 'Va E A , if there is some a' E A such th at a < a' , t hen t here exists a least b E A for which a < b (b is called t he s uccessor of a a nd is denoted s(a)); E A , if t here is some a' E A suc h t hat a' < a, t hen t here exist s a maximal b E A for which b < a (b is called t he pred ecessor of a; obvio usly a = s(b)).

(ii) 'Va

Acco rdingly we ca n dist ingu ish 4 classes of (infinite) discrete linear orders: 1. t he class of discrete linear orders wit h a leas t , bu t no las t element (like

(N , :S;)); 2. t he class of discret e linear ord ers with a las t , but no least eleme nt (for instance, N with resp ect to t he relation reversing its usu al order); 3. t he class of (infinit e) discrete linear orders with both a least and a last element (like the disjoint union of two discret e linear orders (A, :S; ), (E , :S; ), the form er in 1, th e latter in 2, with a < b for all a E A and

s « B );

4. t he class of discrete linear orders without end points (like (Z , :s;)) . Each of t hese classes is elementary. Moreover one ca n show t hat its t heory has t he eliminat ion of qu an tifiers in a suitable lan gu age extending L , and is com plete even in L. Here we limit ourselves to prove, for simplicity, t hese resu lts in t he case 1, in other words for discrete linear orders wit h a least but no last element .

CHAPTER 2. QUANTIFIER ELIMINATION

48

Accordingly conside r the language L' = {::; , 0, s} where 0 is a constant (to be interpreted in t he least element) and s is a 1-ary operation sy mbol (to be interpreted in the function mapping any element into its s uccessor). It is easy to write down a first order set of axioms for our class in L'. Let dLO+ denote th e corresponding t heory. By t he way, notice that suitable formulas in the restrict ed langu age L define th e minimal eleme nt a nd t he successor function in every model of dLO+. This implies that the axioms of dLO+ ca n be rewrit ten also in L , at t he cost of some more complicat ions (and quantifiers). For instanc e, expressing the existence of a minim al element requires the L-sentence :3wVv(w::; v) instead of

Vv(O ::; v). Bu t we momentarily prefer to treat dLO+ in L'. Ob serve th at: • (N ,::;, 0, s) is a model of dLO+; • if A is another mod el of dLO+ , then A contains a substructure ({ s" (OA) : n EN} ,::; , OA , sA) isomorph ic to a (N ,::; , 0, s), and moreover some furt her copies of (Z, ::; , s) (as OA is t he only eleme nt wit hout an y predecessor) . Theorem 2.2.1 dLO+ has the elimination of quantifiers in L' .

Proof. Take a formula 'P(v) in our langu age L' ; we look for an equivalent formula 'P' (v) without qu an t ifiers. Our first ste p is to show t hat we ca n ass ume that 'PCv) is of th e for m :3w

1\ ai( v, w) i
where each ai(v, w) is a n at omic formula , or it s negation, and w actually occurs in ai ( v,w) for ever y i ::; r . We wish to underline that this step is qui t e general, and does not depend on our part icular language L' . Let us see why. F irst of all, we can tacitly ass ume that 'P (v) is of the for m

where the Qj's (1 ::; j ::; m) denote qu an ti fi ers V or B, a(v, w) is a quantifier free formul a , a nd even a disjunct ion of conj uncti ons of atom ic formul as and

49

2.2. DISCRET E LINEAR ORDERS

negations (and wabridges (Wl, . . . , W m ), of course) . Th e st rategy at this point is first to eliminate Qm, and then to repeat the procedure and remove t he quantifier st ring completely. We recall t hat V is equivalent to -,:3-, and consequently ag ree that it is enough to deal with the case when Qm is :3, namely with

:3w a(v, w) where a is a disjunction of conjunctions of atomic formulas or negations, W = and v is possib ly en lar ged to include Wl , ... , W m - l ' As :3 is distribu tiv e with respect t o V, namely :3w(a' V a" ) is equivalent to (:3wa' ) V (:3wa" ), there is no loss of generality for our purposes in assu ming that a is ju st a conj un ction of atomic formulas or negations. In conclusion we ar e dealing with Wm

:3w

1\ ai(v, w)

i
where each ai(v, w) is an atomic formul a , or its negation . vVe ca n also assume t hat W actually occurs in ai(v, w) for every i ~ r: oth erwise let j ~ r deny t his condit ion and notice t hat our formula

:3w

1\ ai(v, w) i
is equivalent to

a j(v)

1\

:3w

1\ ai(v, w); i=h

at t his point it suffices t o elimina te t he quantifier :3 in the lat ter part of t he formul a

:3w

1\ ai(v, w).

i =j:j

T his completes our preliminary st ep. As alr eady said , this doe s not dep end on our particular fram ewor k. Now let us wor k with our formula

:3w

1\ ai(v, w) i
and our language L' . We wond er which is the form of any ai. A look at L' shows that ai is t = t' , or t ~ t' , or the negat ion of one of t hese formul as, where t and t' are terms in 0, w , v (and w actu ally occurs in tor t'). Reca ll th at -, (t ~ t') means t > t', and so on. Deduce that ai is, with no loss of generality, eit her t = t' or t > t', with t and t' as before. Notice t hat t and

50

CHAPTER 2. QUANTIFIER ELIMINATION

t' are of the form sP(11,) where p is a non negative integer and 11, ranges over 0, w, if. Now recall that s is injective and deduce that the formula under exam ensures that there is a solution w for a finite set of conditions saying that w or a successor sq(w) (q a non negative integer) is equal, or bigger, or smaller than a term sP( 11,) where p is, again, a nonnegative integer and 1J, ranges over w, 0, if: as s is injective, we can assume that, in each of these equations and inequations, s occurs only in one side (on the left, or on the right). Our aim is to translate this formula into an equivalent one avoiding wand simply stating quantifier free conditions on if (and 0). To obtain this, proceed as follows. Trivialities like w = w or w < sP (w) for a positive p can be ignored and deleted (they can be preliminarily listed and hence are easily recognized); if nothing else occurs, then replace the whole formula by 0 = O. On the contrary, when meeting a condition that cannot be satisfied by any w, like w = sP (w), or 0 = sP (w) for a positive p (also these conditions can be preliminarily listed), then replace our formula with -.(0 = 0) (or with -'(Vi = vi) if you like and if is not empty). Otherwise, as soon as you meet one equation like w = SP(Vi), delete wand :3, and replace w with sP(Vi) throughout our formula. Proceed in the same way if an equation w = sP(O) occurs. Similarly, when meeting a condition sq(w) = Vi, consider any further occurrence of w in t he formula and, again using the injectivity of s , represent it as sql (w) for a suitable nonnegative integer q' 2:: q;finally delete wand :3 and replace each occurrence sql (w) by

sq-q'(Vi).

At last, assume that only disequations occur. Again using the injectivity of s, we can suppose that all of them concern the same term sq(w) in w. So our formula states that sq(w) is smaller that certain terms to, ... , th in 0 and if , and larger that some other terms th+i, ... , tk. We obtain an equivalent formula avoiding wand its quantifier in the following way. List (in a suitable disjunction) all the possible orderings of to, ... , tk in ::; according to which to, ... , th precede th+l , ... , tk; for every ordering, let t , t' denote respectively the greatest element among to, ... , it. and the least among th+l, ... , tk; add s(t) < t' (in order to provide sq(w) with suitable room). This concludes the elimination procedure. .. Corollary 2.2.2 dLO+ is model complete (in L') and complete (both in L'

and L).

Proof. Clearly dLO+ is model complete in L'. Moreover (N,::;, 0, s) 1= dLO+, and (N, ::;,0, s) is embeddable in every model of dLO+ . As dLO+ is

2.2. DISCRETE LINEAR ORD ERS

51

mod el com plet e, all t he corres ponding embeddings are elementary. Accordingly all t he mod els of dLO+ are elementarily equivalent t o (N, ::;, 0, s) and hence t o each oth er. T his shows t hat dLO+ is complete in L' . Now recall t hat both the minimal element a nd t he successor fun ction (so t he int erpretations of t he symbols in L' - L ) are 0-definable by L-for mulas in t he mod els of dLO+. T hen it is an easy exercise t o deduce t hat dLO+ is complete in L , t oo . .. Corollary 2.2.3 dLO+ is decidable (in L' and in L ).

Proof. Reduce any sente nce of L' into an equivalent qu an tifier free statement. This is a Boolean combinat ion of formulas sm (o) ~ sn(o) where m a nd n a re non-n egative integers, and dLO+ ca n eas ily check its membership . This procedure work s even for L-sentences. .. Corollary 2.2.4 Let A 1= dLO+. Th e subsets of A defi nable in A (in L or in L') are j ust the fin ite unions of (open or closed) in terva ls in A (possibly having +00 as a right endpoint) .

P roof. Let cp( v , w) be a £I-formula. As dLO+ has t he elimination of qu an tifiers in L' , we ca n ass ume t hant cp(v, w) is qu an tifier free; owing to Theorem 2.2.1 (and its proof) , for every ii in A, cp (A , ii) is a union of int ersections of inter vals , and hence a union of intervals. ..

T his accomplishes our a na lysis of discrete linear ord ers wit h a last bu t no leas t elements. How to deal wit h t he other t hree cases of infinite discret e linear ord ers list ed before? They can be handled in a similar way t o get qu an tifier eliminat ion and consequent ly completeness. In particular it turns out that th e four cases exha ust all the possible comp let ions of the th eor y of infinite discrete linear ord ers; in other word s, these com plet ions are fully cha racterized by saying if t he corres ponding models adm it or lack a least a nd a greatest elem ent. Act ually t he case withou t end points deserves some more comments . In fact , in t his fra mework, t he enla rged language L' needs no natural " constant " sym bol (just becau se end poin ts a re lacking), a nd takes t he only addit iona l operation sy mbol s . Accordingly, properly spea king, t he elimination of qu an tifiers fails in t his extended language, becau se we have no constant t o build atom ic sentences . For inst an ce t he (t rue) sente nce 3w(w = w) ca nnot be t ra nslated into an equivalent quantifier free sentence; the same happens for t he (false) sentence 3w(s(w) = w). So t he right statement here is as

52

CHAPTER 2. QUANTIFIER ELIMINATION

follows: an elimination set for the theory of discrete linear orders without end points is the set of atomic formulas plus a unique sentence (such as "there is no least element", or " t here is no last element"). We do not discuss the proof here. In fact, we shall treat this case in detail when considering dense linear orders in the next section. Finally, notice that decidability can be shown (in L) in the 4 possible cases. Consequently the (incomplete) theory dLO of infinite discrete linear orders is decidable, too; in fact , a sentence

2.3

Dense linear orders

Now we deal with dense linear orders. The plan here is exactly the same as in the discrete case. We use the language L = {~} and we distinguish 4 possi ble cases: 1. there is a least element , but no last element (just as among nonnegative rationals with respect to the usual order);

2. there is a last element, but no least element (now non-positive rationale provide an example); 3. there are both a least element and a last element (look at the rationale, or even at the reals, in the closed interval [0, 1]); 4. there are no end points (this is the case of (Q,

~;)).

In 1, 2, 3 one shows elimination of quantifiers in a language with one or two additional constants to be intepreted into the end points; 4 deserves a more specific treatment, because quantifier free formulas need an auxiliary single L-sentence to form an elimination set (even in L): we provide full details below. In all these cases it is easy to deduce completeness in L. This imp lies that these 4 classes exhaust all the possible completions of the theory of dense linear orders. As already said , here we limit our analysis to dense linear orders without end points. We just met their theory in Chapter 1; we called it DLO- and we observed that it is ~o-categorical , hence complete. 'Ve treat now quantifier elimination (in L), and in this way we provide an alternative and detailed proof of its completeness. Theorem 2.3.1 The quantifier free formulas of L together with a single sentence of DLO- (such as 3v(v = v)) are an elimination set of DLO-.

2.3. D ENSE LINEAR ORDERS

53

Proof. We follow th e sa me approach as in th e discrete case . Bu t now t he successor symbol do es not make sense , our lan guage is smaller and hence our setting is simpler: L-terms are ju st variables (no constant arises because there are no endpoints) . Accordingly what we have to do is to eliminate the qu an tifier in a formula 3w 0'(ii, w )

where 0'(il, w) is a conj unct ion of condit ions say ing t hat w is equal, or smaller, or lar ger t ha n some v in il. To obtain t his, proceed as follows. Again ignore trivialities like w = w (th ey ca n be preliminarily listed and eas ily recognized) ; if nothing else occurs, just repl ace our formula with 3v( v = v). On the cont rary, when meeting a condit ion that ca nnot be satisfied by an y w , like w < w (also th ese negative st ate ments ca n be prelimin arily listed) , replace our formul a with ,3v(v = v) . Otherwise, as soon as you meet one equation w = Vi, delete wand 3 , and replace w wit h Vi t hroughout ou r formula. At las t , if only disequ ation s occur and hence our formula states t hat w is smaller t ha n certain varia bles (V I , ..• , Vh wit h no loss of generality) and larger t hat oth ers (Vh+l , ... , Vk), t hen get t he required quantifier free formula in th e following way. List (in a s uitable disjunction ) all t he possible ord erings of VI, ..• , Vk in :::; according to which VI, • •• , Vh pr ecede Vh + l , ... , Vk ; for every ordering, let v, v' denote respectively t he maximal element among VI , . .. , Vh and the least among Vh + l , ... , Vk ; V < v' and th e densit y ass umpt ion are sufficient t o ensure t hat an interm ediate w exists. Wh en h = 0 or h = k , one uses t he lack of end poin ts. This conclu des t he elimination procedure. .. Corollary 2.3.2 DLO- is mod el complete and com plete . Proof. Mod el complete ness is a straightforwa rd consequence. Co mplete ness can be dedu ced as follows, using model complet eness. First notice that an y two den se linear ord ers with no end points (A , :::; ) and (B , :::;) embed in a common exte nsion (for inst ance, t heir sum (A + B , :::;) , where A + B is t he disjoint union of A and B , :::; enla rges t he ord erin gs in A and B and , in addition , satisfies a < b for every choice of a E A and b E B ). As DLO is mod el complete, eac h of t hese embe ddings is eleme ntary, in particular (A, :::;) and (B , :::;) a re elementarily equivalent t o t heir sum, a nd hen ce to .. each other.

As already recalled, completeness was also observ ed in the pr evious chapter via ~o-categoricity and the Vaught crite rion. By th e way, notice that th e

54

CHA PTER 2. Q UA NTIFIER ELIMINAT I ON

Vaught T heorem pro vides a com plete ness pro of even when end poin ts a rise. In fact, also t he remaining classes of dense linear orde rs (wit h least and/or las t element) have an ~o-c ategorical t heory. T he decidability of DLO - can be easi ly shown. Ind eed , by proceed ing as in t he discrete case , one sees t hat even t he t heory of arb it ra ry dense ord ers (with or wit hout end poin ts) is decidable. Now let us deal wit h definabili ty. Corollary 2.3.3 Let A F DLO -. Th e subsets of A definable in A are j ust the finite union s of (open or closed) int ervals, possibly with infin ite endpoints.

Proof. Proceed as for dL O+ .

2.4

..

Algebraically closed fields (and Tarski)

Tarski obtained his celebrated qu an tifier elimin ation proced ures for t he complex field and t he ord ered field of reals in t he t hirties. O wing to t he stop d ue to t he World War, he pu blished his results only in 1948. We cons ide r here t he complex case, and we delay t he real one to the next sect ion. We shou ld underlin e t hat Tarski dealt wit h theories of single st ruct ures (t he complex field , t he ord ered field of reals) rather t ha n on axiomatizable classes (AC F , RCF) . Bu t a careful analysis of t he proofs singles out which kind of algebraic condit ions a re necessary to ensure t he qu an tifier elimination result: so one realizes t hat what ma kes t he machin ery wor k is ju st being algeb raically closed in t he co mplex case, and t he int ermediate value pro perty for po lynomials in t he real case. This is a crucial result , specially towards t he aim of findin g a nice axio mat ization for t he t heory of t he complex field , or of the ordered field of reals. As prom ised, her e we consider t he com plex case, but we prefer a n a pproach dealin g wit h t he whole class of algebraically closed fields. Theorem 2.4.1 (Tarski) T he theory ACF of algebraically closed fields has the elimination of quantifiers in the language L = {+ , " - , 0, I} .

Proof. Take a formul a
2.4. A LGEBRA ICA LLY CLOSED FIELDS (AND TARSKI)

55

where a(V, w) is a finite conj unc t ion of at omic for mulas and negations, all containing ui. In our langu age, at omic form ulas are j ust equa lit ies of t erms, hence equa t ions. Using - , one can express each of t hem as

p(v, w) = 0 where p(if, x) is a polynomial with integer coefficients. Accordin gly
where the Pi'S a nd th e qj's are polynomials with integer coefficients, all havin g a positi ve degree, ti; and m j respectively, in x, hence with respect t o w. Basic field t heory te lls us that a sequence of eleme nts in a fi eld excludes 0 if and only if its product is not O. Accordingly, we ca n ass ume t hat at most on e inequation occurs in
-,(q(v, w) = 0). where q(if, x) is the pr oduct of t he polyn omials qj (if, x) when j ::; h; let m denote the degree in x of q(if, x ). At this point one might wonder whether we ca n reduce t he numb er of equations in our formula
Pi(if, x) =

L

r. Pi,r(f!) X

r :::;ni

Take two of t hese polynom ials, for instance Po a nd PI , and suppose for simplicity no ~ n l . We clai m t hat t here is qu an tifier free formul a in v yielding, whenever PI (v, x) is not t he null polynomial (in x), the coefficient s in x of the quotient and t he remainder of the division of Po (v, x ) by PI (v, x ). To get this formula , just follow the usual division procedure for polynomials.

56

CHA P TER 2. Q UA NTIFIER ELIMINA T ION

This is a te dious bu t st ra ight forward exe rcise. Fo r inst an ce, t he first ste p is t o write that eit her PI , n l (v) = 0 or th e coefficients of Po(v, x) a nd

PI

(v, x) satisfy

where P (v, x) is a polynomial of deg ree < no in x, and, in t he lat ter case, t he requi red quotient admits PO, n o ( V

~) (

( ~) ) - 1

PI,nl V

X

nO - n l

as a coefficient of maximal degree. At t his point , recall t he Euclidean algorit hm of repeated divisions , yielding the greatest com mon div isor (in x) of our polyn omi als Pi (ij, x) (i ::; k) as t he las t nonzero remainder in a finite sequence of successive divisions. Again, a suitable quantifier free form ula in v determ ines the coefficients in x of our greatest common divisor , whenever t he Pi(V, x) 's (i ::; k) are not all zero . In conclusion, we ca n ass ume that our formul a
:3w --, (q(v, w) = 0),

3. :3w(p(v, w)

= 0 1\

--,(q(v, w)

= 0))

whe re p and q are as before . F irst cons ide r 1. In a ny field, 1 is eq uivalent to say t hat, if v annihilates all t he coefficients of p(fl, x) in x of posit ive degree in x, t hen v ass igns t he value 0 also t o th e term of degree 0 in x ; this ca n be writ ten as a suitable quantifier free for mula in v. Now conside r 2. In any infinite field , 2 is equivalent t o say t hat v do es not an nihilate t he coefficients of t he polynomial p(fl, x) in x. Again, the lat ter statement ca n be expressed by a quantifier free form ula in ii. F ina lly let us deal with 3. We cla im that, in any algebraically closed field, 3 is equivalent to the statement

(*)

p(v, x) does not divide q(v, z}";

recall t hat n is the degree of p(fl, x ) wit h resp ect to x, and not ice t hat (*) ca n be expressed as a qu antifier free formula in v (just use t he previous remark s

2.4. ALGEBRAICALLY CLOSED FIELDS (AND TA RSK I)

57

a bout divisibility, and write that t he re mai nder of the division in (*) is not 0). The dir ection from left to right is true in every field K : if, for a given sequence bin K , the annihilator of p(b, x) is not included in t he annihilator of q(b, x ), then p(b, x ) cannot divide q(b, x) and (q(b, x) )n. Con versely, take K and b as before; assume t hat every root of p(b, x) annihilates q(b, x), t oo; for K algebraically closed , this imp lies t hat every linea r factor of p(b, x) divides q(b, x ) and hence th at p(b, x ) divides q(b, x)n. This accomplishes our proof. • Now let us comment t his quant ifier elimination result , and propose some noteworth y consequ ences. First of all , we want to em phasize t hat the quant ifier elimin ation property characterizes t he algebraically closed fields among infinite fields. In fact , it is a profou nd result of Macintyre, McKenna e Van den Dries that a n infinite field whose t heory eliminates the quantifiers in t he language L = {+, - , " 0, I } must be algebraically closed. An obvious consequence of quantifier elimin ation is the following. Corollary 2.4.2 AC F is model complete.

Clearly ACF is not complet e. In fact , for every prime p, t he sentence p = 0 is true in every algebraically closed field of characterist ic p and false in ever y alge braicall y closed field of cha racteristic -# p. However , as we alr eady showed in C ha pter 1, Corollary 2.4.3 For every p = 0 or prime, th e th eory AC Fp is com plete . P roof. In Chapter 1 we provided a proof founded on Vaught 's T heorem . An alternative approac h, using qu antifier elimination (ind eed model completeness) , is t he following. F ix p. There is a minimal alge braically closed field IC p of characterist ic p: t his is the algebraic closu re of t he prime su bfield. IC p is embeddable in every algebraically closed field of the same characteristic. Owing to the mod el com pleteness of AC F p , a ll t hese embedding are elementary. In particular, all t he algebraically closed fields of characteristic p are eleme ntarily equi valent t o IC p , and conse quentl y to each other. ..

As we already observed in sect ion 2.1 , the theories AC Fp exhaust all t he possible com pleti ons of AC F in L when p ran ges over t he primes and 0 (furthermo re, each of t hem has the qu antifier eliminat ion in L , just becau se it extends ACF). An a pplicat ion of the Compactness Theor em let s us say even mor e. In fac t , we have seen t hat t he theory of the complex field is just ACFo, and so is axiomatized by ACF an d , in addition , by t he infinit ely many sentences stating

58

CHAPTER 2. QUANTIFIER ELIMINATION

(1 be any sentence in ACFa. Compactess tells us t hat (1 is a con sequ ence of AC F and finit ely man y sentences concerning the characterist ic. Hence (1 is true in every algeb raically d osed field of prime characterist ic p for all but finit ely many p' s. So we have shown t he following res ult.

-,(p = 0) for every prime p . Let

Theorem 2.4.4 Let (1 be a se n tence of th e language L . (1 is true in some (equivalently every) algebraically closed field of characte ristic 0 if and only if (1 is tr ue in some (equivalently every) fiel d of characte ris tic p for all but finitely many primes p . Hence what is true in the complex field (and in any algeb raically closed field of characteristic 0) is satisfied by the algebraically dosed fields of characteristic p for almost all primes p. We 'll see later in this section a nice application of this model theor etic tra nsfer principle to Algebra. Now let us deal briefly with decision problem s. As already said , decidability follows in a very simple way from qu antifier elimination. Corollary 2.4.5 Th e theory AC F of algebraically closed fi elds is decidable.

Proof. It suffices to decide if a given quantifier free sentence (1 of L is in AC F or not . With no loss of generality, (1 is a conjunction of disjunction s of atomic se ntences and negat ions. As a conjunction is in a theory if a nd onl y if eac h co nj unct is, we can write (1 (up t o equivalence, using -) as

(V m ; = 0) V (V -,(nj = 0)). z

j

where the m;'s an d th e n j's are positive int egers. So our sent ence just says t hat the characteristic divid es f1 m ; or is coprime with some n j (or suitable variants when no equation , or inequation arises). This ca n be easily chec ked in the fixed framework . .. We shall add some more comments about t he decid ability of AC F in the last section of this chapter. Now let us deal with definability. We hav e seen in Ch ap ter 1 th at , in any fi eld /C , const ructible set s (in particular algebraic vari eties) are definabl e. Theorem 2.4.1 implies that , within algebraically d osed fields , the converse is also true. Corollary 2.4.6 In an algebrai cally closed field /C, for eve ry positiv e integer n , a subset of K " is definable if and only if is consiructible.

2.4. ALGEBRAICALLY CLOSED FIELDS (A ND TARSKI)

59

Proof. It suffices to show t hat, if X ~ K " is definable, then X is constructible. Let
wh er e n is the length of V. Using qu antifier elimination , we ca n repl ace
q(v, ill) = 0 wher e q(x, ff) is a polynomial with coefficients in the s ubring generated by 1. Hence X =
•

Not ice th at in every field K t he s ubsets of K " definable by q ua nt ifier free formulas are con structible. Qu antifier elimination ensures that , when K is algebraically closed, no further definable set ari ses. The following proposition underlines the geom etrical content of Tarski's Theorem.

Theorem 2.4.7 (Chevalley) Let K be an algebrai cally clos ed field, n be a positive integer, X ~ I
If
•

Now let us consider l-a ry definabl e sets in an algebraically closed field K . In t his restrict ed fram ework, t he following proposition hold s.

Corollary 2.4.8 Let K be an algebraically closed fie ld, X in K. Th en X is ei ther fin it e or cofini le.

~

I< be definable

Proof. Fo r every q( x , il) E I<[x], q( v , il) = 0 defines I< if q( x , il) is zero, a nd a finit e set otherwise. A finit e Boo lea n combinat ion of finite or cofinite sets is still finit e or cofinite. • Actually we can say even more. Ind eed , in any (po ssibly non- algebraically closed) field K , a subset X of I< definable by a quantifier fre e formula is either

60

CHAP T ER 2. Q UA N TIFIER ELIM INAT I ON

finit e or cofinit e. Qu an tifier elimination exten ds t his proper ty t o arbitrar y l -a ry subsets when J( is alge braica lly closed. To conclude t his section , we wa nt t o propose a nice a pplication of Model T heory t o Algebra within algeb raically closed fields. This is t he so called inj ecti vity-implies-surjecti vity Theorem , du e t o J. Ax [3]. Com pact ness , an d t he co nsequent rem ark t hat t he se ntences t rue in t he com plex field a re just t hose satisfied by t he algebraically closed fields of characterist ic p for almost all prim es p , ar e used to ded uce Theorem 2.4.9 A ny inj ective morphism f from an algebraic variety V over the complex field in to V itself is surjective.

Proof. We already noticed that a ny algebraic va riety is a definable set , and is even defined by a finite conj unction of eq uations (possibly with paramete rs) . In part icular let t he for mul a

1\ Pj (il, ii) =

0

j9

give V in t his way (t he pj's are polynomials with int eger coefficie nts, an d ii denotes a seq uence of com plex par am et ers). Analogo usly, a morph ism between varieties is a map defined by a finit e conj unction of equations . Acco rd ing ly let qi (V, e, ii) = 0

1\

i«:s

yield f (t he qi's are again polyn omi als with int eger coe fficients; we can freely use her e t he sa me param et ers ii as before; if necessary, we extend ii t o include new com plex numbers). At t his point it is a n eas y exercise to wri te a first ord er se ntence in t he la ngu age L (without param et ers) saying :

w,

Z) = 0 defines a morph ism from th e for all z, if A i<s qi (v, variety given by A j
2.5. TARSKI AGAIN: REAL CL OSED FIELDS

61

for instance of the algebra ic closure F p of t he field F p with p elements: a positive answer in F p for sufficientl y many p implies a positive answer for the complex field. So take an algebraic variety V over F p and an injective morphism f from V to V over F p. Use the algebraic fact t hat F p is locally finit e a nd represent V as t he union of it s intersections with F" where F ran ges over t he finit e subfields of F p (containing t he param eters definin g V a nd f ). Recall t he t rivial prin ciple t hat any injecti ve fun ctio n from a finit e set t o itself is also surjective. Ded uce t hat t he restrict ion of f t o V n F" is s urjective for every F . Ex t end t his resul t to f: f is surjective, as requi red .

.

2.5

Tarski again: Real closed fields

In t his section we deal wit h t he qu antifier elimination t heorem for real closed fields . This is t he ma in res ult of Tarski in t his fra mework , not only becau se, as we sha ll see , t he proof is deeper an d mo re cornplicat ed t han in t he cornplex case , bu t also becau se t he ordered field of reals is intrinsecally related to geom etry. It is certainly needless to recall t hat, for instan ce, in t he Euclidean plan e equipped with some fixed Cartesian axes, every poin t is essent ially an ordered pair of reals, every straight line is the variety given by a polyn omial with degree 1 and 2 unknowns over th e reals, and so on. Accordingly, statements a bout poin ts, lines, ... ca n be eas ily t ra nslate d into st atements a bout reals, addit ion, multipli cati on (oft en in a first order way). In pa rt icular, a decision algorithm a bout th e t heory of t he ordered field of reals (t he elementary algebra accor din g to Tarski's te rminology) should work for (first ord er ) Euclidean geometry as well. Act ually Tarski's qu an tifier elimin ation procedure dealt with t he reals ra th er than with t he t heory RC F. Bu t , ju st as in th e complex case, one can realize that t he basic ingredients of the proof concern a r bit rary real closed fields. So we state (and show) t he resul t in t his (seemingly enlarged) setting; but we shall dedu ce quickl y t hat RCF is complete a nd hence equ als t he th eory of (R, +, - , ,,0, 1, ~) . We follow t he elega nt approac h of Cohen [27] rather t han Tarski 's or igin al proof. Theorem 2.5.1 Th e theor y R C F of reals closed fiel ds has the elim ination of quant ifiers in th e language L = {+, - , " 0, 1, ~ } .

Proof. By proceeding as in the case of algebraically closed fields, one pr elimin arily reali zes that the heart of th e matter is to eliminate th e quantifier

62

CHAPTER 2. QUANTIFIER ELIMINATION

:3 in a formul a

:3Wct( w , v) where ct(w, v) is t he conjuncti on of at most one equation p(w, v) = 0 a nd a finite (possibly empty) set of disequations qj( w, v) > 0 (with j :::; m), where p( x, Y) a nd qj (x, if) (j :::; m) are polynomials with integer coefficient s. So let us op en a (long) parenthesis and examine an arbitrary polynomial f( x) = L i 0 a nd s uppose our claim true for every natural value < t , in order t o extend it to t. The idea here is t o relate the zeroes of f( x) t o th e roots of its derivative and th e sign of f( x) in these roots. Hence build t he form al derivative f'( x) of f(x) with respect to x

f ' (X ) =

,"",, LJ

a
Z'fi X i - I •

Preliminaril y, notice t hat f'( x) = 0 if and only if (fl " ' " fd = (0, . . . , 0). Except this case , ind uction equips us with qu antifier free formulas defining (with resp ect to (fa , ... , ft) via (fI , 212, ... , tft)) 1) t he funct ion cou nting, for every sequence (fa, ... , ft) with (fI , ... , fd (0, .. . , 0), how man y roots f'( x) ad mits,

i=

2.5. TARSKI AGAIN: REAL CLOSED FIELDS

63

and , for 1 ::::; r ::::; s < t , 2) the set of the sequences (fo, .. . , ft ) in K t+l such t hat (fI, . . . , ft) is not zero and f' (x) has exact ly s roots, 3 ) the function mapping an y (suitable) non-zero (fo , . . . , fd int o the r-th root of f' (x ) . Now ord er th e roo ts of f' (x) PI

< .. . < Ps·

The intermediate valu e property, holding in every real closed field, ensures that f' (x) cannot change its sign between two successive roo ts. Can we dedu ce th at J( x) is monotone (increasing or decreasing according to the sign of f'( x» in t he same interval ? Cert ainly yes in th e case of th e real field: thi s is a well known resul t in elementary real an alysis. But a complete algebraic (although non trivial) proof can be done for polynomials by using only t he axioms of RC F. Consequent ly, in every real closed field K , f( x) is mono ton e (increasing or decr easing according t o th e sign -positi ve or negative- of its der ivative) in each int erval (pi, Pi+I), 1 ::::; i < s. Now look at f(P i) and

f (Pi+ d· (i) If t hey are not 0 and t heir sign is t he same, th en (pi, pi+d do es not contain any root of f( x) becau se f( x) is monotone in th e int erval (notice th at t he cases when exact ly one between Pi and Pi+l annihilates f (x ) ca n be handled in a similar way ). (ii) If f(P i) and J(Pi+d admit opposite signs, then (pi, pi+d do es con tain a root of f( x) by the interm ediate value property. The uniqueness of this root might follow from Rolle 's T heor em (two distinct roots of f( x) Pi < a < b < Pi+! determine a new intermediate root of f' (x), and t his is impossible) . Elementary analysis ensures t hat Rolle's T heor em is certainly true for t he reals; but , again, one can give an alternative algebraic and non t rivial proof (for polynomials) holdin g in every real closed field. (iii) Assume at last f (Pi) = f(P i+d = O. The argument in 2 again exclud es any additional intermediate roo t of f( x) . This machinery lets us count th e roots in t he interval [PI , Ps]' But what ca n we say in (-00, pI) and (Ps, + oo)? T he same a rguments as before ensure

CHAPTER 2. QUANTIFIER ELIMINATION

64

that f(x) is monotone, and at least one root occurs in each of these halflines. But our setting changes when we examine the existence of this root. For, every interval (pi, Pi+!) (1 ~ i ~ 5) is bounded, while our half-lines are not. However the following algebraic fact helps us. Let f(x) = I:i 0)

1\

j~m

or

(b)

:3w

1\

qj(w, 17)

>0

j~m

where p(x, jJ) and qj(x, jJ) (j ~ m) are polynomials with integer coefficients. Each of them can be written as a polynomial with coefficients in Z[yJ in the following way p(x, jJ) = LPi(y)X i ,

i
(a) is quickly reduced to (b) because its formula is equivalent to

65

2.5. TA RSKI AGA IN: REAL CLOSED FIELDS

(/\ Pi(V) i5,t

= 0 1\ :3w

1\

j5,m

Vl
qj( w , v)

> 0) V

has s roots" 1\

" t he r -t h root Pr(v) satisfies !\ j5,m qj(Pr(v) , v) > 0") ,

where t he latter disjunct ca n be expressed by a quantifier free first ord er formula. So look at (b) . For every j ::; m a nd for every Sj ::; tj , t here a re qu antifier free formulas defining , for every real closed field 1< , the set of th e sequences bsuch that q(x , b) has Sj roots in x , and listing t hese roots

One can cornpu te the sign of qj (x, b) in t he intervals

(-00 , Pj, 1 (b)) , (pj,i(b), pj,i+db))

(1::; i < Sj),

(Pj, Sj (b) , +(0) in a uniform way (ind ependent of 1< and b) by looking at t he (sign) valu e of

qj(pj ,l(b) -1 , b) .(Pj,i(b) + (pj,i+l(b) b)

qJ

2

'

qj(p j,sj(b) + 1, b) resp ectively. List all the possible ord erin gs of t he roots (in x) of t he qj (x, b) 's when j ranges over the natural numbers ::; m, and divide in every case 1< int o finit ely many inter vals such that the qj (x , b) 's have a constant sign (with resp ect t o x) in eac h of t hem; check these signs (in the way suggested before) a nd for m a suitable disjunct ion picking the intervals where all t hese signs a re positive. This procedu re is ind epend ent of 1< a nd b and provides the required quantifier free formul a . • Corollary 2.5.2 RCF is model complete. Corollary 2.5.3 RC F is complete; in particular, RC F is the theory of the ordered field of reals (as well as of ever y real closed field).

66

CHAPTER 2. Q UANTIFIER ELIM INAT I ON

Proof. There is a min imal ordered real closed field , embedded in any mod el of RCF. This is the ordered field R a of real algebraic numbers. The model completeness of RCF ensures that every real closed field is an elementary extension of R a. In particular all the real closed fields are element a rily eq uivalent to R a and , con sequ ently, t o eac h other. '"

This is t he first completenes s proof we give a bout RCF; in fact Vau gh t 's criterion does not apply because RC F is not cate gorical in any infini t e power . We have see n th at real closed fields eliminate quantifiers in t heir lan gu age L = {+ , -, ' ,0, 1, ~}. Notably, they are fully charact erized by t his property : for, Macintyre, McKenna and Van den Dries showed t hat an ordered field, whose theory has the quantifier eliminat ion in L , must be real closed . We also noti ce that RCF doe s not pr eserve quantifier elimination in t he restricted lan guage L' = {+ , -, " 0, I} wit hout ord er . Actually on e can rem emb er t hat , even in checking solva bility of the popular equation ax 2 + bx + c = 0 wit h degree 2 and 1 unknown ove r t he reals (or over any real closed field), on e needs a disequ ation b2 - 4ac ~ 0 to ens ure roots, and henc e to eliminate 3 in the formul a 3w (V2 W2 +VIw +va = 0) . More form ally, recall that , with resp ect to the theory of the real field , th e formulas

',o( v):

v~ O,

',o' (v ) : 3w(v

= w 2)

a re equivalent. As RC F is com plete a nd hen ce eq ua ls the t heory of t he real field , t he same hold s in every real closed field . Consequently t he £I-formula (wit h t he quantifier 3) defines th e set of non-n egative eleme nt s in ever y real closed field. However ',o(v) cannot be equivalent in RCF to any quantifier free £I-formula ',o" (v ). In fact ',O (R ) is the half-line [0, + 00) of R , and so is both infinite a nd coinfini te , whil e ',O" (K ) is either finite or cofinite for ever y field K : see the proof of Corolla ry 2.4. 8. Now we discuss decid abili ty: as already said , t his was t he main conseq uence of elimina tion of qu an tifiers, according to t he ge ne ral feeling in t he fourties . Corolla r y 2.5 .4 RCF is decidabl e. Proof.

Ow ing to q uantifier elim ination, every L-sentence

(J

is equivalent in

RCF to a Boo lean combination of se nte nces m = n or m < n where m and

2.5. TARSKI AGAIN: REAL CLOSED FIELDS

67

n are integers. This quantifier free statement can be easily checked in our

framework.

..

We shall comment this result later in 2.9. Now we examine another remarkable consequence of quantifier elimination, namely definability. Recall that, in an ordered field K, every semialgebraic set (in other words, every finite Boolean combination of sets of solutions of disequations

q(x) 2:: 0 with q(x) E K[X]) is definable. Corollary 2.5.5 In a real closed ordered field K, the definable sets are exactly the semialgebraic ones.

Proof. Let n be a positive integer, X S;; K" be a set definable in K . So there are a formula
Owing to Tarski's quantifier elimination theorem, we can assume that
q(c, w) 2:: 0 with q(x, if) E Z[x, Y'J. Consequently X is a finite Boolean combination of sets of solutions of disequations q(il, if) 2:: 0,

and so is a semialgebraic set.

..

Here is a geometric restatement of Tarski's Theorem. Theorem 2.5.6 (Tarski-Seidenberg) Let K be a real closed ordered field, n be a positive integer, X S;; J(n+l, X' be the projection of X onto the first n coordinates. If X is semialgebraic, then X' is semialgebraic, too.

This formulation is due to Thorn, who also coined the name semialgebraic set. It has a more geometric flavour. Some mathematicians might appreciate this alternative terminology, for instance because it allows to state several results in a (seemingly) more agreable way (avoiding logic). Nevertheless, Tarski's original approach (via quantifier elimination and formulas) often provides quicker proofs, even in this geometric framework. Let us quote the following example from [168].

68

CHAPTER 2. Q UANTIFIER ELIMINATION

Exa m p le 2 .5.7 Consider the following statement: For a semialgebraic function f from R n+! to R , the set A of the sequences x in R n+! s uch that limy--too f(x , y) is in R is sem ialge braic.

If on e replaces everyw he re sem ialge braic by d efinable, this proposition may lose part of it s (m athem atical) gla mour. Bu t , using logic, on e obtains a short proof: A is definable via t he formula


:3w'liE(E > 0 --+ :3rVy(y > r --+

--+ :3z(J(v, y) = z A [z -

wl < E)))

(recall that both f and the ab solu t e valu e a re definabl e) . A direct approach via se mialge braic sets and projections is longer. In a real closed field K , the definable subsets X ~ J( have a very sim ple form. Corollary 2.5.8 Let K be a real closed fi eld, X be a definable subse t of K. T hen X is a finit e un ion of inter vals (clo sed or open , possibly wit h infinite en dpoints) .

Proof. Let q(x) E J([x]. We know th at q(v) = 0 defines J( if q(x ) = 0 a nd a finit e set (that is, th e set of th e roots ao < .. . < as of q(x ) in K) oth erwi se. On the other hand , q(v) > 0 defines 0 if q(x) = 0; otherwise q(v) > 0 defines t he union of some int er vals among ] - 00, ao[, ]ao, al[, ..., ]as, -l-oo] (recall that K sat isfies t he int ermediat e valu e proper ty for polynomials) . Hen ce an y definable (equivalent ly, semia lge braic) set X ~ J( is a finite Boolea n combina tion of intervals, and so a finite union of intervals. •

2.6

pp-elimination of quantifiers and modules

In this section we deal wit h (left) modules over a (countable) ring R with iden ti ty. In Chapter 1, we introduced a suitable lan gu age LR = {O, +,- , r (r E R)} for t hese structur es, and we saw how t o axiom atize their class by first ord er sente nces in LR . Let nT denote the corresponding t heory. A qui ck look at the axioms of nT shows t hat each of them is a univ ersal sentence Vva( v) wh ere a(v) is an at omic formula of LR; this confirms that the class of t he models of n T (namely of t he R -mod ules) is closed under

2.6 . PP-ELIMINATION OF QUANTIFIERS AND MODULES

69

s ubstructures. Now we wond er whether nT has qu an t ifier elimination in LR. A t rivial example shows th a t t his is false even in t he simple case when R is t he ring Z of inte ger s. In fact , just consid er Z as a module over itself. In Z the formula


rv = 0 for some non-negative int eger r . This form ula define s in Z {O} if r =1= 0 and Z otherwise. No Boolean combinat ion of t hese sets can equal 2Z. Therefore no qu antifi er free formula

2. Let

70

CHAPTER 2. QUANTIFIER ELIMINATION

3. Let SO( v) , 'Ij; (if) be pp-form ulas of L R with n free variables , and let k be a positive integer. It is simple t o writ e a sentence in LR ensuring th at , in a given R-module M , t he ind ex of SO (Mn )n 'lj;(Mn) in SO (M n) is ~ k ; in det ail t his sente nce says

We will denote it by (SO : 'Ij;) ~ k. Any sentence of this form is called a n invariant statement (we will see later the reason why) . Notice th at the finite Boolean combinations of invariant st at ements include t he sent ences sayin g: • t he ind ex of SO (Mn ) n 'Ij; (Mn ) in SO (Mn ) is k (" = k" means " ~ k" but " 1. k + 1" ) ; we will denote this formula by (SO : 'Ij;) = k;

• t he ind ex of SO(M n) n 'Ij; (M n) in SO (M n) is :::; k (" :::; k" means " 1. k + 1" ); we sha ll den ot e this formul a by (SO: 'Ij;) :::; k .

At this poin t we can state and show the following fund amental theorem (of pp- elimin ation of quantifiers for modules). Theorem 2 .6.2 (Baur - Mo nk) Let R be a (countable) ring with iden ti ty. Th en the pp-formulas of LR togeth er with the in var iant state ments fo rm an elimination se t for RT in L R. M ore precisely: for every formula a( v) of L R, there are a Boolean com bination j3 of invariant state me nts and a Boolean com binati on ,,(if) of pp-formulas su ch that

'v'v(a(v)

+-7

(3/\ ,,(if) ) E RT.

We shall use in our proof the following result of group th eor y. Lemma 2.6.3 (B . H. Neuma nn) Let g be a group, a, ai E G, H , H i be subgroup s of g (where i ranqes among the naiurals less than some fixed N ), aH ~ U
2.6. PP-ELIMINATION OF QUANTIFIERS A ND MOD ULES

71

ensures that there exist an invariant st ate ment (3' and a Bool ean combination , '(w, V) of pp-formulas such that

VwV v( 0:' (w, V) +-+ (3' 1\ " (w, V)) E RT . pt redu ction: without loss of gen erality, 0:' (W , V) is a disjunction of ppformu las or negations. In fact , V v(o:(v) +-+ (3' 1\ Vw, '(w ,V)) EnT. Accordingly we ca n replace o:' (w, V) by , ' (w, V) , which is a Boolean combination of pp-formulas , and hence is equivalent to a conj unction of disjunctions of pp-formulas or negations. Correspondingly put o:'(w , V) : I\ j<s o:j (w, V) where, for every j ::; s, o:j (w, v) is a disjunction of pp-formu las or negations , Vwo:'(w , v) is equivalent in RT to I\j<sVwo:j( w, v). Then we can handle o:j(w, V) (with j ::; s) instead of o:'(w,V) . 2 n d redu ction: o:'(w ,V) is of th e form B(w ,v) -+ V i
(1)

B(M ,il) ~

U Bi(M ,il).

i
if and onl y if A1 F (3 1\,(il). We kno w t hat, given A1 a nd ii , eit her B(M , il) = or B(M , il) is a coset of th e pp-definable subgroup B(A1 , 0). The same ca n be said a bout Bi(M , il) for every i < N . By t he way, notic e t hat 3wB (w, V) , 3WOi (w, V) (with i < N ) are pp-formulas . (1) is certainly true when ii sat isfies -,3wB(w, v) (the negation of a pp-form ula) in M , and certainly false when il satisfies

o

3wBi(W, v) 1\ III

V -,3wBi (w, v)

i
M . So there is no loss of genera lity for our purposes in assuming

B(M , il) =I- 0 and Bi(M , il) =I- 0 for every i < N . Let 5 be t he set of t he indi ces i

< N satisfying

IB(A1 , 0) : B(M , 0) n Bi(M , 0)1::; N ! . Notice that 5 depends on M (and on ii, of course). However there a re only finitely many poss ible ways of choosing 5, and each of them is described by

72

CHAP TER 2. QUA NTIFIER ELIM INAT I ON

a suit a ble invariant state ment . Let us assume, with no loss of generalit y, that 8 is ju st the set of t he positive integers ~ m for some m ~ N . We can apply B. H. Neuman n's Lemma and deduce t hat (1) is equivalent to

(2)

U Bi(M , ii) .

B(M , ii) ~

i<m

Put K = B(M , 0) n n i< m Bi(M , 0) . As B(M , ii) and Bi(M , ii) for i < m a re union of cosets of K in 1'1,1 , (2) ca n be equi valently writ t en

(3)

B(M ,ii) /K ~

U Bi(A1,ii) /K .

i <m

As B(M ,ii)/K is finite, we ca n use some (hopefully) well known combinatorial arguments and restate (3) in the equi valent form

(4)

n

2)-I)IXII(B(M ,ii) n Oi(M ,ii)) /KI = x i EX

°

where X ran ges over t he subsets of {a, 1, ... , m-I}. For ever y X , put

k(X) = IB(M, 0) n

n

Oi(M , 0) : KI ;

i EX

notice that , when B(M , ii) n n

iE X

Bi (A1, ii) =f- 0,

k(X) = I(B(M ,ii) n

n

Oi(M ,ii)) /KI ·

i EX

More over k(X) ~ N !N . Hence we have shown that M sat isfies a(ii) if and on ly if I:(-I) IX1k (X ) = 0, where the sum concerns all the s ubsets X of { a, 1, ... , m - I} such that M F :3w(B(w, ii) A A iE X Oi(W, ii)), a nd hence if a nd only if M satisfies a convenient disjunction of conj unctions of invari an t statements an d pp-formulas. This is what hap pens for a given 8 . As t here a re only finit ely many possible S 'e, one ca n find some suit a ble f3 and -y (V), valid for every R- mod ule M. • Remark 2.6.4 1. Not ice that the pro ced ure given in the proof of T heorem 2.6.3 is effect ive, and provides exp licit ly for every a( v) the required formulas f3 and -y(0 . Furtherm ore f3 is act ually a finit e Boolean combination of invariant statements conce rni ng pp-for mulas cp( v), 'IjJ (v) (with at most one free var ia ble) .

2.6. PP-ELIMINATION OF QUANTIFIERS AND MODULES

73

2. In particular, when a is a sentence of LR, what the previous procedure produces is just a Boolean combination /3 of invariant statements (concerning pp-formulas
There exist an Ln-formula a( ii, w) and a sequence a In M such Proof. that X = a()\1n , a). We can assume that a(v, w) is a Boolean combination of pp-formulas, and we know that, for every pp-formula
Proof. It is clear that, if M and M' are elementarily equivalent, then, for every
F (
~ k

{::}

M'

F (
The inverse implication follows from Remark 2.6.4,2.

~ k.

'"

The previous result explains why "invariant statements" are called in this way: actually these sentences fully characterize any R-module )\1 up to elementary equivalence.

74

CHA P TER 2. Q UA N TIFIER ELIMINAT ION

Now let us discuss th e content of t he previou s resul ts in some particular case . We deal with a principal ideal dom ain n (this setting includes the ring Z of integers, as well as any field). First let us examin e a gen eric pp-formula of

LR

O' (v) : :3w (Av= Bw).

n

A and B ca n ob t ain a simpler form when is a principal ideal dom ain. For, it is a fact of Algebra t hat , in t his fram ework , t he re are two inver ti ble matrices X, Y wit h coeffi cients in R such t hat t he prod uct B ' = X B Y is diagon al. So 0'(v) is eq uivalent to

a nd, unl ess replacin g 1V by y-1w, A by XA a nd B by B ' , one ca n suppose B diagonal in O' (v). Co nseq uent ly O'(v) becom es of the form :3w l ... :3wm

(

n

1\

(L

aijVj = biiWd )·

j= l

l ~i~ m

Now let us mom ent arily restrict ou r a na lysis to a smaller setting . Case 1: n = K is a field (so we are dealin g wit h vectorspaces over K ). Assume that , for some i with 1 ~ i ~ m , bii i- o. Then we ca n divide t he i-th eq uation in 0'(v) by bii and consequ ently ass ume bii = 1. At this point it is easy t o show t hat O' (v) is eq uivalent to

1\ l ~ i ~ m , b i i= O

n

(LaijVj = 0), j= l

which is a conj unction of atom ic form ulas . Com bine t his obser vation and Baur-Monk 's T heo rem, and dedu ce t hat every LK-formula is equivalent in KT t o a conj unction of a quantifier free formul a and a Boolean combination of invariant state me nts. Moreover t he pp-formulas with a uniqu e free vari abl e v red uce to r v = 0 for some rE I<, a nd hence to eit her v = 0 when r i- 0 or to v = v when r = o. Co nsequent ly t he only pp-definable subgroups of a vecto rspace V a re {O} and V . Owing to Corollary 2.6.5, t he subsets of V defina ble in V are just t he finite Boolean combinations of t he cosets of t hese subgro ups, an d so redu ce t o t he finit e or cofinite subsets. Now let us exa mine invariant statements . In par t icular we dir ect our attention on t he sentences of t he form

2.6. PP- ELIMINA TION OF Q UA NTIFI ERS AND M ODULES

75

wher e k is a positi ve integer. In any given vectorspace V over IC , t hey witn ess if the size IVI di V is finit e or not , a nd, in t he positive case, its value. We claim that they ca n even determine th e =-type of t he vectorspace. Let us see why. First ass ume J( infinite. We know t ha t , in t his case, all t he nonz ero vect orspaces over IC are eleme ntarily equivalent (for, t heir t heory is complet e). In oth er words, when J( is infinite , t here are only tw o =-classes of ICvectorspaces: t he former contains all t he nonzero vectorspaces, and t he latter red uces to t he zero space . Bu t a vect orsp ace V is {O} if and only if V 1= (v = v : v = 0) = 1. In particular , t he statements (*) determine t he = -typ e of any vect orspace . Now assu me I< finit e (say of size q). Now we meet infinitely many =-classes of IC-vecto rsp aces. In fact, there is a class for every natural n , consist ing of the (pa irwise isom orphic) vect orspaces of dimension n over IC (hence size qn), while infinit e vectors paces form again a uniqu e class. T he sente nces (v = v : v = 0) ~ k can obviously dist inguish t hese classes. On e can ded uce t hat every invaria nt statement in LK is a Boolean combi nation of sentences (v = v : v = 0) ~ k where k ranges over t he positi ve integers. In conclusion, given a fi eld IC , t he atomic form ulas of LK t ogether wit h t he inva rian t state ments (v = v : v = 0) ~ k, with k a positiv e int eger , form an eliminat ion set for K;T . In particular t his yields qu antifier elimination in LK for th e t heory of infinit e IC-vectorsp aces. Case 2: Now let us enlarge our setting t o a rbitrary prin cipal ideal dom ai ns R: F irst let us examine a pp-for mul a O'(v) . We cannot ex pect any longer t hat, wheneve r b« i= 0 in t he i-t h equation of 0'(v), one can divide t he whole equation by bii a nd so o bt ain bii = 1. However O'(v) is equ ivalent in n T to a conj unction of form ulas (1)

·I:/l=i aijVj = 0 (for bii = 0) ,

(2) :JW(:L5=i aijVj

= qw ) (where q = bii i= 0).

T he lat ter ones are divisibility condit ions : we ca n a bbreviate each of t hem by

Of course, when q is a unit in R: t his is a trivial condit ion and ca n be forgo t ten . In the remaining cases, recall that q decomposes (uniquely) as a product of powers of pairwise distinct primes in n , and q divid es an element

76

CHAPTER 2. QUANTIFIER ELIMINATION

r E R if and only if all th ese pri me powers divid e r . So there is no loss of generality in ass uming that, in (2) , q is a prime power . T he situation becomes clearer if we restrict our analysis to formu las having only one free variables. In fact, in this case, our pp-formula a(v) gets equivalent in nT to a conjunction of form ulas

(1) ' rv = 0 (a torsion condit ion), (2) ' pi I sv (a divisibility condition) ; here p, r, s E R , p is a prime and I is a non-negative integer. Again, simple algebraic fact s about principal ideal domains let us ass ume t hat s itself is a power of p , s = ph for some non-n egative integer h < l. Every ppform ula in at most one free variable is a conj unct ion of t orsion and divisibility conditions as before. This result helps also the analysis of invari ant statements (

(3) (pv

= 0 A pn- 1lv : pv = 0 A pnlv) 2':

(4 ) (pv

= 0 A pn Iv : v = 0) 2':

k,

k,

(5) (pn-1 1v : pn Iv) 2': k, (6) (v

= v : rv = 0) 2':

k.

The reader may check their truth (at least when R is the ring of integers) in some familiar abelian groups, like Z j qhZ , t he Priifer groups Z j qOOZ , th e localizations of Z at q (when q ranges over the primes, and h over the natural numbers), and the addit ive group of rationals, and realize in this way their meaning. Ind eed Wanda Szmielew (a student of Tarski's) showed t hat , for every R module M, the = -type of A1 is fully determined by the invariant statements (3)- (6) satisfied by M . In concl usion , owing to Baur-Monk's theorem , the form ulas (1)-(6) are an elimination set for the t heory n T when R is a principal ideal domain.

2.7

Strongly minimal theories

We saw in 2.4 that the only (l-ary) definable sets in an algebraically closed field a re the finite and cofinite ones. 2.6 told us that the same happens, for

2.7. STRONGLY MINIMAL THEORIES

77

instance, in every (infinite) vectorspace over a fixed countable field. One can also check that even pure sets (in a language with no symbols besides equality =) enjoy this feature; again , every definable set is either finite or cofinite (this was implicitly shown when we treated definable sets in 1.7, in particular when we provided an example of an infinite coin finite non definable set). So we find some non-trivial algebraic structures A whose definable l-ary subsets reduce to the ones definable in the pure set A (with the equality relation =) by quantifier free formulas. Let us name these structures in the following way.

Definition 2.7.1 An infinite structure A is said to be minimal if and only if the only definable subsets of A are those finite or cofinite. A complete theory T is said to be strongly minimal if and only if every model A of T is minimal. Hence any algebraically closed field is a minimal structure, and any theory ACFp (with p = 0 or prime) is strongly minimal. The same is true for infinite vectorspaces, or pure sets. It should be underlined that the minimality of a structure is not preserved by elementary equivalence. In other words there are minimal structures A such that the theory of A is not strongly min imal , and so admits some non-minimal models. Here is an example.

Example 2.7.2 Consider the theory dLO+ of discrete orders with a least element 0 but no last element. We know that dLO+ is complete, and has quantifier elimination in a language L with a constant (for 0) and a l-ary operation symbol (for the successor function s) in addition to the relation symbol ~ . Consequently every definable subset of a model of T is a finite Boolean combination of intervals (possibly with an infinite right end point). Therefore (N,~, 0, s) is a minimal model of T, because every interval in (N ,~, 0, s) is either finite or cofinite. However no other model A = (A , ~, OA, sA) of T is minimal. In fact, let a E A satisfy a =J. (sA) n (OA) for any natural n . Then both [OA , a[ and [a, -l-oc] are infinite intervals in A. We shall examine again strongly minimal theories in the next chapters. In particular, with respect to algebraically closed fields, we will prove a theorem of Macintyre showing (among other things) that the only integral domains with indentity having a strongly minimal complete theory are just the algebraically closed fields .

78

2.8

CHAPTER 2. QUANTIFIER ELIMINATION

a-minimal theories

Turning now to linearly ordered st ruct ur es A = (A, ::; , . . .), we met in th e previous sect ions som e exam ples where t he definable subsets of A redu ce to the finit e unions of int ervals (possibly with infinite end points) in (A, ::; ). This is wh at happens in real closed fields (as observed in 2.5), but also in dense or discrete (infinit e) linear ord ers (see t he sections 2.2 and 2.3) . This suggests t he following definition . Definition 2.8.1 An infi nite linearly ordered struct ure A = (A , ::;, . ..) is called o-minimal if and only if every subs et of A definable in A is a finit e union of intervals (closed or open , possi bly with infinite endpoints) . A complete theory T of infinite linearly ordered structures is called o-minimal if and only if every model of T is o-minimal.

"0" abridges " order", of course. This o-rninim al setting clearl y reminds minimalit y. In fact t he minimal st ructures (and t he strongly minim al theories) are t he ones where every definab le (l -a ry) set is already defined by a qu ant ifier free formula involving th e only (language) sy mbol =. Similarl y t he o-m inim al struct ures (and t heori es) are ju st those admit t ing a total ord er relation j; such t hat every definable (l-ary) is already defined by a quantifier free formu la involving the only (lan guage) symbol ::;. In t his sense the o-minimal structures and th eor ies are th e simplest ones in th e presence of a t ot al ord er relation . Nevert heless th ey includ e several non- trivial algebraic exa mples. We will st udy in mor e detail t hese st ruct ures and t heories in the las t cha pter of t his book. Bu t it is worth emphasizing since now that , in s pite of t he simila rit ies underlin ed a bove between minimality and o-m inim ality, a relevant difference ari ses. In fact , we not iced t ha t th e th eory of a minimal structure may adm it some non-minimal models, and so fail to be st rongly minimal. This do es not ha ppen in the o-minimal setting. In fact the following t heorem hold . Theore m 2.8.2 (Knight - Pillay - St einhorn) 1fT is the theory of a lin early ordered o-minimal structure, then every model of T is o-minimal.

Accordingly, we ca n spa re the adverb " st rongly" in defining a t heory with o-rninim al mod els. Com ing back to real closed fields, we would like t o mention her e a result quite sim ilar to t he one reca lled at t he end of t he previous sect ion on algebraically closed fields. In fact , it was shown by Pillay and Steinhorn that t he only

2.9. COMPUTATIONAL ASPECTS OF Q. E.

79

ordered rings (with identity) having an o-minimal theory are the real closed fields. The proofs of these theorems will be provided in Chapter 9.

2.9

Computational aspects of q. e.

In this section we shortly discuss the quantifier elimination procedures with respect to effectiveness and fastness. Actually these criteria did not correspond to the spirit of the forties (and some decades later), when the main quantifier elimination results were proved. For , those times lived the influence of Godel incompleteness and undecidability phenomena; so, according to that feeling, any decision algorithm (such as Tarski's method for real elementary algebra), or even a decidability theorem simply ensuring the existence of such a procedure without explicitly exhibiting it, were exactly the best answer one might expect. But later, in the seventies, the birth of modern computers and the beginning of their science changed this setting and inspired a prevalent interest in quickly running algorithms. Hence complexity theory introduced

* *

the class P of the problems having a fast procedure to find solutions, the class N P of the problems having a fast procedure to verify solutions (namely to check that a solution works).

We agree that a problem has a solving procedure when there is a Turing machine handling it and that an algorithm is fast when it runs in a polynomial time with respect to the length of the input . To realize the difference between finding or verifying solutions, look at the problem of factoring integers. To decompose a natural number 2:: 2 into its prime factors -more precisely, to find these factors- can be significantly slower (at least with respect to the currently available algorithms) than to check this decomposition when done. Just to quote a famous historical example, F. Cole announced during an AMS meeting in 1903 that the Mersenne number 26 7 - 1 is not prime. Factoring 26 7 - 1 is not easy (and certainly it was not in 1903, when computers were not available). But Cole 's proof is quite short to write and needs only one line 26 7

-

1 = 193797721

and can be checked very quickly.

X

761838257287

80

CHAPTER 2. QUANTIFIER ELIMINATION

Coming back to P and N P, it is trivial to observe that P ~ N P, because a procedure yielding solutions implicitly confirms these solutions. A fundamental problem in complexity theory (and, more generally, in the area linking computer science and mathematics) asks whether P = N P, hence whether, whenever a problem has a fast procedure verifying solutions, then it admits a (possibly slower but still) fast (=polynomial) procedure finding solutions. According to the new spirit, what is primary even in quantifier elimination, mainly towards decidability, is to get fast methods. Let us discuss this feature for the elimination of quantifiers of real closed fields: this is perhaps the most interesting case, owing to its connection with elementary geometry. However a devastating result of Fischer and Rabin shows Theorem 2.9.1 Any algorithm deciding, or even verifying membership to RCF for sentences in the language of ordered fields requires a running time at least exponential with respect to the length of the input sentence.

Algorithm still means Turing machine. Notice that Fischer-Rabin's Theorem only refers to the additive structure of the reals (R, +, -); recall that, in this restricted language, the reals inherit in an obvious way a structure of Qvectorspace, because the scalar multiplication by any rational can be defined using +, and so form a structure elementarily equivalent to (C, +, -). In this sense, the theorem applies also to the complex field, yielding the same negative lower bound for the decision procedures concerning algebraically closed fields. By the way, it is not known any decidable theory with infinite models and a decision procedure running in polynomial time. Indeed, if such a theory exists, then P = N P . In particular, Tarski's original elimination procedure is very inefficient and slow. However, recently faster and more powerful elimination methods for real closed fields have been introduced. We wish to quote the Collins procedure, called cylindrical algebraic decomposition CAD, working in the worst cases in a doubly exponential time with respect to the number of variables in the input formula. Implementations of CAD, and other real quantifier elimination methods are discussed , for instance, in [33]. But now we want to treat briefly another intriguing relationship between complexity theory and quantifier elimination (for arbitrary theories and structures). A few lines ago we have said algorithm = Turing machine, in accordance with the Church-Turing Thesis. But the Turing model of computation has an intrinsic discrete character, so that its applications to

2.9. COMPUTATIONAL ASPECTS OF Q. E.

81

a continous fra mework (like R or C) seem laborious a nd unna t ural (however, see [121] for a discussion of this point). Consequently, new models of computation , including real and complex numbers as possible inputs, and even working in arbit rary structures, hav e been introduced. We quote the Blurn-Shub-Smale B SS model ([14], [13]), or also t he Poizat approach [133]. T hese new perspectives extend t he Turing point of view: t he classical compu t abili ty is ju st the comput a bility over the field F 2 with 2 elements; but now computability over ar bitrary st ructu res is allowed . In part icular , for ever y structure A , one ca n define in a su itable sense the classes P and N P (over A), one can cornpare t hese classes and check if P = N P over A . To introduce t hese matters in detail would require a long time, so we refer t he interested reader t o the bibliography quoted at t he end of the chapter. Rema rkably, qu antifier elimination arises in t his setti ng. In fact , Poizat obs erved Theorem 2.9.2 If P = N P over A , then the theory of A eliminates the quan tifiers. It is comparat ively easy to realize why. Let us refer for simplicity t o t he Cole exa mple quoted before. To prove that 267 - 1 is composite req uires to find a non trivial divisor , and hence to obtain some witn esses of the (exist enti al) sentence 3u3v(2 67 - 1 = u v). But, after Cole , we have simply t o check t he quantifier free sentence

267

-

1 = 193797721

X

761838257287.

In ot her words, what P = N P as ks here is a procedure (ind eed a quick proced ur e) of eliminat ion of qua nt ifiers . T heorem 2.9.2 provides several exa mples of structures for which P -I N P. For instance, recall t he MacintyreMcKenna-Van den Dri es theorems characterizing the infinite fields whose theory eliminates the quantifiers in the language for fields (they are t he algebraically closed fields) , or the ord ered fields whose theory elimin ate s th e quantifiers in the langu age for ord ered fields (the real closed fields) . One easily dedu ces Corollary 2.9.3 1. P -I N P over the fie ld of rationale, or over the field of reels (without the order relation).

2. P

-I N P

over the ordered field of rationale.

On the other side, quantifier elimin ation is only a necessary condit ion towards P = N P over a given structu re. There do exist quantifier elimina ble

CHA P T ER 2. Q UA NTIFIER ELIMINA T ION

82 struct ures A such t hat P nice resul t of Mee r.

=1=

N P over A : t his is the conte nt of the following

Theorem 2.9.4 P =1= N P over (R ,

+, - ).

In fact t he t heory of (R, +, -) is essent ially t he t heory of nonzero vect orspaces over the rational field , a nd so admits t he elimination of qu an tifiers in t he lan gu age LQ and even in {+ , -} because t he action of any ration al is eas ily defined by the addit ive st ruct ure without using quantifiers. Wh a t ca n we say about t he complex field, or t he orde red field of reals? T heir t heor ies eliminate t he qua nt ifiers in t he corres ponding language. Nevert heless P = N P is still an open que stion over these st ruct ur es. Notabl y, in t he complex case, a key (N P-complet e) problem tow ards a definitive answer is related to t he celebrated Hilbert Nullstellensatz (a classical algebraic resul t closely related t o Mo del T heory, as we will see in t he next chapter) : in fact, it as ks a quick proced ure checking t he solvability of a given finit e system of polyno mials over C (in arbit rarily man y varia bles) . One shows t hat P = N P over t he corn plex field if a nd only if t his fas t pro ced ure exists. N P -com plete problems over the ord ered field of reals a re discussed in t he references quoted below.

2.10

References

Van de n Dri es [168] and Doner-H od ges [34] are two excellent and enjoyable exposito ry papers, explaining Tars ki's work on t he qu a ntifier elimination an d, mor e generally, t he hist or y of this matter . T hey also includ e a rich list of references . Here let us mention [87] and [154] on t he pioneeristi c contributions of Lowenh eim a nd Skolem t o the elimination of quantifiers. Lan gford 's elimination met hods for dense or discrete ord ers are in [81] a nd in [82], while Tarski 's subsequent cont ribut ions in t his sett ing a re in [160]. Tarski 's elimina t ion pro cedures for the real field an d t he complex field are given in [157], while Cohen's method in the real case is in [27]. [98] contain s t he Macintyre-McKenna-Va n den Dries t heorems saying that algebraically closed fi elds are t he only infini te fi elds whose first orde r t heory eliminates t he quantifiers , a nd real closed fields a re t he only ord ered fields wit h t he same feat ure. Let us menti on t hat R. Thorn coined t he word "semialgebraic" in [162]. Some more details a bout pp-elimination in modules (and a pro of of Neuman n's Lemma 2.6 .3) can be found in M . Prest 's book [136]; t he Ek lof-F isher

2.10. REFERENCES

83

paper [40] deals with t he particular case of a belian groups (and modules over Ded ekind domains). T he computat iona l aspects of the qu antifier elimina t ion for the real field ar e discussed in [33], while t he cylindrical algebraic decomposition algorit hm C AD is in the Collins paper [29]. The Fisher-Rabin t heorem ensuring that no decision algorith m for the real field run s in polyn omial t ime is in [47]. [13], [14] describ e th e new Blum-Shub-Smale model of computat ion; [133] provides Poizat 's a pproach to this th eme. K. Meer 's th eorem th at P =J N P over (R , +,-) (although the corres ponding t heory eliminates t he quantifiers) is in [112] .

Chapter 3

Model Completeness 3.1

An introduction

We already defined model completeness in Chapter 1: a t heory T is called model com plete if every em bedding bet ween models of T is elementary. We dealt with this notion also in Ch apter 2, where we considered it s connection with quantifier elimination and completeness. Bu t now we wish to examine model completeness in a closer and more dir ect way, t o discu ss its genesis and motivations, as well as its importance and applications. Model completeness deals with embeddings between st ruct ur es. This perspect ive might look slight ly oblique with res pect t o t he fu ndamental pu rpo se in mod el t heory, nam ely to connect sentences a nd str uct ures via t ruth; under this poin t of view , the most genuine relation among struct ures is elementary equivalence (that is , to satisfy the same sente nces). Nevert heless some basic theorems in mod el theory, such as t he Lowenheim-Skolem t heorems, involve pre t ty naturally extensions, subst ruct ures, embeddings , and so d raw attenti on t o thi s subject . Furthermore, as we will see in Section 3.2, th ere are oth er possible ways of introducing mod el completeness. The first one st ill deals with embeddings and says that a theory T is model complete when each embedding betwe en mod els of T preserves existential formulas. But another cha rac terizat ion is quite sy ntactical an d resembl es the way we defined qu an tifier eliminat ion; it says that a t heory T is mod el complete exactly when any formula !.p(v) in the language of T is equivalent in T to an appropriate existential formula !.p'(v). The main motivations leadin g t o model completeness com e from algebra. For instance, consider field th eory. Given a field K , one looks at t he irreducible polynomials f( x) E I<[x]. Algebra build s richer a nd richer extensions of 85

86

CHAPTER 3. MODEL COMPLETENESS

equ ipping t hese polynomials with a single root , or all the po ssible roots. Eventually, one reaches the algebraic closure J( of J( : a minimal extension where every nonconstant polynomial f(x) in I< [x], a nd even in I< [x ] it self, splits into linear factors, and so get s its own roots. Notice that, from a logical poi nt of view , adding a root of a polynomial f( x) means t o satisfy the se nt ence 3w(f (w) = 0) with param et er s from I< (the coefficients of f( x)) . J( is alg ebraically closed wh en it equals J( a nd hen ce wh en it is a ble to sati sfy a ll t hese sentences when f( x) ran ges over t he non con stant polynomials over I< it self. Pursuing t his logical a ppro ach, on e ca n gen eralize a nd look at a rbit rary L-structures A instead of pure fields, towards t wo possible objects: J( ,

*

*

to enla rge A to a rich er A sat isfying every existent ial sent ence 3wa( w) (with a qu antifier free a(w)), or even ever y se nt ence in L(A), or (why not?) in L(A) , t oo; to examine closely t he structures A.

This program recalls A . Weil's notion of universal domains in [176]. Weil's idea (for t he class of fields) was t o determine large and rich structures, embedding every fi eld under con sid er ation. Of course , in the cas e of fields , universal domains are just alg ebraically closed fields of infinite transcendence degree. This strategy has now fallen into disuse within Algebraic G eom et ry, but it is still alive in Model Theory (and certainl y it was in t he sixt ies) . Model com pleteness a rises quite naturally in this fr am ework: for , in a model com plet e theor y T , for eve ry model A a nd for every L (A)-sen t en ce

"Sym bolic Logic ca n produce useful tools for th e developments of actual math ematics, more par ticularly of Algebra and, it would

3.1. A N INTRODUCTION

87

app ear, of Algebraic Geometry. This is the realization of an ambition ... ex pressed by Leibniz in a let ter t o Huyghens as long ago as 1679 ". This poin t of view is de velop ed one year later in [140]. T he algebraic theorem s recalled before do corroborate t his program . Other deep confirmations (also concern ing Geom etry) will be prov ided in the next cha pters. Let us conclude t his sect ion by recalling some connect ions between model comp let eness, elimination of qu an t ifiers and complet eness . F irst of all, remember t hat elimination of quantifiers impl ies mod el completeness. T he converse is not true. For instance, we saw in t he last chapter that the t heory of real closed fields R C F loses t he quant ifier eliminat ion proper ty if one removes t he relation symbol for ~ from it s lan guage L: act ually the order ~ is definable in the restrict ed language L o = {+ , - , " 0, 1}, as v ~ 0 is equivalent in RC F t o :3w( w 2 = v), bu t any possible definit ion needs qu an tifiers. However t o forget ~ doe s not affect model completeness: in fact every embed ding f : A -+ B of real closed fields in the rest ricted lan guage L o enlarges naturally and involves ~ (beca use t he nonnegative elements mus t eq ual t he sq uares) , so is elementa ry in both L an d L«. On t he other side, mod el com pleteness can yield complet eness und er some suit a ble additional hypotheses. We saw t hat this happens, for inst an ce, for real closed fields (or also for discrete linear orders) . The reason was that R C F has a "minim al" model , emb edd abl e in every real closed field : the orde red field of real algebraic numbers (hence t he real closure of t he rat ionals) . To extend this exa mple t owar ds a general set t ing, we need t he following Definition 3.1.1 Let T be a theory. A model of T is prime if it is em beddable in every model of T. Examples 3.1.2 of ACFo.

1. T he (comp lex) algebraic numbers are a prime model

2. The real algebraic numbers are a prime model among real closed ordered fields. 3. (N , ~, 0, s) is a prime model of dLO+. Proposition 3 .1.3 Let T a model complete theory . 1fT has a prime mod el A , then T is complete.

88

CHAPTER 3. MODEL COMPLETENESS

Proof. Just adapt the argument of R CF. For every model B of T , there is an embedding of A into B. Owing to model complete ness, t his embedding is elementary. In pa rt icula r B is eleme ntarily equivalent t o A. Hence all th e models of T are elementarily equivalent to each other. Co nsequently T is '" complet e. Another useful criterion deducing completeness from mod el completeness is t he following one (we used it when dealing with dense linea r ord ers in t he last cha pter). Proposition 3.1.4 Let T a model complete theory. A ssume that any two models A and B of T admit a com mon extension C in M od(T). Th en T is complete. Given A a nd B , form a common ext ension C. Owing to mod el Proof. completeness , t he em beddings of both A and B in C ar e elementary. So A and B are elementarily equivalent t o C, a nd consequent ly to eac h oth er . '" However , be careful: model completeness does not imply completeness in general. Just to avoid a ny temptation a bout t his point, recall algebraically closed fields. AC F is mod el complete, but it is not complet e: one needs t o specify the characteristic to get a prime mod el and hence t o ded uce, even via model completeness, that ACFp is complet e for every p = 0 or prime.

3.2

Abraham Robinson's test

Let T be a t heory in a language L . We know t hat T eliminates t he qu an tifiers if a nd only if every L-formula is equivalent in T to a s uitable quantifier free formula (with the same free variables). Model completeness can be characterized in a similar way. Indeed one can show that T is model complete if and only every L-formul a is equ ivalent in T to an existential formula , or also, if you like V rat her th an 3, to a universal formula (in fact , assume th at every L-formula ip(v) admit s an exist ent ial L-formula equival ent in T ; apply this property to 'ip( v) and yield the corresponding existential form ula ip' (v); conclude that ip (v) is equivalent in T to ' ip'(v) , which in its t urn is obviously equivalent to a universal formula; the converse ca n be shown in t he sa me way) . There is another rem arkable related characterization of model completeness via embeddings. Recall th at T mod el complete just means that every embedding between two mod els of T is eleme nt ary. But, notably, this is also

89

3.2. ABRAHAM ROBINSON'S TEST

equivalent to require that every embedding between two models of T is existential (at a first sight, a weaker condition). This is the con tent of the so ca lled Abraham Robinson Test for model completeness . We will apply t his criterion t o some algebraic set t ings in the next section. Now we want to show Robinson 's Test and , at th e sa me t ime, the previou s cha ract erizat ions of model completeness in terms of existential , or universal formulas. Theorem 3.2 .1 (A. Robinson) Let T be a theory of L. Th e following propo-

sitions are equivalent: (i) T is model complete; (ii) every embedding from A into B, where A and B are models of T , existential;

(iii) for every L-formula
ip' (if)

is

equiv-

alent to
Proof. (i)=>(ii) is clear , and (iii){:} (iv) was alr ead y established . Let us consid er now (ii)=>(iii) ,(iv) . We preliminarily show that , if (ii) holds, t hen every existent ial formula

90

CHA P TER 3. MOD EL COMPLE T ENESS

(otherwise, for every mod el M of T and every m in cr(Mn), m E rp (Mn ), and so V'v(cr (V) -+ rp(v)) is in T ). Recall t hat S is closed under conjunctions a nd use compactness t o dedu ce t hat

T U {cr(C) : er(V)ES}U{. rp (C)} has a mod el. In oth er words , t here a re a model A of T a nd a sequence ii in An (th e int erpretation of c in A) such t hat

ii E

n a (iI)ES

cr(An )

bu t ii ~ rp (An ). Put ii = (aI, . . . , an)' Now take any qu an tifie r free formul a O(V, w) in L and s uppose that t here is a sequence b in A for which A F O(ii, b). Let h denote th e lengt h of w (and of b); in particular put b = (bl , . .. , bh). Then V'W.O(V, w) is a universal L-formula, but ii cannot sat isfy it . Hence V'W.O(V, w) ~ S . It follows t hat , in some mod el A* of T , a suitable sequence a* in rp (A*n) doe s not sat isfy V'w.OCu, w); hen ce, for some b-;, A* F O(a* , b-;). We can ex press t his fact in t he following way. Conside r t he language L (A) ; t o avoid any dan ger of confusion, distin guish t he elements of A and t heir nam es in L (A ), a nd deno te by Ca th e constant sy mbol corresponding to a, for every a E A . Wh at we have j ust shown is t hat

TU {rp (caIl . .. , can)} U {O (cap . . . , Can' Cb I' .. . , Cb h )} has a mod el. Again using compact ness and t he fact t hat t he quantifier free formul as of L are closed und er conj unction, we see t hat t he union of TU {rp (Ca! , . . . , Can)} and of th e whole collectio n of t he for m_ulas O(Ca!, ... , Can ' Cb!, .. . , Cbh ) wher e h ranges over non negati ve integers, b = (b1 , ... , bh) over Ah, O( v, w) over qu an tifier free L-formulas and A F O(ii, b) , has a mod el. Let A' denote th e restriction of this st ruct ure to L -hence a model of T -, and let a' = ct for every a E A. Therefore, for every quantifier free Lformu laO(v, w) and every bin A, A' F O(;;} , hi) if and only if A F O(ii, b) . So a f---7 a' for every a E A embeds A into A'. (ii) impli es t hat this embedding is existe nt ial. Con sequ ently, as A' F rp (al ), one dedu ces A F rp( ii) -a cont radi ction- . So a univ ersal er( v) equivalent to rp (v) exists . The converse redu ction (from universal to exist enti al formul as) ca n be han dled by passing to negations. At t his point , the proof is stra ightforwa rd . Take a n a r bit ra ry L-formula rp (v). We are looking for a n equivalent existenti al, or uni versal formul a. If rp( V) has no quantifier , t hen t here is nothing t o prov e, a nd we are don e. Other wise write rp( v) as QI ZI ...

Qkzka (V,

Zl , .. . ,

Zk)

3.3. MODEL COMPLETENESS AND ALGEBRA

91

where k is a positive int eger, the Q j's (1 ::; j ::; k) are qu antifiers (\7' or :3) and a(v, Zl, ... , Zk) is qu antifi er free. P roceed by ind uct ion on k, using what was preliminarily shown : t he det ails a re a n easy exer cise. At last, let us show th at (iv) *(i) . This is quite simple. We know t hat , if f : A -+ B is any emb edding between model s A , B of T , then, for ever y un iversal L-formula
3.3

Model completeness and Algebra

Model com pleteness of AC F a nd R C F was shown in the last chapter as a consequence of elimi na tion of quantifiers. Robi nson's Test provides a direct proof (in these and in other relevant case s). Here we wish t o illust rate this new a pproac h. However , the main object in this sect ion is to emphasize t he role of model complete ness towards some noteworthy applications to Algebra . We und erlined in 3.1 A. Robinson 's pro gram , and his hope t ha t a progress in mod el th eor y could su pply Algebra with new important a nd fruitful tools and te chniques. Mod el complete ness really exemplifies this project. In fact , we will see that , just using the mod el complete ness of ACF and RCF , A. Robinson found neat and elega nt proofs of classical res ults, such as t he Hilbe rt Nullst ellensat z and the solution of Hilbert's Seventeenth Problem (a theorem of Art in) . But we want to underline that the model th eoretic a pproac h can provide not only alternative proofs of previously known theorems, but also , and more not ably, new a nd original answers to some formerly open famou s algebraic probl ems, for instance Artin 's Conjecture on p-adi c fields (this will be treated in 3.4). But now let us show t he mod el complet eness of ACF and RCF via the A. Robinson Test , as promised. Theorem 3.3.1 (A . Robinson) ACF is model complete .

92

CHAPTER 3. MODEL COMPLETENESS

Proof. Suppose not. Owing to the Robinson Test, there is some embedding between algebraically closed fields which is not existential. Let f : K -7 1-l be a counterexample. With no loss of generality, we can assume that K is a su bfield of 1-l and f is just the inclusion (otherwise, replace K by its isomorphic copy inside 1-l). There are a quantifier free formula cv( V, w) in the language L = {+, - , " 0, I} and a sequence ilin K such that :3wcv(v, w) is true in 1-l and false in K. Incidentally, recall that cv( V, w) is (equivalent to) a disjunction of conjunctions of equations and inequations; however, as :3 is distributive with respect to V, we can assume that cv( V, w) is directly a single conjunction of equations and inequations. Let b satisfy cv(il, b) in

1L Form the extension K(b) of K by i, its algebraic closure K(b) embeds itself in H. and satisfies :3wcv(il, w) because it includes b. So there is no loss of generality in replacing 'H. by K(b) and hence in assuming that 1-l = K(b) has a finite transcendence degree (> 0) over K. Now take a transcendence basis tl, .. . , t s of 1-l over K, and split the embedding K ~ K(tl , .. . , t s ) by

where K, = K(t 1 , ... , ti) for every i = 1, ... , s. There is some i = 1, ... , s such that :3wcv(il, w) is true in K, and false in Ki-l. We can replace K ~ H. by Ki-l ~ x; and to assume

1-l

= K(t)

for a single transcendental element t over K. So :3wcv(il, w) is true in K(t) and false in K. Now take any algebraically closed extension K' of K having transcendence degree 2 1. Hence K' enlarges K (u) for some transcendental element u over K. Steinitz's analysis of algebraically closed fields tells us that K(t) and K (u) are isomorphic via a function enlarging the identity of K and mapping t into u. Then K (u) and, consequently, K' satisfy :3wcv( ii, w): our sentence is true in every algebraically closed extension K' of K with transcendence degree 2 1. Equivalently, in the language extending L(K) by a new constant c, :3wcv(il, tu) is a consequence of Th(KK) plus the infinitely many sentences ensuring that c is transcendental over K, i. e. does not solve any nonzero polynomial with 1 unknown and coefficients from K. Use compactness and deduce that finitely many sentences suffice (in addition to Th(KK)) to imply :3tucv( ii, w); in particular, c can be interpreted by a suitable element of K (out of the roots of a finite system of polynomials in K[x]). In conclusion, K satisfies :3wcv(il, w): a contradiction. Hence ACF is model complete. ..

3.3. MODEL COMPLETENESS AND ALGEBRA

93

Which are t he main algebraic ingredients of thi s Robinson proof? Basically, Steinitz 's analysis of algebraically closed fields . More specifically, two key points should be underlined: 1. every field has an algebraic closure , and this is unique up to isomorphism enlarging t he identity in the ground field ;

2. if K is an algebraically closed field and t is transcendental over J( , th en t he isomorphism class of K(t) over K is uniquely de t ermined (in t he sense that two extensions of this kind are isomorphic via a map extending the ident it y of J(). One can rea lize t hat real closed fields satisfy simila r propert ies: 1. every ord ered field has a real clos ure (a min imal real closed extension), and this is algeb ra ic over t he ground field , and un ique up to isomorphism enla rging th e identity in the gro und field ;

2. if K is a real closed ord ered field and t is t ra nscende ntal over J( , th en the isomorphism class of K(t) over K is fully charact erized by t he cut t det ermines over J(. So, when dealing with model completeness for R C F, one can reproduce the proof of the algebraically closed case (with some complicat ions du e to th e order) and deduce Theor em 3 .3.2 (A . Robin son) R CF is model complete . Robinson 's Test ca n also be used t o prove th e mod el completeness of sever al theories we met in t he previous chapter : discre te linea r orders , den se linear orders and so on . The reader may check this, as an exer cise. But here we prefer to discuss some very noteworthy applications of t he model complet eness of ACF and R C F to Algebra. T hey provide new elegant proofs of known algeb raic facts . First let us deal with algebraica lly closed fields and Hilbert 's Nu llstellensatz . Theorem 3.3.3 (Hilbert Nullstellensatz) Let K be an algebrai cally closed field, J be an ideal of the rin g K[X] (whe re x abridges, as usual, th e sequen ce of unknowns (Xl , ... , x n ) ) . Th en , fo r eve ry polynomial f( x) E J([X] , f(ii) = 0 f or every d E K " su ch that g(ii) = 0 fo r all g( x) E I if and on ly if

94

CHA P T ER 3. MODEL COMPLET ENESS

for some positive integer m f m(x ) El . Proof. The direction from right t o left is clea r. Converse ly, su ppose t owards a cont ra dict ion t hat t here exists some polynomial f( x ) in K[x] such that

f (ii) = 0 for every ii E K " s uch t hat g(ii) = 0 for all g(x) E I but

f m(x) rJ. I for every positi ve int eger m . Let J be an ideal of K[X] s uch th at J ;2 I , no power 1m (x) of 1(x) (with m a positive int eger ) lies in J, and J is maximal with resp ect t o th ese conditions (Zorn 's Lem ma ensures th at such a J exists). We claim that J is prime . In fact, take two polynomials go (x) , gl (x) in K[X] - J; t hen t he ideals

Jo generated by J and go(x) , J 1 generat ed by J and gl (x)

st rict ly includ e J ; accordingly t here are two positi ve int egers mo , m 1 such t hat f mo (x) E Jo, I": (x) E J 1 . So t here exist two polynomi als qo (x ), q1(x) E K[X] such t hat

F" (x) - s.(x) qi(x) Co nsequent ly

E

Ji' Vi

= 0 ,1.

r-r: (x)

is in t he ideal gener ated by J an d go(x ) . gl (x ), a nd so go(x) . gl (x ) rJ. J . Hence J is prime, and R = K[X] / J is an int egral domain exte nding K by th e fun ction mapping an y a E K t o a + J. Th en K em beds into the algebraic closure F of the field of quotients of R . As AC F is model complet e, this embedding is elementary. Now take any (finit e) set of generators 10(x), ... , 18(X) of I (I is finit ely gene rated becau se K[X] is Noetherian), an d noti ce that the L(K)-sentence

::Iv (/\ fi(V) = 0 1\ -, (J (v) = 0)) i~8

is t rue in F (owing t o t he seque nce (X l + J, ... , X n + J )). Consequent ly t his sentence is t rue also in K . So t here exists some ii in K " such t hat ii satisfies 10(x ), ... , f8(X) and consequent ly all the polyno mials in I , bu t ii do es not a nnihilate f (x ) -a contradictio n-. ..

3.3. MODEL COMPLE TENESS AND ALGEBRA

95

Now we deal with RCF, and with Hilbert 's Seventeenth Problem . This was solved by Artin in 1927. Indeed Artin himself and Schr eier developed t he algebraic notion of real closed field just to answer Hilbert 's qu estion. Later A. Robinson proposed a very nice a nd simple proof, founded on th e model com plete ness of RCF. Here we want t o report A. Robinson's a pproac h. First let us introduce Hilbert's problem in detail. Ind eed the seventeenth question in the celebrated Hilbert 1900 list ju st concerns ordered fields (more properly, the ordered field of reals ) . Recall t hat, in a ny ordered field K , a rational function f (x) E K (x) is said t o be sem idefinite positive if a nd only if, for every sequence ii in K (such that f(ii) is defined) , f(ii) ~ O. Of course, t he s ums of sq uares in K(x) a re semidefinite positi ve. I-Tilbert 's Seventee nt h Problem conjectures t hat the converse is also true when K is t he ordered field of real numbers. As already said, Art in solved positively t his question ; indeed he extended th e result to a rbit ra ry real closed ordered fields K . Now we pro vide A. Robinson 's proof of t his t heorem. Theorem 3 .3.4 (Ar tin) Let K be a real closed ordered fiel d, f (x ) a se midef inite positive rational fun ct ion in K (x). T hen f (x) can be expressed as a sum of squa res in K(x). Proof.

We need t he following algebra ic fact .

Fact 3 .3.5 Let K be a fiel d, an d assume that, fo r eve ry natural t an d for every choice of aa, .. . , at E K , if L i
96

CHA PTER 3. MODEL COM PLETENESS

mod el complet e, t his embedding is elem entary. Bu t K( x) , and consequentl y R , satisfy t he L (K)-sen t ence

3iJ(f (iJ) < 0) (owing t o t he sequence x). Accordingly also K sat isfies 3iJ(f(iJ) < 0). Bu t • t his cont ra dicts our ass umption that f( x) is semidefinite positive.

3 .4

p-adic fields and Artin's Conjecture

Real numbers complete the rational field with resp ect to the usual metric t opology. For every prime p, p-adic numbers complete th e rationals with res pect t o a n alt ern at ive t opology, called p-adic . Let us short ly remind th is t opology. First write (uniquely) an y nonz ero rational a as a = ph: s

where h, rand s are int egers, s is positive a nd h, r , s a re pairwise coprime; t hen define a fun ction v p from t he multiplicati ve g ro up Q* of nonzero ra tio na ls into t he ordered additi ve group of int egers by pu t tin g, for a as befor e,

On e gets in th is way a gro up homomorphism fro m Q* int o t he integers, satisfying t he addition al condit ion

vp (a + b)

~

min{ vp (a), vp (b)} 'tIa,

se Q*

(this is st raightforward to check) . v p can be form ally extended to 0 in some a rt ificial way; putting vp(O) = 00 is a reasona ble choice, as ph divides 0 among th e integers for every natural num ber h. Now pu t , for every positive integer h, Oh = {a E Q : vp ( a) ~ h} a nd take the Oh'S as basic op en neighbourhoods of 0, hence their translations Oh + b wit h b E Q as basic ope n sets. One gets a new topology of th e ration als: t he p-adic topology. As already said, t he set Q p of p-adi c numbers is t he complet ion of Q with resp ect t o t his t opology. In orde r to realiz e as well as possible what a p-adi c numb er is, and so to introduce Qp in a more det ailed way, one can follow seve ra l eq uivalent ways: see [36] for a gen eral ou tli ne of t his poin t . Here we limit ourselves to sketch

3.4. P-ADIC FIELDS AND ARTIN 'S CONJECTURE

97

some basic ideas. A possible approach to p-adic numbers might use the well known fact t hat every positive integer a has a unique p-adic decomposition

where n and t he ai's are natural numbers, an =1= 0 an d a; < P for all i ::; n. Under this perspective, vp(a) is just the least i ::; n for which a; =1= o. Hence, for every nonzero natural a n +l < p, it is t rivial to realize

So, when considering Cau chy sequences of integers with respect to the p-adic t opology and equipping such a sequence with a limit , one naturally bu ilds infinite sums

where the ai'S are non negative integers < p. Enlarging the a nalysis from positive int egers to arbitrary nonzero rationals leads to conside r general infinite sums 00

(*)

L

aipi

i=N

where the ai'S a re, as above, natural numbers < p, aN =1= 0 but N is now a ny fixed -eve n negat ive- integer: in this sense (*) exhibits a typical p-adic number. 0 can be easily recovered in this fram ework as the infinite sum whose coeffi cients are constantly O. Hence, for every prime p , Q p is t he set of t hese infinite su ms . On e defines in a suitab le way addition + an d multiplication· in Q p, extending the usual operations in Q. Bu t here we have to be careful: sum and product are not computed componentwise, but as suggested by t he algebraic framework. For inst an ce, the trivial iden tity 1 + (p - 1) = P must be read (1

+

0 . p)

+

((p - 1)

+ 0 . p)

= 0

+

1 · p.

However t hese operations equip Q p with a field structure, extending the rational field; v p can be enlarged to Q p in very simple way, as, for a = I:~N aipi as before (and aN =1= 0) , vp(a) is just N. One sees t hat

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CHAPTER 3. MODEL COMPLETENESS

is a local s ubring (t he ring ofp-adic integers), and

Ip ={aE Q p : vp(a»

O}

is its maximal ideal. On e easily checks that the quo tient field Z p/ Ip is isomor phic to the field F p with p elements . We want to und erline t wo fur th er basic properties of Q p. (a) The first one is quite trivial, and simply points out that vp (p) = 1. (b) The other is mor e subst a nt ial a nd concerns the so called Hensel's Lemma. This is a key result in locating root s of polynomials in Q p, in fact it states t hat,

if f(x) is a monic polynomial in Z p[x ], then any decomposition of f( x) modulo I p in F p[x ] as a product of two relat ively coprim e monic polynomials lifts to a decomposit ion of f( x) as a product of two mon ic polynomials in Z p[x]. This concludes our short sum ma ry about th e algebraic struct ure of Q p for every prime p. What we have sket ched suggests some simila rit ies with the reals . Actually both Q p a nd R have a common topological genesis from t he rationals (and, und er this point of view, a common topological st ructure of locally compact field); mor eover th ere do admit some reasonable crite ria to locat e roots of polynomials (t he Sign Cha nge prop er ty for t he reals , Hensel's Lemma in t he p-adic case ). Now we want to discuss another example, closely recalling Q p. We take any field K (but below we will be pr imarily interested in the field F p with p elements for any prime p). We look at the formal La ur ent series 00

L: ai

a(t) =

i= N

where a; E K for every i ~ N and N is a given integer. T he cor responding set K((t )) inh erits a fi eld st ruct ure extending K , provided that we define th e addition + componen twise and the mult iplication . in the obvious way enlarging t he product in K . Again th e set K [[t]] of formal power serie s 00

L: ai i= O

3.4. P -ADIC FIELDS AND ARTIN'S CONJECTURE

99

is a local s ubring, who se maximal ideal just contains the power series with ao = O. One eas ily deduce that t he quotient ring is (isomorphic to) K. The fun ction VK from the mul tiplicative group K ((t ))* (where ", as befor e, mean s t o exclude 0) into t he (ordered) additive gro up of integers sending an y nonzero Laurent series a(t) as before (with aN i= 0) to N again yields a g ro up homomorphism sharing with vp and Q p th e following property

VK(a(t ) + b(t)) 2 min{ vK(a(t)), vK(b(t ))} for all a(t) and b(t) in I<((t)) *. Now assume K = F p for a given prime p . In this restricted fram ework (a) VFp (P)

i= 1.

Notice t hat th is distinguishes Fp((t)) a nd Q p.

(b) Howeve r Fp((t )), just as Q p, satisfies Hen sel' s Lemma. An abst rac t not ion including p-adics as well as formal Laurent series and other relat ed examples t owards a common gener al t reat me nt is the concept of val ued field: this is a str ucture (K , 9, v) where K is a field , 9 is an ordered a belia n group, a nd v is a gro up homomorphism from th e mu lt iplicative group K * in 9 satisfying the furth er ass umpt ion

v(a + b) 2 min{ v(a), v(b)} 'tfa, s « 1(*. The fun ction v is ca lled the valuation m ap . A genera l algebraic analysis promptl y confirms some basic proper ties observed in th e previous examples : in par ticu lar , for every valued field (K, 9 , v), {a E I< : v(a) 2 O} is a local s ubring (the val uation ring) , and its maximal ideal is ju st {a E I< : v(a) > O} ; th e corr esponding quotient field is called t he residu e field of (K , 9, v ), and will be denoted by K below (hence all throughou t t his section K denotes resid ue field rather than alge bra ic closure). A valued field is called Henselian when it satisfies Hensel's Lemma, hence when the following holds: Let f( x) be a monic polynomial in R [x] and let l( x) denote its projection in K[ x]. Th en any decomposition of](x) in K[x] as a produ ct of two relatively coprime monic polynomials f( x) = ,(x) . A:(x) lifts to a de composition of f( x) into the product of t wo monic polynomials in R[ x] f (x ) = g (x ) . k (x ) wh ere g (x )

= ,(x ) and k(x) = A: (x ) .

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CHAPTER 3. MODEL COMPLETENESS

For every prime p, both (Qp , Z, v p) and (Fp((t)), Z, VF p ) are Henselian valued fields (although they do not admit the same characteristic, and hence their valuation of p is not the same). Now let us come back to our original framework and hence to p-adic numbers. Here Artin proposed a famous conjecture. Conjecture 3.4.1 (Artin's Conjecture) Let p be a prime. For all positive integers nand d with n > d 2, every homogeneous polynomial f(Xl ' .. . , x n ) E Qp[Xl' ... , x n ] of degree d has a nonzero root in Qp. The conjecture is inspired by the underlined resemblance between Qp and Fp((t)) for every prime p. Actually in the valued field Fp((t)) the claim is true for every choice of p, nand d, as proved by Lang. But the behaviour of the p-adics in this setting is not the same, in fact Artin's Conjecture fails for some p, nand d. This was observed by Terjanian, who in [161] did exactly what one is expected to do in disproving a statement, and exhibited a counterexample for some suitable p, nand d. However an asymptotic form of the conjecture is true: for any choice of n and d, Artin's statement is satisfied by all but finitely many values of p. This was the content of a celebrated theorem ofAx, Kochen and Ershov in 1965, which, combined with Terjanian's counterexample, provides a sufficiently complete answer to the question. The Ax-Kochen-Ershov approach is essentially model theoretic. Indeed , they developed a general analysis of the model theory of valued fields, and deduced the asymptotic form of Artin's Conjecture as a consequence, using compactness and transfer techniques. Let us briefly survey their work. First of all, we have to clarify how to handle valued fields from a model theoretic point of view: which language to use, and so on. The more convincing approach views a valued field as a two-sorted structure, in other words as a structure with two sorts of variables, where

* *

the former sort of variables concerns the elements of the field, the latter sort of variables is devoted to the elements of the valuation group;

moreover there are the usual field symbols for the former sort , and (disjointly) the symbols of ordered groups for the latter; finally a l-ary operation symbol v is reserved for the valuation map. Valued fields are easily axiomatized in a first order way in this language. This approach emphasizes, within valued fields (!C, 9, v), the role of three underlying structures:

3.4. P -ADIC FIELDS AND ARTIN 'S CONJECTURE

101

the original field K , the ordered abelian group Y, and finally, to capture the valu ation map v, t he residu e fi eld K. Ax , Kochen and E rs hov show t hat t hese st ruct ures rul e the beh aviour of t he whole valued field (K, y , v) with resp ect to elementary equivalence . In fa ct , t heir main general result says Theorem 3.4.2 Let (K , y , v ) be an Hen selian valued fi eld, whose residu e field has characte ris tic o. Th en the complet e theory of (K , y , v) is fully det ermin ed by th e complete theories of the ordered valued group Y and the residue fi eld K . Warning: T he theorem do es not a pply directl y t o Q p or t o Fp((t)) for an y prim e p becau se t hese valued fields do not respect t he hypothesis on t he characterist ic of the residue field . Nevert heless t he Ax-Koc hen-Ers hov main t heore m is enough t o t hrow a bridge between t he valued fields of Lauren t se ries Fp((t)) and t he p-adic valu ed fields Q p with res pect to Ar t in 's Conjecture, and to deduce its asymptotic solution. Theorem 3.4.3 (Ax-Kochen-Ershov) For all posit ive in tege rs n , d with n > d 2 , for all but fi n it ely ma ny primes p, eve ry homogen eous polynomial f( XI' . . . , x n ) E Q p[x }, . .. , xn J of degree d has a nontrivial root in Q p.

Proof. (Sketch) We can limit ourse lves t o treat t he case n = d 2 + 1. Other wise pu t Xi = 0 for n > i> d 2+1 and work wit h X l , •• • , xd2 +l : up to rearranging t he ind ices of our unknown s, we can ass ume t hat what we get in this way is a homogeneou s polyn omial of degree d in X l, . . . , X d2 +I , and every nontrivial zero of this polynomial clearly produces a nontrivial root of f(XI ' ... , x n ) . Needl ess t o say, for a ny fixed d (and n), Ar tin 's Conjecture becomes a first order sente nce Cid in t he lan gu age of valued fields: t his is a rou tine exe rcise, easy to check. We know from Lan g t hat t his se ntence Cid is t rue in t he valued field (F p( (t)) , Z , V F p) for all primes p. This fact , in pa rt icular it s uniform validi ty for every p, is not ewor thy ; act ually, we are in a sit uation qui t e similar t o t he one described in C hapter 2 , § 4, for algebraically closed fields. In part icular , one ca n apply th e same transfer machiner y, combine the Ax-Kochen-Ershov main theorem a nd Lan g's resul t , and in conclusion deduce that Cid is true in every valued field (K , y, v) such that

102

CHAPTER 3. MODEL COM PLE TENESS

*

K is I-Ienselian ,

*

9 is elementarily equivalent (as a n ord ered group) t o t he integers (9 is called a Z-gro up in t his cas e) ,

*

t he residue field K is a pseudofinite field of characterist ic O.

Of course, this does not concern directly any p-ad ic field Q p, becau se its residu e field has cha racterist ic p . Bu t , in proving ad, only finite ly man y sentences about the characterist ic 0 of th e residu e fi eld a re necessary. This impli es that a d is true in Q p for almost all prim es p, as claim ed . '" Owing to the Terjanian counterexample quoted befor e, this Ax-Koch enErshov an swer t o the Artin Conject ure is the best possib le. None of these results refers directly to mod el complete ness. Nevertheless the spirit and the techniques of t he mod el theor etic a pproac h ofAx, Koch en and Ershov clearly owe to A. Robinson 's ideas, and a re int imately related to his dr eam of linkin g Algebra and Mat hematical Logic via Mod el T heory. On e should also rememb er t hat t he Ax-Kochen-Ershov main t heorem do es not a pply t o t he p-adic fields Q p, because, as already observed, the char act eristic of t heir residu e fields is not O. So one may wonde r, for every pr ime p , how to cha racterize Q p, and even its twin Fp((t)), up to eleme ntary equivalence, in other word s how t o ax iomatize in a first orde r way t heir complet e theories (incidentally recall t hat Th( Qp) -I Th( Fp((t)), becau se t he involved charact eristics are not eq ual) . Here mod el cornplet eness sounds useful. Indeed , we alread y underlined t hat p-adics and reals sha re seve ra l relevan t similarit ies: model complet eness, and a prec ise first ord er axiomat iza t ion, a re a mong them . In fact , t he following theorem holds. Theorem 3.4.4 (Ax-Kochen-E rshov) For every prime p, let T p be the the-

ory of the valued field s (K, g, v) such that

* * *

K is Henselian and has characte ristic 0,

9 is a Z -group, the residue fi eld K is (elem en tarily equivalent to) the field with p eleme nts,

* v( p)

is 1 -th e least positive element in G- .

Th en T p is model complete and complete. In part icular T p is the theory of the valued fi eld of p-adic num bers .

3.5. EXISTEN TIA LLY CLOSED STR UCT URES

103

What ca n we say about definable set s in Q p, or also abo ut qu antifier elimination for p-adics? Maci ntyre showed quantifier elimination in a very natural language with addit ional relation sy mbols for t he valuation ring a nd the set of n-th powers for every n . This pro vides, of course, some mor e information on definable sets; in particular, one can see that the p-ad ics sat isfy Theorem 3.4.5 (Macintyre) Eve ry infinite definable subset of Q p has a nonempty int erior. Notice th at t his is exact ly wh at happens for the reals. F inally, let us remind that Tp is not t he t heory of Fp((t)); actu ally, the qu estion of determining the first ord er theory of this valu ed field seems still open .

3.5

Existentially closed structures

Example 3.5.1 T he class of fields and t he subclass of algebraically closed fields satisfy t he following properti es:

(i) every field emb eds into an algebraically closed fields (for instance , into its algebraic closure);

(ii) every embedding between algebraically closed fields is eleme ntary (in other word s, the theory ACF is mod el complete ). Con sequ entl y (in our language L for fields)

(iii) for every embedding of an algebraicall y closed fi eld /C int o a field 1l , and for every qu antifier free for mula

:Jw(J(w) = 0) (true in some extension of /C) . (iii) says that this still holds when we replace few) = 0 by any qu antifier free L (I<)-formula

CHAPTER 3. MOD EL COMPLETENESS

104

From a n algebraic point of view, (iii) implies that any finite sys te m of eq uations 10(x) = 0, .. . , 1t(x) = 0, with 10 (x) , . . . , ft( x) E K[X), has a solut ion in our algebraically closed field /C whenever it finds a zer o in some extension of /C : to see t his, ju st apply (iii) to
1\

i
Example 3.5.2 A similar analysis ca n be develop ed in t he langu age L of ordered fields with respect to t he su bclass of real closed fields. In fact

(i) any orde red field has some real closed extension (for instance, it s real closure) ;

(ii) any embedding between real closed fields is eleme ntary (as t he t heory ReF is model complet e).

(i) and (ii) imply, j ust as in th e pre vious case , that , in the language L of ordered fields,

(iii) for every embedding of a real closed field /C into an ordered field 'H. and fo r every quantifier free formula
10(x)

= 0, . .. , 1t(x) = 0,

90(X) > 0, ... , 98( X) > 0,

with 10 (x) , . . ., 1t(x), 90( X) , . . .,98( X) E K[X), adm itting a solution in som e o rdered field extending /C , do es have some solut ion even in /C . So (iii) is not a minor property of real closed fields but includes their definition it self, or, mor e pre cisely, their characterization saying that real closed fields are ju st the ord ered fields /C with no proper ordered algebraic extension: if 1( x) E K[x] has a root in some ordered extension of /C , then it finds a solut ion also in /C. Now consider a ny class K of structures examples suggest the following notion .

ill

a language L . T he previous

3.5. EX ISTENT IALLY CLOSE D STR UCTURES

105

Definition 3.5.3 A structure A in K is existentially closed (e. c.) if and only if every em bedding of A in som e B E K is existential (i. e., for every quantifier f ree L(A)-form ula a(w), if B F :3wa(w), then A F

:3wa(w)) .

Let E(K ) denote t he class of e. c. struct ures in K. Clearl y, a mong fields, e. c. struct ures are ju st t he algebraically closed fields, and , am ong ord ered fields, e. c. st ruct ures are ju st t he real closed fields. Giv en a class K =1= 0, first we ca n wonder whether t here exist some e. c. struct ures in K , a nd hence whether E(K) =1= 0. T he a nswer is posit ive, at least under some simple condit ions on K. For inst an ce, one can see what follows. Example 3.5.4 Let K be a class of struct ures . Assume th at , when ever (I , ::;) is a to t ally order ed set and each i E I ind icates a st ruct ure A i E K in such a way t hat, for i ::; j in J , Ai is a substructure of A j , t hen t he union A = U iE! A i -as defined in Cha pt er 1- is still in K . In t his case , every A E K embeds in some structu re of E(K ); in particular, E(K) is not empty. Notice also t hat, in t he examples above, K is elementary, as well as E(K) . Accordingly one ca n wonde r whether , in general, if K is eleme ntary, t hen E(K ) is . The an swer is negative. Examples 3.5.5 (a) If K is t he -element ary- class of groups, t hen E(K) is not elementary a ny more (thi s is a resu lt of Eklof a nd Sa bbagh ). (b) If K is t he -elementary- class of com mutative rin gs, t hen E(K ) is not elementary any mor e (as shown by Cherlin) . The pr oofs mix compactness and some algeb raic facts. We shall provide t heir det ails at t he end of t his section. It should be emphasized that , while e. c. fi elds a re fully characterized in a first ord er way, e. c. rings a re not . T he same ha ppens for gro ups. Conversely, a mong ab elian groups, e. c. structures are a n elementary class: t hey are exactly t he divisible ab elian groups admitting infinitely many elements of period p for every pr ime p ; t his is shown in [41], where e. c. closed modules over suitable rings are also discussed . However , take an element a ry class K ; let T denote its t heory. If E(K) is element ary, t hen T * = Th( E(K)) is said t o be a model companion of T . Clea rly T ~ T * . Mo reover t he following result holds (generalizing wha t was observed in our start ing examples).

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Theorem 3 .5.6 (Eklof - Sabbagh) Let K be an elem entary class of Lstructures, T = Th(K), T* be an L-theory containing T . Then T* is a model companion ofT (and £(K) = Mod(T*)) if and only if

(i) for every A E K, there exists some B E M od(T*) where A embeds zn; (ii) T* is mod el complete. Existentially closed structures, and model companions, were intensively studied in the sixties and in the seventies. Several -elementary- algebraic classes K were considered , to check whether existentially closed structures in K were or not an element a ry class, to provide a satisfactory characterization in the positive case, and to analyse their complexity in the negative one. This was not (and is not) a barren and unproductive exercise: in fact, it brought to light some very interesting classes of (existentially closed) structures, such as differentially closed fields. We shall treat them (and other key examples) in the next sections; but we want to emphasize since now that th e notion itself of differentially closed field -a quite algebra ic conce ptarises for the first time within the fram ework of e. c. structures: no algebra ic treatment preceded the model theoretic approach. Moreover, it should be pointed out that new interesting elementary classes of existentially closed structures hav e been considered quite recently, for instance among fields with a distinguished a ut omor phism. This fram ework , too, will be explained later in this chapter. Also in the negative case, when existentially closed structures cannot form an elementary class , their analysis has some intriguing features. The main purpose here is to understand the reasons of nonelementarity and, hence, in some sen se , to measure how complicated the class of e. c. objects is. To illustrate this point , we show, as promised, that existentially closed groups are not an elementary class, and the same is true for existentially closed commutative rings (with identity), and we discuss briefly these negativ e results. First let us deal with groups. Theorem 3.5.7 (Eklof - Sabbagh) Let K be the -elem entary- class of groups. Th en £ (K) is not elem entary.

Proof. We work in the language L = {-, facts. 9 denotes an e. c. group.

-1 ,

I}. We need two preliminar y

Fact 3.5.8 For every positive integer n, 9 admits some elements of period n.

3.5. EXISTENTIALLY CLOSED STRUCTURES

107

Proof. Tak e t he direct product 9' of 9 a nd a cycli c group of order n. Clearly 9' has some eleme nts of period n , in other words it sat isfies the existenti al sentence 3w(w n = 11\!\d/ n,d=/=n'(Wd = 1)). As 9 is e. c. , 9 satisfies the same condit ion, hen ce it has some elements as required. Fact 3.5 .9 Two eleme nts of infini te period in

9 are conjugate.

Proof. This is a more delicate point , and refers to a theorem of Higm an , B . H. Ne umann a nd H. Neu mann , building, for a ny two eleme nts a a nd b of infinite period in 9 , a group 9' extendi ng 9 a nd a n eleme nt t E C' such that taCl = b. So 9' F 3w(wa w- l = b). 9 inh erits this property because it is existentially closed . No w ass ume £(K) eleme ntary, £(K) = M od(T) for a suitable L-theory T . Fact 2 says that , in a lan guage L' extending L by t wo new con stant symbols c and d , T U {,( en

= 1),

,(dn = 1) : nE N, n > O}

F 3w(wcw- l = d).

By com pactness t here is some posit ive int eger N for which

T U { , (en = 1), ,(dn = 1) : 0


~ N}

F 3w( wcw- l = d) .

Hence take an e. c. group 9 and two elem ents a, b in G of period N + 1, N + 2 respectively : owing to Fact 1, this can be done. Then a and b are conjugate in C: but this clearly contradicts the fact that their periods a re differe nt.

'"

Actually ex istent ially closed gr oups are a ver y complicated class: [91] contains several results illustrating t heir wildn ess. In particular, we like t o mention a noteworthy connection wit h t he word problem for groups. Theorem 3.5.10 (Macintyre, Ne uma nn) A finit ely generated group can be

presented with a solvable word problem if and only if it is embeddable in all e. c. groups. Now let us deal with rings. Theorem 3.5.11 (Cherlin) Let K be the -elem entary- class of commutative

rings with identit y. Th en £(K) is not eleme ntary. Proof. T hroughout t he proof, ring a b brev iates com mutative ring with identity. Accordingly we work in the langu ag e L = {+, -, " 0, I}. Again , we need two pr eliminary facts. R denotes an e. c. ring.

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Fact 3.5.12 Th e nilradical of R (i. e. the ideal of nilpotent elements) is 0-definable in R in a un iform way (valid in any e. c. ring) . P roof. We claim that, for every a E R , a is nilpo t ent if and only if a doe s not divid e any non zero idempotent element . This is clearl y enough for our pu rpo ses, because the latter condition can be easily written as a (n existent ial) first ord er formula in L :3w(v · w

= v 2 • w 2 1\ ,(v · w = 0)).

The im plication from the left to the right is quite simple. For, let a divid e some idem pot ent e E R ; as an = 0 for some integer ti 2:: 2, it follows o = en = e. Conversely, ass ume that a is not nilpotent. Form the quotient ring R' = R[x]/I where I is t he ideal generated by the polynomial ax-a 2 x 2 • Then R' ext ends R in the obvious way, by the embedding of R in R[x] and the project ion of R[x] onto R' . In R' t he image I + a of a divides t he idempotent 1+ ax; moreover 1+ ax =f 0, otherwise in R[x]

(*)

ax = (ax - a 2 x 2)f( x)

for some f( x) = L i
a nd , event ually,

= a f i+l

Vi = 1, . .. , n - 1,

a 2 in = O.

In particula r a n +2 = 0 -a contrad ict ion-. So t he image of a in R' divides a non zero idem potent ; as R is e. c., a satisfies t he same cond ition in R . Fact 3.5.13 For every positive integer n, R contains some nilpotent a satisf yi ng an+! = 0 and an =f O. Proof. Again enla rge R t o suitable quo tients of the polynomial ring R[x] . This ti me, for every positive integer n, form R' = R[ x]/ I where J is the ideal generated by x n +1 • On e easily checks t ha t R embeds in R' by t he map a 1---7 I + a for every a E R ; moreover (I + x )n+l = I , while (I + x )n =f I. Hence R' satisfies :3w(w n+1 = 0 1\ ,(wn = 0)). As R is e. c., t he same is true in R.

Now we ca n conclude our proof. Again we use compactness. Enlarge L by a new const ant symbol c a nd, in t he new language L ' , look at T ' = Th([(K)) U {,(cn = 0) : n E N , n > O}U

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3.6. DCFo

Uf 3w(c· w = c2 • w 2 1\ --,(c· w = O))}. Every finite s uch t hat

T~ ~

T ' has it s own mod el: for , th ere is a positive integer N

a nd t he latter set of sentences do es admit a mod el: it s uffices to take a n e. c. ring and interpret c in a nilpotent eleme nt a satisfying aN =f:. 0 (Fact 2 ensures that a exists ) . By compact ness T' has a mod el: this is a ring sat isfying the same sente nces as e. c. rings , but it is not e. c. , because it contains a nonnilpotent element (the interpretation of c) div iding some nonz ero idempotent. In conclusion, E(K) is not eleme nt a ry, because its theory has non e. c. mod els. .. Notably, nilpotents ar e t he key obstacle t o the non elementa rity of the class of e. c. rings. In fact , for th e restrict ed class K of redu ced rings (i. e. rin gs without nonz ero nilpo tents) E(K) is elementary. On t he other side, if one allows nilpot ent elements, even of bounded ex ponent (for inst anc e one conside rs rings whose nilpo tents a satisfy a 2 = 0), t hen t he element arity of e. c. object s gets lost [164].

3.6

D C Fo

A differen tial ring is a com mutative ring K with identi ty, having an additional 1-a ry operation D (called derivation ) such th at

D(a+b ) = Dc v Db, D(a · b) = a· Db + b · Da for every a and bin K. A differential field is a differential ring which is also afield . So a suitable first ord er language L' for differential fields enlarges our language L for fields by a new 1-ary operation symbol (to be represented by D). Differential fields include

• (K , D) where K is a ny field and D = 0, bu t also more significant and int erestin g st ruct ures, like

• (K(x ), l;J where K is any field , K (x ) is t he field of rational fun ctions wit h coefficients from K in t he unknown x , an d op eration ;

txis t he deri vative

HO

CIIA PTER 3. MODEL COMPLETENESS • for any nonempty connected open subset U of C, the field of meromorphic functions from U to C , with respect to the usual derivative.

Differen tial fields and rings were introduced by Ritt in the thirties, and differential algebra developed greatly since then . Now classical and excellent references, like [76] or the nimbl er [67], expound its foundations. For our purposes in this section, we just need a few algebraic crumbs on differen tial fields (/C, D) . First of all , notice that the usu al derivation rules for power s and quotients, like D(a n ) = na n - 1 Da for every element a in I< and every int eger n , still hold and can be easily deduced from the definition. Moreover, the elements a E I< such that Da = 0 form a subfield of /C , called the constan t su bfield and denoted C (/C). Instead of the usual (algebraic) polynomials f(x) E I<[x], now differential polynomials are considered: they are algebraic polynomials in the unknowns

x , Dx , .. . , Dnx, ... where n is a natural, and form a differential ring /C{x} with respect to the obvious op erations. For every non-zero differential polynomial f (x) E I< {x} we can define • the degree of f( x) (with respect to x , or Dx, a nd so on) but also • the order of f(x): thi s is the maximal natural n such that Dn x occurs in f(x) , if there is some n with this property, and -1 otherwise, so when f(x) = a E K (clearly one agrees DOx = x). Differential polynomial rings in more variables /C{i} are introduced in the same way. What is the role of model theory in this setting? As we shall see in a few lines, it is quite relevant , mainly with respect to existentially closed structures. Actually differential algebra did not provide, before model theory, any notion of differentially closed field -something resembling algebraically closed fields among fields, or real closed fields among orderedfields-. But the interest in existentially closed structures , and hence, particularly, in existentially closed differential fields , led A. Robinson to consider this question from the model theoretic point of view.

3.6. D CFo

111

Hence take t he -elernenta ry- class of differential fields in our language i: We wonde r if t he class of existentially closed objects is element ary, in oth er words if t he t heory of differ ential fields has a model companion. At t his poin t it is advisa ble for us t o distinguish in our t reatment t he characterist ic 0 case from t he prim e charact eristi c case . We shall devote t he next sect ion to t he lat ter ; hence, now we limit ourselves t o differ ential fields of characterist ic O. In t his restrict ed framework, A. Robinson poin ted ou t Theore m 3 .6 .1 (A. Robinson , 1959) Th e theory of differential fi elds of characte ris tic 0 has a model com pani on, and thi s eliminates the quantifiers in the language i:

This mod el companion was denoted DCFo; its mod els were called differen ti ally closed fields (of charact eristic 0) . Unfortun ately, A. Robinson 's appro ach was not a ble t o determine any incisive first ord er ax iomat izat ion for D C F o. Actu ally, wh at was lacking at t hat time was a suit a ble background . In fact , in t he paradigm ati cal cases of AC F and RC F, th e notions t hemselves of algebraically closed field and real closed field clearl y prec eded A. Robi nson 's t reatment, a nd t he ex iste nce and uniqu eness of a n algebra ic clos ure a nd a real closure played a key role in t he model completeness proofs. On t he cont rary, in t he different ial case, t he concept of differenti ally closed fields was ju st rising wit hin t he model t heoretic a pproac h, a nd, consequent ly, nothing resem bling a differen tial clos ure of a differenti al field (as a minim al different ially closed extension) was known , even in cha racterist ic O. Acco rdingly one had to wait for new significa nt progr ess in model theory, mainl y du e to Michae l Morley, befor e over comin g t his algebraic gap. We shall refer in det ail Morley's ideas, and their effects for differentially closed fi elds, in C ha pte r 6. But we wish to anticipate here a short repo rt on th e end of th e affair (so far) . In particular , we want to recall t hat in 1968 L. Blum , in her PhD thesis, found an elega nt and nice axiomatization of DCF o. What she showed was Theo rem 3.6 .2 (1. Blum) A mong differential fi elds (K , D) in characteristic 0, the existe ntially closed objects are exact ly tho se satisf ying the follo wing property :

(*) fo r every choice of f (x) an d g (x ) in K{ x} - {O} such that the order of f is larger tha n the order of g , there is some a E [( fo r whic h f (a ) = 0 but g (a ) f. O. It is easy t o express (*) in a first order way by infinitely ma ny sentences in L', L. Blum 's work is described in Sacks's book [146]: she confirmed

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112

qu an tifier elimination, model com plet eness (a nd completeness) of D C F o; but her analysis went fa rt her a nd , com bined with a quite abstract mod el t heoretic result of Shela h, yielded Theorem 3.6.3 Any differential fi eld (K, D ) of charact eristic 0 has a diff erential closu re (a minimal differentially closed exte nsion), and this is unique up to isom orphism fi xing I< pointwise. We shall refer in mor e det ail on this poin t in C hapter 6. Bu t we wan t to emphas ize that , at last , differen tial closures do exist and are uniqu e (in cha racteristic 0) , and t he model th eoretic approac h of A. Robinson and L. Blu m was the very first not only in introducing differentially closed fields, but also in showing th ese existence and uniqueness result s. Of course, some myster y still per sists. Most not abl y, it is surprising, a nd perh aps regrettable, to lea rn t hat , presentl y, no exp licit example of differenti ally closed field is known . However , we conclude t his section by discussing some minor bu t useful points. F irst of all, notice that every different ially closed field (K , D ) of cha racteristic 0 is algebraica lly closed : t o see t his, j ust a pply Blum 's t heo rem t o t he par t icular case when f (x ) E I< [x ], g (x ) = 1. Similarly t he constant field C(K) is algebraically closed (we will see why in Chapter 6) . Second ly, what ca n we say a bout definabl e sets in a differentially closed field (K , D ) of cha racterist ic O? Here

(*) t he basic definabl e sets a re, of course, t he sets of roots of finite systems of different ial polynomials in I< {i} ; t hey are j ust t he closed sets in a Noether ia n topology (t he Kolchin topology) in K " (where n is t he length of i);

(*) the Kolchin constructible sets -i. e. th e finite Boolean combinat ions of Kolchin closed sets- are st ill definable, of course;

(*) owing to the elimination of qu an tifiers, no further definabl es occur; in oth er words, definabl e j ust means Kolchin constructible.

3.7

SCFp and DCFp

As promi sed , we want to deal here wit h differenti ally closed fields in pri me cha racteristic p. Bu t our t reatment needs a pr elimi na ry remark. Indeed we

3.7. SC Fp AND DCFp

113

und erlin ed in the previo us section t ha t differentially closed fields of characterist ic 0 are algebra ically closed as well. This ca nnot hold any more in the prime characterist ic case. In fact , t ake any differential field (K , D) in cha ract erist ic p: notice that I

Va E

I<.

Now a ny algebr aica lly closed field of characteristic p is p erfect ; in other words, every element can be (un iqu ely) expressed as a p-th power. It follows that no differential field of characterist ic p ca n be algebraically closed, except the trivial case when th e der ivation D is identically O. In particular we cannot expect that the underlying field of a differentially closed field is st ill algebraically closed. This remark threatens some major complications with resp ect to the cha ract erist ic 0 case; a nd a nyhow suggests a preliminary analysis on some possible weak closure notions for pure fields K in cha racteristic p, compatible with I< =1= to: Accordingly, t ake a field K (in a ny cha racte rist ic) . A polynomial f (x) E I< [x] of degree 2: 1 is said to be separable if and only if f (x) has no multiple roots (in the algebraic closure of K). When K has characterist ic 0, every irr ed ucible polynomial f (x) E I<[ x] is also separable. Otherwise a multiple root a of f( x) annihilates also th e formal derivative f'( x) of f( x) -still a polynom ial in I<[x]-, an d hence t he greatest com mon divisor q(x) of f( x) and f'( x) (in I<[x]). So q(x) is not constant; however the degr ee of q(x) is st rict ly smaller than the degr ee of f(x ), becau se it is less or equal to the degree of f'(x). Hence f( x) is reducible. Now assume t hat K has a prime characteristic p. Recall t he Frobeniu s morph ism Fr in K (and in every extension of K) , the one sending any element a into its p-th power a", Fr is injective. This t ime, I< [x] may include some irreducible non- separable polynomials: for instance , given a E I< - I

is irreducible, but has a uniqu e root in the algebra ic closure of K becau se Fr is 1-1; notice t hat f'(x) = o. We say that K is separably closed if and only if every sepa ra ble polyno mial f( x) in I<[x] has a root in K . Separably closed fields K form an elementary class in the langu age L of fields; in fact , they are axiomat ized by t he infinit ely many sentences saying th at every polynom ial f( x) with a non-zero derivative has a root in K .

CHAPTER 3. MODEL COMPLETENESS

114

However, in characteristic 0, separably closed just means algebraically closed. On the contrary, in a prime characteristic p, there exist some separa bly closed, non-algebraically closed fields; indeed one shows th at a field J( is algebraically closed if and on ly if J( is sepa ra bly closed a nd

Fr is onto (that is,

J( = J(P is perfect).

If this is not th e case, we are led to consider the field extension J( :2 J(P a nd hence J( as a vectorspace over J(P . A set B <;;;; K is ca lled a p- basis of J( over J(P if the monomials (when n , eo, ... , en range over natural numbers, aa, .. . , an a re pairwise distinct element s in Band e; < P for every i ::; n) form a basis of J( over J(P (as a vectorspace) . One shows that a p-basis B always exists and it s cardinality depends on ly on J( : it is called the imperfection degree of J(. The model th eory of sepa ra bly closed (possibly non algebraically closed) fields J( of prime ch aracteristic p was investigated by Ershov in 1967 [46]. Let se Fp denot e t heir t heory. Of course, se Fp is not complete, because t he im perfect ion degr ee of these fields may change, and suitable first order formu las in the language offield s ca n express it s valu e, when finite, or wit ness its infinity otherwis e. But E rshov observed that this imperfection degree is the on ly obstruction to completeness; in fact, he showed Theorem 3 .7 .1 (Ershov) Let J( be a separa bly closed field of prime characteristic p . Then the elemen tary equivalence class of J( is fully determined by its imperfection degree, if finite , or by the fact that this degree is infinite, oth erwise.

Ershov's proof shows (an d uses) model completen ess in a suit a ble enriched language capturing the notion of p-b asis. Notice that sep arably closed field s do have some noteworthy algebraic connect ions with differential fields. For instance, given any sepa ra bly closed field J( and a p-basis B of J( , one can see t hat a ny function 0 from B to K enlarges uniqu ely to a derivation D of J( , a nd so equips J( with a structure of differential field . This relationship between separa bly closed fields and differential fields in prime characteristic p gets stronger if we enter t he model theoretic framework a nd we consider diffe re ntially closed fields. But, before providing more details, we have to explain what a differentially closed field in characteristic

3. 8. A CFA

115

p is. Ind eed , we still have t o clarify if, even in this se t t ing , existe ntially closed objects a re an elem en t ary class: so far , we have simp ly realized t hat the an alysis should be more complicated t han in characteristic 0, becau se no existe nt ially closed differ en ti al field ca n be alge braically closed . Well, the answer is again positive: t he theory of differential fields of characteristic p do es admit a model companion. This is the cont ent of a theorem of Carol Wood , who also found a nice axiomatizat ion of existe nt ially closed objects.

T he orem 3. 7. 2 (C . Wood ) A differential fi eld (/C , D ) in characte ristic p is existentially closed if and only if

(i) C(K) equals K P, (ii) fo r every choice of two differential poly nomials f (x ) an d g (x ) in K { x } - {O} such that the order of f (x ) is greater than th e order of g (x ) and the formal derivat ive of f (x ) with respect to D" x is not 0, th ere is some a E K such that f(a) = 0 and g(a) =1= o. The model com panion of the t heory of differen ti al fields in characteristic p is usually deno t ed DC Fp; it s models -h ence t he exist ent ially closed differ enti al fields- a re again ca lled differen tially closed fields. Notably, t hey a re se parably closed: t his is implicitly said in (ii), pro vided t hat we restrict to pol ynomi als f( x) of ord er 0 and we t ake g( x) = 1. Moreover their imp erfection degree is infinite, because t he dimension itself of /C over C (K ) = K P is infinite: t o see this, just use (ii) again and take, for eve ry positive int eger rn , an eleme nt m- 1 X =1= 0; noti ce t hat t he xm's a re X m in K satisfying D m x m = 0 and D m linearly ind ep endent over C( K) (this is an easy exe rcise) . DC Fp is mod el com plet e a nd com plete, bu t does not eliminate t he qu an tifiers in the language L' : quantifier eliminat ion needs a larger setting, with a further op eration ext racting p-th roots when possibl e, and valuing 0 otherwise. Again, ge neral resul t s of pure model t heory ens ure t he exist ence a nd uniqu eness of a differen ti al closure for differen ti al fields (/C, D ) satisfying C( K) = K P.

3 .8

ACFA

In t his section we deal with difference fields. These are struct ures (/C, 0" ) where /C is a field and 0" is a distingui sh ed (su rjective) a utomorphism . More

CHAPTER 3. MODEL COMPLETENESS

116

generally, difference rings ca n be introduced in t he same way. Difference fields include som e t rivial examples like

(!C, id) for every field !C bu t also

(!C , Fr ) where !C is a perfect field of pr ime cha racterist ic p and Fr is t he Frobeniu s morphism in !C: for every a E I< , F r( a) = al', Notice t hat t hese examples include an y finite fi eld !C. At a super ficial sight, this set t ing resembles different ial fields. In both cases t he underlying field is enr iched by a l -ary op eration, and the on ly dissonance concerns the rules that this new function has t o sat isfy: the derivation laws in th e differenti al case, a nd t he morphism laws pr esently. Of course, t his connection is shallow, a nd sharp distin ctions arise in examining t hese structures in a deeper way. Nevertheless some (minor) simila rit ies persist . Fo r inst an ce, given a difference field (!C,
Fix(
x,
3.8. ACFA

117

But our main qu estion is how t o characterize existent ia lly closed difference field s. Is their class axiomatiza ble? In other words, do es the theory of differ ence fields admit a model companion ? We sho uld em phasize t hat this interest in existent ially closed difference fields is com paratively rece nt. Ind eed , at t he begi nning of the nin eties, Macintyre, Van den Dries a nd Wook showed t heir eleme ntarity an d fou nd a nice (although no n trivial) axiornatization: t his is report ed in [95]. In detail Theorem 3.8.1 A difference fiel d (K , a) is existen tially closed if and only if

(i) K is algebraically closed; (ii) for every irreducible affine variet y U over I< and fo r every subvariety V of U X a( U) su ch that both the projections of V in to U and a(U) are dense with respect to the Z ariski topology, there is a point a of U (o ver I<) for whic h (a, a (a)) is in V . Expressing (ii) in a first or der way is not im medi ate, becau se it requires, after all, to qu an t ify with res pect t o irreducible va rieti es. However this can be done, owing t o general bou ndedn ess resul t s for (algebraic) polynomials over fields [172]. The t heory of existent ially closed difference fields is denoted ACF A ; their models are called algebraically closed fields with an autom orphism, although this name is a lit tl e misleading, and we have to be ca reful a bout it. So recall that they are no t sim ply algebr aically closed fields enriched by any arbitrary automorphism , bu t j ust existent ially closed structures among fields wit h a n automorphism: (ii) has t o be resp ected . Of course ACF A is model complete, as a model companion . But ACFA is not com plete. However one shows t hat the key features fully det ermining t he elementary equ ivalence class of a model (K , a) of ACFA are

(*) t he characteristic of the underlying field K , an d

(*) the action of a on the algebraic closure of t he prime su bfield (up to isomorphism). The interest in ACF A is also related t o its role t owa rds a model theoretic proof of classical qu estions in Algebraic Ge ometry, like t he Manin-Mumford Conjecture. We will discuss t his point in Ch apt er 8. However t hese intriguing connections led to a syste mat ic st udy of AC F A, mainly pursued by

CHAPTER 3. MODEL COMPLETENESS

118

Chatzidakis and Hrushovski in [20], and later by Chatzidakis, Hrushovski and Peterzil in [21]. Here we limit ourselves to some very basic information . First of all, it turns out that , for any algebraically closed field with an automorphism (K, 0"), the fixed subfield Fix(O") is pseudofinite, so it satisfies the axioms of finite fields, but (consequently) it is not algebraically closed. What about definable sets? • First one meets the zero sets of finite systems of difference polynomials in K(X';; they are, again, the closed sets in a Noetherian topology for K" (where n is the length of x). Fix(O") is an example, because it is defined by O"(v) = v. • Secondly, one has to include the constructible sets in this topology, i. e. the finite Boolean combinations of closed sets. • However, this is not enough, because quantifier elimination fails for ACFA in its original language. So definable does not imply constructible. Here is a possible counterexample. We work inside an "algebraically closed field with an automorphism" (K ,O") of characteristic 0 and suitably large cardinality. Due to this assumption and existential closedness, we can find inside K two elements a and b transcendental over the prime subfield Q, a and b in Fix(O") , and two square roots a' and b' of a, b, respectively, such that

O"(a') = a',

O"(b') = -b'.

a and b satisfy the same quantifier free formulas, because they are fixed by 0" and f (a), f (b) i= 0 for every nonzero polynomial f (x) over the rationals. Nevertheless

O"(V) = v

A

:3w(w 2 = v A --'(O"(w) = w))

holds for b and not for a, and hence defines a non-constructible set. Of course, owing to model completeness, the projections of constructible sets exhaust the definable ones. Of course, owing to model completeness, the projections of constructible sets exhaust the definable ones. A final important remark. We saw that no example of differentially closed field (even in characteristic 0) is known so far. This is not the case for ACFA. In fact, Hrushovski and, independently, Macintyre built some models (K, 0") of ACFA explicitly; they are obtained by considering suitable pseudofinite fields with a rather natural extension of the Frobenius morphism.

3.9. REFERENCES

3.9

119

References

An excellent introduction to mod el completeness is in Macinty re expository pap er [94]. Bu t also t he A. Robinson book s [141] and [143] a re still an enjoyable reading. Hilber t Nullstellensatz and Hilbert Seventeent h Problem a re described in [94]; p-adi c numbers ar e also t he mat t er of a (particular ) chapter in [36] ; the Ax-Kochen-E rsov theorems are in [4, 5, 6] and [44, 45], while Macinty re's qu an tifier elimination results on p-adics are in [93]. A very goo d refer ence for existentially closed st ruct ures (and mod el complet eness, too) is Ch erlin 's book [23]. See also the Eklof-Sa bbagh paper [41]. Macintyr e's analysis of existe nt ially closed groups is in [91], and Cherlin 's treatment of e.c. commutative rings in [22]: in th e latter set t ing, the role of nilpotent elements is also discussed by Carson [17], Lipshitz-Saracino [86], a nd Macinty re again [92], who observed that the t heory of commutative rin gs without nonzero nilpo tent elements do have a model companion. See also [164]. Differenti ally closed fields are introdu ced in [142], but t heir mod el th eor y (in characteristic 0) is principally develop ed in L. Blum 's Ph.D . Thesisi [12], also rela t ed in t he Sacks book [146]. By t he way, both [76] and [67] pro vide complete t reatme nts of t he basic differenti al algebra . T he Ers hov model t heoretic ana lysis of sepa ra bly closed fields is in [46], while C. Wood 's t heorem on differenti ally closed fields of prim e cha racterist ic is in [179] a nd , wit h resp ect to prime mod els, in [180]; t he Marker a nd Mess mer contributions t o [110] provide a complete a nd excellent ex posit ion of t hese matters. T he Maci ntyre - Van den Dri es - Wood axiomatization of existent ially closed fields with an automorphism is related in [95]; t heir lar ge mod el theoretic a na lysis du e to Chat zidakis , Hrush ovski and (later) Pet erzil is in [20] and in [21]. T he Hru shovski and Macintyr e explicit example of an e.c. field with a n a ut omorphism ca n be found in [88] and [58].

Chapter 4

Elimination of imaginaries 4 .1

Interpretability

Let A be a structure for a language L. We already dealt with definability in A when we introduced definable sets and , mor e generally, definable struct ures in A (see exa mple 1.7.3 , 4) . The latter are th e st ruct ur es A' for a language L ' (possibly different from L) such that both t he universe A' of A ' a nd t he interpretations of sy mbols of L ' in A' are definable in A . As in t he case of sets, we can introduce also the concep t of X -de finable str uct ure for X ~ A. In t he quoted exa mple, we observed t hat (N, + ,. ) is a st ruct ure definabl e in (Z , +,.). Here we provide some fur th er examples. Examples 4. 1. 1 1. Let L = { x , e,- l } be th e la nguage for groups, 9 be a group. The centre Z(9) of 9 is t he set of elements of G commuting with any element of G, and so it is 0-definable as a set, by th e formula

Vw(v x w = w

X

v) .

But Z( 9) is also a subgroup of 9, an d hence a gro up wit h respect to t he restrictions to Z (9) of t he op erations of 9. Clearly t hese operations a re 0-definable in 9 . It follows t hat Z(9) is 0-definab le in 9 also as a group (and hence as an L-structure).

2. Let L = {a, 1, +," - } be t he language for fields, IC be a field. Let us conside r, in t he la nguage L ' = { x , e,- l } for groups, th e linear group of degree n (n a positi ve int eger) over IC

GL (n,IC ), 121

122

CHAPTER 4. ELIMINATION OF IMAGINARIES in other words the group of n X n matrices with entries in J{ and determinant i= 0, with the usual row-by-column product . Then GL(n, K) is a structure 0-definable in K. In fact, the n X n matrices with entries in J{ can be viewed as tuples 2 if = (aijkj in J{n • So the element s of GL(n, K) are exactly those 2 tuples in J{n satisfying -,(det if = 0) (here if abridges (Vij kj): recall that det is a homogeneus polynomial of degree n with coefficients in the substructure of K generated by 0, 1. The multiplication is defined by ifxw=z

n

1\

(zij=LvihWhj) , Is i,jsn h=l

the identical matrix In by

1\

Vii

l
= 1 /\

1\

Vij

Is i,jsn,i#j

= 0,

and, finally, the inverse operation by if

= w- 1

:

"if

X

w

= In"

.

But Algebra deals not only with substructures, but also with quotient structures. For instance, in the examples quoted before, one can observe what follows. 1. Look at the quotient group 9/Z(9). As Z(9) is 0-definable in equivalence relation in G

9, the

whose equivalence classes are just the elements of 9/Z(9) is also 0definable. It follows that 9/Z(9) , as a quotient set, but also as a quotient group, "lives" in 9. 2. The special linear group of degree n over K

SL(n,K) (a subgroup of G L( n, K)) is 0-definable in K: just consider the formula det(if) = 1.

123

4.2. IM A GINARY ELEMENTS

Accordingly, ju st as in t he prev ious exam ple, t he quotient group GL (n, K ) j S L(n, K ) ca n be recovered in K by a suitable 0-defin abl e equivalence relation. On th e ot her side G L (n, K) j SL (n , K ) is isom orphic to t he mult iplicative gro up K * of non zero elements of I< , and K * itself is dir ectl y 0-defina ble in K (as a gro up, wit ho ut involving a ny quotient construction) . These remarks int rodu ce the following Definition 4.1.2 Let A be a structure f or L . A structure A' f or L' is said to be interpretable in A if and on ly if th ere exist a positive integer n , a su bset S ~ An defi nable in A , an equivalence relation E over A n definable in A such that A' = S ] E and

(i) f or eve ry costant sy m bol c of L', { a A;

E An : alE = cA'} is defina ble in

(ii) for every k-ary function sym bol f of L' , {(al , " " a~ , a) E A n(k+l) : fA' (aIIE, ... , aklE) = alE} is definable in A ; (iii) for every k-ary relation symbol R of L' , {(al , . . . , ak ) E A nk : (aIIE, . . . , ak lE) E RA' } is defi nable in A . T he conce pt of X-interpretable struct ure (for X ~ A) is defined in t he usu al way. Eve ry structure defina ble in A is interpret abl e in A (t hrough t he equality relation) . Example 4.1.3 GL(n , K ) j SL (n , K ) is 0-interpretable in K .

4.2

Imaginary elements

The examples of t he previous section show that, in a given L-structure A , on e meets not only

real elements (t hose of t he do main A) , but even

im aginary elem ents, t. e ,

eq uivalence classes of 0-definable equivalence relations E .

T he following technique, essent ially d ue to Shela h, shows how to expand in a natu ral way t he str ucture A (and its lan gu age L) in order to make t he imaginary elements real.

124

CHAPTER 4. ELIMINATION OF IMAGINARIES

Definition 4. 2 .1 U q is the language obtained by L taking, f or eve ry equivalence relation E over A n (f or some positive integer n) 0-definable in A • a 1-ary relation symbol A E (and among these also A =);

• an n-ary fun ction sym bol 7rE .

Definition 4 .2 .2 A eq is the structure for

u q su ch

that

• the universe of A eq is the disjoint union of the interpretations in A eq of the relation symbols A E; • f or ever y equivalence relat ion E, AE is interpreted as the quotien t set An j E (in particular A= as {{a} : a EA} , which can be canonically identifi ed with A) and 7rE as the natural proj ection from A n onto A njE; • the sym bols from L have the same inte rpretation as in A (aft er iden tifying A and the interpretation of A=).

On e ca n check that , if A and B are st ructures for L and A == B, then A eq == B'", This allows t o define , T ' " = Th (A eq ) when T = Th (A). Many significant properties of T a re preserved under passing from T to T '", On t he other hand , in T '" = Th (A eq ) t he imaginary elements of A get real. For example, t ake a positive integer n and the relation = in A n; = is 0-defin able, and hence, for every a = (a1 ' ... , an) E A n , t he class {a} of a modulo =, is a rea l element of A eq; so we ca n view a ny n-tuple in A n as a real element. Indeed it is a common agreement in Model Theory to consider the tuples from a struct ure A as elements of A: t hey are imaginary elements in A , and hence real elements in A eq. Som etimes imaginary elements ca n be naturally ident ified with real elements of A (or wit h finit e seq uences of real elements of A ), and hence referring to A eq is no more necessa ry. Let us pro pose a simple exa mple. Example 4 .2.3 Let K be a field. We alr eady pointed out that GL(n , K)j SL(n , K) is isomorphic to the multiplicative group J(* (0-definable in K). It follows that t he imaginary elements SL(n , K)a with a E GL(n , K) can be ident ified with the real elements det( a) of J( * , by th e determinant fun ction (more precisely by the isomorphism from GL (n , J()j SL (n , K) to J( * induced by det). In general:

125

4.2. IMAGINARY ELEMENTS

Definition 4.2.4 A structure A for L has the elimination of imaginaries if and only if, for every equivalence relation E over An (n a positive integer) 0-d efinable in A and E An, there are a formula r.p(v, w) of Land a unique sequence b in A such that

a

A structure A for L has a uniform elimination of imaginaries if and only if, for eve ry E as above, there exist a fo rmula r.p (ii, w) of L and, for each a E An, a unique sequence b in A such that

Theorem 4.2.5 Let A be a structure for L . Th en the follo wing proposit ions are equivalen t.

(i) A has a un iform elim ination of im aginaries; (ii) for every positive integer ti and equivalence relat ion E over An 0definable in A , there is a map Fi; 0-d efinable in A, with domain An and range ~ A m (for some posit ive integer m), such that, Va, a' E An, a, a' are equivalent in E

~

FE(a) = FE(a' ).

Hence, if A has a uniform eliminat ion of imaginaries , th en , for every equivalence relation E as a bove, t he equivalence classes of E ca n be thought as tu pies of real eleme nts of A , provided we ide nt ify, Va in An,

alE

wit h

FE(a).

It follows that referring t o A eq is not necessary, because what is 0-interpretable in A is even 0-definable in A.

Proof. (i)=?(ii) For each a E A , define FE(a) as the unique element that alE = r.p (An, b) . (ii)=?(i) Take r.p (v, Z) : "FE(V) = i" e b = FE(a). .,

b s uch

T he t heory of elimina t ion of imaginaries was essentially developed by Poizat. Actually Poi zat's treat ment is slight ly different from ours. They do coincide for st ruct ures admitting at least two constants, or even two distinct definable elements (see [131], Theorem 16.16). As we are mainly interested in fields, and fields clearly sat isfy thi s condit ion, we ca n proceed with no a nguishes.

126

4.3

CHAPTER 4. ELIMINATION OF IMAGINARIES

Algebraically closed fields

Notably, many famili ar struct ures admit a uniform eliminat ion of imaginaries. Let us propose some exa mples. The first case we deal with concern s algebraically closed fields: t he aim of this section is ju st t o prove T heo r e m 4 .3.1 E very algebraically closed fi eld IC has a uniform elim ination of imaginaries.

The proof is a direct consequence of the following two lemmas. Lemma 4 .3 .2 If IC is an algebraically closed field, then IC has the elimination of imaginaries. Lemma 4. 3. 3 Let L be a language with (at least) two cons tant symbols, IC be an L-structure interpreting these two constant sym bols in different elements . If IC has the elim inati on of imaginaries, then IC has a uniform elim in ation of imaginaries.

Obviously an algebraically closed field IC satisfies also t he hypotheses of lemma 4.3.3: th e langu age for fields has two constant sy mbols 0, 1 int erpr eted in IC as two differ ent elements . Hence IC uniformly eliminates imagina ries (provided it eliminates t hem) . Now we show Lemma 4.3 .2. The proof we pro vide here uses t he minim ality of an y algebraically closed fi eld IC , hence t he fact t hat every definable s ubset of 1< is finit e or cofinite . An alternative a pproach (working even for differentially closed fields) will be produced in Chapter 6. Proof. Let n be a positive integer , E be an equi valence relation 0-defin abl e in K" : Let E (if, ill) indicate t he formula defining E . F ix a E K ", Con sid er the formula El (VI , ill) : :JV2. . . :JvnE (if, ill) .

El(lC , a) is a definable set, and hence by Corollary 2.4.8 is eit her finit e or cofinite . If E l (IC , it) is finit e, th en it contains only elements algebraic over

(th e subfield generated by) a; in fact t wo elements t ra nsce nde nt al over (t he subfield generated by) a a re linked by some au tomorphism of IC fixing every element of a; hence if one of th em occurs in E l (IC, a) , t hen t he latter is in E l (IC , a) as well, and hence all t he elements of IC t ra nsce ndental over a ar e in E l (IC , a), a nd E l (IC, it) is infinite. On t he other side, if E l (IC, a) is cofinite, t hen E l (IC, a) con tains a t least an algebraic element, because IC has finitely man y elements algebraic over t he subfield generat ed by a.

127

4.3. ALGEBRAICALLY CLOSED FIELDS

In a ny case we can choose

Cl

=

Cl

(it) E

E l

(1( , a) algebra ic over

a.

Now let

E 2 ( Cl , /C , it) is either finit e or cofinite, a nd as a bove we can pick out in E 2 ( Cl , /C , it) an elem ent algebraic over (a, Cl ), a nd hence over a. Repeating th e procedure we build c= ( Cl , • .• ,Cn) E E (J{n, a) such that Cl , ••• , Cn are algebraic over a. We can even form a finit e set X (a) ~ K " a-definable such th at c E X (a); it suffices t o consider t he set X (a) defined by

1\

"fi (V ) = 0"

l < i< n

where, for each i = 1, ... , n , fi(X) is t he minimum polynomial of Ci over t he su bfield generated by a. W ithout loss of generality, X(a) ~ E (J{n,a) (oth erwise substit ute X(a) for X(a) n E (J{n, a)). Sup pose X(a) = {c(O) , . .. ,c(m ) } where c= c(O) and c(j) = (Cl (j) , .. ',C n (j)) for every j :::; m. Then we have

E(J{n, a) = E(IC , c(j)) for every j :::; m . Recall th at we a re looking for a formula t.p (V, Z) and a uniqu e seq uence b such t hat

Then consider the following polynomial f (y , x ) E /C[y , X] (with x = (Xl , ... , Xn )):

n

f(y , x ) =

IT (y - L j Scm

c ; (j ) Xi ) .

i= l

Let b be the sequence of coefficients of f(y , x) (with respect t o some preestablished ordering) . It is clear that b is uniquely defined by X(a) = {c(O) , ... , c (m) } . Conversely, as /C[y ,X] is a un ique factorizat ion do main , given the sequence bof th e coefficients of f(y , x), in other word s given f( y, x), we know that f(y , x) deco mposes in at most one way as n

IT (y - L jScm

Ci (j) Xi)

i =l

(up t o permu ting factors) . So b lets recover X( a) = { c(O) , . . . , c (m ) } (alt hough it may not provide the single seq uences c (O ), .. . , c( m)) . Then we can

CHAPTER 4. E LIMINATION OF IM A GIN A R IES

128

pick out b as the sequ ence of coeffi cients of !(y , i) , and define 'P (71, i) in such a way that

'P (71, b) is th e formula n

Vw( "(y-

L Wi Xi ) i= l

Obviously 'P (71, i) depends on a.

divides j' (y, i)"

-7

E (71, w)).

...

Let us prov e now Lemma 4.3.3. Proof. Let 0, 1 be two differ ent constant symbols in L such that K F= .(0 = 1). We know t hat, if E is an equivalence relation over K" 0-definabl e in K , then for every a E K " there exist a formul a 'PCi (71, i) and a unique sequence b= b( a) in K such that

By compactness there exist a natural number h and t hat, for eac h a E K " , K

V ::J! zi V71(E(71,a)

F=

H

aa ,... ,ah

E K " such

'Pai (71,ii) ).

i< h

Wi thou t loss of generality, we ca n ass ume that Zo , ... , Zh have the same length m (oth erwise add some O's t o the shorter sequences) . Consider t he formul a 'P' (71, il, Z) (where ii = (uo , . . . , Uh) and Z has length m) conj unct ing: • the formula saying that , Vi :::; h , i :::; h such that Ui = 1;

Ui

E {O, I} and there exists a un ique

• the formu la saying t hat, Vi:::; h, u; = 1 if and only if 'Pai(71,i)) , z is the unique sequence Esuch that 'Pai uc-, Z) = 'Pai itc», i) and, Vj :::; h with j < i, it is not tru e t hat t here is a un ique sequence t such that 'Pai (K ", Z) = 'Paj (K ", i) . It follows t hat for ever y that

aE

F=

K'" , there exists a uniqu e

b in

I< m+h+ 1 such

'P' (71, b)). The formula 'P' (71, ii, i) depend s only on E. Hence K has a uniform elimin aK

t ion of imaginaries.

...

V71 (E(71, a)

H

129

4.4. REAL CLOSED FIELDS

4.4

Real closed fields

T he aim of t his section is t o show t hat real closed fields un ifor mly eliminat e imagina ries. Theorem 4.4.1 In every real closed field K imaginary elements can be uni-

formly eliminated. Proof. First of all, let us em phasize a relevan t proper ty of definabl e set s in real closed field s, esse nt ially related t o t heir o-minimality. Take any for mul a O(v , w) in t he la ngu ag e L = {+ , " - ,0 , 1, ~} of ordered fields. We kn ow t hat 'l9 (v, w) is (equivalent in ReF to) a finite bool ean combinati on of formulas

fn(w)vn+ .. . +f1(W) v+fo(w) ~ 0 where n is a natu ral number a nd foCi) , . .. , fn Ui) a re polynomials wit h int egral coefficients. Let q(y , x) = I:i
(*)

a mong

] - oo, so(v)[, ]sO(V) , S1(V)[ , . . . , ]Sr-1( V) ,+00[. It follows that in a real closed field K , for eve ry a in J( , 'l9 (K , a) is a finite union of intervals whose end po int s a nni hilate so me polynomials q(y , a); hence these intervals a re a-d efin abl e by a formul a onl y depending on 'l9 (v, w); even t he number of t hese intervals is uniformly bounded wit h resp ect t o 'l9 (v, w). Let b denote the seq uence of these end po ints: b dep end s on a, as said before, indeed every element in b is a-definable. At this point we may wonder how to determine, for a given a, the zeros so(a), . . . , sr-da) (wit h r = r(a) ~ n) belonging to O(K , a) a nd, a bove all, t he intervals a mong

] - 00, so(a)[, ]so(a), s1(a)[, .. . , ]Sr-1(a) , +oo[ contained in O(K , a) . Of course, this depends on t he sign (+ , - or 0) of the poly no mials q(y , a) in each point of b, a nd insid e t he intervals. T his can be chec ked directly for the roots, a nd choosing in each int erval a b-de finable (hence a-definable) witness c to test . T he latter operation ca n be eas ily done for every possibl e choice of the end points -00 ~ d < e ~ +00. For, take

4¥

c= if c = d + 1 if c = e - 1 if c=0 if

-00 < d < e < +00, -00 < d < e = + 00, -00 = d < e < +00, -00 = d < e = + 00.

130

CHAPTER 4. E LIMINAT ION OF IM A GIN ARIES

and notice t hat C is a-defi nable when d and e a re a-d efinabl e (or infinit e) . In particular, obser ve that we ca n pick a unique a-definabl e c in (} (/C , a): just choose t he lowest root, or t he wit ness in t he left most inter val. Hence, what we have seen so fa r is t hat , given a formula () (v, w) , one can build a for mula 0- (v, Z) and , for every /C F RC F and every sequence a in I< , a unique sequence b in I< for which O' (/C , b) = (} (/C , a): b is t he ord ered t uple of t he roo ts of t he polynomials q(y, a) (an d so is a-defina ble) , o'(v, b) specifies which roots a nd which intervals with end points a mo ng b and ± oo form (} (/C , a). A patient reade r may write down 0'(v, Z) in detail as an exercise. Now let us come back t o elimination of imaginari es . Let E be an equival ence relation on K" ; and ass ume E 0-definabl e in /C. Let E (V, w) the formul a defining it. We are lookin g for a formula 1. Pu t 3V1 .. . 3vn E(v, w).

Again using our preliminar y wor k, and applying it to (}1 (VI , w), we deduce t hat there a re a formula 0'1(VI , zi ) an d, for eac h a E K"; a unique sequence b~ in I< such that 0'\ (/C , b~ ) = (}1 (/C , ii) . Moreover there is a formul a X1(V\ , zi ) such t hat X1(V\, b~ ) picks a unique elemen t Cl in 0'1 (/C , b~ ) = (}1 (/C , a). Now consider

As before we obt ain t he existe nce of a form ula 0'2(V2, Z2) and, for every a E K "; of a unique sequence b~ in I< such that

as ab ove, t here is a for mula X2 (V2 , b~ ) select ing a unique element C2 in O'2(/C ' b~ ) = (}2(C1 , /C , a). Co nt inuing t he pro ced ure one defines a formul a X(v, Z) a nd , for eac h a in I< , a unique sequence b such t hat

x (/C n, b) consists of a uniqu e eleme nt if = (Cl , . . . , Cn) a nd

4.5. THE ELIMINATION OF IMAGINARIES SOME TIMES FAILS 131

K Henc e

F E(c, a), hence

e.

t.p ( Z) : ::Iw (x( e , Z)

is the required formula.

4.5

E(K n, a) = E(Kn , C). 1\

E (V, w))

It

The elimination of imaginaries sometimes fails

Differenti ally closed fields elimi nate imagin ari es. T his will be shown in Chapter 6, in the section devoted to developing in detail their mod el t heory. Similarly, algebraically closed fields with an automorphism (i. e. models of ACFA) eliminate imaginaries [20], as well as separably closed fields of finit e imp erfection degr ee in an enr iched langu age capt uring p-b ases (see [110)). But there do exist struct ures wit hout t his prop erty. This is the case, for instance, of the separa bly closed fields themselves in the pure langu age of fields (again, see [110)). Let us propose here a simpler example. Proposition 4.5.1 Let K be a finit e field with at least 3 elements and V be a vectorspace of dimension 2: 2 over K. Then V does not eliminate the . . . unaquiaries. Proof.

Suppose yes. As K is finite, one ca n consider in L K the form ula

E (v , w) :

V

(v=kw)

kEK - {O}

This formula defines in Van equivalence relation E , identifying t wo eleme nts a, at E V if and only if a, at generate the same subspace. Take a E V , ai-D . Then t here exist a formula t.p (v, Z) of LK and a unique sequence b in V such t hat E (V , a) = t.p(V, b). As I< has at least three elements, fi x k E I< , k i- 0, 1; for every x E V,

Since th e mult iplication by k is a n automor phism of V , we have, for every x E V, x E E (V , ka) ~ x E t.p (V, kb). T hen E'(V , a) = E(V , ka) = t.p (V , kb). It follows b = kb; bu t k i- 1, hence necessarily b = O. In other word s, we can assume t ha t = 0 and t.p (v, Z) reduces to a LK-formula t.p(v) with at most one free variable. The th eory

z

132

CHA PTER 4. ELIMINATION OF IMA GINA RIES

KT has t he quant ifier eliminat ion in LK , and hence
o.

Bu t V has dimension at least 2, therefore no suc h boolean combination ca n eq ua l E (V , a). •

4.6

References

An exhaustive int rodu ction to A eq can be found in Hodges' book [56]. T he elim ination of imaginaries was introduced by Poizat [130]; a complete treatment can be found in P oizat 's book [131], recently t ranslated in english [134]. The results on the elimination of quantifiers for separably closed fields can be found in [110], while the cor responding analysis for e.c. fields with an automorp hism is provided in [20].

Chapter 5

Morley rank 5.1

A tale of two chapters

In 1965 , M. Mo rley's work [116] proposed new ideas, new tools and , altogether , new fertile persp ect ives in Model T heory. Actua lly Morley's main t heorem is very simple to st ate: for , it says t hat a t heory T categorical in som e uncount a ble power is, consequentl y, categorical in every unc ou ntab le power. This is noteworthy, but perhaps not so dramatic and releva nt . However th e germs of Morley's ideas went much fur ther , and their richness permeat ed the deve lopment of Model T heory for several yea rs. Acco rdingly, t his chapter , and t he following one , will be devoted t o prepa ring a proof of Morley 's th eorem (alt ho ugh t his pro of will be do ne only in chapter 7); but our main intent t hroughout t hese pages will be to introd uce, to discuss and t o apply several Mo rley t ools: types, sat urated models, algebra ic a nd definable closure, totally transcendental t heories. We will deal also wit h some relat ed qu esti ons , both model t heoret ical (like pri me models) and algebraic (such as differenti ally closed fields, or w-stabl e gro ups and fields ) .

5.2

Definable sets

Defina ble sets were int rodu ced in Cha pter 1, and were a constant leit moti v t hroughout t he subsequent pages. T hey arise qui te nat ura lly within a given st ruct ure A from t he basic operations and relat ions in A , a nd fully chara ct erize A. Ind eed t he st ructure A could be t hought as a non empty set A plus t he collecti on of its defina ble sets . T his new outlook is less form al t ha n t he t radit ional definiti on given in Cha pter 1, a nd may sound a lit t le pu z-

133

CHAPTER 5. MORLEY RANK

134

zling. However it is quite free a nd eas y. Just to refer to our basic examples, it is instinctive to view an algebraically closed field as naturally endowed with the collect ion of its const ruct ible sets (including va rieties), or , in the same way, a real closed field with the collection of its semialgebraic sets , or a differenti ally closed field with the famil y of it s Kolchin constructible sets. Besides, sh ould you like to fix exact ly this new persp ective in introducing st ruct ures , you could take note that the definable sets in a given A can be characterized in a form ally pr ecise way, as follows.

Theorem 5.2.1 Th e sets definable in a structure A are the sm alles t family D of subs ets of Un>o A n such that

(i) A n is in D for every positive integer n ; m ore gen erally for i ::; j ::; n positive integers, {5 E A n : a; = aj} is in D ;

(ii) eve ry relation of A , as well as th e graph of ever y operation in A,

is

in D ;

(iii) D is closed under union, intersection and compleme nt; (iv) D is closed under projections: if X ~ An is in D , then the im age of X by th e projection of A n onto any i ::; ti coordinates is also in D;

(v) D is closed under fibr es: if X

~ A n is in D , i is a posi ti ve integer smaller than nand b E A n-i, then th e set of th e sequen ces 5 E A i fo r which (5, b) E X is in D .

Just a few words to comment . A careful a nd st raightforward check easily confirms that t hese condit ions a re satisfied by the definable sets in A and even characterize them . Indeed , (i)-(v) allow sets defined by at omic formulas (i)-(ii), Boolean combinations (iii) , quantifiers (iv) , par am eters (v) and nothing else. In order t o realize how complica te d a struct ure or, more generally, a class of structures -for instan ce, the mod els of a given theory T- is, one ca n try to measure the com plexity of its, or t heir, definable sets. A possible way to accomplish this program is to assign , to every definable, a value (such as an ordinal, or something similar) satisfying some reasonable assumptions, like monotonicity (for C ~ D definabl e sets, t he measure of D should not be smaller th an the measure of C), and so on. To prepare this assignment , let us consider again definables. Fi x a set X of parameters. Correspondingly one ca n form the Boolean algebra of Xdefinable sets , as we saw in Chapter 1. In introducing our measure, we

135

5.2. DEFINABLE SETS

may refer to t his algeb raic framewor k. But we have to be careful; for , t he Boolean algebra of X -definable sets, as sketched in C hapter 1, seems to depe nd on the structure A where X lies, and clea rly, even among the models of a given complete t heory T , the choice of A may vary in an essential way. Also , if we enlarge X to a bigger set of pa ra met ers X', the X-definable sets become au to matically X '-definable, but in this extend ed setting t hey ga in more complexity beca use t hey have to be compared wit h more object s; so a finer analysis must be expected . To cla rify t hese dou bt s (at least t he form er one) , cons ider ou r set X, a st ruct ure A containing X an d a positi ve int eger n. Take a model Bf(X) of Th(Ax); accordingly B is a st ruct ur e for Land j is an elementary fu nction from X into B , in particula r j( X) is a subset of Band Ax == Bf( x ). As already said, for every positive integer n we wish to com pare th e algebras Bn(X , A ) and Bn(J (X ), B) int roduced before. So consider any L (X) formula

In fact, for every ii E An, if A F
F


2. If
CHAPTER 5. MORLEY RA NK

136

and 'IjJ (v) are logically equi valent wit hin Th(Ax) in the usual sense

Vv( cp(V) H 'IjJ (V) ) E Th(Ax). Let rv de note t his (equivalence) relation. Eleme ntary mathematical logic tells us th at the quot ient set of L(X)-formulas cp (v) with resp ect to rv gets a nat ur al Boolean alge bra st ruct ure, pro vided we put , for cp(v) and 'IjJ (v) in

L(X) :

*

the meet of t he <-classes of cp(v) and 'IjJ (v) is the rv-class of t heir conju nct ion,

* the join of the rv-classes of cp (v) and 'IjJ (V) is the -v-class of t heir disjunction , * t he com pleme nt of t he <-class of cp (v) is the <-class of its negation , so t he bottom element is th e rv-class of --.( VI = vd or a ny contradict ion, a nd t he top eleme nt is the rv-class of VI = VI or an y tautology; in part icula r the -v-class of cp (v) is smaller or equal to the rv-class of 'IjJ (V) if and only if
Vv(cp(V) --+ 'IjJ (V) ) E T h(Ax ). T herefore it is easy t o check that this quotient algebra is isomorphic t o Bn(X, A) via t he fun ction mapping t he rv-class of cp (v) into cp (An). T he same applies to Bn(J(X) , B), of course , a nd hence, in conclusion , Bn(X, A) and Bn(J(X) , B) ar e isomorphic, as claimed, by t he function mapping cp (An) into cp(Bn ) for every cp(v). Hence t he isomorphi sm type of Bn(X , A) depend s on t he t heory of Ax rather t ha n t he mere struct ure Ax. Moreover we ca n view t he elements of Bn(X, A) not only as X-definable subsets of An, but also as -v-classes of formulas in L(X) . We will oft en confuse t hese different points of view below , an d even we will directl y t hink of t he eleme nts in Bn(X, A) as L(X)-formulas (ident ifying <-equivalent formul as).

5.3

Types

People acquainted with Boolean algebras B know what a n ultrafilt er is and remember t hat t he ult rafilters in a given B ca n be naturally endowed with a topology which makes t heir set a Boolean (i.e. Hau sdorff, compact and tot ally disconnect ed) space: the dual space of B. Of course this pro cedure applies to t he Boolean algebras of definable sets Bn(X, A). Even in t his particular set t ing one ca n look at the ultrafilters: they a re called complete

5.3. TYPES

137

n-types over X and their space is usually denoted Sn(X, A). That's all about types. But who is not familiar with Boolean algebras may appreciate some more details. It is worthy satisfying her (him), also in order to realize explicitly what a type is. This is the aim of this section. Let us begin with a simple example. Everybody knows how real numbers are introduced starting from the rationals and their usual order. But let us summarize very briefly their construction (in our style using formulas and structures). Accordingly consider the language L = {~} and the Lstructure (Q, ~). Partition Q in two non-empty subsets A, B such that, for every a E A and b E B, a < b, A has no maximum and B has no minimum. Then consider t he following set of L(Q)-formulas p

= {a < v:

a EA} U {v

< b: b E B}.

No element r E Q can satisfy all the formulas in p. However, for every finite conjunction
a

Definition 5.3.2 A complete n-type over X in Th(Ax) is a consistent n-type maximal with respect to inclusion. Sn(X, A) denotes the set of complete n-types over X in Th(Ax) .

In the sequel "n-type" will abbreviate "complete n-type", unless otherwise stated.

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CHAPTER 5. MORLEY RA NK

Example 5 .3 .3 Consider a mod el BJ(x) of Th (A x ). Accordingly B == A a nd f is an element a ry function of X into B. For simplicity, assume t hat X ~ B and f is t he inclusion of X into B . Let b E B " ; p be t he set of th e L( X )-formulas Cf/( v ) such t hat B F Cf/(b). It is easy to check t hat p is a complete n- type over X . Let us see why.

• For every finit e conjunction Cf/ (v) of formul as of p, B B F :JVCf/( V) and A F :JVCf/(v).

F Cf/ (b), whence

• Enla rge p by a new formul a 'IjJ (if); 'IjJ (v) rf- p impli es B ~ 'IjJ(b) and so B F -,'IjJ (b) a nd -,'IjJ (v) E p ; hence no set of formu las extending p U { 'IjJ( if)} is a consistent n-t ype any mor e, becau se it contains two formu las - 'IjJ (if) and its negation - whose conjunction cannot be satisfied by any tu pie in A n.

p is called th e type of b over X a nd is denoted tp(bj X) . The next t heorem shows t hat a ny n-type over X ca n be obtained in t his way. T h e orem 5.3.4 Let p be a set of L (X )-formulas Cf/(v) (where v abbreviates (V I , .•• , V n )

as before). T he follo wing propositions are equivalent:

(i) pE Sn(X , A ); (ii) there are a model B of T h(A) such that X ~ B and the inclusion of X into B is an elem entary fu nction, and a tuple b E B " such that tp (bj X ) = p.

Proof. (ii)=} (i) was shown before. (i)=}(ii) Enlarge L( X ) by a t uple c of n new constant sy mbols. Let L' be th e lan guage obtained in this way. Conside r the following set of sentences in L' T' = Th(A x) U {
139

5.3. TYPES

a mod el of T h (A ) a nd f is an elementary fun ction of X into B , but th ere is no loss of gener ality in suppos ing X ~ B , f = inclu sion of X int o B ). Let bE B " interpret c in B'. In Bx , p ~ tp( bjX). Owing t o the maximal ity of p, p = tp( bj X) . • When p = tp(bj X ) for b E B " ; one says t ha t b is a realiza tion of p (or also t hat b realizes p), and one writes b 1= p. Notice t hat t he argument in (i)=;.(ii) actually wor ks also when p is a consistent n-ty pe over X , wit h t he only exception of the las t point (deducing p = tp(bjX) from p ~ tp(bj X)); accordingly, if p is a cons istent n- type over X , then there are B a nd bsuch t hat p ~ tp( bjX), and we can state: Theorem 5.3.5 Let p be a consisten t n-type over X . T hen p enlarges to a comp lete n -type over X (possibly in several ways) . In the particular case when X = A , Theorem 5.3.4 says that every complete n-type over A is realized in some elementary extension of A. Notice also t hat, when X ~ A, any complete n-type over X ca n be viewed as a consistent n-type over A an d ca n be enlarged to a compl ete n-t yp e over A, whence it is realized in some eleme ntary extension of A. Theore m 5.3.4 (t oget her wit h the definition of complete type) also imp lies: Corollary 5.3.6 Let p E S n(X , A) ,
(i) If
E

P (and each case excludes the other).

Just to summarize, we might say t hat , as t he ration al order (Q, ::;) implicitl y contains t hro ugh it s sections all th e real numbers -even t he irrational onesas ideal elements, simila rly, for any A , X and n, t he n-types over X in Th( Ax) te ll us which new n-tuples of eleme nts ca n arise in the structu res B j(X) == A x , in particular in t he elementary extensions of A. Under t his point of view, the notion of n-ty pe seem s to deserve a good deal of attent ion . So let us exp lore it. First we cons ider t he set Sn (X, A ). We already linked types an d to pology at the beginni ng of this section. Let us examine t his co nnection in mor e detail. For every L(X)-form ula
= {p E S n(X, A) :
140

CHA PTER 5. M ORLEY RA NK

Notice t hat, for cp(v) , 7j; (v) L (X )-formulas , U
n U 'Ij;( v) =

U
In oth er words, for every t yp e p E Sn (X, A) ,

cp(v) E p, 7j; (v) E P

<=>

cp(v)

1\

7j; (if) E p.

In fact (=» is ju st (i) in Corollary 5.3.6, while (~) is a sim pIe consequence of (ii) . Conse quently t he sets U
= 0,

U V1=Vl

= Sn (X , A).

Furth ermore, for every L (X)-formula cp(v) ,

S; (X , A ) -

U
(as implicitl y stated in Corollary 5.3 .6, (iii)). So t he topological space Sn( X, A ) is: • Hau sdorff (in fact, for p, q E Sn (X, A ) a nd p =1= q, choose cp( v) E q - p a nd obser ve -,cp(v) E p, whence p E U-,
Sn(X, A ) is also compact . In fact , take an op en covering of Sn(X, A). W ith no loss of generality we can ass ume t hat every open set of t his cover ing is basic. So for some suitable set I of indexes our covering is just { U
has no mod el. So, by Compactness Theor em , t here exists a finite subset l a of J for which T h(Ax) U {-,cpi(C) : i E l a} has no model, equivalently, {U
Sn(X , A ). By th e way, t his to pological application of Co mpact ness T heorem is ju st the reason of it s name. But the topologica l fram ework also suggests t he followin g definition.

5.3. T YPES

141

Definition 5 .3.7 A type p E S n (X, A) is said to be isolated if and only if p is isolated as a point of the topological space Sn(X , A) , and so if and only if there is some L(X)-formula cp(iJ) such that p is the only elem ent in U
q be another typ e over X including cp (iJ) . For every formula 'l9( iJ) E p, cp (iJ) A 'l9 (iJ) is also in p, and so, owing t o t he choice of cp (iJ), cp (An ) n 'l9(An) = cp (An ), nam ely cp (An ) ~ 'l9(An). Hence 'l9(iJ) E q. It follows p ~ q, a nd so p = q by the maximality of p.

(ii) If X = A a nd p E Sn(A , A) is isolated , t hen p is algebraic. Let cp(iJ) be an L(A)-formula isolating p. p is realized in some elementary extension B of A ; consequently B F ::IiJcp(iJ). As an element ary substructure, A sat isfies ::IiJcp(iJ) as well, and so includes a realizat ion a of p, an d p = tp(a/A). But tp(a/A) contains t he form ula iJ = a a nd so a is its only realization. However we will see wit hin a few lines that an isolated t ype over an arbitrary subset X may be non- algebraic. Bu t , more gener ally, let us pro pose now some specific examples of types. Examples 5 .3.9 1. Let L = 0, A be a n infinite set (viewed as a struct ure for L) , X ~ A. For ever y a EX, tp( a/ X) is the only l-'type containing v = a, and so is both isolated and algebraic. Now t ake two elements a a nd a' in A - X ; t here does exist a bijection from A onto A , hence an a ut omorphism of A , fixing X pointwise and mapping a in a'. So, for every L(X)-formula cp(v ), A F cp (a) if and only if A F cp (a' ); in other words, tp( a/ X ) = tp( a'/ X). This shows t hat all t he elements in A - X realize th e sam e type over X. Notice th at , for a finit e X = {xo, .. . , x n } , this typ e is isolated , for instance by Ai
142

CHAPTER 5. MORLEY RANK however , it has infinitely many realizations (t he elements in A - X)' whence it cannot be algebraic.

2. Now take t he language L = {+, ., - , 0, I} for fields and an algebraically closed field /C . Let X be a subset of K. For simplicity we ass ume t hat X is t he domain of some su bfield 1i of /C . This is not so restrictive. Indeed , not ice that eac h eleme nt in t he field 1i generated by X is X -definab le, in other words it is t he only poi nt in /C satisfying a suitable L(X)-formula. Consequent ly there is no great loss of generality, when discussing t he types over X , in rep lacing X by H a nd hence in ass um ing t hat X = H is t he do main of a subfield 1i. Owing to qu an tifier elimination, every n-ty pe p over H is fully determined by its for mulas f( if) = 0 where f( x) ranges over polynomials in H[X] . Indeed every Lwformula is equivalent in ACF to a Boolean com bination of eq uations over 1i. Notice t hat t he polynomials f(x) E H[X] for which f(if) = 0 E P form a proper prime ideal in 1i[X]. T his is easy to check . J ust take a realization ii of p in a suitable extension , and notice t hat, for f( x) , g(x) , h(x) in H[ X] , if f(ii) = g(ii) = 0, t hen f(ii) + g(ii) = f(ii) . h(ii) = 0; moreover, if ii is a root of f(x) . g(x), th en it ann ihilates f(x) or g(x). Conversely, let I be a (proper) prim e ideal in 1i [X] and look at the set of formu las

{f (if) = 0 : f (x) E I} U {--, (g (V) = 0) : 9 (x) E H [X] - I} . This defines a (complet e) n-type over H , t he one of I + x viewed as an element of t he extension of 1i provided by t he field of quotient s of t he integral domain 1i[x]/ f. In t his way we obtain a con nection (indeed a bijection) between ntypes over H and prime ideals in 1i[x]. We shall prov ide a more detailed analysis of this point in Cha pter 8. Now let us examine closely t he case n = 1. Wh en I is t he ideal generated by some monic irr ed ucible f( x) E H[x], t hen we obtain t he ty pe defined by t he single equation f(v) = 0 (a nd its conse quences). So the roots of f( x) share a com mon I-type over H . This type p is clea rly realized in /C because /C is algebraically closed; actually /C contains all t he realizations of p. p is isolat ed by f( v) = 0, a nd algebraic , because f( x) has only finitely many roots. Ot herwise I = O. Now t he corresponding I-type p conce rns all th e elements which are transcendental over H . When the transcendence degree of /C over 'H is bigger than 0, t hen p has (infinite ly many)

5.1. SATURATED MODELS

143

realizations in K , in particular it is not algebraic. Otherwise, if K is ju st the algebraic closure of 1{ , t hen there is no room to satisfy p in K . On e eas ily sees t hat p is not even isolated. 3. Now take the language L = {+, ., -, 0, 1, :S} for orde red fields and a real closed field K . We discu ss directly t he types over K , for simplicity. Owing to qu antifier elimination, every n-type over K is fully det ermined by its equa t ions and disequations f(v) = 0, g(iJ) 2: 0, where f(x) a nd g(x) range over polynomials in n unknowns with coefficients in K. In particular, when we look at a I-type p over K , we have to consider polynomials in K[x]. But then , as explained in Ch apter 2, § 5, p is completely determined by its for mulas v :S a, b :S v (and negations) , wit h a and b in K. So a non algebraic p is given by the cut it defines in K. 4. Let L = {:S} , consider the L-st ructure A = (R, :S) and the subset X = Q . If a, a' E R and a f a' (say a < a'), there is r E Q su ch that a < r and r < a'. So v < r E tp(ajQ), but v < r rt. tp(a' jQ). In conclusion, there are at least 2~o-types over Q in (R, :S). In general , for a countable language , there a re at most 2 m ax{~o ,IXI } n-types over a set X for every posit ive integer n . Now let us mention a nother relevan t t echn ical fact about types; it will be useful several times later. Let A be a mod el ofT , X ~ A, f be an element ary function from X into a model B of T (for instance, let f be t he rest rict ion to X of some isomorphism between A and B). For pE S(X , A), put

f(p) = {
5.4

Saturated models

Let T be a complete theory with infinite models in a countable lan gu age L. A model A of T may not realize all the l-typ es over a subset X: a

144

CHAPTER 5. MORLEY RANK

t rivial counterexam ple is provided , when X = A, by th e consiste nt I- typ e {--,( v = a) : a E A} , which cannot sat isfied by any eleme nt in A. The sat urat ed models of T are, roughly speaking, those realizing as many types as possible over their subset s. In part icula r, for an infinit e cardinal A, the Asat ura ted models of T a re those able to satisfy any I- typ e over an arbitrary subset of power less than A. Definition 5.4.1 Let A be an infinit e cardinal. A model B of T is sai d to be A-saturated if and only if, for every s ubset X of B or po wer < A, B real ize eve r y i -type ov er X.

Of course this definition makes sense when A ~ IBI (as the previous counterexa mple shows); it is also clea r t hat, if A 2:: f-L 2:: ~o are ca rdinal numbers a nd B is A-saturated , t hen B is f-L-saturated , t oo. The first quest ion one may raise at this point just concerns the existence of A-satu rated mod els. T he an swer is positive, as the next theorem shows. Theorem 5.4.2 Let A be an infi nite cardin al. T he n every model A of T has a A-saturated eleme n tary ex te nsion . Proof. We provide a comparatively simple argument working when A = ~o . For an uncountable A this ap proach does not work an y more, and one needs a more complicated proof of a qui te different style (but t he res ult remains true). So take a mod el A of T , we are looking for an elementary extension A' > A realizing a ny l -type over a finit e subset of A'. We do know that every I-type over any (finit e or infinit e) subset of A is satisfied in some elementary extension of A . Accordingly, well ord er the set P of l-t ypes over finite subsets of A , P = {Pv : v < o:} and associate wit h any v < 0: an eleme ntary extension A(v) of A realizing p. Do this in such a way that , if v < fL < 0: t hen A(v) is an eleme ntary substr uct ure of A(f-L) (the reader may check t he det ails as an exercise). Now use t he Eleme ntary Chain Theorem (1.3.19) and dedu ce th at A* = Uv
5.4. SATURATED MODELS

145

a finit e subset of A'; there is some n for which X ~ A n; consequent ly every I-typ e over X is realized in A n+! and , t hrough it , in A'. eTNow let us show some basic properties of A-saturated mod els. Theorem 5 .4.3 (Weak Homo geneity Theorem) Let B be a A-saturated model of T . Let A be a model of T, X be a su bset of A of power < A and f be an elemen tary funct ion from X into B. Th en, for eve ry a E A there is an elemen tar y fun ct ion of X U {a} into B enlarging f . (A model B with th is property is called weakly A- ho m o g e ne o u s) .

Let p = tp(aj X), so f(p) is a com plete n-type over f(X) . As f is < A. As B is A-satu rat ed , B contains some realization b of f(p); hence, for every L(X) -formulal'(v, i), Proof.

1-1 , If (X)1

A

F l' (a, i)

?

l' (v, i ) E P <=> l' (v, f(1)) E f(p) <=> B

F l' (b, f(1))·

So enla rge f to a function g of X U {a} into B put ting g(a) = b. Clearl y g is what we are looking for. .. By definition , a A-saturated mod el B of T realizes any I- typ e over a subset of power < A. Bu t actually B satisfies every ty pe (even with more t ha n 1 varia ble) over such a subset . Let us see why. C o ro lla ry 5 .4.4 Let A be an infinite cardinal, B be a A-saturated model of T, X be a subset of B of power < A, n be a posit ive in teger. Th en ev ery n -type p over X is realized in B.

Proof. We know that p is realized in som e mod el Ax of Th( Bx), say by ii = (al ,"" an) E An . As Ax == Bx , t he identity map of X is an element a ry function from X into B. Using the Weak Homogeneity T heore m over and over again, one exte nds this map to an elementary function g from X U {aI , .. . , an} into B . Let b = g(ii). For every L(X)-formu la r.p(iJ, i),

A Hence tp(bj X) = p .

F r.p(ii, i)

<=> B

F r.p(b, i ).

eT-

Theorem 5 .4.5 (Universali ty Theorem) Let B be a A-saturated model ofT , A be a model ofT of power < A. Th en the re is an elementar y em bedding of A in to B (and so A is an elementar y substructure of B up to isom orphism) . In th is case B is called A- universal.

146

CHAPTER 5. MORLEY RA NK

Proof. Well order A = {a v : v < a} and , for v < a , put A v = {a jj : Il < a }. As A == B, th e empty map is a n elementary fun ction from A o = 0 (viewed as a subset of A) into B. Using t he A-saturation (actua lly t he wea k A-homogeneity) of B , extend t his map to elementary fun ctions from Av into B for every t/ < a, whose domain progressively includes every eleme nt in A . On e even tually gets an elementary fun ction (and so a n elementary embedding) of A into B. ..

Now we deal with t he "most saturated" models of T; as obser ved before, t hey are t he mod els saturated in t heir own power. Definition 5.4.6 A model B of T is said to be saturated if and only if B is IBI-saturated (and conseque n tly realizes eve ry i-type over subsets o] B of power < IBI). Such a model B is (also) wea kly IBI-homogeneous a nd IBI-universal. F irst let us obser ve t hat, for every infinite cardinal A, t here is at most one sat ur ated mod el of T of power A (u P to isomorphism). Theorem 5.4.7 (Uniqueness Theor em) Let Band B' be saturated models of T o] the same power. Th en B ~ B'. Proof. Recall the Weak Homo geneity Theor em: as both B and B' are saturated, if X ~ B , X' ~ B' , IXI < IBI and I is an elementary fun ction of X into B' wit h image X' (whence IX'I = IXI < IE'I a nd I-I is a n elementary function from X' into B), t hen

(i) for every a E B , one ca n enla rge f t o a n eleme ntary function g fro m X u {a} into B' , (ii) for every a' E B', one ca n en large I- I t o an elementary function g from X' u {a'} into B. Now observe t hat, as B

== B' , the empt y map is an elementary function of

o~ B int o B' (and conversely) . St art from t his fun ction , well ord er both B

a nd B' and use alte rnatively (i) an d (ii) in a suitable (possibly t ransfinite) indu ction procedure. On e event ua lly gets a n isom orphism bet ween B and

B'. Let us provide the details of this construction. Let A deno te th e common power of B and B' ; view A as an initi al ordina l; let {bv : t/ < A }, {b~ : v < A }. List B , B' resp ectively. For every v ~ A, one build s t wo elementary fu nctions lv, I~ , t he form er from a subset of B including all t he bjj's wit h Il < A and having power < A into B' , t he lat ter from a s ubset of B' including

5.4. SATURATED MODELS

147

all t he b~ 's with J.l < A and havin g power < A into 8 , in such a way that , for u ::; v < A, f v enla rges I; and f~-1 and f~ enla rges f~ and t;' , a nd eve nt ually 1>. and f~ a re tw o isom orphism s, t he one inverse of t he other on e, between 8 and 8'. We proceed by induction on i/, When t/ = 0, we know wh at t o do: fa and f~ are just t he empty fun ction . For a limit t/ , put I; = Uj.l. and f~ as required. ...

t:'

As a con sequence we obtain: C oro llary 5.4.8 (Strong Homogeneity Theorem) Let 8 a saturated model of T , X be a subs et of B of power < IBI, f be an ele me ntary function of X into 8. T he n f can be enlarged to an automorphism of 8 (an d then 8 is called homogeneous). Proof. In t he lan guage L (X ) conside r t he str uct ures 8 X and 8 f (X) ' As f is an eleme ntary fun ction , 8 x == 8 f (x ), a nd so Bx and 8 f (x ) are models of t he same co m plete theory. As IXI = If( X)1< IBI, 8 x a nd 8 f (x ) are saturated also in L (X ). Fin ally 8x a nd 8 f (x ) have t he same power. Accord ingly t here is some iso mo rphism (in L (X )) between 8x and 8 f (x) , who se restriction to L det ermines an automorphism of 8 enlarging f . ...

As a particular case, conside r X ~ B, IXI < IBI and two t uples a, a' in B" having the same n- type over X . The function fixing X pointwise and mapping a into a' is eleme ntary, henc e can be enl arged t o an automorphism of 8. As said , this automorphism acts ide ntically on X a nd maps a into a'. Now we wonder for which ca rdinals A a complete theory T may have a sat urate d model of power A. This is a quite delicate a nd deep question, and t he a nswer is not easy. Of course, st ronger ass um pt ions on T may somet imes ens ure t hese existenc e results. For instan ce, on e ca n see : • if T is A-categorical in some infinite cardinal A, t hen t he unique model of T of power A is saturated (we will see why for a countable A in the final section of C hapter 7, where we will also discuss t he uncount abl e case; t he exam ples at t he end of t he pr esent section partl y conce rn t his point ).

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Furthermore the (complete) theories having a countable saturated model can be characterized in the following way. • T has a saturated model of power ~o if and only if T has at most countably many n-types over 0 for every positive integer n (the reader may check this as an exercise). But what we can say from a general perspective, when dealing with arbitrary complete theories T? In this abstract framework, proving the existence of saturated models seeems related to some deep set theoretic assumptions, like the existence of large cardinals. In fact one shows: Theorem 5.4.9 Let A > ~o be an inaccessible cardinal. saturated model of power A.

Then T has a

The existence of an uncountable inaccessible cardinal is a quite delicate matter. But assume momentarily that such a saturated model n exists (for a given inaccessible A > ~o). Recall that n is unique up to isomorphism. Furthermore

• n is A-universal:

every model of T of power < A can be embedded as an elementary substructure in n;

• n is

A-homogeneous: if X is a subset of n of power < A and a, a' are two tuples in n having the same type over X, then there is an automorphism of n fixing X pointwise and mapping a into a'.

It is a general agreement (and habit) in Model Theory to assume that such a model n exists. This makes things easier and, on the other hand, is sufficiently plausible; in particular, one can check that everything is shown inside n, so assuming that n exists, can be proved (at the cost of some major complications) even avoiding any reference to n. Which are the benefits of working in As we said, the cardinality of is very large, and so one can reasonably suppose that all the models we expect to handle have a smaller power. But this implies that they actually are elementary substructures of n of a smaller size (up to isomorphism). Under this perspective, the subsets of models of T can be directly viewed as subsets of n with power < 1nl: we will call these subsets small subsets of n, just to tell them from the other subsets of n admitting its same inaccessible cardinality. As already said, referring to small subsets of n instead of su bsets of arbitrary models makes our life, and also our notation simpler. For instance, take a

n?

n

5.4. SAT URA T ED MODELS

149

small X and a posit ive int eger n . W hen definin g Bn(X , A ) an d Sn(X , A ), we had t o explicitly refer to a model A of T (now an elementary substructu re of Q) where X is embedded . We also em phasized that Bn(X , A ) an d Sn(X, A ) do not de pen d directly on A , but on ly on the t heory of Ax : t he cho ice of A is qui t e arbitrary wit hin these bounds. But t hen it is convenient to refer to A = Q , as a uni versal mo del where X is embedded . T his is what we will do fro m now on. Ind eed we will write, when t here is no danger of misunde rstan di ng, Sn(X) to mean Sn(X, Q) for every small Xj so S(X) will denote t he union of t he spaces Sn(X) when n ra nges ove r posit ive int egers. At last , let us propose some algebraic examples concerning more or less sat urated st ruct ures.

Examp les 5 .4.10 1. Which algebraically closed fields K are sat urated in a given card inality A 2:: ~o ? To an swer , we may recall t hat, for any p = 0 or prime, t he t heory ACFp is categorical in every uncoun t abl e powe r; so, according to what we said before, eve ry uncou ntable algebraically closed field (in any characteristic) is sat urated. To confir m t his from t he algebraic point of view and to discuss t he existence of saturated mod els in t he co untable case, we ca n refer to Steinitz 's analysis of algebraically closed fields K and recall that such a K is the algeb raic closure of Ko(S) , where Ko is th e prime subfield and S is a t ranscendence basis of K , so a maximal algebraica lly independent subset. The isomorph ism ty pe of K is fully determ ined by it s characteristic and it s transcendence degr ee, i. e. t he power of S. Moreover, when S is infini t e, t his transcendence deg ree eq ua ls 1/(1. Now let 1/(12:: A and take a subset H of K of power < A. As we saw in t he last section , H ca n be replaced by t he subfield generated by H j t his does not change its size, except when H is finit e; however, even in t his case, eac h point in t his subfield is /i-definable. Every alge braic l -fype over H is clearly realized in K becau se K is algebraically closed . So t he point is: ca n we realize the remaining l-nype, the on e of t he eleme nts which are transcende nt al over H , for a ny H? If K has an infini t e t ransce nde nce degree, t hen t his degree is ju st 1/(12:: A > IHI, so it is st rictly larger t han t he t ransce nde nce degree of t he field ge nerated by If. T his mea ns t hat we can realize our type insid e K for every H . Otherwise, let H be generated by a finite t ra nscendence basis S of K , Clearly IHI < A, but now t here is no way to realize our ty pe inside K . In conclusion , for

1/(12:: A, K

is A-sat urat ed if and on ly if it s transcen-

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CHAPTER 5. MORLEY RANK

dence degree is infinit e. In particula r every algebra ically closed field satisfying t his condit ion is satur ated (in it s own power ). So every uncountable algebra ically closed field is saturated (t his a pplies also t o t he complex field), while t he only countable saturated algebra ically closed field (in a fixed characterist ic) is t he algebraic closure of /Co(S ) where /Co is t he pri me subfield and S is a countable (infinite) algebraically ind epend ent set. 2. The real ord ered field is not ~o-saturated . This is perhaps remarkabl e, if one remembers t hat t he reals are a complete ord ered field (in t he sense t hat every Cau chy sequence has a limit) , and even the only complete ordered field up to isomorphism: in oth er words, given a Cauchy sequence (rn)n>o in R , the l -type defined by 1

Iv - rnl < - , n

when n ra nges over positi ve integers, has a (unique) realization in t he real field. However , look at t he ty pe of a positiv e infinitesim al non zero element . This is defined by t he cut 1

0
for n as before. So it is a ty pe over t he empty set. Any finite portion is satisfied among t he reals, so this infinitesim al element lives in some element ary extension of t he real field. Nevert heless t he ty pe ca nnot be realized in R. Furt her exa mples will be discussed in th e next section within mod ules.

5.5

A parenthesis: pure injective modules

Saturated models are very larg e, powerful and rich. Wi thin algebraically closed fields, t hey remind , and actually coincide with t he uni versal dom ain s int rodu ced by Andre Weil in his " Foundations of A lge braic Geometry" [176] ; in fact, as we saw in t he las t section, t hey a re ju st t he algebraically closed fields wit h a n infinite t ransce nde nce degr ee (over t he prim e subfield) . Hence it is not surprising t o realize t hat t here do exist in other par ts of Algebra some notions resembling saturation, perhaps in a wea ker form. This is t he case of pure injecti vity wit hin module t heory.

5.5. A PARENTHESIS: PURE INJECT/VE MODULES

151

So, fix a (coun table) ring R (with identity), and consider (say left) Rmodules. We want to recall in this section what a pure injective R-module is, and why these modules are so important. First we need introduce a particular class of embeddings concerning modules. Definition 5.5.1 Let M and N be R-modules. An R-module homomorphism f from M into N if called pure if, for every pp-formula
Notice that a pure homomorphism is 1-1 and hence is an embedding. To see this, just apply the definition to the formula Vi = V2. Moreover the implication =? in (*) is trivial, hence the qualifying point in the definition of purity is~. When M is a submodule of N, M is called pure in N if its inclusion embedding is pure; in this case, one says also that N is a pure extension of M. The algebraic content of purity is the following: if a linear system A·il = B·x with parameters ii from M admits a solution in the extension N, then it admits a solution already in M; here A and B a re, of course, matrices with entries in R and suitable sizes. Examples 5.5.2 1. If M is an elementary substructure of N, then M is pure in N. Indeed, (*) holds for arbitrary formulas.

2. If M is a direct summand of N , then Ai is pure in N. In fact, any solution in N of a linear system as before projects itself to a solution inM. 3. Let R = Z, so let us deal with abelian groups. We saw in Chapter 2 that, in this particular framework , pp-formulas reduce to torsion or divisibility conditions. Hence it is not difficult to realize that, over Z, M is pure in N if and only if r M = M n r N for every integer r , as usually required in the definition of purity in the handbooks of abelian group theory. Incidentally, let us quote a noteworthy result of Sabbagh. Theorem 5.5.3 Let M and N be R-modules, M be a submodule of N . Then M is an elementary substructure of N if and only if M is pure in N and M and N are elementarily equivalent.

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CHA PTER 5. M ORLEY RA NK

Now we ca n int rodu ce pu re injectivity: t his notion restricts in some sense the usual injectivity requirement to pure embeddings. Definition 5.5.4 An R-module M is called p ure injective p . i. (but someon e prefers to say algeb raically com pact) if and only if on e of the fo llowing equivalent conditions holds :

(i) For every choice of R -modules N and N ', for which N is a pure subm odule of N ', every hom omo rphism f from N to M lift s to a homomorphism l' from N ' to M . (ii) M is a direct summand of every pu re extension. The equivalence bet ween (i) and (ii) is pro ved in the references quoted at the end of t he cha pter. (i) and (ii) chacterize pure injectivity from t he algebraic point of view. But t here is a t hird equivalent definit ion , disclosing a model t heoretic flavour and showing that pure injectivity is directly related to saturation . Theorem 5.5.5 An R -module M is pure injecti ve if and only if, for every countable subset X of M and every (incomplete) i-type p of pp-formulas
Hence pure injectivity is j ust a weak form of sat uration, restricted to incom plete l-types of pp-formulas (they are usu ally called pp-types) . Conseq uently every ~ l-sat u rated R- module M is p. i.; t his fact , and Theorem 5.4.2 clearly im ply t hat a ny R-module enla rges to a p. i. elementary extension. But something much stronger holds. Theorem 5.5.6 Let M be an R -module. T hen M has a p.i. pure M which is minima l in the following sense: fo r every pure and p.i. N of M , N contains M as a pure su bmodule up to isomorphism. M is unique up to isomorphism fixing M pointwise. Finally , eleme n tary extension of M.

extension extension Moreover M is an

For a proof, see t he refer ences at th e end of th e chapter. These exist ence and uniqueness result s j ustify th e specific symbol A1 to denote this p.i. extension of A1 and , furthermore, a special name to dub it; actually, M is called "the" pu re injective h ull of M .

5.5. A PARENTHESIS: P URE INJECTIVE MODULES

153

Up to eleme ntary eq uivalence, p. i. R- modules represent t he whol e class of R-modules, hence their algebraic a nalys is ca n help a description of the possible com plet ions of RT. But wh at ca n we say a bout a possibl e classification of pure injective R-modules? A usual and general t echnique in studying a given class of modules is (a) to look for a possible (and hopefully unique) direct su m decom positi on of an y module of th e class into inde comp osable object s, a nd t hen (b) t o classify the indecomposable modules of the class up to isomorphism . By the way, recall t hat an R-module M is call ed indecomposable if M =J. 0 but there is no way to express M as a direct sum of t wo nonz ero submod ules. T his procedure is not always successful within arbitrary classes of modules , but is sufficient ly sat isfactory when applied to p. i. modules. Let us explain why . First of all it is comparatively easy t o check t hat pure inj ectivity is pr eserved by direct summands . Mo reover (a) do es wor k wit hin p. i. modules, du e to t he following result of F isher and Ziegler.

Theorem 5.5.7 (Fisher-Ziegler) Let M be a p.i. R-module. Th en M decomposes uniquely (up to isomorphism) as ffiiUi ffi E ,

where E has no indecomposable direct summand, denotes p. i. hull and , fo r every i, Ui is an indecomposable p.i. R-module. Moreover M is elem entarily equivalent to ffi# i . Incidentally, pure injectivity is not preserved under infinit e direct sums . To summarize wh at we have seen so fa r: • every R-module is an elementary su bst ruct ure of its p.i . hull ; • every p.i. R-module is eleme ntarily equivalent to a direct sum of indecomposable p.i. 's, and even isomorphic to th e p .i. hull of this sum up to a summand with no indecomposable direct factors . So what we have to do now is t o study isomorphism classes, or elem entary eq uivalence classes of indecomposabl e p.i. R-modules. Cl assification up to isomorphism is som etimes easy. For instan ce, when R is a field, so when we are dealing with R-vectorspaces , t he only ind ecomposable object is just R (as a vectorspace over itself) and is pu re injective.

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CHA PTER 5. M ORLEY RA NK

But over oth er rings t he situation cha nges dramat ically and in some cases one has t o conclude t hat no classification is possible. This is what ha ppens, for inst an ce, when R = K (x , y) is t he ring of polyno mials over a field K in tw o non-commuting variables x an d y , so when one considers K -vect orspaces wit h two additional linea r operators, corresponding to t he action of x, y resp ecti vely; in fact , t his class of mod ules enco des in some way t he word pro blem for groups, and t here are some good reasons to believe t hat t his forbids a ny class ification of ind ecomposa ble p.i. ob jects. See [136] for a det ailed discussion of this point. As a consequence, even modules over K[x , y] (wit h two com mut ing unknow ns x and y) a nd, in another directi on , over K (X l, ... , x n ) wit h n > 2 inh erit t he same "wild" sit uat ion, excluding a ny possible classification. So t he key point is t o realize for which ring s R indecomposable p.i. modules ca n be classified up to isomorphism . In t his perspect ive we would like to quote a beau tiful resu lt of Ziegler, using model t heory to equip in a nat ural way indecomposable p.i. mod ules over any fixed ring R wit h a to pological space structure. In fact , let RZg denote t he set of (isomorphism classes of) indecomposa ble p.i. R-modules. For every choice of pp-formulas rp (v) and 'IjJ (v) (in one free variable) in LR , let (rp / 'IjJ ) be the set of the elements U in RZg such that rp (U) properly includes rp(U ) n 'IjJ (U). T hen the (rp / 'IjJ )'s are a basis for a topology of RZg , which is always com pact and seldom Hausdorff. RZg with t his topology is called the (left) Ziegler spectrum of R. Again , see [136] or d irectly [181] for more details. W hat we can say in t he restricted fra mewor k of these pages is t hat t he knowledge of t he Ziegler spectrum of R is a sort of fixed co urse towards t he solution of several sign ificant mode l t heoretic (an d also algeb ra ic) pro blems about R- modules. In particula r, a successfu l analysis of the Ziegler spectrum ca n provide some useful informat ion on t he eleme ntary equivalence class of any R- module M . Let us see very briefly a nd roughly why. We saw in Cha pter 2 t hat t he complete t heory of M is fully det ermined by t he values (modulo 00) of [rp( M) : 'IjJ (M )] where rp (v) and 'IjJ (v) range over pp-form ulas in one free variab le in L R. Now, up to elementary equivalence, M ca n be replace d by a direct sum of indecomposabl e p. i . R-modules G7iUi , where

is th e canonical decom posit ion of the pure injective hull A1 of A1 according to the Fisher-Ziegler Theorem . Consequently, for rp( v) and 'IjJ (v) as before,

[rp(M) : 'IjJ (M )] = [rp(G7iUi) : 'IjJ (G7iUi )]

5.5. A PARENTHESIS: PURE INJECTIVE MOD ULES

155

modulo 00 (in the sense t hat th ese values are eit her finite and equal or both infinite) . Now it is easy t o realize that

[

modu lo 00 . So, when in t he Ziegler spectrum t he poin ts (Le. t he inde composable p.i. R-modules) are classified as well as th eir basic open neighbourhood s, th e element ary equivalence t ype of M can be det ermined by saying which U ER Zg a re involved in th e decomposition EBiUi and how many t imes t hey occur. For instance, if U is known to be isolat ed -say by (

where p is a prim e, l < k are positive int egers and r is a n integer. An effect ive list of indecom posable p. i. Z-modules is known , a nd includes (up to isomorphism ) exactly th e following object s: • the finit e modules Z/ pnZ , • th e Priifer grou ps Z/ pco , • th e p-adic int egers

Z (p) ,

• the additive group of rationals Q where p ra nges over primes and n over positive int egers; t he p-adic integers are j ust th e p. L hull of the localization Z (p) of Z at p . It is not prohibitive to realize t hat t he pr evious examples are ind ecomposabl e and p. i.; for instance, t he pure injectivity of th e Priifer groups and t he rationals comes directl y from th eir divisibili t y. Bu t t he poin t is to show t hat t heir list ex ha usts all t he possib le cases: see [181] for a discussion , and a complete proof of t his. Now we have to study t he t opology of t he Ziegler spectrum z Z g. Here is its Cant or-Bendixson a nalysis.

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CHAPTER 5. MORLEY RANK

• The isolated points (those having rank 0) a re t he finit e mod ules Z /pnz ; in fact, eac h of t hem is t he only eleme nt in t he op en set ((pn- 1Iv 1\ pv = 0) / (pnlv 1\ pv = 0)). • Now forg et the Z/pnZ 's; t hen a Priifer gro up Z / p= gets isolat ed by ((pnlv 1\ pv = 0) / v = 0) for a ny positive n , while t he p-adi c int eger s are isolat ed by (pn- 1Iv / pnlv) for a ny n; hence all t hese gro ups have ra nk l. • The rational gro up is t he only remaining point , a nd so gets rank

2. This a nalysis was pu rsued by E klof-F isher in t he particular case when R is a principal ideal domain , a nd supports a nd clarifies the previous work of Wand a Szmielew (sket ched in our C ha pter 2) for ab elian gro ups . In particul ar, as a consequence, it impli es t he decidability of t he t heo ry z T. Ziegler 's a pproach was inspired by t hese par ti cular cases, bu t is fully genera l and covers any rin g R. 2. Hence what has bee n said on Z act ua lly enlarges t o princip al ideal dom ains and , partly, to Dedekind dom ain s. For instance , it applies t o t he ring K [x] of po lyn omi als over a field K wit h a single unknown: also in t his case one can accomplish a quite satisfacto ry descripti on of t he Ziegler spectrum , esse nt ially rep eating t he analysis for the int egers wit h t he necessary varia nt s . Compa re t his a nd what was obser ved befor e when t he nu mb er of unknowns increas es.

5.6

Omitting types

Sat ura te d models realize man y ty pes . But non isolated t ypes are not easy to sat isfy in ar bit rary models. Let us see why. We conside r a complet e theory T, and we denote it s uni verse by n. Recall that we met isolated types in 5.3: a ty pe p over a set X ~ n is isolated when p is t he onl y type over X containing some L(X)-formula cp(v). It is an easy exercise t o check what follows. Fact 5.6.1 For p E S(x), t he followin g propositions a re eq uivalent . 1. P is isolat ed ;

2. t here is some L (X )-formula cp (v) E p suc h t hat p is ju st t he set of L (X )-formulas 7f; (v) for which cp(n n) ~ 7f; (n n);

157

5.6. OMITTING TYPES 3. there is some L(X)-formula tp(v) such that tp(Dn) t he set of the L(X)-formulas 'IjJ (if) for which tp(Dn)

=f. 0 and ~

p is just

'IjJ (Dn).

Actually what distinguishes 2 and 3 is th at in 2 we require t he additional condition tp(v) E p. Bu t it is easy to see t ha t tp(v) (j. p forces ,tp(v) E p, hen ce tp(Dn) ~ ,tp(Dn) , a nd so tp(Dn) = 0, a cont radict ion. Now an isolated t ype p over X is trivially realized in every mod el M of T h (Dx ). For , D do es cont ain some tuple satisfying p, a nd in particular the formul a tp(v) isolating p. So :3vtp(v) E Th(Dx). On the contrary a non-isolated type p is not always realized. This is the content of t he following result. Theorem 5.6.2 (Omitting Types) Let T be a complete theory in a language

L , p be a non-isolat ed type over 0. Th en there exists a (countable) model M of T om itting p (in the sense that p(M) = 0).

Proof. For simplicity, assume p E SI (0) (the reader ca n eas ily adapt the following argument to more variables, if he (she) likes). We use some usu al techniques of classical Model Theor y. In detail , first we exte nd our language L by countably many new constant symbols co, Cl,

. .. , C n , . ..

(n natural).

List the sentences of the enla rged language L'

= L U {cn

:

n E N}

Now we build an increasing sequence of consistent theories in £I

To ~ Tl

~ ... ~

T; .. .,

all enlarging T and satisfying the following conditions, for every n: 1. T n is axiomat ized by To and a unique further £I-sentence ()n;

2. eit her tpn or ' tpn is in Tn+I; 3. whenever tpn or ' tpn is in T n+l a nd has the form :3v'IjJn(v) , then 'ljJn (cm ) E T n+l for some new constant Cm not already occuring in any tpi, ()i for i ::; n; 4. finally, t here is some formula O"n( v) E P for which 'O"n( cn) E T n+ l

.

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CHA PTER 5. M OR£EY RA NK

To is ju st T, or , mo re precisely, t he £I- theory ax iomatized by T. Now, ass ume T n given, and let us build Tn+l' Let Cia ' ... , Ci h _ 1, C be t he constants in £' - L occuring in ()n ; replace t hese constants by new varia bles Wo, ... , Wh- l , v chosen out of ()n ; place before :Jwo, ... , :JWh-l and get a for mula ()~ (v) in L , with t he only free variable v . Take a model M ~ of T n ; as o; E T n , ( M~) is not em pty (it includes cM~) . As p is not isola ted , t he condit ion 3 in Fact 5.6.1 ensures t hat, for some formula un(v ) E p ,

e;

Pu t ,Un (Cn ) in Tn+l' This gua ra ntees 4. Now look at 'Pn. If T n U {,Un(cn), 'Pn} has some mod el, let T n+1 include also 'Pn . Oth erwise ' 'Pn is a consequ ence of T n U {,Un(cn)}. T his gives 2. When 'Pn (or ''Pn) is of t he form :Jv'Pn(v), pick t he least m suc h t hat Cm does not occ ur in 'Pi, ()i for i ~ n, an d pu t 'ljJn(cm ) in T n+ 1 • This ensures 3. Let T n+1 be t he t heory obtained in t his way. Clearly Tn+l is consistent and

satisfies l. Now form T ' = Unr; T ' is consistent. Take a ny model l v1' of T'. By 2, T ' is com plete , T ' = T h( M '): t he sentences :Jv'ljJ (v ) M' satisfies are just those occu ring in some T n . Owing to 3, we ca n apply the Tarski-Vaught Theorem and dedu ce that {C~' : n natural} is the domain of a (countable) eleme ntary substruct ure of M' , an d hence a countable model of T' . Its restricti on t o L is a countable model of T , and omits p because , for every n , C~' can not satisfy un(v). ..

5.7

The Morley rank, at last

Let T be a complet e theor y in a count able lan gu age. We remind the program outlined in 5.2: we aim at measuring t he complexity of definable sets X in n (equivalently, of r-v-classes of formul as with par am eters in n) an d, more generally, of sets of formulas with par am et ers in n, including ty pes over small subsets of n. A reason abl e way to get t his is to define some function assigning, if possible, to every definabl e, or formula , or type, an ordinal value , according to some reasonabl e condit ions, like monotonicit y. An ax iomatic introd ucti on to t his complexity measu res ca n be found in [8] or [131, 134]; t hese fun ctions a re usu ally called ranks. T here a re several possible ways t o define a ra nk , according to some pa rt icula r algebraic or model t heoretic featu res of t he theory T . Morley 's ra nk was t he very first ,

159

5.7. THE MORLEY RANK, AT LAST

a nd fund amental, exa mple in t his direction. Historically, it was inspired by t he Cantor-Bendixso n analysis of t opological spaces S . Recall t hat t his eq uips an y poin t in S with a rank , whose value is eit her a n ordinal number or 00 (where one assumes t hat 00 is gr ea ter than a ny ordinal) ; in parti cul ar the Ca nt or-Bendixson ra nk ass igns the valu e 0 t o t he isolated points of S, th e valu e 1 t o t he poin ts in S t hat get isolated wit hin t he rela ti ve t opology once th e isolated poin ts are forgotten , and so on. Of course one can a pply this Cant or-Bendixson machin er y t o t he topological space S(C) , where C is a s mall subset of n , an d hence to the types over C . Using t he Stone du ality between topological Bool ean spaces and Bool ean algebras , one can define a Cantor-Bendixson rank also for th e eleme nts of t he du al algebra of S (C) , a nd hence for C -defina ble sets. However , if one translates literally th e Cantor-Bendixson analysis into our particular set t ing, then the sa me definabl e object may obtain several possible ranks, according to which basic set of param eters C we refer t o; in particular , if we replace C by a larger C', t hen we ca n expect t hat, over C', C -definable sets get a stronger complexity and conseq uent ly a bigger Cant or-Bendixson rank : some examples of t his phenomenon will be provided lat er in t he present section. Mo rley's rank adapts t he Cant or- Bendixson a pproach t o ob viate this difficulty. The recip e is ju st to refer to our universe n an d so t o evaluate any definabl e within ar bit rary n -d efinabl e sets. Bu t it is t ime t o give, at last, t he exact definition of t he Mo rley rank; we will introduce also a related notion

(Morley degree). vVe consider a complete t heory T in a co untable lan gu age L: n denotes, as usu al, t he univ erse of T . Let X be a definable subset of n n (for some positiv e int eger n), we wan t t o define the Morley rank of X RM(X). First let us say what

RM(X) ::::: mean s for every ordinal

0:.

0:

We proceed by induction on

0: .

Defi nition 5 .7.1 Wh en 0: = 0, put RM(X) ::::: 0: if and only if X i= 0. Wh en 0: is a limit ordinal, put RM(X) ::::: 0: if and only if RM(X) ::::: 1/ fo r every ordinal 1/ < 0: . Fi nally, when 0: = 1/ + 1 is a successor ordinal, put R M(X ) ::::: 0: if and only if th ere are infi ni tely many pair wise disjo int defi nable subse ts X i (i E N) of X su ch that RM (Xi ) ::::: 1/ for every i EN.

It is easy t o observe t hat, if

0:,

R M (X ) :::::

{3 are ordinals and

0:

impli es

Accordin gly it makes sense to pu t:

0: :::::

{3, t hen

RM (X ) ::::: {3.

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CHAPTER 5. MORLEY RANK

Definition 5.7. 2

1. If X =

0, then RM(X)

= - 1.

t.

2. Let X#-0 and suppo se that there is some ordinal a suc h that RM(X) a (so RM(X) (3 for every ordinal (3 2: a); the least ordinal aD with this property is a successor; if aD = v + 1, then put R.A1(X) = t/,

t.

3. Finally suppose X RM(X) = 00.

#- 0,

RM(X) 2: a for every ordinal a. Then pu t

(Of course we assume - 1 < a < 00 for every ordinal a) . When <.p( v) is a formu la (pos sibly with parameters from n), define

(where n is the length of v). E xamples 5 .7.3

1. RM(X) = - 1 if an d only if X =

0.

2. RM(X) = 0 if and only if X is finite and non-empty. For instance, in algebraically closed fields, t he (definable) zero set of a given polynomial of degree 2: 1 in one unknown is finite and non-empty, and hence has Morley ra nk O. ~t

RM(X) = 1 if and only if X is infinite, but ca nnot partition as the union of infinitely many pairwise disjoint infinite definable subsets. For example, in algebraically closed fields, the only definable infinite 1-ary sets are cofinite. This implies t hat their Morley rank is 1. Notice that the same is true for infinite vectorspaces over a countable field.

4. Let T = DLO - be the theory of den se linear orders without end point s. Fix a < b in n a nd consider the int erval I =]a, b[= {x En: a < x < b}. I is definable, and RM(J) = 00. In fact, for every a, b as before and ordinal a , RM(]a, bD 2: a. This can be eas ily checked by induction on a. The cases a = 0 and a limit are trivial. So suppose a = v + 1 for some u , As the ord er S; is dense, we can choose c E ]a, b[, and observe that ]a, b[ includes the disjoint intervals ]a, C[, ]c, b[, both having Morley rank 2: u, because of th e induction hypothesis. Repeating this procedure, one finds infinitely many pairwise disjoint open intervals in ]a, b[, all of Morley rank 2: u , Hence RM(]a, bD 2: t/ + 1. Notice that even [a, b[, ]a, b], [a, b] (for a < b) have Morley rank 00.

Let us point out now some simp le properties of RM :

5.7. THE MORLEY RANK, AT LAST

161

P roposi tio n 5 .7.4 Let X , Y be two definable subsets of

itive integer n , (i) If X

~

nn for some pos-

Y , then R M(X ) :S RM (Y ).

(ii) For every automorphism f of n, RM(X) = RM(J(X)) .

(iii) RM(X U Y) = ma x{RM(X) , RM (Y )}. Proof. (i) Eve ry definable subset of X is clea rly a definable subset also of Y . So a simple induction argument shows t hat, for every ordinal a, if R M(X) 2:: a, then R M(Y ) 2:: a . (ii) It suffices to show RM(X) :S RM(J(X)) (as th e opposite relation RM(X) 2:: RM(J(X)) can be obtained ju st by reversing t he roles of X a nd f(X ) and replacing f by f-l) . In oth er words, we have to check t hat, for every ordinal a, R M (X ) 2:: a impli es R M (J (X )) 2:: a. Proceed ag ain by induction on a (a nd observe th a t , if Z is a defina ble subset of X , t hen f(Z) is a definable s ubset of f(X)) . (iii) 2:: follows from (i) . To check :S, it suffices to prove th at, for every ordinal a, RM(X U Y) 2:: a impli es eit her RM(X) 2:: a or RM(Y) 2:: a . As befor e, we ca n pro ceed by induction on a . If a = 0 and R M (X U Y) 2:: a, t hen X UY

i= 0 and

consequent ly eit her X or Y is not empty. Now sup pose

a limit a nd RM (X U Y ) 2:: a, namely RM(X U Y) 2:: t/ for every ordinal t/ < a. If th ere exists some v < a s uch th at RM(X) 2:: J1 for every ordinal J1 sat isfying v :S J1 < a , t hen RM(X) 2:: a . Otherwi se, for every ordinal v < a, th ere is some J1 2:: t/ such t hat J1 < a a nd R M (X) J1 . By indu ction , RiV1 (Y ) 2:: J1 , whence R M (Y ) 2:: u , It follows R M (Y ) 2:: a . At las t , take a

t

s uccessor a = v+ 1 and suppose R M (X UY ) 2:: a . T hen t here a re infinitely many pairwi se disjoint infinite defin abl e subsets Z, (i E N ) of X U Y , all having RM 2:: u , Look at t he sets

(i E N) . For every i, eit her R M (Xi) 2:: t/ or R M (Y;) 2:: u , Acco rdingly R M (Xi) 2:: u, or R iV1(Y;) 2:: v for infinitely many i 's. In t he form er case R M (X) 2:: a; in t he latter RM(Y) 2:: a . • Now we are goin g to associa te with a ny definable subset X ~ nn (whose Morley rank is an ordinal) a positi ve int eger: t he M orley degree of X

G M (X ).

162

CHA P TER 5. MORLEY RA NK

nn

Proposition 5 .7.5 Let X ~ be a definabl e set whose Mo rley rank is an ordinal 0:' . Th en there is a maxim al positive in teger d suc h that X partit ion s as the union of d pairw ise disjo int definable su bsets of M orley rank 0:' .

Proof. Suppose no t. T he n, for eve ry po siti ve int eger d , one can partition X as t he union of d defin abl e subsets of Morley rank 0:' X(d ,O), .. . , X (d , d - 1). We ca n assume t ha t, for every d an d i ::; d , ther e is some j < d such th at X (d + 1,i) ~ X(d,j). In fact, X(d+ 1,i) decomposes as th e disjoint union of it s sub sets X (d + 1, i) n X (d, j) for j < d. Owing to Proposition 5.7.4, (iii) , at least on e of these subsets has th e same Morley rank 0:' as X(d + 1, i). By Proposition 5.7.4, (i), all th ese subse ts have Morley rank ::; 0:'. Accordingly we can repl ace X(d+1 , i) by a subse t X(d+l , i )n X (d, j) (to whi ch one possibly adds othe r subse ts X (d , j) of Morley rank s, 0:') . Now a com binatorial argument (K6nig's Lemma) ens ures t hat t he re are positi ve int eger s (m E N ) do < d l < .. . < d m < ... a nd natural number s

. .

.

ZO,ZI , · · · ,Z m, · · ·

s uch t hat, 'IIm EN , i m

(m E N)

< dm a nd

We ob t ain in this way infinitely many pairwise disjoi nt definabl e su bsets of X X(d m , i m ) - X(d m +l , i m +l ) (wh er e m ranges over N ), all having Morley rank 0:' + 1, a nd t his is a contradiction. '"

0:'.

Consequently RM(X) 2:

Owing t o t his proposition , we can at las t introduce t he Morley degree as follows.

nn

Defi n it io n 5 .7 .6 Let X be a defi nable subset of (for some posit ive integer n) whose Morley rank is an ordinal 0:'. Th e M orley degree of X (den oted G M (X)) is the ma ximal posit ive integer d su ch that X can partition as the union of d defi nable subsets of Morley rank 0:'.

5.7. THE MORLEY RANK, AT LAST

163

1. If X is finite a nd non-empty (in other word s RM(X) = 0) , GM(X) is ju st the size IXI of X. In particula r, when X is the zero set of a polynomial f(x) of positive degr ee in one unknown over an algebra ically closed field /C , then the Morl ey degr ee of X is ju st the number of roots of f( x) in K.

Remarks 5.7.7

2. A definable X having Morley rank 1 and Morley degree 1 (so an infinite definabl e X admit t ing no infinit e coinfinite definable subset) is called strongly minimal. A t heory T is st rongly minimal if it s univ erse (so the formula v = v) is: we met this notion in Chapter 2, and we sha ll deal again with it in the next sect ions. Recall t hat algebraically closed fields , as well as infinite vectorsp aces over a countable field, admit a st rongly minim al t heory. 3. Let X , Y be two definable disjoint subsets of nn suc h t hat both RM(X) and RM(Y) are ordina ls, and RM(X) < R M(Y). Then GM(X U Y) equals

G M (X ) + GM(Y) if RM(X) = RM(Y) , GM (Y) otherwise (the read er may check this as an exercise). When
2. RM(X) = 1 if and only if X is infinite, bu t X can not contain infinitely many pairwise disjoint infinit e definable su bsets. As before, suppose

164

CHAPTER 5. MORLEY RA NK

x =
degree of any M -definable set X. Proof. Put X =
165

5.7. THE MORLEY RA NK, AT LAST

o because .....,:3 W (ipi(W, Vi)

/\ ipj(w ,vj) ) is in tp (aa , . .. , a"k l a) . Not ice also t hat, Vi :s; k , ai , b~ correspond t o each other by some automorphism of n fixin g a poin twise, and conseq uent ly ipi( nn, ai) and ipi( nn, b;) hav e the same RM ~ u , F inally, for every i :s; k , ip(nn, b~ ) is M -defina ble becau se b~ is in M . In conclusion, for every natural k , X contains at least k + 1 M definabl e pairwise disjoint subsets of Morley rank ~ u, So, t he combi nat orial a rg ument in Proposition 5.7.5 ca n be repeat ed and shows our claim . Now let RM (X) = 0' where 0' is a n ordinal , d = G M(X) . We have to show t hat X ca n partition as the union of d M -defin abl e s ubsets of Morley rank 0'. Again we know that X does admit such a decomposition in definabl e subsets ipo(nn, a"O) , .. . , ipd-I (nn, ad-I) with aa, .. . , ai~1 in n. On t he oth er hand tp(aa, . .. , ad-=-I la) ca n be realized in M , say by b~ , ... , bd-I, becau se M is ~o-saturated . By proceeding as before, one sees that

p rovid e a partition as req uired.

•

Now we wa nt t o define t he Mo rley rank and degr ee of a complete ty pe p over a small subset A of n: t he Morley ra nk of p R M (p) is a n ordinal or 00, while t he Morley degree GM( p) is defined onl y when R M (p) is a n ordin al , an d is a positi ve int eger. T he read er ca n eas ily observ e t hat t his ass ignme nt of a Mo rley rank a nd degr ee ca n be eas ily extended to arbit ra ry set of formul as over A (alt hough, in t his enla rged framework , the Mo rley rank may ass ume a lso the value -1 -for inconsistent set of forrnul as-). D efi nitio n 5 .7.10 Let A be a small subset of n, p E S (A ) . The Morley rank of p RM(p) is the least Morles) rank of a formula in p. If RM(p) is an ordinal 0' , GM(p) (the Morley degr ee of p) is the least Morley degree of

a formula in p of Morley rank

0' .

Accordingly, with any p E S (A ) it is associated a for mul a ip (v) E p such that RM(ip(v)) = RM(p) and , when RM(p) is an ordinal , G M(ip (if)) = GM(p) . Of course, one may wond er whether t his formul a ip (v) is unique . To clarify t his qu estion , first notice: P roposit io n 5 .7. 11 Let p E S(A) satisf y R M (p)

<

ip(v) be a for mula in p having the same Morleu rank and degree as p. Let ip'(v) be any formula with parameters f rom A . Then ip'(v) E p if and only if R M (ip (n n) ip'( n n)) < R M (p). 00 ,

166

CHAPTER 5. MORLEY RA NK

First suppose
Proof.

ip' (v))

At t his poin t one ded uces: Corollary 5.7.12 Let pE S(A) satisfy R M(p) < 00,
Proof. Let r.p'( v) E p. Owing to Proposition 5.7.11, R M( r.p(nn) _
=aidentifying two

is a n equivalence relation. According ly, when RNf (p) is a n ordinal a, a for mula r.p(v) E p with t he sa me Morley rank and degr ee as p is uniquely det ermined (in p) up to

=a.

1. Let A ~ B be small subsets of n, p E S(A) , q E S(B) , p ~ q. Then RM(p) ;::: RM(q) a nd, when RM(p) = RM(q) , GM(p) ;::: GM (q). For , all t he formul as in p belong t o q as well.

Remarks 5.7.13

2. Let p E S(A). Then p is algebraic if an d only if RM(p) = 0 (recall t hat p is algebraic if and only if t here is a formul a r.p( if) E P such t hat
167

5.7. THE MORL EY RA NK , AT LAST

subset of n has Morley ra nk 1. The same is true for every strongly minimal t heory. So we have just introduced Morley ra nk and degree for types p E S(A) by referring t o Morley rank and degr ee of formulas . Conversely, for every L (A)formul a
RM(
=L

GM (p)

p

where p ranges over the n- types over A containing
is consistent. In fact , let 'l?o(v), . . ., 'l?s(v) be L( A)-fo rmulas of rank < 00; by Proposition 5.7.4 , (iii) , also V i<s'l?i(V) have rank < 00 . Moreover
qi = {
CHA PTER 5. M ORLEY RA NK

168

By proceedin g as in t he previou s case, one checks t hat qi is consist ent, a nd so enla rges to a type Pi E Sn(A). Furthermore t his ty pe Pi is unique: in fact , if fl( v) is an L(A)-formula and RM(
5.8

Strongly minimal sets

As said before, a defina ble set is strongly minimal if it is infinite, but adm its no par ti tion int o 2 disjoint infinite definable subsets in D. A t heory is called strongly minimal if the dom ain of its universe is. Recall also from Ch apter 2 that a struct ur e A is sa id t o be strongly minimal if its th eory is, and minimal when its dom ain A ad mits no partition into 2 disjoin t A-definable infinit e subsets. Of course , st rongly minim al struct ures a re minimal , bu t t he conve rse is not t rue (see, again, Exam ple 2.7.2) . Clea rly strongly mini mal sets a re t he simplest infinit e definabl e sets and hence provide a matter worthy of some int erest. Incident ally, du e to t he new approach t o struct ures pro vided in 5.2, any strongly minimal set , and , more genera lly, an y defina ble D in D ca n be nat ur ally regarded as a structu re in it s own right. It suffices to ass ume as definable sets in D t he t races in D of the definable sets in D, hen ce D" n X for every definable X ~ Dn. It is easy t o check that t he res ult ing collection satisfies t he condition in T heo rem 5.2 .1. Wh en D is strongly minim al , t he resulting new struct ure is strongly minim al , to o. Accordingly, one ca n examine strongly mini mal structures a nd t heor ies instead of st rongly minim al sets. We already met some examples in t his area. In fact we know that infinit e vectorspaces over a countable fields , as well as algebraically closed fields (in a fixed cha racteristic) , a nd even mere infinite sets, ad mit a st rongly minim al complete t heory. Let us analyse closerl y these examples.

5.8. STRONGLY MINIMAL SETS

169

E xample s 5.8.1 1. Let K be a countable field and conside r the theor y KT ' of infinite vectorspaces over K. Recall t hat eac h of t hem is fully determined , up to isomorphism , by a cardinal number (it s dim ension over K ). In view of a possible generalization, let us remind very quickly why. So let V be any vectorspace over K. For a E V , X ~ V defin e

a -< X

(a linea rly dep end ent on X )

if and only if a is in t he subspace (X) of V spanned by X (in oth er words, a is a finit e linear combination of eleme nts in X U {O}). On e introduces in this way a subset -< of V X P(V) (the linear depen dence relation). Moreover one sees that -< satisfies th e following properties: for a, b in V and X , Y ~ V , (D 1 ) if a E X , t hen a

-<

X;

-< X , t hen t here is some finit e X o ~ X such th at a < X o; (D 3) if a -< X a nd, for every x E X , x -< Y , t hen a -< Y ; (D4) if a -< X U {b} but a -f< X , then b -< X U { a} . (D2 ) if a

C hecking (D1) and (D2) is t rivial, while (D3), (D4) ju st need some simpIe calculations. Now define a subset B of V linearly ind epend ent if a nd only if, for eve ry b E B , b -f< B - {b}; and say th at B ~ V is a basis of V if and only if B is linearly ind ep end ent and , for every a E V, a -< B . Using (D1) - (D4) a nd Zorn 's Lemma (wit h no specific reference t o t he algebraic fr am ework of vectorspaces), one shows t hat V admits some basis B and t hat two bases of V have the same power. Accordingly one defines the dim ension of V over K as the power of any basis of V. Finally one proves that two vectorspaces over K a re isomorphic if and only if they have the sa me dimens ion. This t ime, t he proof needs also th e following fact : if V is a vectorspace over K , X ~ V , a, b E V a nd both a and b are linearly ind ependent of X , t hen t here is a n a utomor phism of V fixing X pointwise and mappin g a into b (in other word s , a and b ha ve t he same ty pe over X ). 2. Let p = 0 or p be prime, conside r now t he t heory ACFp of algeb raically closed fields of cha racteristic p. Also in this case it is known t hat every model K of ACFp is fully det ermined up t o isomorphism by a cardinal number: it s tran scendence degr ee (over t he prime subfield) . Unde r

170

CHA PTER 5. M ORLEY RA NK

th is point of view, t he setting is qui t e similar to what we saw in t he prev ious example. In fact, for every algebraically closed field K of characteristic p, one int rodu ces a subset -< of J( x P (J( ) (now called t he alge braic dep enden ce relation ) in t he following way : for a E J( and X ~ J( , one puts a

-<

X

(a alge braically dep enden t on X)

if and only if a is algebraic over t he subfield generated by X. On e checks that (D1) - (D4) still hold for a, b E J( a nd X, Y ~ J( (although t he proofs, especially t hat of (D3) and (D4) , require now some major algebraic difficulties) . Any how, as in th e previou s example, we ca n define a subset B ~ J( alge braic ally indep enden t if and only if, for every b E B , b -< B - {b}; a nd we can say t hat B is a transcendence basis of K if B is algebr aically ind epend ent and, for every a E J( , a -< B. T he existe nce of a t ranscende nce basis of K and t he fact t hat t wo transce nde nce bases of K have t he same power are shown exactly as in t he previous exam ples, despi t e t he different algebra ic fr am ework , by using only (D 1) - (D4) (an d Zorn 's Lemma) . T he cardinality of a t ra nsce nde nce basis of K is ca lled t he transcen de nce degree of K. As in t he previous exam ples, one shows t hat an algeb raically closed field K of characterist ic p is fully determined up to isomorphism ju st by its t ra nsce nde nce degr ee (so by a ca rdinal nu mber). T he reason is, again, t he fact t hat, if a, b E J( , X ~ J( and a, b -I< X (so both a a nd b are t ranscende ntal over t he subfield gene rated by X ), t hen t here is a n a ut omor phism of K fixing X pointwise a nd mapping a into b (whence a and b have t he same ty pe over X ). 3. Let us propose a mor e trivi al exa mple. Let L set . For a E M , X ~ M , put

a -<X

{:}

= 0,

M be an infinite

aEX ;

t his again defines a subset -< of M x P (M ) clea rly satisfying (D1 ) (D4) . In t his case t he only possible basis of M is ju st M, and actually t he power of M fully determines t he isom orphism class of M a mong L-structures (i. e. nonem pty sets) . Before cont inuing ou r analysis of strongly mini mal t heories and sets, in order to exclude a ny possible misun derst a ndin gs a nd t o prepar e the next Chapter

5.8. STRONGLY MINIMAL SETS

171

7, let us point out a key remark: we ca nnot expec t t hat, for any complete t heory T , every model of T is determined up to isomorphism by a single cardi nal invari ant. Here is a simple " counterexample" . Example 5.8.2 Let T be the t heory of th e st ruct ures (A , B) where B ~ A a nd bot h B a nd A - B are infinite. Not ice t hat T is com plete, for exa mple becau se it sat isfies the assumpt ions of Vaught 's Theorem: it has no finite models a nd is ~o-categorical , as the only countable model (up to isomorphism) must satisfy IBI = lA- BI = ~o . In order to characterize a mod el (A , B) of T up to isomo rphism , one needs an ordered pair (IBI, lA - BI) of ca rdinal numbers.

Of course this example can be easily gene ra lized . Just think of a structure A = (A , (B n)n
notice that this sequence is infinite when N = w . Now let us restrict again our attent ion to st rongly minimal t heories. We wond er if t he similar behaviour observed in Ex a mples 1, 2 and 3 in 5.8.1 for infinite K-ve ctorspaces, algebraically closed fields an d infinite sets is found ed on som e common basis. To clarify this poin t , let us give a fu rt her glance at these examples. Examples 5.8.3 1. Let V be a vectorspace over K , a E V , X ~ V . Recall that a -< X mean s that there are n EN , k o, ... , k n E K , Xo , . .. , Xn E X such that a = Li
2. Let K F AC Fp , a E K , X ~ K. Now a -< X if and only if th ere is a polynomial f(t) E K[t]- {O} with coefficients in th e subfield generat ed by X such that f(a) = O. There is no loss of generality in ass uming that t he coefficient s of f(t) a re also in the subring generated by X (so in the L-su bstructure generated by X). Accordingly every coefficient of f(t) gets X-definabl e, and "f (v) = 0" can be written as an L(X )formula. Of course there are only finite ly many elements satisfying t his formul a , and a is one of them .

CHAPTER 5. MORLEY RANK

172

3. Let A be an infinite set, a E A , X ~ A . Now a -< X simply means a EX , hence that a is the unique element in A sat isfying the L(X) formula v = a (a is in X!) .

5.9

Algebraic c losure and definable clo s ure

What we observed at the end of the previous section suggest s the following definitions. Let M be a st ructure in a language L , X ~ M. Definitio n 5 .9 .1 1. The algebra ic closur e of X (d enoted by acl(X)) is th e union of all th e finit e X -definable subsets of M (and so is th e set of the elements a E M such that, for some L(X)-formula
2. Th e definable cl osure of X (d enoted by dcl(X)) is th e se t of all th e X -definable ele men ts of M (and hence of th e elemen ts of M s uch that, for some L(X)-formula
C lea rly dcl(X) ~ acl(X) for every X and A1. Sometimes dcl(X) = acl(X). For instance, this is the case when M expands a linear order :S and X is a ny subset of M. In fact , let a E acl(X ) ,

A

w < v).

However there do exist some st ru ctures ;\,1 and s ubsets X dcl(X) f. acl(X), as we will see in the next lines.

~

M s uch that

Remarks 5 .9 .2 1. Let M, N be structures for L such that M is an elementary substructure of N, X ~ M (hence X ~ N) . Of course, we might expect to hav e to form two algebraic closu res of X , the former in M and the latter in N. However these t wo sets coincide . Let us see why. First notice t hat, for every L (X) -formula
5.9. ALGEBRAIC CLOSURE AND DEFINABLE CLOSURE

173

rp(M) = rp(N). In conclusion, the algebraic closure of X is the same

in M and in N.

Clearly the same can be said about the definable closure. Consequently, for a complete theory T with universe n and for a small X ~ n, ael(X) and del(X) do not depend on t he model of T containing X one refers to . 2. Let T be a complete theory, X be a small subset of

n, a E n.

Then

a E ael(X) {:} there are at most finitely many elements f(a) when f ranges over the automorphisms of

n fixing

X point wise.

In fact, take a E ael(X) . Let rp(v) be an L(X)-formula such that a E rp(n) and rp(n) is finite. For every automorphism f of n fixing X pointwise, f(a) E rp(n) as well. Accordingly the images of a under the automorphisms of n fixing X pointwise form a finite set . Conversely, let a ~ ael(X), so, for every L(X)-formula rp(v), if a E rp(n) , then rp(n) is infinite. Let ao, .. . , an be different elements, all realizing tp( a/ X) . For every formula rp(v) E tp( a/ X), there is some b E rp(n), b f. ao, ... , an' As tp( a/X) is closed under finite conjunction, tp( a/ X) U {-,( v = ai) : i ::; n} is consistent and can be enlarged to a complete type over X U {ao, ... , an}: any element an+! real izing this type satisfies an+l 1= tp(a/X) , an+! f. ao, . .. , an. So there ar e infinitely many realizations of tp(a/X) in n, and each of them is the image of a under some automorphism of n fixing X pointwise. A similar argument shows

a E del(X) {:} f(a) = a for every automorphism f of

n fixing

X

pointwise. Now let us come back to the Examples 5.8.1 of the last section. We want to examine ael and del in their frameworks. E xamples 5 .9.3 1. Let V be a vectorspace over a countable field K, X ~ V . Then del(X) = ael(X) coincide with t he subspace (X) of V spanned by X.

In fact, we know that del(X) ~ ael(X) in general, and we have already seen that (X) ~ del(X) . Hence it suffices to prove ael(x) ~ (X). Notice that, if a and b are two elements of n, and both of them are linearly independent of X, then there is some a utomorphism of n fixing X pointwise and mapping a into b. So, for a ~ (X) , every element

CHAPTER 5. MORLEY RANK

174

in n - (X) is the image of a und er some a ut omor phism of identically on X ; accordingly art ael(X).

n act ing

2. Now consider a n algebraically closed field IC (of som e given characte ristic p). If X ~ J(, then ael(X) coincide wit h the algebraic closure (in t he usu al field theoretic sense) of t he su bfield of IC generated by X. In fact , we already saw t ha t :2 holds . Conversely, if a, bEn are t ra nscendental over the subfield generated by X , then there is som e a ut omorphism of n fixing X point wise and mapping a into b. Just as in 1, this excludes that a, or b, or any eleme nt transcendent al over t he subfield generated by X ca n belong to ael(X) . This t ime, del(X) i- ael(X) . The reason is very simple: in fact , if a, b E J( are two different roots of the same irr ed ucible polynomial with coefficient s in the subri ng generated by X , th en one can build an automorphism of IC fixing X poin twise a nd mapping a into b. Hence a E ael(X) , but a rt del(X). Ind eed one ca n check t hat, if IC has characteristic 0, th en del(X) is ju st t he subfield of IC generated by X , while , if IC has a prime characteristic p, then del(X) coincid es with th e closu re of t he subfield of IC generated by X under the inver se function of t he Froben ius morphism F r (the one taking any a E J( to Fr (a) = aP ) . 3. F ina lly let M be an infinit e set, X

~

M . Now ael(X) = del(X) = X.

In fact ael(X) :2 del(X) :2 X is clear. So it suffices t o show ael(X) ~ X. Owing to the Remark 5.9.2.1, we can ass ume IXI < IMI. Any two elements of M - X correspond t o each other by a permutation of M fixing X poin twise. Hence a ny X -definable subset of M overlapping M -X includes M -X, and so is infinit e. In conclusion , ael(X) ~ X , as claim ed . Anyhow, in t hese examples, we recogniz e a com mo n feature: in fact , for every small subset X of n and a E n,

a -< X

{:}

a E ael(X) .

But , before examining closerly this point, let us propose a couple of furth er examples. Examples 5.9.4 1. Let T = DLO- , X be a small subset of n. As T is concern ed with linear orders , ael(X) = del(X) . We claim ael(X) = X.

5.9. ALGEB RAIC CLOS URE AND DEFINABLE CLOS URE In fact , take a E n bEn - X such that

-

175

X. Owing to Theorem 2.3.1 , every element

b «: »

{:::}

a<x

for every x E X sati sfies the sa me L(X)-formulas as a. Hence, if a E 0 wher e f( x) a nd g(x) are nonzero polynomials with coefficients in the subfield generat ed by X. As a must be the only element sat isfying t hese condi tions, at least one equa t ion occurs, whence a is algebraic over the subfield generated by X , as claim ed . The read er may check th e details of t he last exa mple, and also ca lculate acl(X) , dcl(X) in other familiar cases, for inst ance when X is a small subset of th e universe n of dLO+. Now let us come back to a general framework . Accordingly, let A be a struct ure for a given langu age L . For a E A, X ~ A , pu t

a -< X

{:::}

a E acl (X ).

We wond er whether thi s relation < sat isfies t he conditions (D1) - (D4) in 5.8 (just as in th e t hree main examples before) . (D1) - (D3) st ill hold in this general set t ing. Let us see why.

(D'l ) For every a E A and X

~

A , if a E X , t hen a -< X.

In fact a is the only element satisfying v = a (a nd so is even in dcl (X )).

(D2) For every a E A and X ~ A , if a E acl(X) , th en t here is a finite subset X o of X such that a E acl(Xo). In fact take a E acl (X) ; th ere is some L(X)-formula
176

CHAPTER 5. MORLEY RA NK

(D 3) For every X ~ A , acl(acl(X)) ~ acl(X) . Let a E acl(acl(X) ), cp(v , X) be a formula with parameters F .e- (xo, . . . , x m ) in acl(X) such t hat a E cp(A, x) and cp(A, x) is finite. Put Icp(A , x)1= k. For every i ~ m , there is a for mula t9i(V, Y"i) of L(X) such that t9 i (A , Yi) is finite and includes X i . Put fj = (Yo , . . . , Y;") and consider the formula

o; v, fj)

: :.Jwo ... :.Jw m

( ./\

t9i( Wi, Yi) !\ :.J!kz cp(z , w) !\ cp(v ,

10))

t<m

where w abbreviates th e tuple (wo, .. . , w m ) . Clearly a E t9(A, 17) . Furthermore t9(A,17) is th e union of the sets cp (A , ; ') when X ' = (x~ , . . . , x~) satisfies x~ E t9i(A, Yi) and Icp(A , Xl ) 1 = k. Accordingly t9(A, 17) is a finite union of sets of size k , an hence is finite . Now let us wonder if even (D4) holds in any st ruct ure A. Recall what (D4) states. (D4) For every a,b E A and X ~ A , if a E acl(X U {b}) - acl(X ), th en b E acl(XU {a} ).

Examp le 5.9.5 Let A = (N 2 , J) wher e for every X, YEN , f (x , y) = (x , 0). Pu t

a= (0,0) ,

f

is the 1-ary function such t hat,

b= (0,1 ).

Then a is t he only element in N 2 sat isfying v = f(b) , whence a E acl(b). Moreover a rt acl(0) becau se an y element (x,O) with x E N is the image of a in some automorphism of N 2 . But b rt acl(a) becau se any element (0, y) with yEN - {a} is the image of b = (0,1) under som e automorphism fixing a = (0,0). Accordingly (D4) fails in general. However the following proposition holds. Pro p osition 5. 9 .6 Let A be a minimal structure. Then (D4) holds in A: if a,b E A, X ~ A and a E acl(XU {b}) - acl(X) , then b E acl(XU {a}).

Recall that A is minimal if and only if any su bset of A definabl e in A is either finite or cofinite. Algebraica lly closed fields, vectorspaces (over coun t abl e fi elds) and pure infini te sets a re minimal structures. On t he contrar y, t he structure A of t he las t example is not minimal: for , t he formul a f (v) = (0, 0) defines a n infinite coinfinite set.

5.9. ALGEBHAIC CLOS UHE A ND DEFINABLE CLOS URE

177

Proof. As t he minimality of A is preserved under adding or forg etting parameters from A , there is no loss of generality in assuming X = 0 (otherwise replace A by Ax). Suppose towards a contradiction that t here are a a nd b in A such that

a E acl(b) - acl(0),

b f- acl(a).

As a E acl(b) , t here is a n L-fo rmula ~ (v, w) for which ~ (A, b) is finite a nd includes a; let k be the size of ~(A , b) . Accordingly b sat isfies the formula

'l9( v, a) : ~(a , v) !\ 3!k z ~ (z , v). As b f- acl(a), 'l9(A , a) must be infinite, hence cofinite. Let m denote the (finite) cardinality of A - 'l9(A , a); so a sat isfies

As a f- acl(0), also ')' (A ) must be infin ite. Pick k + 1 pairwise different elements aa , . .. , ak in ')' (A). For every i ::; k, t here are exactly m elements in -,'l9(A , ai) . Exclude all these elements for i ::; k. We ca n st ill find some b' E A satisfying

A

1=

1\ 'l9(b' , ai),

i< k

hence

A

1=

1\ ~ (ai , b') !\ 3!k

Z

~ (z, b').

i< k

Consequ ently aa , ... , ak E ~ (A , b') , contradicting 1 ~(A,b')1 = k. This accom plishes th e proof. '" Hence, when T is a strongly minimal t heory, -< sat isfies (D1)-(D4) in every mod el A of T. T his allows to generalize the notions already introd uced in the main examples: for B ~ A , one can say that

• B is independen t if and only if, for every b E B , b f- acl(B - {b}), • B is a basis of A if a nd only if B is ind epend ent and A = acl(B) . Using Zorn 's Lemma, one can st ill deduce the existence of a basis of A, and the fact t hat two bases hav e the same ca rdinality; in t his way A is naturally ass ociated with a ca rdinal nu mbe r, t hat is t he power of an y basis. T his is ca lled the dim ension of A. Of course , all this directly depends on (D1)-(D4). However the fac t that this ca rdina l det ermines A up to isomorphi sm refers to another basic property of st rongly minimal theories, ensuri ng that , for

178

CHAPTER 5. M ORLEY RANK

every X ~ A , in pa rticular for X = 0, t here is a unique non- algebraic l -type p x over X : t he one containing every formula
-< satisfies (Dl) -(D4);

(b) t here is a unique non-algebraic l-type over a ny small s ubset X of

n.

To und erline once again the relevance of (b) , let us mention th e behaviour of a-m inimal t heories. Recall that t hey are t he complete theories of t he infini te st ructures A expanding linear orders in such a way t hat th e only defina ble subsets of A are t he finite unions of singletons and open intervals (possibly wit h infinite endpoints) . We will see in the Chapter 7 t hat t here are good reas ons to agree t hat t he models of t hese theories ca nnot be classifi ed up to isomorphism . In spit e of t his , in an a-minimal structure A -< ca n sat isfy (Dl) - (D4)' hence (a) holds (but (b) do es not ) . As (Dl) - (D3) a re always true, we have to check (D4) . Here is th e proof. Proposition 5.9.7 Let A be an o-minimal structure expanding th e linear order (A , <). a, b E A , X ~ A. If a E ael(X U {b}) - ael(X) , then b E ael (X U {a}). P roof. As the o-min imality of A is preserved under adding or forgetti ng paramet ers from A , t here is no loss of generality in ass uming X = 0. Let a, b E A satisfy a E ael(b) - ael(0) , we have to show b E ael(a) . As ael and del coincide in linea rly orde red st ruct ures, a is in del( b) a nd so t here is an L-formu la
5.9. A L GEBRAIC CL OSURE A ND DEFINABL E CL OSURE

179

are {a }-definabl e. Co nseq uently, if b eq uals one of them, t hen b E acl(a). So suppose t hat b is in the int erior of some interval among 1o, . .. , Ik. Let I de note this interval, d1 < d2 be its end points (where d1 , d2 may possib ly eq ua l - 00, +00 respectively). If I is finite, then it is easy to deduce again s « acl(a). Acco rdingly assume I infinite. Notice that d1 cannot equal - 00 , otherwise a is t he only element in A satisfying :3wYz(z ::; w -+ f (z) = v), whence a E acl(0). Simila rly, d2 < +00. For a suitably la rge positive integer n, let D n denote t he (0-defina ble) set of t he left end points d E A of some open interval J having size ~ nand satisfying t he following furt her ass umptions: f is (defined and) constant in J , and one ca nnot enla rge J to t he left keeping this cond it ion. Associate any d E D n with the right end point g (d) E AU {+oo} of a maxim al int erval J. By o-minim alit y, g(d) is well defined: in fact, t he elements> d lying in the domain of f an d having the same value in f as immediately after d form a defina ble set, and so a finite union of intervals (in t he broader sense said before) . Mo reover , for d < d' in Dn , ]d, g(d)[n]d', g(d' )[= 0. Recall t hat D n is definable, so , by o-m inimality, D n must be finite . As d 1 E D n , d1 E acl(0) = dcl(0). As a is t he image of any element in ]d 1 , d2 [ by J, a 'is in acl(0) as well, and th is contradicts our hypotheses . So I ca nnot be infinite, and we are done. ., Before concluding th is section and the whole chapter , we would like to underline two more points. The form er specifically concerns a minimal, or o-rn inimal L-struct ure M. Both t hese prop erties (minimality a nd o-m inimality) are preserved under adding parameters from a subset X of M, so und er pas sing from L to L(X) and from M t o M x: M x remains mini mal , or o-minimal. Indeed , if the t heory of M is st rong ly minimal, t hen T h(Mx) is: in fact its models are t he struct ures Nf (x ) where N is an L-st ruct ure elementarily equivalent to M -so a mini mal struct ure-, and f is a n elementary function from X into N . T he same happens for an o-m inimal T h(M ). So, for M as before, we ca n int roduce for every X ~ M a new de pen de nce relation <x in M X P (M ) by put t ing, for a E Me S ~ M, a <x S if a nd only if a E acl(S) in M x , in other words if a nd only if a E acl(X U S) in M. Moreover <x st ill satisfies (D1) -(D4) . Secondly, when our relation -< on M X P (M ) satisfies (D1) - (D4) , we can eq uip not only M, but also every subsets S of M wit h a dimension dimS : this can be formally int rodu ced as the minimal power of a subset B of S such t hat S ~ acl(B) (in fact , it is easy to check that such a B is independent ). Of course, t his also concerns the framework sket ched a few lines ago: for

CHAPTER 5. MORLEY RANK

180

x

~

M, the dimension of X over X dim(Sj X) is just the dimension of S

with respect to <x A final remark: in order to pursue and possibly accomplish our analysis of strong minimality, we can wonder how general the examples we proposed (pure sets, vectorspaces and algebraically closed fields) are. Do there exist a ny" new" significant instances, or do th ey exhaust all the possible cases? In Chapter 7 we will discuss the relevance of this question (due to Zilber) and we will provide its solution (due to I-Irushovski).

5.10

References

The papers where Morley developed his ideas and, in particular, introduced his rank are [116, 117, 119, 120] and [118]. The duality between Boolean algebras and Boolean spaces is in [52]. The proof of Theorem 5.4.2 in the general setting, for any cardinal A can be found in Poizat's book [131], which also includes full details on saturated models. Inaccessible cardinals are described in [66] or [78] . Now let us deal with modules: both [181] and [136] treat the matters of Section 5.5 (pure injectivity, Ziegler topology, and so on). Tame and wild classes of modules are also discussed in [136]. The classification of pure injective objects among abelian groups, and more generally among modules over a Dedekind domain, is given in [40]. An abstract treatment of dependence relations, including the main examples in Section 5.8, was developed in [173]. The details on the real closure of an ordered field can be found in [65]. An axiomatic introduction to ranks, and a comparison of Morley rank with other rank notions, are in [8]: see also [131].

Chapter 6

w -stability 6.1

Totally transcendental theorie s

We cont inue her e our t reat ment of Morley's Theor em a nd related ideas. All t hroughout t his section, T is a complete t heory wit h no finite mod els in a countable language L , and S1 denotes a big saturated mod el of T . In t he last cha pter we defined Morley rank and Morley degree as a complexity measu re for definabl e sets. In parti cular we st udied t he simplest infinite definable sets wit h res pect t o t his measure , i. e. t he stro ngly minimal sets, t hose having Mo rley rank 1 a nd Mo rley deg ree 1; we cons idered also st rongly minim al theories, i. e. t he complete t heories whose universe is strongly minimal. More generally, one ca n examine t he t heories in which every non empty definable set gets a (n ordinal) Morley rank. T hese were called by Morley totally transcendent al. Defi ni tion 6.1.1 T is called totally transcendental if and only if, for every non empty definable set X ~ S1n (wit h n a positive integer) , R M (X ) is an ordina l num ber.

Hence totally transcend ental theories exclude t he th eor y DLO- of den se linear orderings without end point s, as observ ed in Example 5.7.3,4, but include, for inst ance, t he strongly mini mal t heories, which sat isfy RM (S1) = G M (S1) = 1; ind eed , as we will see within a few lines, strong min imality impli es R M (S1 n) = n , G M (S1n) = 1 for ever y n , a nd so RM(X) :S n for every definable set X ~ S1n. Totally transcendental theories ca n be cha racterized in t he following alt erna ti ve a nd equivalent way.

181

182

CHAPTER 6. w -STABILITY

Definition 6.1.2 T is cc -stable if and only if, for every countable A th e space 5(A) of typ es over A is also countable.

~

D,

On e ca n equivalently ask that , for every countable A ~ D a nd positive integer n , 5 n(A) is countable. Actually it suffices to require 51 (A) countable. In fact let n be a n inte ger bigger than 1, pE 5 n(A) , if = (aI , . . . , , an) realiz e p in Dn. Then p is fully determined by:

(i) the (n - l.j-type of (a1, "" an-I) over A (equivalently, the orbit of (a1,"" an-d with res pect t o t he A-au tomorphisms of D);

(ii) the l-type of an over A U {aI , . .. ,an-d (hence over a countable set ) up to A-automorphism . So , when 5 1(A) is countabl e, a simple ind uct ion proves t hat 5 n(A) is countabl e for every posit ive integer n . Ob serve also t hat T is w-stable if and only if, for every countable model M of T , 51 (M) is countable. In fact , for every countable A ~ D, t here exists some countable model M of T including A (just a pply the Lowenh eimSkolem Theore m t o the t heory of DA). Every l-type over A extends t o some (possibl y non unique) I-type over M, and con versely restricting to L( A) a l-type over M determines a l-type over A . So 15 1(A)1 ::; 15 1(M)I . Now let us point out: Proposition 6.1.3 A stro ngly m in imal theory T is eo-stable. Proof. Let A be a count a ble subset of D. The language L( A) is countable, as well as the set of the L(A)-formulas
Example 6 .1.4 DLO- is not w-st abl e. In fact (Q ,::;) is count a ble, but it is easy t o realize t hat t here are 2~o I-typ es over Q: just observe that , for every r, s E R with r < s , there is some a E Q such that r < a < s. Accordingly tp(r/Q ) f tp(s/Q) as the former typ e contains v < a , and the lat t er do es not. As said before, w-stability is j ust equivalent to total tran scendency. Indeed the following theorem holds. Theorem 6.1.5 T is totally tran scendental if and only if it is eo -stable.

6.1. TOTALLY TRA NSCEND EN TAL THEORIES

183

Proof. Let T be t otally t ranscende nt al. Then every ty pe p over a small set A ~ n is fully determined by an L (A )-formula
The set A of the parameters needed t o define th e X a 's when a ranges over {O , 1} <w is countable. For every a E {O , I} W, th e L (A)-formulas " v belongs to X u1n" (with n a nat ural number) form a consiste nt type, which ca n be extended to a (complet e) typ e Pu E St(A) . Of course different sequ ences a , a ' E {O, 1} <w yield different typ es Pu =1= p~ . Accordingly ther e are at least 2 No I-types over A , and this cont radicts t he fact that T is w-stable. .. A consequence of th is th eorem is that t ot al transcend ence (in fact w-stability) is preserv ed by int erpret ability. C oro llary 6.1.6 Let A be a struc ture for L , A' be a structure for a possibly different language L ' suc h that A , A' are infi nite and A' is in terpretable in A. If Th (A) is w-stable, then T h( A ') is.

Proof. (Sketc h) We know t hat, for a suitable choice of a positi ve integer n , a subset S of A n definable in A a nd a n equivalence relation E in A n

CHAPTER 6. w -STA BILITY

184

definable in A , A' is ju st t he quotient set SI E and t he whole L'-structure A' ca n be definabl y recover ed by A. Furthermore, if D is a big saturat ed elementary extension of A , t hen t he machin er y definin g A ' inside A singles in D a big saturated elementary extension D' of A' (of t he same power as D) an d every set X' definabl e in D' is t he quo t ient set of some X definab le in D wit h respect to E. At t his point a simple arg ument of ord ina l induct ion shows t hat t he Morley rank of X ' in T h(A' ) is smaller or equal t o the Morley rank of X in Th(A) . Co nseq uent ly, when RM(X ) is a n ordinal , R M (X') is a n ord inal, too . In conclusion , if T h(A ) is w-stable, th en T h(A' ) is. ., Now let us deal again with Morley rank and degree for a st rongly minimal theory T. We already seen t hat

RM(D) = 1,

GM(D) = 1.

Recall t hat t here is a uniqu e l -f.ype of Morley rank 1 over a ny small subset X of D, t he one of t he eleme nts ou t of acl(X ). Furth ermore, for a, S in D,

a <x S

<=?

a E acl(X U S)

determines a dep end ence relation satisfying (Dl) - (D4) . Mo re gene ra lly, one ca n see t hat , for every positi ve integer nand n-t uple ii in D, RM (tp(ii]X)) equals t he dimension of ii wit h respect t o <x - In par ti cular R M(tp (iil X)) ca nnot exceed n; indeed t here is a unique n-ty pe over X of Morley rank n , t he one of an n-tuple ii --<x -independe nt. Furt hermore as already said at t he beginn ing of t his sect ion. The proofs of t hese fac ts require patience rat her t ha n ingenuousness, and can be found in t he references qu oted at the end of t he chapter. Not ice t hat , in t he particular case of algebraically closed fields, the Morley rank of a t uple d over a subfield F ju st equ als t he t rascende nce degree of F(ii) over F.

6.2

w-s t a b le groups

A paradi gmatic al example of w-stable t heory is t he t heory ACFp of algebr aically closed fields of a fixed characteristic p = 0 or pri me. Bu t consequentl y every structure defina ble, or interp retable in a n algebra ically closed field IC has a n w-stable t heory. T his is t he case of t he groups G L(n, IC ) and,

6.2. w-S TA BLE GRO UPS

185

mor e generally, of th e alge braic groups over K , as we will see in Chapter 8. A common feature of t hese exa mples is t he presence of a binary operation (th e addition in K , t he usu al row-by-column multiplication in GL (n , K )) makin g t hem a group. Accordingly we propose t he following definitions. Defi nition 6.2.1 An w-st a b le structur e is a struc ture having an w-stable theory ; an w-s tab le group is an w-stable structu re with a binary operat ion making it a group (and possibly oth er operations and relations). Then a ny algebraically closed field K is an w-stable group (wit h respect to the sum operation) , as well as any group GL(n , K) a nd, more generally, any algebraic group over K (as we will see in Chapter 8). Other exa mples ca n be found among abeli an groups; for instance, we know that the theory of non-z ero vectorspaces over Q is strongly minimal and consequent ly w-stable. Accordingly every non-z ero Q-vect orspace is an w-stabl e group (with resp ect to addit ion), and so every divisible torsionfree a belia n group A is a n w-stable group (for, A can be naturally equipped with a st ruct ure of Q- vect orspace). The aim of t his section is to begin t he analysis of w-stabl e groups. In fact, pa rt of its results will be accomplished, and hence fully und erstood and ap preciated, only in t he next C ha pters 7 and 8, where mor e powerful t ools will help our st udy. However t he present pag es will develop severa l basic noti ons in t his area. Most of t hem clearly owe t o t he t heory of algebra ic gro ups , and par t of t hem refer to t he study of finit e groups. Of course, we a re also int erested in classifying w-stable groups an d in und erstanding t heir connections with algebraic groups, or finit e groups, or a belia n groups; bu t we will examine in a closer a nd deeper way th ese t hemes in C ha pte r 8. First let us observe what follows. Let 9 be an w-st a ble group, H be a definabl e subgro up, a E G. Notic e t ha t, if X is a subset of H definable in g, t hen aX = {ag : g E X} is a definable subset of th e coset aH

a nd X a = {ga : g E X} is a definable subset of Ha .

It follows that, for any a, both aH and Ha have the same Morley rank a nd degree as H. Con sequ ently, if H ~ K are definable subgroups of 9 and Il i- K , t hen R M (Il ) ~ R M (K ) a nd , if = holds, GM( Il) < G M (K)

(indeed K is a union of cosets of H ). Now we show a cha in condit ion for definabl e subgroups of w-stable groups.

186

CHAPTER 6. w -STABILITY

Theorem 6.2.2 Let 9 be an eo-stable group. For every natural n, let H n be a definable subgroup of 9 such that, for every n , H n +1 is a subgroup of H n . Th en there is natural n such that, for eve ry m ~ n in N , H m = H n .

P roof . Otherwise, for every natural n , there exists mEN such t hat m > n and n.; #- n.. Hence RM(Hm ) < RM(Hn ) a nd, when R M (H m ) = RM(Hn ), G M(H m ) < GM(Hn ) . In part icular , fixed n , there is some s E N tale che s > n e RM(Hs ) < RM(Hn ) . But in this way one forms a st rictl y decreasing infinite sequence of ordinals, and this is a cont radiction . '" Corollary 6 .2.3 Let 9 be an eo-stable group an d, for every natural n, let K; be a definable subgroup of g. T hen n nEN K ; is definable.

Proof. Fo r every nat ural n, pu t H n = n i < n Ki . T hen H'; is a definable subgroup of 9 and H n +1 ~ H n for every n. Con sequ ently t here exists mEN such t hat, for every i :s: m , H i = H m · So n nEN K n = n nEN H n = H m = n n<m tc; an d n n<m «; is definable. '" Corollary 6.2.4 Let 9 be an ea -st able group , f be a defi nable group homomorphism from 9 int o 9 having a fin ite kernel. Th en the image f(G) of G has a fin it e index in g .

Proof. Otherwise one ca n build a st rict ly decreasing infinit e sequence of s ubgr oups G ~ f( G) ~ f 2(G) .. . ~ r(G) ...

contrad ict ing T heorem 6.2 .2. In fact , as f is defina ble, r (G ) is a definable subgroup of 9 for ever y natural n. Fur th ermor e, if f(G) has an infinite index in g, t hen, for every n EN , (G) has an infinite index in (G), a nd so rH(G) #- r(G). Let us see why. The case n = 0 is ju st our assumption. So t ake n > 0, and suppose th at the index of (G) in (G) is infinite. Notice t ha t, for a and b in r(G) , f (a) - f(b) E r(G ) if and only if a - b E (G ) + K er f . As K er f is finit e and r(G) has an infinite 1 index in (G), t here ar e infinitely man y elements in (G) pairwise 1 inequivalent modulo r(G) , and so (G) has infinite index in r (G ).

r

r:'

r

r:'

r:'

r+

'"

r:'

r+

Now take any st ruct ure 9 exp anding some given group (G, .) (so 9 is not necessaril y a n w-stable group). As said before, if X is a definable s ubse t of di G, t hen also aX={ag :gEX } ,

Xa={ga :gEX}

187

6.2. w-STABLE GROUPS

are for every a E G, as well as X - i = {g- i : g E X}. Notice also t hat, for every ty pe p over G and a E G , we can form the sets ap, pa of the formulas cp(v) of L (G ) such that cp(av) E p, cp(va) E p respectively. It is easy to see that ap, pa are types over G. More prec isely, if x denotes any realization of p, then ap is t he ty pe of a-ix over G and pa is the type of xa- i over G . The functions fro m G X S (G) to S (G) mapp ing any ordered pai r (a, p) of G X S(G) into ap and pa define two actions (a left action and a right action respecti vely) of G on S(G). Correspondingly we ca n cons ider, for every pES (G), t he left stabilizer {a E G : ap = p} of p in G and t he right stabilizer {a E G : pa = p} of p in G . It is well known t hat bot h are subgroups of g. In a similar way, for every type p over G , we ca n form t he set p-i of t he formulas cp(v) in L(G) such t hat cp (v - i ) E p. Eve n p- i is a t ype over G; ind eed , if x is a ny realization of p, t hen p- i is ju st t he type of x-lover G . Now let 9 be an w-stable gro up. As w-stability is preser ved by =, t here is no loss of generality for our pu rp oses in rep lacing 9 by a su itably saturated elementary extension, and so in ass uming t hat 9 itself is saturated in some uncountable power (and so is the universe of its com plete theory). Let X ~ G be definable. As observed before X has the same Mo rley rank and degree as aX and X a for every a E G, and also as X-i . Conseq uently, for every type p E S(G) , RM(p) = RM(ap) = RM(pa) per ogni a E G and RM(p) = RM(p-i) . Moreover

9 be an w-stable group, p be a type over G. Then both the left and the right stabilizers of p in G are defina ble subgroups of g.

Lemma 6.2.5 Let

Proof. We treat t he left case (t he right one ca n be han dled in a simila r way). For every L-formul a cp(v, ill) let G(p, cp) denote the set of the elements a E G such t hat, for every 9 and h in G, cp (hv,9) E p

{:}

cp (hv, 9) E ap.

It is easy to check that G ip, cp ) is a subgroup of g. In fact , let a, bE G(p, cp); t hen, for every h an d 9 in G,

cp(hv, if) E P {:} cp( hv, if) E ap {:} cp(hav, if) E P {:} cp(hav, if) E bp {:} {:} cp(habv, if) E P {:} cp( hv, if) E abp; whence ab E G(p, cp). Furthermore

188

CHAPTER 6. w -STA BILIT Y

whence a-I E G(p,

or


(t hat is
Clearly th e left stabilizer of p is th e intersection of the subg roups G(p,
9 be an to-stable group. T hen there exists a uniqu e

subgroup GO of 9 which is definabl e, connected and of fi nite index in g. Every definable subgroup of fin ite in dex of 9 includes GO.

Proof. Suppose that 9 has no definable connected subgrou p of finite ind ex . Acco rdingly every definab le subgro up H of 9 of finite ind ex has in its turn a proper subgroup definable in 9 and of finite index (in H and consequently in G) . Th en one can build a strictly decreasing infinite sequence of defina ble subgroups of 9

each of finit e ind ex in its predecessor , and so in g. Bu t t his cont radicts Theorem 6.2.2. This shows the existe nce of a s ubgroup as required . It s uniqueness was alread y observed. Of course t his unique subgroup do es deserve a specific symbol (Go , for instance) . Fin ally notic e that , if H is a definable su bgroup of finite inde x and H 1. GO, then If n GO contradicts the fact that GO is connected. ..

189

6.2. w-STABLE GROUPS

GO is ca lled the connected component of 9 . GO is a normal subgroup of 9. In fact both the right and the left multiplication by a given element a E G are defina ble. Hence, for every a E G, a- 1GOa is a definable subgroup of 9 of the same index as GO, a nd consequently equals GO . But we can say even more. Proposition 6.2.7 L et

9 be an w-stable group. The connect ed compon ent

GO of 9 is an 0-definable subgroup; in particular it is in variant under any automorphism of 9 .

Proof. GO is definable, and so th ere are a formula

1\

defines a s ubgroup
The form er claim is almost trivial. Let k still denote t he ind ex of GO in G. Of course G is the union of th e k cosets of GO in G. Each coset is of th e form aGo for some a E G , and so has t he sa me Morley rank as GO. Henc e RM(G) = RM(GO). W it h respect to the latter claim, again it is easy to observe that , if 9 is an w-stable gro up of Morley deg ree 1, then 9 has to be connected . But showing t hat in general, for any w-stable G , GO has Morley degree 1 is not so simple. Indeed we have to prove t hat GO does not decompose as the un ion of two disjoint subsets of the sa me Morl ey rank, and so that there is a uniqu e 1type p over G containing the formu la <po(v) defining GO and having the sa me Morl ey rank as G . This needs some further mor e sophist icated ideas and tools (to be introduced in Chapter 7) . So we postpone th e full details of Theorem 6.2.8 to Theor em 7.5 .10. Here we limit ourse lves to a prelimin ary result , saying t hat , if p is a type over G as requ ired before (so cont aining t he formula <po (v) defining GO and having the same Mo rley ran k as G), t hen GO equals both t he left and the right st a bilizers of p.

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CHAPTER 6. w -STABILITY

L e m m a 6 .2 .9 Let 9 be an cc-stable group, G O be it s con nected component, p be any t ype containing th e formula vP( v) defining G O and ha ving th e same Mo rley rank as G. Th en GO equals th e left and th e right stabilize r of p. Proof. It suffices t o handle t he left case . Let S deno t e t he left st abilizer of p. Then S is a definable subgro up of g. Moreover, for every a in G , ap has t he same Morley rank as p ; as t here are only finit ely many ty pes over G ha ving t he same Morley rank as G, there are onl y finit ely many differen t types ap when a ranges over G . On t he other side, for a a nd bin G , ap = bp hold s if and onl y if a and b are in th e same left coset of S in G . So S has finit e index in G , whence S ;2 GO. Conversely take a E S. For every realizat ion x of p in n, ax sat isfies p as well. In particul ar both x a nd ax lie in (l (n ); hence a = ax x- l sat isfies
To summarize, if 9 is a n w-st able group (of Morl ey ra nk a ), GO is it s connect ed component and 1 = aD, al , . . . , ak-l are a set of representatives of th e cosets of G O in G , t hen G = Ui
n is said

t o be generic over A if and onl y if its typ e

Clearly there are only finitely many generic t yp es over any small A <;;;; n. Moreover, if x is generic over A, th en also x- l , ax and x a (for a in A) are. It is worthy underlining th at every g E G can be written (in n) as t he product of t wo elem ents generic over G : in fact , if x E n is generic over G, t he n also g x- l is, and g = (g x -l) x . Let us conclude th is section by proposing another fund am ental tool in t he a na lysis of w-st a ble groups, in particular of w-stable gro ups of finit e Morley rank (i. e. of Morley rank < w): t his is t he so called Zilber's Indecomposa bility Th eorem. We need t he following definition . Definitio n 6 .2.10 Let 9 be a structure expanding a group (G, .), X be a (non empt y) definable su bset of G . X is sai d to be (left) indecomposable

191

6.2. w-STABLE GROUPS

if and only if, for every definable subgroup H di g, either X is included in a unique left cosel of H in G, or X overlaps infinitely many such cosets. Right indecomposability is defined in a specular way. It is easy to see that, for every definable X , X is left indecomposable if and only if X-I is right indecomposable. In particular, when X = X-I , X is left indecomposable if and only if it is right indecomposable. Now we can state and prove Zilber's Indecomposability Theorem . Theorem 6.2.11 (Zilber) Let 9 be an w-stable qroup of finit e Morley rank (where finite means < w). For every i in a set J of indexes, let Xi denote a definable left (right) indecomposable subset of G containing the identity element 1 of g. Then the subgroup of 9 generated by UiEI Xi is definable and connected in g . Moreover only finitely many Xi's are sufficient to generate it. Proo]. Assume for simplicity Xi left indecomposable for every i: the right case can be handled in a similar way. As G has a finite Morley rank, we can choose a subset B of G such that B is a finite product of sets Xi or X i- 1 with i E J (and hence is definable) and B has a maximal Morley rank. Accordingly, for every i E J, RM(XiB) = R 1W.(Xi- 1 B) = RM(B); in fact 1 E Xi n X i- 1 and hence B ~ XiB, B < X i- 1 B. Let p be a type over G containing the formula "v E B" defining B and having the same Morley rank as B. Moreover let S denote the (left) stabilizer of p . Recall that S is definable (Lemma 6.2 .5). We claim that S is just the subgroup generated by the Xi'S when i ranges over J , and that only finitely many Xi's are sufficient to generate S. First notice that, for every i E J , Xi ~ S. It suffices to show that Xi intersects at most finitely many left cosets of S in G; in fact, in this case, as Xi is left indecomposable, Xi is wholly included in a single left coset of S in G, which must coincide with S as 1 E Xi. Accordingly take a E Xi; the formula "v E X i- 1 B " defining X i- 1 B is in ap. We know that RM(ap) = RM(p) = RM(B) and that "v E X i- 1 B" occurs in at most finitely many types having its rank. Hence there are only finitely many pairwise different types of the form ap with a E X, otherwise RM(Xi- 1B) > RM(B), which contradicts what we have observed before. Consequently there are only finitely many left cosets of S in G of the form with a E X (recall = bS {:} b- 1a E S {:} bp = ap for every a and b), as claimed. So S is a subgroup containing each Xi. Now we show S ~ BB-I. Let a E S, x FP. SO even ax realizes p, and both x and ax satisfy the formula "v E B " . Consequently a = axx- 1 satisfies

as

as

CHAPTER 6. w -STABILITY

192

"v E BB-I " in n and hence in g. In par t icular S is gener at ed by t he X i's (act ually, by a finit e subfa mily of them ). At t his point it rem ain s t o show t hat S is connected in g. So take a subgroup H of S definable in 9 and having a finit e ind ex in S . Accord ing ly, for eve ry i E I , X i overl aps onl y finit ely many left cosets of H , whence X i ~ H. Bu t t his for ces 1I = S. ..

6.3

w-st a b le fields

T he aim of this section is t wofold . F irst we want to a pply wh at we hav e seen a bout w-stable th eori es a nd , more particularly, w-stable groups to prove a beautifu l t heorem du e to Macintyre a nd characteri zing the fields having an w-stable theory. We already know t hat they include the algebraically closed fields ; bu t now we will show t hat no further exam ples arise, so t he w-st abl e co m plete theories of (pure) fields ar e just the t heories ACFp where p is 0 o r a prime. Secondly, we will provide a new proof of t he fact t hat algebr aically closed fields eliminate the imaginaries; t his alternative approach mainly refers t o t heir w-stability and , owing t o t his feature, a pplies t o other w-stable settings, including differen ti ally closed fields of cha racteristic 0 (as we will see lat er in t his chapter) . Now let us state Macintyre's T heo rem . Theorem 6.3.1 (Maci ntyre) Let I( be an infinite in tegra l do main with ident ity 1, and let I( ha ve an w -s table theory. T hen I( is an algebraicall y clo sed fi eld. Proof. First let us see t hat I( is a field. Take a E J( , a i= O. Let C denote prop er inclu sion. If «« C J( , t hen a n+1 J( C an J( for every positive integer n because I( is a dom ain and so, for b E J( - o.K, anb E anJ( - a n+ l J(. So I( is a n w-stable group (with resp ect to the sum op er ation) with a st rict ly decreasing infinite seq uence of definabl e subgroups J( J aJ( J a 2 J( J ... , whi ch cont radicts Theorem 6.2.2. Accord ingly o.K = K , whence there is some c E J( s uch t hat ac = 1. So I( is a field. Now suppose towards

a contrad iction th at I( is not alge braically closed . Hen ce I( has a G alois extension F of finite degr ee > 1. Conseq uently t here is some intermedi at e field E extending I( an d included in F such t hat t he Galois group of F over E is (cyclic) of pri me or de r q. O n t he other side E is an extension of finit e degree of 1( , a nd hence is definable in 1(: in fact , let d de note t he dimension of L as a vectors pace ove r 1( , t hen E ca n be viewed as I( d eq uipped with

193

6.3. w-STABLE FIELDS

s uitably defined field op erations. Hence E has an w-stable t heory, a nd an extension F with a (cyclic) Galois group of pri me order q. Wi th no loss of generality replace K by £ and hence ass ume that K it self has a Galoi s extension F of prime degree q. F ield t heory tells us t hat, in t his setting, F = K (a ) where t he minim um polynomial a over K is eit her xq -

a

wit h

a

E K , q =1=

ca r

K

or xq

-

x - a with a E K , q = car K .

To get a contradiction, we show that every poly nomial of this form must be redu cible . For this purpose, first recall th at K is an w-stabl e group with respect to addition +. Moreover J( coincid es with its connected component (with resp ect to +). In fact , take a E J( and look at t he multiplication by a. T his gives a n a ut omor phism of t he addit ive group of K , and indeed a n a ut omorphism definabl e in K. Consequent ly t he multiplication by a fi xes t he connected component So a J(° = J(o for every a E J( , in oth er words aJ(o is a n ideal of K . Hence J( o = J( as J( is infinite and so ca nnot eq ual {a } . Now conside r J(* = J( - { a}; J(* is an w-stable gro up wit h res pect to multipli cation . As J( is infinite, J(* has t he same Mo rley ra nk a nd degr ee as K , in particular J(* has degree 1, a nd consequent ly J(* eq ua ls its connect ed com ponent . Notice t hat t he fu nction of J(* int o J(* mapping any element k E J(* into k q is a definabl e group homomorphism having a finite kern el {a E J( : aq = I} . Owin g to Corollary 6.2.4, t he image of th is fun ction has a finite index in 1<* a nd consequent ly equals J(* as J(* is connected. Mo reover aq = a and so k f---t k q defines a surjective function from J( ont o J( . In particul ar, for every a E K th e polynomial x q - a is reducible. In t he same way, when q = c a r K, th e function h of J( int o J( mapping any element k E J( int o k q - k is a definable endomorphism of t he additive grou p K a nd has a finit e kern el {a E J( : a q = a} . Accordingly its image has finit e ind ex in J( , and so coincides with J( because J( is connected . Then h is surjective and, for every a E J( , even t he polyn omi al x q - x - a is redu cible. T his yields t he required cont radict ion. '"

«».

x»

It is worth und erlini ng once again t hat Macinty re's T heore m deals not only

wit h infinite fields , but more gene ra lly wit h integral domains wit h identi ty; mo reover, as algebraically closed fields are st rongly min imal, it classifies even t he strongly mini mal examples in t his framewor k.

CHAPTER 6. w -STABILITY

194

Now let us deal with the second matter of this section: elimination of imaginaries. Theorem 6.3.2 Every algebraically closed field I( eliminates uniformly the . . . tmaquuiries. Proof. Our new approach needs the following non trivial algebraic preliminaries, regarding arbitrary fields I( and ideals I in I([X] (where x still abbreviates (Xl, ... , x n ) ) : their details can be found in the references quoted at the end of the chapter.

1. The first concerns the fields of definition of I: these are su bfields of I( containing the coefficients of the polynomials of some generating su bset of I. One can see that there is a (countable) field of definition I( (1) of I satisfying the following additional condition: for every automorphism a of 1(, a (more exactly, its natural extension to I([X]) fixes I setwise

if and only if a acts identically on K

(1).

2. Incidentally one can notice that, for every automorphism a of I , if a(1) 2 I then a(1) = I. 3. Now consider prime ideals in I([X]. More particularly, take pairwise different prime ideals 10 , ... , I m of I([X] in the same conjugacy orbit: so, for every j < h S; m, there is some automorphism of I( taking Ij to h. Correspondingly one can find a subfield 1(0 of I( such that, for every automorphism a of 1(, a permutes 10, ... , I m if and only if a fixes [(0 pointwise. It suffices to form I = nj<m Ij and choose as 1(0 the field of definition 1((1) associated with I as in 1. The key point here is that an automorphism a fixes I setwise if and only if it permutes the Ij's. This is a consequence of two facts: the former is that, owing to 2, I = nj <m Ij is an irredundant decomposition of the (radical) ideal 1 as an intersection of prime ideals, and the latter is that this irredundant decomposition is unique up to the order of the involved prime ideals. Let us deal at last with algebraically closed fields I( and with Model Theory: we want to show that I( uniformly eliminates the imaginaries. As observed in Chapter 4, it is sufficient for our purposes to prove that any such I( eliminates the imaginaries and so to find, for every 0-definable equivalence relation E = E (v, w) in K" and i1 in K";

195

6.3. w-S TA BLE FIELDS

* *

a formul a
such t hat E(Kn, a) =
E( K n, a) =

n 'Ij; (Kn).

'ljJ (V)Ep

Oth erwise t here is some l E K " sat isfying P and -, E (d~ a). T his rem ains t rue for every ({1 E K " having t he same type as lover K o; in fact , as K is

CHA PTER 6. w -STA BILIT Y

196

~l -saturated and 1(0 is countable, d~ d! corresp ond to eac h other by some a utomor phism (Y of K fixing 1(0 poi ntwise, hence E( K n, ii) setwise. T hen tp(d/I(o) impli es -,E (v, ii) in K. Use compactness an d deter mine B(if) E tp (d/I(o ) such t hat B(K n) ~ -,E (K n, ii ), equivalent ly -,B(K n) 2 E (Kn, ii ). Bu t t his forces -,B (v) E p ~ tp (d/I(o ), a cont radict ion. Co nsequent ly E (v, iD is a consequen5e of p. Again use compactness and find a formula
K

F= VZ(1J (Z) 1\ -,(z =

b) -+ -,(Vv(
D:

So, unless replacing
6 .4

P rime m o dels

T his section t reats a fund am ent al feature of w-stable t heories, nam ely the ex istence and uniqu eness of prim e mod els over s ubsets. T his property will be useful in 7.8, in provin g Morley's T heorem on uncoun t abl y ca tegorical t heories; bu t a remarkable a pplication will be pro vided also in 6.5, when dealing with differ ential fields; in fact we will observ e t hat D C Fo is w-stable, a nd so the machinery of this sect ion will apply and yield both existence and uniqueness of prime mod els over subsets among differentially closed fields of characterist ic 0 - in algebraic t erms, exist ence and uniqueness of differenti al closures among differential fields of characterist ic 0 -. T he first ste p of our t reat ment is clearly to define what a prime mod el is. Let T be a ny (possibly non w-stable) complet e t heory wit h infinite mod els in a count a ble lan guage L . n still deno tes a big saturated mod el of T . D efi n itio n 6 .4.1 Let X be a small su bset of n. A model A of T is called prim e over X if and only if:

(i) th ere is an elemen tary functio n f of X in to A ,

197

6.4. PRIME MODELS

(ii) for eve ry model B of T and elementary f un ction g from X into B , there is an elem entary em bedding h of A into B fo r whic h h f = g.

T wo problems spring qui t e naturally in t his setting. Let X

~

n be given.

(a) (E xistence) Is t here any mod el A of T prim e over X? (b) ( Uniquen ess) Is t his model unique? More precisely, let A o and A l be two mod els of T prime over X - via t he elementary funct ions f o, !l respect ively -. Are A o a nd A l isomorphic by a map h satisfying hfo = i l ? Intuitively sp eaking, a mod el of T pnme over X is a "minimal" mod el exte nding X by some elementa ry function . Of course a sha rp definition in a genera l set t ing fatally invol ves all the details given before. In particular uniqueness has to be required up to isomorphism , in th e way stated some lines ago. On t he other hand , t his generalit y has its positi ve sides, and even allows some well acce pted freedo m. For inst ance, ju st owing to (b), t here is no loss of generality in assuming t hat a mod el A prime over X , if a ny, do es extend X (a nd so f is an inclusion). We will see t hat existence and uniqueness of prim e mod els a re not gu ar anteed in genera l, for ar bit rary t heor ies T and over a rbit ra ry sets X . However t hey do hold when T is w-stable. But now let us propose some exam ples. Examples 6 .4.2 1. Let T = ACF or , if you like to keep our completeness ass um pt ion on T , let T = ACFp for some p = 0 or prime. When X is an integr al dom ain wit h identit y in cha racterist ic p , a mod el of ACFp prime over X is t he algebraic closu re of t he field of quotients of X (in t he field th eor et ic sense), and is unique (as t he Algebra handbook s relate).

2. Let T = RCF. A mod el of RCF prime over an ord ered field X is it s real closure, and is uniqu e up to isomorphism . Again , any handbook of Algebra (such as [65]) provid es t he details. 3. Now let T = D C Fo. Take a differential field X of cha racterist ic O. Now Algebra does not te ll us anyt hing similar t o t he previou s exa mples; in fact it clarifies neith er t he existence nor t he uniqueness of a differenti al closure of X , ind eed Algebra ca nnot eit her define what a differential closure is. So t he existe nce an d uniqueness probl ems of prime models overlap some non trivia l (unsolved?) algebraic qu estions.

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CHAPTER 6. w -STABILITY

Now let us summarize briefly how we plan to organize this section . First we introduce a nd discuss some notions closely related to prime models: atomic sets and constructible sets. Indeed we will see that over countable sets a model is prime if and only if it is atomic and countable, and if and only if it is constructible (warning: here constructible has a specific meaning, having nothing to do with what we introduced in Chapters 1, 2 and 3 in the particular framework of fields). In this perspective, a theorem of Ressayre stating the uniqueness of a model constructible over a given subset is very noteworthy, as it implies, among other t hings , th e uniqueness of a prime model over a countable set. At t his point we will discuss the existence problem of prime models. We will give a condition of topological flavour characterizing the theories T such that every small subset X of n admits a prime model over itself: they are exactly t hose where the isolated types over every X are den se in S(X). This will accomplish the general analysis for arbit rary theories. At this poi nt we will ass ume T w-stable, and we will show a theorem of Mo rley proving that, in this setting, the pr evious topological condit ion is satisfied and , consequently, t he existence of prime models is always guaranteed.Fin ally we will treat a nice and deep theorem of Shelah showing th e uniqueness of prime models in the w-stable framework. Actually what Shelah proves is that , for an w-stable T, a model prim e over a set X is also constr uct ible over X; so Ressayre's Theorem applies and gives the required uniqueness res ult. As already said, Shelah's Th eorem is quite complicated, and needs some subtle notions and tools to be introd uced later in this book. So we will postpone its proof unti l Chapter 7. To conclude, we will see th at both existence and uniq ueness of prime models fail when the w-st ability assumption is dropped . This is th e sketc h of this section . Now let us begin our report by introducing atomic and constructible sets, as promised. We ass ume som e basic acquaintance wit h ordinal numbers. X, Y , Z , ... denote small subsets of n, a, b, c,.. . tu ples in n. For simplicity we sometimes ident ify a tuple a an d t he (finit e) set of th e elements in a. Definit ion 6 .4.3 Y ~ X is atomic over X if and only if every tuple in Y has an isolated type over X .

Recall th at every model A of T extending X via a n elementary functio n realizes all the isolated types over X; on the contrary, non-isolated types can be omitted by su it ably chosen models, as t he Omitting Types Theorem shows. So a mod el prime over X, being elementarily embeddable in every

199

6.4. PRIME MODELS

model A extending X as before, should not be far from being atomic over X. Here is a list of simple and useful facts concerning atomic sets and isolated types. Remarks 6.4.4

1. If tp(b, clX) is isolated, then tp(bl X) is.

In fact, let
c be in Y.

Then Y is atomic over X U c.

In fact , for every bEY, tp(b, clX) is isolated. So 2 implies tp(blxuc) isolated, too. Definition 6.4.5 Y :2 X is constructible over X if one can list the ele-

ments ofY by ordinal indexes bv (with v < 0', 0' a suitable ordinal) in such a way that, for every v < 0' , b; has an isolated type over X U {bJ1 : fL < v}. In this case the sequence (b v : u < 0') is called a construction of Y over X.

200

CHA P T ER 6. w -STABILITY

We put for simplicity X; = X U {b/-l : fL < v}. So X o = X , for a limit v X; = U/-I
Proof. Let ba, bl , ... , bn , ... (n natural) enumerate Y - X . It is eno ugh to show th at , for every n , th e typ e of bn over X n = X U {bm : m < n} is isolated. As X n - X is finit e, t he previou s Remark 6.4.4.4 ensur es that Y ., is isolat ed over X n , a nd so that tp(b n / X n ) is isolated , as required. So at om icity sometimes implies construct ibility. But a stronger and deeper resul t holds in t he opposite dir ecti on: in fact , as we will see wit hin a few lines, if Y ;2 X is construct ible over X, t hen Y is atomic over X. Now we rela te const ruct ibility an d prime mod els. Fi x a small X ~ 12 a nd t ake a mod el A of T exte nding X by some given elementary function. As already said, there is no loss of gener ality for our purposes in ass uming th at X is ju st a subset of A a nd that t he corresponding inclu sion is elementary. Theorem 6.4.7 If A is construc tible over X , then A is prim e over X .

Proof. Let B be a model of T a nd 9 be an elementary function from X int o B. We a re lookin g for an elementary embedding h of A into B extending 9. Let (bv : v < a ) be a construct ion of A over X. T he st rategy we devise to build h is t he following: we progressively exte nd 9 to larger a nd larger elementary function s 9v from X; into B (for v < a ); at last , we check that Uv
6.4. PRIME MODELS

201

9v be the resulting function. Then the domain of 9v is just Xv = Xjt U {bjt}; moreover, owing to the choice of b~, 9v is elementary. At last form f = Uv
a-:

a-:

Lemma 6.4.8 Let X ~ Z ~ Y, Z be closed. Then the theory of Dz is axiomatized by Th(Dx) U {
Th(Dxuz,J

is axiomatized by Th(Dx) U {
This is clearly enough, as the case J-L = Cl: is just our thesis. Now it is obvious that Th(Dx) U {
n.

202

CHA PT ER 6. w -STABILITY

because b; and consequent ly a: are in Z . Notice xJ.Lnz = XUZJLl X vnz = X U Z; and deduce that T h (Ox uzl') is axiomat ized by T h (Ox uzJ U {
n.

Bu t owing to t he induction hypothesis T h (OxuzJ is axiomatized in its turn by Th(Ox) U{

~

Y and Z is clos ed, th en Y is at omic over X U Z .

Notice t hat t his applies also to Z = X and impli es Corollary 6.4.10 IfY 2 X is constructi ble over X , th en Y is atomic ov er

X. Proof. (of the lemm a) Let hbe in Y , we have to show that t he ty pe of h over X U Z is isolated . Extend h to a larger and closed tuple c in t he following way. For every b E h, pick 1/ < 0: for which b = bv, and consider the formul a

Now we ca n st at e and prove Ressayre's Uniqueness T heore m . Theorem 6.4.11 (Ressayre) Let X be a sm all subse t of 0 , A o and A i be t wo models of T con taining X via some elem entary inclusions, and construc ti ble over X. Then th ere is an isomorphism h of A o onto Ai fi xing X po intwis e. Proof. Look at th e triples (Yo , Yi , f) where, for every i = 0, 1, X ~ Yi ~ Ai , Yi is closed with resp ect to some given const ru ct ion of Ai over X a nd f is a n element ary fun ction of Yo int o Ai acting identically on X and having Yi as image. The collection of th ese triples is not empty because it cont ains

203

6.4. PRIM E M ODELS

(X , X, i dx) (in fact , X is closed) ; moreover it is parti ally orde red by t he relation ::S such th at (Yo, Yl , J) ::S (Y~ , Y{, 1')

if and only if Yi ~ Y/ for every i = 0, 1 a nd f ~ 1'. A st ra ight forward check ensu res t hat Zorn 's Lemma applies to ::S an d yields a maxi mal t riple (Yo, Yl , ] ). We claim t hat Yi = Ai for every i = 0, 1, so ] is the required isomorphism. Suppose not . For inst an ce, ass ume Yo f= A o. Use t he same tec hnique as in th e pro of of Lemm a 6.4.10 and enlar ge Yo t o a closed Yoo 2 Yo U {b} wit h YoO - Yo finit e. Owin g to Lemm a 6.4.10 t he sequence Co of t he element s in Yoo - Yo has a n isolated type over Yo. Call it p. Not ice th at even ](p) is an isolated type (over YI) becau se] is elementary. In par ticular ] (p) is l a nd get a new set YlO 2 Yl in reali zed in AI, say by Cl . Enlarge Yl by C Al . Clearly ] can be extended t o a n elementary fun ction f O from Yoo into . Al wit h image Yd: j ust map Co in C l ' If Yd is closed wit h resp ect to t he fixed constructi on of Al over X , t hen we are done; in fact , we have fou nd a counterexample (Ycf , YlO, f a) to t he maximality of (Yo, Yl , ] ); t his is a cont radiction, and so shows Yo = A o as claim ed. Otherwise we apply t he previous pr ocedure to YlO. Accordingly we enlarge YlO t o a closed Yl 2 YlO with Yl - Y? finite, a nd correspondingly we build Ycf ~ Yd ~ A o and a n l elementary func tion i' from Yd in A l with image Yl. If Y o is closed , th en we are done; ot herwise we cont inue our pro cedure. In t he worst cas e, we get for every natural n a t riple (Yon , Yr, f n) such t hat , for every nand i = 0, 1, (Yo, Yl , ] ) ::S (Yon, Yr , I" ) , X ~ Y~ C Yi n+ l , I" is a n element ary map fro m Yon into Al wit h image Yt, Yi n - Yi is finite a nd both Y02n+l a nd y 12n+2 a re are closed . So closed . Owing to t he last cond it ion both UnYon and Un (UnYon, UnYt , Unr) again cont radicts t he choice of (Yo, Yl , ]). In conclusion Yo = A o, a nd, simila rly, Yl = A l . •

yr

Bu t now it is time to come back t o t he existence and uniqueness problems of prim e mod els. As a warm-up, let us first discuss the case X = 0. Here we show uniqueness a nd we characterize existence by some eq uivalent conditions. Not ice t hat t he ass umption X = 0 is not so rest ricti ve as it may look. Indeed it impli citl y includ es t he seemingly mor e general case when X is countable: it suffices to repl ace L by t he (countable) lan gu age L (X ) and T by t he t heory of Dx , a nd t o observ e th at a mod el of T h(Dx) prim e over is ju st , as a st ruct ure of L , a mod el of T prim e over X . Not ice also t hat a mod el of T prim e over 0 is eleme ntarily embedda ble in every mod el B of T ; in fact, as T is complete, the inclusion of 0 in B is element ary. Th e

o

204

CHA P T ER 6. w -STABILITY

uniqueness of a model prime over Theorem and the following result.

0 is

a direct conseq uen ce of Ressayr e's

Theorem 6.4.12 Let A be a model of T. Th en the following propositions

are equivalent: (1 ) A is prim e over

0,.

(2) A is coun table and atomic over 0,.

(3) A is constructible over 0. Proof. (1) =} (2) As already said, a model A of T pr ime over 0 elementarily embeds it self in any mod el ofT . This clearly implies that A is countable, as T do es admit some countable mod els. Furthermore ever y non -isolated ty pe over 0 is omitted in some countabl e model of T (by the Omitting T yp es Theorem) a nd consequ entl y in A: so A is atomic over 0. (2) =} (3) Just apply Lemma 6.4.6 t o Y = A and X = 0. (3) =} (1) This is a particular case of Theorem 6.4.7. .. At this point Ressayre's Uniqueness Theor em applies and show s: Theorem 6 .4.13 Any two models of T prime over

0 are isomorphic .

In fact t hese mod els are const ructible over 0. Now let us discuss th e existe nce of a mod el of T prime over 0. The following t heorem pro vides a nice equivalent condit ion.

0 if and only if, for every positi ve integer n, the isolat ed n -types over 0 are dense in the topological space S n (0) (thi s means that , for every L-formula

Proof.

(=})

Assume that A is a mod el ofT pri me over

0. As T is complet e,

cp(A )n is not empty , in oth er words t here is some tuple ii in A sat isfying cp(v) . As A is atomic over 0, th e ty pe of ii over 0 is isolat ed (and includes
205

6.4. PRIME MODELS

(a) Pn is an n-typ e over

0 (in t he free vari ables (Vi, . .. , vn)

= v(n));

(b) if 3w n
(2) Two models ofT exte nding X by elementary inclusions are isom orphic by a fu nct ion fixing X pointwise.

206

CHA PTER 6. w -S TA BILIT Y

Bu t t he pr eviou s a nalysis discloses some useful information even in t he general case, when X is a ny, possibly uncountable , small subset of D. In particular, it clarifi es unde r which condit ions on T a ny X admits a prime model. This is wh at the following t heorem explains. Theorem 6.4.16 T he following propositions are equivalent.

(1) For every X, there is a model of T prim e over X.

(2) For ever y X and positive integer n , the isolated n-types over X are dense in Sn(X) . (3) For every X , the isolated I -types over X are dense in SI(X).

(1)' For every countable X , there is a model ofT prime over X. (2)' For every countable X and positive integer n , the isolated n-types over X are dens e in Sn(X) . (3)' For ever y countable X , the isolated I- types over X are dens e m

SI( X ). Proof. (I)::::} (1)', (2) ::::} (2)' a nd (3) ::::} (3)' a re trivial, as well as (2) ::::} (3) and (2)' ::::} (3)'. Moreover (1)' {:::? (2)' is just Corollary 6.4.15. So, in order t o accomplish the proof, it suffices to show (3)'::::} (3) , (3) ::::} (2) an d (3) ::::} (1).

(3)' ::::} (3) Suppose towards a contrad ict ion t ha t there are X , a tuple ain X a nd a formula

6.4. PRIME MODELS

207

which both 1jJ(v) 1\ r.p(v, if) 1\ ()(v, b) and 1jJ(v) 1\ r.p(v , if) 1\ -,()(v, b) enlarge to consistent l-types over Xl' For any L(Xo)-formula 1jJ(v) which cannot isolate any l-type containing r.p( v, if) even in 51 (X o), ()(v, b) can be also found in L(Xo ) ; otherwise ()(v, b) is produced by the previous procedure. We repeat this machinery and we build, for every natural n, a countable X n ~ X enlarging if in a way such that, for any n, no L(Xn)-formula can isolate a l-type over X n+! containing r.p(v, if). At this point, consider X' = Un X n. X' in countable. Moreover, for every L(X')-formula 1jJ (v), there is a suitable n for which 1jJ(v) belongs to the language L(Xn) and hence cannot isolate any l-type containing r.p(v, if) over X n+ l , and consequently over X'. (3) ::::} (2) We proceed by induction on n. As the case n = I is just our hypothesis, we have simply to see how to pass from any n to n+1. Let r.p(v, w) be an L(X)-formula with n+1 free variables (v, w) (w = (WI, ... , w n )) and r.p(nn+!) =I- 0. So :3vr.p(v, w) lies in some n-type over X, and hence in some isolated n-type p over X. Let b realize p, so n p= :3vr.p(v, b), and r.p(v, b) is in some I-type over X U b. By hypothesis, there is some isolated l-'type q over X U b containing r.p(v, b). Let a realize q, so p= r.p(a , b). Look at the type of (a, b) over x: it contains r.p(v, w) and is isolated because both tp(ajXUb) and tp(bj X) are. (3) ::::} (I) Fix X, we have to find a model of T prime over X . As shown in Theorem 6.4.7, it suffices to build a model A of T constructible over X. Let us work in a model M of T containing X via an elementary inclusion. To obtain A inside M, we extend progressively X and form larger and larger sets XJ.L 2 X in M (for It an ordinal), all constructible over X and elementarily included in M, until we meet in this increasing sequence the domain of an elementary substructure of M and, in this way, a model of T constructible over X. Start putting X o = X. For a limit It, set X J.L = UV<J.L XV, as it is right to expect: indeed this preserves constructibility and the other assumptions on X w The crucial step concerns successor ordinals It = v+ 1. Suppose that Xv is not the domain of an elementary substructure of M (otherwise we are done). By the Tarski-Vaught Theorem, there is an L(Xv)-formula r.p(v) such that :3vr.p(v) is true in M - in other words r.p(M) =Ibut X; contains no realization of r.p(v). Use (3) and get some isolated l-type p over Xv including r.p(v). There is some element av E M satisfying p. Put Xv+! = XvU{a v}; Xv+! is constructible over X because p = tp(avj Xv) is isolated; moreover the inclusion of X v + l in M is elementary. Now notice that the length of this procedure cannot exceed IMI. Accordingly XJ.L = XJ.L+! for some It ~ IMI; but this means that XJ.L is the domain of the required model. ..

o-

208

CHAPTER 6. w -STABILITY

Let us underline one more tim e t hat what we get in (3) =} (1) is actually a model of T constructible over X . Now assume, at last , T w-stable. We show t hat, under this assumption, th e existence of pr ime models is ensured over any X. This is a resu lt of Morley and is obtained as a direct consequ ence of t he previous theor em . Theorem 6 .4.17 (Morley) Let T be ta-stable. Th en , for every X , there is a model of T pr im e over X .

Proof. Owing to Theorem 6.4.16 , it suffices to show t hat, for every countable X , t he isolated I-types over X are den se in 51 (X) . Use w-stability and ded uce that, for a countable X, 51 (X) is countable, to o. Now refer t o topology and specifically recall a theorem of Cantor and Bendixson saying t hat in every count able compact Hausdorff space - like 51 (X) - isolated points are den se. This is ju st what we need . .. Indeed th e previou s t heorem says even more: in fact , for an w-st a ble T , over an y X we ca n find a constr uctible mod el. Finally let us deal with uniqueness in th e w-stable set t ing. Theorem 6.4.18 (Shela h) Let T be w-sta ble, X be a small subset of rl , Aa and A 1 be two mod els of T prime over X . Th en there is an isom orphism h betw een Aa and A 1 fixing X pointwise.

Actually what Shelah proves is t ha t , for an w-stable T , a mod el A prime over X must be con structible over X as well. So Ressayr e's Theor em applies and ensures the uniquen ess of t he prim e mod el over X . However , Shelah 's tools in th e proof ar e qu ite sophist icate d and require new pro gress in t he general theory. We will develop t hese preliminari es in Chapter 7, so we have to delay th e full details of the proof to t hat chapter. We concl ude t his section observing that neither exist ence nor uniqueness of pr ime models a re guarant eed when T is not w-stable. In particular we propose a simple counterexample to existe nce. Uniq ueness is also cont radicted by some suit a ble th eori es T , but t he corresponding exam ples are more t echnical and com plicated, and may make this section still hea vier. We omi t them. Example 6.4.19 In the language L = {S; , P} , where P is a l -ary relation symbol, look at the st ruct ure (R , S; , Q). T his is a dense linea r order without end points (R , S;) with a subset Q both dense and codense in R. All th ese

6.5. D C F o REVISITED

209

proper ties can be easily written as first ord er sentences in L. Let T be t he L theory axiomatized in t his way. A simple arrangement of Cantor 's Theorem on DLO- shows that T is ~o-c ategorical and so, by Vaug ht's Test , complete. Hence T = Th (R , ::;, Q). Of course T is not w-stable (just as DLO- ). Now suppose t hat T has a prim e mod el over Q inside (R, ::;, Q ) . Up to isomorphi sm this model will be of t he form (A, ::;, Q) for some countable s ubset A of R properly including Q . Let a E A - Q. If we take a away from A , t hen we get a new mod el (A - {a} , ::; , Q) of T ; t his contains th e ration als, but cannot embed (A , ::;, Q). In fact t he cut a fills in A over Q is not realized in (A - {a}, ::;, Q).

6 .5

D CF o revisited

We pursue here th e analysis of existenti ally closed differential fields of cha racterist ic 0 begun in Chapte r 2. We said at t hat t ime t hat their class is elementary and we pro vided (wit hout proof ) t heir nice first order cha racterization D CFo due t o Lenore Blum, according t o which t hey a re th e differenti al fields K of characterist ic 0 such t hat, for every non zero j (x ) an d 9 (x) in [( { x } such t hat t he ord er of 9 (x) is smaller t ha n t he ord er of j (x ), t here is some a E [( such t hat j (a ) = 0 a nd g(a ) =I 0 (let us moment aril y call t hese fields differenti ally closed fields; actua lly we are going to show t hat t hey a re ju st t he existent ially closed differenti al fields). Now it is t he right mom ent to pro ve at las t t hat result , as well as th e related fact , mentioned several t imes, th at every different ial field of charact erist ic 0 has its own differential closure. In fact, we will show t ha t D C Fo is w-st a ble, a nd so t he machinery develop ed in t he last section will apply and produce a prime model of D CFo (so a differential closure) over any differen ti al field of cha racterist ic O. The w-stability of DCFo will be also used to prove, in a way closely resembling that pursued for ACF in 6.3, th at D CFo uniformly eliminates the imaginari es. This is the plan of t his section. In order to begin our t reat ment and , in particular, to focu s existent ially closed differential fields, let us preliminarily examine how a differential field K ca n be enlarged and which is t he st ruct ure of a differ ential extension H. of K. In par t icular, for a E H - K; we wan t to single out in 'H. t he smallest differential subfield K (a) containing [( and a (i. e. t he int ersection of all t he differential subfields of}{ cont aining both [( a nd a); t his requires t o clarify a nd und erst and how a is related t o K , in oth er words which differenti al polynomials over [( a sa t isfies. Of course t his is basic (Different ial) Algebra rath er t han Mod el T heory; a nd in fact t here is

210

CHA PTER 6. w -STA BILI T Y

a straightforward a nd genera l algebraic te chnique to a pproac h t his question , t hat is t o conside r t he function Fa from the differential domain K{ x } int o 1£ taking a ny differenti al polyn omial f (x ) E K{ x} to its value f (a ) in a . T his is a different ial ring hom omorphism , and its kern el [ (aIK ) = {f (x ) E K { x } : f (a) = O} is a prop er pri me ideal of K { x }, and even a differen tial ideal (i. e. an ideal closed und er derivation D , as it is easy t o check) . Incidentally define a differenti ally algebraic over K if [( al K ) =1= 0 (so if f (a ) = 0 for some nonzero differenti al polynomial f( x) E K{ x} ), and differen tially transcenden tal otherwise. When a is differenti ally transcende ntal over K , t here is nothing t o add. For, Fa is a differential ring isom or phism between K{ x } a nd a suitable differential subring of K containing K and a . It is straightforward t o dedu ce that the differential fi eld K(a) we are looking for is isomorphic to t he field of quotients of K{ x} (which inh erit s a natural st ruct ure of differenti al field regulated by t he usu al derivation rul es for quo tients) . Bu t t he most cr itical poin t of our ana lysis conc ern s a differentially algebraic element a over K. In fact , on t he one hand it is easy to realize t hat even in t his case t he quotient ring K{x }I[ (a ] K ) gets a natural struct ure of differenti al dom ai n, j ust because [ (a] J() is a prim e differenti al ring and so it makes sense to put , for f (x ) E J({x },

D (J (x ) + [ (al K )) = D f (x ) + [ (aIK ); t he field of quotients of K { x} I [ (o] K ) inh erits in its tu rn an obvious struct ure of differential field , a nd K (a) is isomorphic to t his field . Th is is qui te ge nera l, a nd formally satisfactory . Bu t we aim at underst anding t he internal struct ure of K (a ), and t his clea rly requires t o st udy t he prim e differenti al ideal [( al J() and to provide an intrinsic description of its polynomials. F irst let us observe t hat every prime differenti al ideal 1 =1= {O} , K {x } of K{ x } can be represent ed as I = [(a I K) for some suitable 1£ and a E II (differentially algebraic over K) . This is stra ightforward a nd quite gener al. In fact , look at the differenti al dom ain K{ x }I L, form it s differen tial field of qu oti ent s Q, noti ce t hat Q extends K provided one identifies each k E K a nd its class I + k in Q, and t ha t a = I + x ju st a nnihilates t he differenti al polynomials in I over K. So take a ny non t rivial pri me differenti al ideal I of K{x}. Choose a non zero polynomial f (x ) E I having

*

a minimal ord er n ,

then

6.5. DC Po REVISITED

*

211

a minimal degr ee in D" (x),

a nd finally

*

a minimal total degree.

As I is prime, f( x) is irreducible as an algebraic polynomial in ,'C[Dm(x) : m natural]. We won der whether I eq ua ls by chance the differential ideal (J(x)) gen er at ed by f( x) , in other word s t he algebraic ideal of ,'C[Dm(x) : m natu ral] generated by f(x) and it s derivatives. T his is not true in general, a nd I is somewhat mo re com plicated . Bu t cla rifying this point requires a further notion. Definition 6.5 .1 Let f(x) E K {x} have order n. The formal partial derivative of f( x) with respect to Dn(x) is called the separant of f(x) and denoted

sf (x). For exam ple, the sep arant of f( x) = (D 3 x) 2+ x .D3x+D x .D 2x is 2D 3 x+ x. Now, given a differen t ial polynomial f( x) E K{ x} , look at t he set I(J( x)) of the polynomials t( x ) E K {x} such that for some natural m

SJ(x) . t(x) E (J(x) ); one sees that I(J( x)) is a differential ideal a nd, wh en f( x) is irr edu cible, it is also prime. Of course the memb ership to I(J(x)) is not eas y and immediate to check ; but one can show t hat, for an ir reducible f( x) of order n , a polynomial g(x ) E I(J( x)) must have order 2:: n and , if it s order is just n , then f( x) directly divide s g(x). This an aly sis im plies th at , if a is a roo t of f( x) , then I(J( x)) = I(ajK). Mo re gener ally, let us come back to our nont rivi al prime differential ideal I a nd to t he polynomial f (x) E I chosen before. One proves Theorem 6.5.2 I = I(J( x)) . This is a noteworthy and deep algeb raic fact; it s proof can be found in t he references quoted at t he end of this chapt er. This also concludes ou r preliminary outline of basic Differential Algebra. Now let us deal at last with Model T heory. In fact , owing to wh at we said about I(J(x)) we are in a posit ion to show: Theorem 6.5.3 Existentially closed differential fields J( of characteristic 0

are differentially closed.

212

CHA PTER 6. w -STA BILITY

Proof. We have to check t hat K satisfies Blum 's ax ioms : for eve ry nonzero polynomials f (x) , g(x) E K {x} such t hat the or de r of f (x ) is larger t han the order of g(x) , t here is et is K sat isfying f( et) = 0 and g(et) i: O. As t he existence of et can be expressed by an existe nt ial sentence wit h par am eter s in K a nd K is existentially closed , it suffices to find such a n eleme nt et in some differenti al field 'H. extending K . Pick an irredu cible factor f' (x ) of f (x ) in K{ x} of t he same order as f(x), a nd form I = I (J' (x )). Clearly f (x ) E I , while g(x) ca nnot belong to I becau se its order is less than t he order of f '(x) . Now enla rge K to t he differenti al field of q uotients H of K{ x }/ I a nd not ice t hat in Il I + x sat isfies f (v) = 0 and ca nnot realize g(v) = 0, becau se f( x ) E I a nd g(x ) rJ. I. .. In order to conclude t hat D C Fo is ju st th e t heory of existent ially closed differenti al fi elds of cha racterist ic 0, we have to show t he inverse impl ica tio n (say ing t hat any differenti ally closed fi eld of characteristic 0 is existent ia lly closed ) and definit ively to check two mo re points: (i) every differential field of characteristic 0 has a differenti ally closed extension;

(ii) DCF o is model complete . Her e ar e t heir proofs. Theorem 6.5.4 Every differential field K of characteristic 0 has a differ-

entially closed extension. T his ca n be shown by using some familiar chain arguments. In Proof. detail , list in so me (possible t ra nsfinite) way t he pai rs (J (x ), g(x)) of non zero polynomials in K {x } such t hat t he order of f (x ) is la rger t han t he order of g(x ). Apply t he sa me tec hnique as in Theor em 6.5.3 a nd , for every pa ir (J (x ), g(x)), enlarge the ground field in ord er t o include a roo t et of f (x) such t hat g(et) i: o. Repeat t his procedure a nd event ua lly ob t ain a differenti al extension K' of K with t he followin g property: for every choice of f( x ), g(x ) in K{ x} - {O} suc h t hat t he orde r of f (x ) is la rger t ha n t he orde r of g(x ), there is some et E K' such t hat f(et) = 0 an d g(et) i: o. However t his does not mea n t hat K' is different ia lly closed (in fact , what ha pp ens for f (x ) and g(x ) in K' {x }?). Bu t now we ca n form a new seq uenc e of di fferenti al exte nsions K n of K , by pu t tin g K o = K a nd, for every n, K n +1 = K ~ (a differ enti al fi eld enlarging Kn and cont aining, for eve ry choice of non zero polynomials f( x) , g(x ) in Kn{ x} such that the orde r of f( x) is larger t han the order of g(x) , a roo t et of f( x) t ha t does not a nnihilate g(x)). tc = U K n

213

6.5. DCF o REVISITED

is the domain of a differential field extending /C. A straightforward check shows that f( is differentially closed . ..

Theorem 6.5.5 DCF o is complet e and eliminate the quantifiers in the language L = {+, " - , 0, 1, D}. In particular DCFo is model complet e.

Proof. Fi rst we show t hat DC Fo is complete, and hence th at a ny two mod els /C , /C' of D C Fo are elementarily equivalent . As every structure has some t:io-sat urated elementary extension , we can assume that both /C a nd /C' are t:io-saturated . As parti al isomorphism impli es element ary equivalence , it suffices to show /C ~p /C'. Accordingly let J be the set of all the isomorphism s between finitely generated differenti al subfields of /C and /C' respect ively. J is not empty because both /C and /C' include the rational field (with a zero derivative) as the minimal differential subfield generated by 0. Now let /Co a nd /Cb be t wo finitely generated differential subfields of /C , /C' resp ectively, corr esponding to each other by some isomorphism h. Of course, if ii is a tuple generating K«, then h( ii) = a generates /Cb ' So, algebraically speaking, any ' element of Ko can be obtained from ii by the usual elementary operations a nd D , and t he sa me ca n be said a bout Kb and O!. Incidentally notice that Ko ~ dcl(ii) and , pa rallely, Kb ~ dcl(al ). We have to control that h sat isfies the back- and-forth properties (i) and (ii) . Clearly it suffices to deal wit h (i), as (ii) can be handled in a simila r way. So t ake a E K - Ko. We look for t wo finitely generated extensions /Cl, /C~ of /Co, /Cb respectively such t hat a E K 1 a nd h ca n be enlarged to an isomorphism between /Cl and /C~. We distinguish two cases, according t o whether a is differentially algebraic, or differentially transcendental over /Co. When a is differentially algebraic, consider the nonzero prime differential ideal I(a/K o) a nd th e (irreducible) polynomial j( x) in l(a/Ko) having minimal order n , then minimal degree in D ":» and , finally , minimal total degree. Then j'( X) = h(J(x)) is an irr educible polynomial of order n in /C o{ x } . Look at t he ty pe p given by th e formulas

j'(v) = 0,

-,(g'(V) = 0)

where g' (x) ranges over t he non zero polynomials of ord er < n in Kb{x}. As Kb ~ dcl(a p ca n be viewed as a type over O!. As /C' is differ entially closed , an y finite '), portion of p is sati sfied in /C'. As /C' is t:io-sa tu rat ed, p itself is realized in K ' - Kb by a suitable element a'. Moreover I (J' (x)) = I (a' / Kb)· So one can easily enlarge h to an isomorphism between /Co (a) and /Cb(a /)

214

CHA P TER 6. w -STABILITY

mapping a in a ' (for , I (a /K o) and I (a' /Kb ) corres pond to eac h oth er by = Ko(a) and K~ = Kb(a' ) yield (i) .

h). Accordin gly K i

The case when a is differentially t ranscende ntal over Ko is simpler. In fact , usin g t he ~o -s atu r ation of K' , one eas ily finds an elemen t a ' E K' algebraically t ra nsce nde ntal over K b ~ del(at) . At t his poin t , one obser ves t hat Ko(a) and Kb(a') a re isom orphic by a function extending h and taking a to a'. T his ensures (i) even in t his case, and eventually acco mplishes t he proof t hat D C F o is com plet e. Now we prove t hat DCFo eliminates t he quan tifiers in L. Owing to completeness , every L-sent ence is D C Fo-equivalent to 0 = 0 when belonging to D C Fo, and to 0 = 1 otherwi se. So we have to elimin ate t he qu an tifiers in t he L-formulas
°

215

6.5. DCFo REVISITED oth er word s

Vv(
f-----+

V
E

DC Fa .

pE P o

As t he lat ter formul a is qu an tifier free, we a re don e.

..

Hence DCFo is t he mod el com pa nion of t he t heory of differen tial fields of cha racterist ic O. But t he pr evious a nalysis says mor e a nd in particular ensures : T heorem 6 .5.6 DC Fa is eo -sta ble.

Proof. What we hav e to check is t ha t , for every differenti ally closed field /C in characterist ic 0, IS1(K) 1 = IK I. Actually we will show somet hing more, resembling what we did for algebraically closed fields. In fact, we will see th at , for every /C , there is a natural bijection betw een l-types over K in DC Fa and (proper) prim e differential ideals in /C {x}: owing to what we saw a bout t hese ideals and t heir st ructure, this impli es 15 1(1<)1 ~ IK{ x}1 = IKI , whence 15 1(K)1 = IKI. F irst take a l-t ype p over K a nd form

I (p) = {j (x ) E K{ x} : j (v) = 0 E p}. C hecking t hat I (p) is a prime different ial ideal in /C {x} is a straightforwa rd exe rcise. Indeed every (proper) prim e differential ideal I of /C{ x } can be o btai ned in this way : in fact , build t he (different ial) field of quo tients Q of /C{ x} / I , enlarge Q t o a differenti ally closed K a nd, at last , take t he ty pe of I + x over (t he isomorphic copy of ) /C (in K): clearl y a polynomi al j (x) E K{ x} sat isfies j (v) = 0 E P if a nd on ly if j (x ) E I . Furth ermore t wo different l-types p =1= q over /C define different ideals . In fact t here is some L(K)-formula

CHAPTER 6. w -STABILITY

216

Corollary 6.5.7 Over any differential field IC of characteristic 0 there is a prime model of DCPo (hence a differential closure), and this is unique up to isomorphism fixing K pointwise.

We already emphasized several times that this approach -using Model Theory, in particular w-stability, Morley's Theorem and Shelah's Theorem- is the very first proof, and virtually the only one known till now, of the existence and uniqueness of differential closures. Again using w-stability we show now the last result of this section. Theorem 6.5.8 DCF o uniformly eliminates the imaginaries.

Proof. We repeat almost wholly the approach followed in 6.3 for algebraically closed fields. Indeed the first two algebraic preliminary steps in that proof concern arbitrary fields and ideals, so apply to our present setting as well. The same can be said about the use of w-stability, owing to Theorem 6.5.6, although now types correspond to (proper) prime differential ideals, and not directly to prime ideals. In conclusion what we have to check is that the third preliminary step in the proof of Theorem 6.3.2 still works in the new framework. Accordingly t ake a differential field IC of characteristic 0, and 10 , • •• , I m prime differential ideals of IC{ x} in the same conjugacy class (with respect to differential field automorphisms of IC) . We look for a(n even algebraic) subfield ICo of IC such that, for every automorphism (J of IC as a differential fi eld, (J

permutes 10 ,

•.. ,

I m if and only if (J fixes Ko pointwise.

This can be obtained by rearranging the approach in the field case. Form I = nj<m Ij. This is a differential ideal, and even a radical ideal in the usual sense: for every f(x) E K {x}, if ft(x) E I for some positive integer t, then f(x) itself is in I. A differential version of the Basis Theorem, due to Ritt and Raudenbush and working over differential fields of characteristic 0, says that every differential ideal of IC{x} -like I, but also 10 , ••• , I m - is finitely generated (as a differential ideal). Fix a finite set of differential polynomials including generators of I, 10 , . . . , I m . Let]-f be their maximal order, so all these polynomials can be viewed as algebraic polynomials over K in the unknowns tr», for h ::; 1 ::; i ::; m. Put t: = IC[Dhxi : h < u, 1 ::; i ::; m], and look at In R , 10 n R, ... , i; n R. Then In R = nj<m(Ij n R), and 10 n R, ..., I m n R are prime ideals in in the same conjugacy class with respect to field automorphisms of IC: in fact , for i , j' ::; m, there is some automorphism of IC as a differential field, hence as a field, taking Ij to

n,

n

6.6. RYLL-NARDZEWSKI'S THEOREM, AND OTHER THINGS

217

Ij , and so t, n R t o Ij n R . Use Step 3 in Theorem 6.3.2 and dedu ce that t here is an algeb ra ic su bfield Ko of K such that, for every aut omor phism 0of K , viewed as a field and in particular as a differenti al field, 0- permut es t he Ij n R's if and only if it fixes Ko pointwise. Bu t eac h Ij n R generates Ij as a differential ideal in K{ x} , so, for i. j' ::; m, o-(Ij n R) = Ij n R impli es o-(Ij) = Ij . Co nsequ entl y, for every aut omorphism 0- of the differenti al field K, 0- permutes 10 , .•. , I m if a nd only if 0- act s identically on Ko. ..

6.6

Ryll-Nardzewski's Theorem, and other things

We conclude our outline of Morley 's ideas in t hese two cha pters with a perhaps oblique a nd superficially related argument. In fact we want t o treat ~o-catego rical t heories. As already und erlined , they behave qui t e autonomo usly and include examples which are not categorical in a ny uncounta ble ca rdina l, a nd neither are w-stable: for instan ce, t hink of DLO-. On t he other hand , there do exist uncountably categorical struct ures which are not ~o-categorical (su ch as ACFp for any fixed p). However ~o-categorical theories have their peculia r and specific propert ies. In part icula r t hey enjoy t he following nice characterization, due t o Ryll-N a rdzewki and others. Theorem 6.6.1 (Ryll-Nardzewski) Let T be a complete theory in a countable language L. T hen the following propositions are equivalent:

(i) T is

~o- categorical;

(ii) for eve ry positive integer n , there are only fini tely ma ny n -types over in T ;

o

(iii) fo r every positive integer n , there are only fin itely many fo rmulas in n f ree vari ables pairwise in equivalent in T. Proof. The equi valence between (ii) a nd (iii) is clear: it can be deduced directly, or using the classical du ality bet ween Boolean sp aces and Boolean algebras (recall t hat, for every n, t he n-types over 0 form the du al space Sn(0) of t he Boolean algebra of 0-definable subsets of o n). So let us compa re (i) an d (ii). F irst suppose Sn(0) infinite for some n. As Sn (0) is cornpact, it contains some non-isolated type p . The Om it t ing Types Theorem ens ur es that p is omitted in some countable model of T; on the other hand , there does exist some countable model of T realizing p , The latter mod el cannot be isomorphic t o t he form er; hence T is not

218

CHA PTER 6. w -STABILITY

~o-categorical. Now ass ume

Sn(0) finit e for every n. Consequently Sn (0) is

a dis crete space, and so every n-type over 0 is isolated. As t his is true for every n , every model of T is atomic over 0, hence every count a ble mode l of T is prime over 0. Then all the countabl e mod els of Tare pairwise isomorphic.

.

Let us un derline a sim ple conse que nce showing in the cou ntable case t he more gene ral fact t hat , for a A-cat ego rical T, t he only model of power A is sat urated (in t he uncountable case, this res ult is much mo re com plicated a nd refers to Morley's Categoricity Theorem) . Corollary 6.6.2 Let T be an ~o -categorical th eory. Th en th e only coun table model M of T is ~o -saturated. Proof. Let a be a tuple (of length m) in M . For every positive integer n , there are only finit ely ma ny n-types over a, ind eed onl y finit ely man y (n + m)-typ es over 0. So every n-type over a is isolated , and conseq uently is realized in M. ..

Now let us propose some examples illustrating Ryll- Na rd zewski's Theorem. Examples 6.6.3 1. We know that dens e linear ord ers (with or ) without end points have a n ~o-catego rical t heory (see C hapt er 1) . Let us con firm t his result via t he Ry ll-Nardzewski T heorem . First take a positive int eger m and ao, ... , am , bo, . . . , bm in the universe of DLO-, with ao < ... < am, bo < .. . < bm . Then t here is a n a utomorph ism of n mapping aj in bj for every j ~ m . In fact t he intervals in n

] - 00, ao[, ] - 00, bo[, ]aj , aj+d, ]bj , bj+d Jam,

Vj

< m,

+00[' ]bm, +oo[

are models of DLO- and ar e saturated in the same po wer as n. Conseq uent ly t hey are isomorphic to each other. In part icular (ao , . .. , am), (bo, . . . , bm ) have the same typ e over 0. This implies t hat, for ever y positi ve int eger n an d sequence (Cl, . .. , C n) E nn, the type of (Cl , . . . , cn ) over t he em pty set is fully determined by t he isomorphism ty pe of t he ordered set ({Cl, ... , cn } , ~) . So we get onl y finitely ma ny n-ty pes over 0 for every n , and the Ryll- Nardzewski Theorem ap plies to confir m t hat DLO- is ~o-categorical.

6.6. RYLL-NARDZEWSKI'S THEOREM, AND OTHER THINGS

219

2. On the contrary, ACFp is not ~o-categorical for any p = 0 or prime. Let us see why by using the Ryll-Nardzewski Theorem. Is suffices to notice that there are infinitely many pairwise distinct 0-definable subsets in the universe n of ACFp, and even that there are infinitely many pairwise distinct finite 0-definable subsets of n, in other words that ad (0) is infinite. But this is well known because ad (0) equals the (field theoretic) algebraic closure of the prime subfield of n. Of course, no (expansion of a) field of characteristic 0 can admit an ~o­ categorical theory, for the same reason as in Example 6.6.3, 2. This applies in particular to real closed fields. Checking what happens for differentially closed fields, or separably closed fields, or ACFA, could be a useful exercise. We conclude this section and the whole chapter by discussing a related matter, again concerning definable sets and types. In particular we wonder which information we can obtain about a complete theory T by looking at the isomorphism types of the Boolean algebras of definable sets of its models. Can the knowledge of these algebras say anything essential about the model theoretic complexity of T? Incidentally, recall that, at least in the countable case, there does exist a satisfactory classification of Boolean algebras up to isomorphism, provided by Ketonen. In this framework, A-categoricity can be replaced, for every infinite cardinal A, by a seemingly weaker notion, called Boolean A-categoricity. A complete theory T is said to be Booleanly A-categorical if and only if all its models of power A have isomorphic Boolean algebras of definable 1-ary sets. Clearly categoricity implies Boolean categoricity in every A. Notably Theorem 6.6.4 (Mangani-Marcja) For an uncountable A, a complete theory T is Boolean A-categorical if and only if is A-categorical.

But this is false when A =

~o.

Indeed

Theorem 6.6.5 (Marcja-Toffalori) Every ~1 -categorical theory is Boolean ~o -categorical.

So algebraically closed fields have Boolean ~o-categorical theories. One can see that the same is true also for real closed and differentially closed fields, as well as for every module. Other connections between isomorphism types of Boolean algebras of definable sets and structural properties of theories are investigated in the papers quoted at the end of the chapter.

220

6.7

CHAPTER 6. w -STABILITY

R eferences

w-stable t heories and structures are deal t with in [8]. The det ails a bout Morley rank a nd degree in st rongly minim al t heories ca n be found in [108], or in the Ziegler con tribu tion to [16]. [132] (and its recent english translation) provide a nice an d clear treat ment of w-stable groups; also [15] is a rich source on this subject . Macintyre's Th eor em on w-stable fields is in [90]; by t he way, let us mention its genera lization t o s uperstable fields in [25] say ing t hat an y superstable field is algebraically closed (supersta bility is a notion enla rging w-stability, a nd will be int roduced in t he next cha pter) . Prime mod els over t he em pty set were st udied by Vaught [174]; t he analysis over arbitrary sets was developed in [118]. Ressayre's contributions to constructible sets, and in particular the Uniqu eness Theorem , were never publi sh ed , while Shelah 's Uniqueness Theor em is in [148]. As a lready sa id, t he t heory of differentially closed fields of characteristic 0 was develop ed in Blum 's Ph .D . T hesis [12] (see also [146]). Go od references are also [110] or [131]. Ryll-Nardzewski 's T heorem is in [145]; see also [43] and [156]. The classification of countable Boolean algebras is in [69] . Boolean Aca te goricity was introduced in [104] and intensively studied in [105] and [165]; actually, Boolean categoricity was called pseudo categoricity in t hose papers. The st udy of t he st ruct ural properti es of t heories by t he isomorphism t yp es of Boolean algebras of definabl e sets of their mod els is also pursued in [106] and [107].

Chapter 7

Classifying 7.1

Shelah's Classification Theory

A cent ra l subject in Mathematics is classifying, t ha t is cha racterizing t he objects of a given clas s up to equivalence relat ions, for instan ce t he structures of a given language, or t he models of a given theor y up to isomorphism. Let us mention some classical examples. Examples 7 .1.1 1. An infinite set (withou t additional structure) is fully characterized up t o isomorphism by its power, so by a (n infinite) ca rdinal numb er.

2. Let J( be a fixed (cou nt abl e) field . A non- zero vectorspace over J( is completely determined up to isom orphism by its dimension over J( , so again by a cardina l number. 3. An algebraically closed field of a given char acteristic is fully det ermin ed up to isomorphism by its transce nde nce degree over its pr ime subfield, so, once again, by a ca rdinal number. Notice t hat t he pre vious examples concern element ary classes an d even strongly mini mal th eories (provided we restrict our attention to infinite vectors paces in 7.1.1, 2). But, of course, the classification purposes touch la rger horizons. 4. It is well known that every finite ly gener at ed a belian gro up A decomposes as a finite direct sum of cyclic grou ps, and even of copi es of Z , Z /pn where p ranges over pr imes a nd n over positi ve int eger s. Moreover t he lat t er decomposit ion is unique up to isom orphism (a nd up 221

222

CHAPTER 7. CLASSIFYING

to permuting the summands); so the isomorphism type of A is determined by saying how many copies of Z, Z/pn (p prime, n positive integer) are involved in this decomposition of A. Notice that finitely generated abelian groups are not an elementary class (why?). 5. Consider torsionfree abelian groups of rank at most n , where n is a given positive integer: they are the subgroups of the additive group (Qn, +). Baer found in the thirties a nice classification of these groups up to isomorphism when n = 1. However it has been a longstanding open question to accomplish a general satisfactory classification for every n 2: 2, and actually no classification is presently known. We will go deep in this question later. 6. Let K be a given field, and look at the ring K(x, y) of polynomials with coefficients in J( and two non-commutings unknowns x and y. Consider K(x, y)-modules finite dimensional over K towards a possible characterization of their isomorphism classes. Well, this problem is generally believed intractable. In fact, finite dimensional K(x, y)modules interpret the word problem of groups, and classifying them as said means solving this problem. Of course, classification problems do not restrain themselves to the isomorphism relation; for instance one can try to classify square matrices by similarity, and so on. Also, it should be underlined that a classification is som etimes (often?) very unlikely to be obtained (as Examples 7.1.1, 5 and 7.1.1,6 witness). However looking for a classification, and even realizing its unfeasibility, can disclose new connections between different areas, generate new techniques and, definitively, enlight new horizons within Mathematics. From an abstract point of view , a natural problem arises in this framework, that is developing a theoretical treatment to recognize which classes and relations admit a classification, and to propose general tools to accomplish this classification when possible, or to measure its difficulty in the hard cases. A closely related and preliminary question is just: what does classifying mean? These issues are intensively studied within Mathematical Logic. For instance, Descriptive Set Theory (and consequently, through it, both Set Theory and Recursion Theory) are concerned with the classification matter. In fact Descriptive Set Theory deals with definable sets of R, viewed as a separable complete metric space, and of similar topological spaces (Polish spaces). This does overlap classification. Indeed the objects that are to classified often belong to a Polish space X , or at least to a Borel su bset of

7.1. SHELAH'S CLASSIFICATION THEORY

223

X , and t he classifying equivalence relation E is a Borel, or a nalyt ic, etc ., subset of X 2 : in fact, t his is the case of the similarity relat ion for square matrices, as well as the isomorphism relation for den umerable struct ures (such as groups a nd fields ) in a given language. In this setting it is sometimes possible t o associat e in a Borelian way to any object of X a point (called "invariant") in another Polish space such that two different objects have the same associated point if and only if they are equivalent in E. The relation E is called smooth when it satisfies these requirements. For example, Jord an 's canonical form is an invariant of the similarity relation for mat rices, which is therefore smooth. In Ergodic Theory, smooth relations are just t hose which are considered act ually classifiable. But not all equivalence relations one meets are smooth . In the latest years a general theory of classification of equivalence relations which live in Polish spaces has been developed within Descriptive Set Theory: it allows to est ablish, for E and F eq uivalence relations, when classifying modulo E is mo re complex than classify ing mod ulo F. In particular it includes some recent and beautiful results of Simo n Thomas (and others) on isomorphism for torsionfree abelian gro ups of ra nk n ; these works show that t his relation is of increasing complexity when n grows, and is not smoot h (and so is likely to be intractable) when n ~ 2. But now let us come back to Model Theory. Of course, also Model Theory is interested in classifying (structures, definable sets, and so on). But its approach is peculiar, and differs from the perspectives and the aims of Descriptive Set Theory : and indeed, very roughly speaking , one could say th at Model Theory and Descriptive Set Theory are complementary in this framework. In fact, one could agree that the latter is mainly concerned in measuring how ha rd and difficult classifying is in the worst "wild" cases , while Model Theory aims at determining t he "tame" settings in Mathematics, and accordingly at clarifying where and why a classification ca n be done. Classification in Model Theory promptly recalls Shelah. In fact, it was Shelah who approached , treated and essentially solved the classification problem in Model Theory since the end of the sixties until the ea rly eighties. So Shelah's formidable work dates back to almost twenty years ago, but is far from having exh austed its st imulus, as we will exp lain later in more det ail. The Shelah strategy was to determine a series of successive key properties (simplicity, stability, superstability and so on) concerning complete theories and having a twofold role: in fact , each of them allows a new significant step towards classifiability, while its negation excludes any hope of classification . At the end of these successiv e dichotomies Shelah exactly determi nes which abstract conditions ensure class ifiability.

CHA P TER 7. CLASSIFYING

224

The aim ot this chapter is to outline Shelah's classification program, its concepts, tools a nd techniques (forking, stability, supe rstability and so on) , as well as its recent developments regarding simplicity and spectrum problem . We will lay em phasis on ideas an d motivations rather than on proofs. So we will sacrifice full details in general. However we will provide a com plete report in t he w-st able case . The first point to be clarifi ed is: wha t do we want to classify? Here th e a nswer is qu ick and ready: we will deal with classes of struct ures . Just to avoid too many complicat ions, we will limit our analysis here to element ary classes , and even to classes M od(T) of mod els of count able complet e first order theories T. We illustrated this choice in Ch a pt er 1. As usual, we will work inside a big saturated model n of T. The second prelimin ary qu estion is: with respect to which relation will we classify? Also in t his cas e the reply is fast and easy: we will classify up to isomorphi sm . Bu t now we have to an swer a more delicate and fundament al quest ion, th at is: what do es it mean to classify the mod els of a complete T up to isomorphism ? By which "invariants" will we acco mplish our classificati on pr ogram? When T is st rongly minimal - just as in t he Exa mples 7.1.1, 1-3 quoted before - classifying mean s ass igning to every mod el a ca rdinal number - it s dim ension - in such a way t ha t two models are isomorphic if and only if t hey a re given th e same ca rdinal. But we already saw that, for some theories, a classification cannot be accomplished by assigning a single cardin al and often requires somet hing mor e complicate d, like pairs of ca rdinals, or even (possibly infinite) sequences of cardinals, and so on. T his is what happens in Example 7.1.1, 4, where act ually the involved class is not elementary; ot her elementary cases were discussed in Ch apter 5. Here is anot her exa mple. Example 7.1.2 Con sider the theory of the structures (A , Eh E 2 ) where E l, E 2 are equ ivalence relations, E 2 ~ E l and • every E 2-class has infinitely many elements, • every El -class contains infinitely many E 2-classes; • t here a re infinitely many E l-classes. T is complete (for instan ce becau se it is categorical in ~o) . In a mod el of T any El -class of power ~ O' is determined up to isomorphism by the function mapping a ny infinite cardinal ~,6 ::; ~O' to the number of t he E 2 subclasses of power ~,6 , a nd hen ce by an ordered sequence of a ca rdinals

7.1. SHELAH'S CLASSIFICATI ON THEORY

225

~ ~ a . Con seq uently, if A = (A, E t , E 2 ) is a model of T of powe r A, th en the isomorphism type of A is completely given by the function associating to any ordered seq uence of (11 ca rdinals ~ ~a ( ~ O' ~ A) t he number of th e Et -classes in A corresponding to this seq uence .

T his con struction can be it erat ed to produce more and more complicated examples , need ing more and more sophist icat ed isomorphism invariants. This suggests t he followin g definition .

Definition 7.1.3 Let C denote th e class of cardinal numbers. For eve ry ordinal (11, we define a class C " by induction on (11 in th e f ollowing way .

• CO is C, • if (11 = {3 + 1 is a successor ordinal, th en we put C " = C f3 U P (C f3 ) ; • if (11 is limit, th en we put C" = Uf3
Definition 7.1.4 Let T be a com plete theory , (11 be an ordinal. T is said to have an invariant system of rank (11 if and only if th ere is a funct ion f associat ing eve ry model of T with an eleme n t of CA such that two models A and B of T are isomorphic if and only if f(A) = f( B) . Examples 7.1.5 of ran k O.

1. A st ro ngly minimal theory has an invariant syst em

2. An ordered sequence of cardinals is an eleme nt of C 2 • So th e theories in Example 5.8.2 have an invariant system of rank 2. 3. The t heory in Example 7.1.2 has an inva ria nt syst em of rank 4. Shelah 's proposal is to assume t hat a complete theory T is classifiable if and only if T has an invariant system of ra nk (11 for some ordinal (11. We will discuss Shelah 's point of view lat er in this cha pter, after describing Shela h' s clas sification th eory. In particular, we will see that th ere are some good reasons to agree wit h it . Once t his is don e, we can deduce:

Theorem 7.1.6 Let T be a complete theory such that, for eve ry uncountable power A, T has 2'\ pairwise non isomorphi c models of cardinality A. Th en T is not classifiable. In fact some cardinal and ordina l computations exclude that such a T has an invariant system of any possible ra nk. T his is quite remarkable, because there are some very fam ilia r t heories sat isfying the ass umptions of T heor em

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226

7.1.6. For instance, this is the case of the theory DLO- of dense linear orders without endpoints: in fact DLO- is ~o-categorical and so has only one countable model (up to isomorphism) but it gets 2'\ pairwise non isomorphic models in any uncountable power >.. Accordingly DLO- is not classifiable. Mo re generally Theorem 7.1.6 applies, as we will see in the next section, to any complete theory of linearly ordered infinite structures. None of these theories is classifiable. In particular, no o-minimal theory is classifiable, although any such theory satisfies the conditions (Dl)-(D4) of the dependence relation a E acl(A) a -< A for a E n, A a small subset of n. This is a little surprising and will be discussed in more detail later in this chapter, and then in Chapter 9. Anyhow, if we (momentarily) agree with Shelah's definition of classifiable theory, then we have to take note that too many models (in other words 2'\ non-isomorphic models in every uncountable >.) exclude classifiability. On the other side, we would like also to determine the key criteria ensuring the classifiability of an arbitrary complete theory: this will be the matter of the forthcoming sections. To conclude the present one, let us spend some more words about the close relationship between classifying theories T (in the Shelah sense) and counting the models of T. This is already explicit in Theorem 7.1.6. More generally, for every countable complete T, one can define a function I(T, . ..) associating to every infinite cardinal >.

I(T, >.) = number of the isomorphim types of models of T of power >..

x f---7

I(T, >') is called the spectrum function ofT. Recall that 1 ::; I(T, >.) ::;

2'\ for every X, owing to the Lowenheim-Skolem Theorem and cardinal computations. The content of Theorem 7.1.6 is just that I(T, >.) = 2'\ for every uncountable >. excludes the classifiability of T. Indeed there was a conjecture of Morley, preceding Shelah's work and, in some sense, originating it, saying: Conjeet ure 7.1. 7 (Morley) Let T be a countable complete first order theory. Then the spectrum function>. f---7 I(T, >.) is increasing among uncountable cardinals: for ~o < < f-t, I(T, >.) < I(T, f-t).

x

Shelah positively answered this conjecture, as a non-minor consequence of his classification analysis. The problem of determining all the possible spectrum functions>' f---7 I(T, >.) when T ranges over countable complete theories (and>. is uncountable) was solved only in 2000 by Hart , Hrushovki

7.2. SIMPLE THEORIES

227

a nd Laskowski, who explicitly listed all these functions . Notably their proof involves some arguments from Descriptive Set Theory. Finally, what can we say whe n A = ~o? As we observed in th e last chapter when talking about ~o-categoricity, this case is a little oblique with respect to the general analysis and requi res peculiar approaches a nd techniques. A classical conject ure in this framework was raised by Vaught . Conjecture 7.1. 8 (Vaught) For a countable complete first order theory T , either I(T, ~o) ::; ~o or I(T, ~o) = 2No (apart from the Continuum Hypoth esis, of course) .

As in the case of Morl ey's Conjecture, this que stion is not only a mer e ca rdinal investigation; what is more relevant is to underst and the structure of the countable models of T. Shelah (together with Harrington and Makkai) positively answered Vaught's Conjecture for w-stabl e theories T . Other partial posit ive answers are known. Indeed in t he latest months the news of a counterexample (a theory with exactly ~l count a ble mod els) due to R. Knig ht [75] has been spreading, but this negative solut ion st ill seems (october 2002) under examination.

7. 2

Simple theories

All throughout this section T is a complete first order th eory in a coun table language L, T has no finite models and n denotes t he universe of T. We aim at determining which key properties ma ke T classifiable. A classification is very easy in the strongly minimal case . In fact, whe n T is strongly mini mal , every mod el of T is lab elled by a cardinal number - its dimens ion - classifying it up to isomorphism ; what ru les this dimension and its assignme nt is a notion of dep endence, based on the mod el theoretic algebraic closure ad . Unfortu nately the ad dependence does not work any more when we enlarge our setting and we leave the strongly minimal fram ework. So we need a more general notion of (in)depend ence , still including the classical cases of linear independence in vectorspaces, algebraic independence in algebraically closed fields a nd , definitively, ad independence in strongly minimal theories, but applying to a wider context. In other words we ai m at defining for any T ii is ind ependent fro m B over A

where ii is a tuple in n a nd A ~ B are small subsets of n. As not ions require a bstract symbols to be presented let us denote by I (I for independence) the set of all the triples (ii, B , A) in n such t hat

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a is independ ent from

B over A

in a sense t o be made more preci se. It is reason able t o ex pect t hat 1 satisfies the following proper ties:

(11) (invariance) for ever y (a, B, A) E 1 and a ut omorphism f of (1 (a), f(B) , f(A)) is still in 1; (12) (local character) for eve ry t hat

(a, E , A) E 1;

a a nd

B , th ere is a count able A

~

n,

B such

(13) (fini te character) for every a, A and B , (a, B , A ) E 1 if a nd onl y if, for all finit e tuples bin B , (a,AUb,A) E I;

a, A and B , t here is a t uple a' having t he same length and t he sa me ty pe over A as a suc h t hat (a', B , A) E 1;

(14) (ex tension) for every

(15) (symm etry) for every a,

(b, A U a, A)

b a nd

(a, A

A,

U b, A ) E 1 if a nd onl y if

E 1;

(16) (tran sitivity) for ever y a a nd A (a, B , A) E 1 and (a, C , B ) E 1.

~

B

~

C,

(a, C, A ) E 1

if and onl y if

Definition 7.2.1 A set 1 of triples (a, B , A) with a in n and A ~ B small subset ofn is called an independence system ofT if and only if I satisfies

(ll) - (16). A n eas y a pplicat ion of (13) and (15) shows t hat, if B , B' :::> A a re s mall subsets of n, t hen

(b, B' , A)

E1

VbE B

if and only if

(b , B , A) E 1 V;} E B'. ' We will say t hat B and B' a re ind epend ent over A when t his happens. Notice also t hat, if a' is a subseq uence of a and (a, B , A) E 1, t hen (a' , B , A) E 1 as well. By (13), it suffices t o check (a' , A Ub, A) E 1 for a ll bin B. We know (a, A U i, A) E 1. By sy mmet ry (15), (b, A U a, A ) E 1 , whence (b, AUa', A) E I by (13), and (a' , AUb, A) E 1 by (15) once again. Let us propose some exa mples of ind ep end ence systems, both t o illustrate t he meaning of (11) - (16) and to confirm t hat t hey pro vide a reason abl e axiomatic ground t o introdu ce a n a bstract notion of indep end ence.

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7.2. SIMPLE THEORIES

Examples 7.2.2 1. First let us check that the old independence notion in strongly minimal theories T corresponds naturally to an indendence system in the new sense. In fact, let T be any theory. For a E nand A <;;:; B small subsets of n put

(a, B, A) E I if and only if a E acl(B) implies a E acl(A) or, equivalently but more transparently,

(a, B, A)

rf. I

if and only if a E acl(B) - acl(A).

We claim that, if the acl dependence relation in T satisfies (Dl), (D2) and (D4) (in particular, if T is strongly minimal, or also o-rninimal), then the triples (a, B , A) in I do satisfy (11) - (16). We underline that we are momentarily dealing with elements a rather than with tuples a in n. (11) is clear. (12) follows from (D2) , which says even more; in fact, it ensures that, for every a and B, if a E acl(B), then there is a finite A <;;:; B such that a E acl(A). (13) says in our particular setting that, for every a and A <;;:; B with a rf. acl(A), a E acl(B) if and only if there exists a tuple bin B such that a E acl(A U b). So the direction from left to right just follows from the definition of acl, and the converse is a direct consequence of (D2). (14) Let (a, B, A) rf. L, so a E acl(B) - acl(A). tp(a/A) is not algebraic, and so is realized even out of acl(B). Take any a' rf. acl(B) such that tp(a'/A) = tp(a/A) and notice that (a', B , A) E I, (15) when restricted to elements a and b in n is just the Exchange Principle (D4). (16) is clear. It is straightforward to see that the previous analysis extends to arbitrary tuples a in n, provided we put , for a = (al' ... , an),

(a, B, A) if and only if, for every i

(ai, B

U

~

E [

n,

{aa, ... , ai-d , A U {aa, ... , ai-d)

E

I,

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The resulting I is an inde pendence sys te m of T. As al ready said , this exa mple includ es both stron gly minim al and o-rninim al theories. In part icular it a pplies to t he following classes of struct ures . (a) (Pure) infinite sets: here , for a, A, E as above, (a, E , A) a dependent from E over A) mean s a E E - A.

et I

(so

(b) Vectorspaces over a given (countabl e) field K: in this case, (a, E , A) et I means th at a is in the subspace spa nned by E bu t is not in t he subspace spanned by A ; hence I recovers linear indepe nde nce. (c) Algebraicall y closed fields: this time, for A and E subfields, (a, E , A) et I means that a is algebraic over E but is transcendental over A ; accordingly I recovers algebraic ind ependence in t his cas e. (d) Real closed fields (such as t he ordered field of reals): recall t hat these fields a re o-minimal ; more over, for ordered subfields A and E , t he model theoretic algebraic closure acl equa ls t he real closure in t he usual algebraic sense; accordingly, for a , A and E as before, (a, E, A ) et I means that a is in the real closure of E but is not in the real closu re of A. But independence systems do go beyond the st rongly minimal and 0minimal set ti ngs, and apply to larger sceneries, for instance to w-stable t heor ies, a nd other theories as well. 2. Let T be w-stable. Define I by putting, for ii, A and E as before ,

(ii, E , A ) E I

~

RM(tp(ii, E)) = RM(tp(ii, A)).

Then I is a n independ ence sys tem of T . The de tails will be provided in Section 7.5 below. We give here ju st a few comments. F irstly recall that RM(tp(ii, E)) ::; RM(tp(ii, A)) for every s and A ~ E . Secondly not ice t hat, for a strongly minimal T and for a E n, RM(tp(a, E)) < RM(tp(a , A)) j ust means a E acl(E) - acl(A) . In other words, when we rest rict our attention to st rongly mini mal theories, we recover t he ind ependence syst em in 1. Bu t , of course, ind ependence concerns now a much larger fram ework , including, for instan ce, differentially closed fields of characterist ic 0 (the differential case will be discussed in more detail in Section 7.10) .

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7.2. SIMPLE TH EORIES

3. Now conside r random graphs. They a re (infinite ) graphs (C, R ) (with R a sy mmetric irreflexive binary relation on C) satisfying t he following addit ional condit ion :

(*) for every m , n E N and aa, . .. , an , ba, ... , bm E C , t here is some g E C satisfying R( g, ad for all i ::; n a nd -,R( g, bj ) for all j ::; m . Clea rly (*) ca n be exp ressed by infinitely man y first ord er sente nces in t he lan guage L = {R} , so random gra phs are an elementary class. T heir t heory T is w-categorical a nd consequ entl y complete ; fur th ermor e it elimina tes t he qu antifiers in L. Notice t hat T is not w-stabl e. In fact, given a ra ndom gra ph (C , R) , for any s ubset 5 of C , th e formulas R( v, s) Vs E 5, -,R(v,g) vs E C - 5 defines a I- typ e over C, and different subset s produce different ty pes; so 5t{C) contains at least 2 1G1 elements. This implies t hat T is not w-stable. Now define, as in t he case of infinite sets, for a in

n an d

A

~

B,

(a, B , A) E fi f a nd only if a E B - A . It is easy to check t hat one determines in t his way a n ind epend ence sys tem f. 4. Any cornpletion of t he t heory ACF A of existent ially closed fi elds wit h a n aut omorphism has a n ind epend ence system : thi s will be described in mor e detail in t he final section of this cha pter , amo ng other algebraic exa mples. However t heories having a n ind epend ence sys t em (such as t hose considered in th e previous examples) do not behave in the same way. For instance, one realizes that th e following a malga mat ion property is a crucial dividing line:

(17) (amalgam ation) let A be a model ofT , B , B' ;2 A be ind ependent over A, b, fi be tuples in n having the same ty pe p over A and satisfying (b, B ,A) E I , (fi, B' ,A) E fr esp ecti vely; t hen t here is some c in n realizing t he same ty pe as b over B and as fi over B' , and satisfying (c, B U B' , A) E J. Definition 7.2.3 A n in dependence system I of T is good if an d onl y if it satisfies (17).

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232

Let us run t hro ugh t he previous examples t o illustrate (17) and it s relevance. 1. Let T be strongly minimal , I be defined as before. E xample s 7 .2 .4 Then (17) holds. Let us check t his when dealin g with t uples of length 1, so for elements of n. If b or b' is in A , t hen b = b' and c = b = b' works . So ass ume b, b' t/. A; ind epend ence impli es b t/. acl(B) a nd b' t/. acl(B') , in other words b, b' realize the only non-algebraic I-typ e over B , B' resp ectively; so any realiz ation of t he unique non-algebraic I-type over B U B' works as c. Not ice that , in this case, (17) is a direct consequence of th e fact th at, for every small subset X of n, there is a uniq ue non-algebraic l -type px over X and consequent ly over every small Y ;;2 X there is a uniq ue non-algebraic l -fype extending ux .

However (17) fails within o-rninimal theories: / is not good in this set t ing. Let us see why in t he particul ar case when T = RCF is th e theor y of real closed fields. Con sider th e ord ered field R of reals and , in a suitably sat urate d exte nsion n of R , 4 positive element s b < a < a' < b'

eac h infinit ely larger t ha n t he previou s ones , and b infinitely lar ger t ha n R . Then a a nd a' a re ind epend ent over R ((a , R U{a' }, R ) E /) , b, b' realize t he same ty pe over R a nd

(b , R U {a} , R ) E I ,

(b' , R U {a'} , R ) E /.

However no c E n can satisfy t he same ty pe as b over R U {a} a nd as b' over R U {a'} because b < a < a' < b' and so c < a excludes a' < c. 2. The independence system assoc iated with an w-stab le T as before is good . T his will be checked in detail in 7.5. 3. Also t he independence system of random graphs enjoys amalgamation: the read er may easily check this. 4. The same is true for AC F A and its complet ions. Definition 7.2.5 T is called simple when T has a good indepe nde nce system . A struct ure A is simple whe n its theory is. T herefore w-stable t heories (and in par t icular strongly minimal theories) are simple; random gr aphs a re simple, as well as an y existent ially closed field

233

7.2. SIMPL E T HEOR IES

with a n a utomor phism: in all these cases , a n indep end ence notion sat isfying not only (11) - (16) bu t also the additional condit ion (17) can be introduced. But we ca n say even mo re. Indeed , for a sim ple T , t his independence not ion is unique; in other words, there is only one good ind ependence system I of T making T simple. I ca n be cha racterized as follows.

Definition 7 .2.6 (Shelah) Let A ~ E be sm all subsets of n, p be a type ov er E . p is sai d to fork over A if and only if th ere are a formula

w, suc h that

(i) th e bll 's have th e same type over A; (ii)

Theorem 7.2.7 (Kim - Pillay) Let T be simple, I be a good independen ce system of T. T he n , fo r eve ry tuple a in n and A ~ E small subsets of n, (a , E , A) E I if and only if tp( aj E ) does not fork over A. So t he only possible good independence system of a simple t heory T is t he set I of the triples (a , E, A) as befo re such t hat tp (ajE ) doe s not fork over A : this is what is necessarily meant when we say t hat a ind ependent from E over A in a simple T; we will write a .J..A E when t his happens. When A and B are a ny small sets (so possibly A 'l:. E) , a .J..A E abbreviates a .J..A A U B ; one usu ally omit t he subscript A whe n A is empty; so .J. ju st means .J..0 ; for B and E' sets, E .J..A E' a bridges b .J. A E ' for every bin B, or equivalent ly f} .J..A E for every f} in E'. For tec hnical convenience, let us rest ate (11) - (17) in t his new notation for a simple T.

(ll) (Invarian ce) for ever y a, E and A as before a nd ever y a ut omorphism f of

n, if a.J..A

B , then f(a) .J.. f( A) feE);

(12) (local character) for every th at a.J..A E ;

a and

E , t here is a cou ntable A ~ E such

(13) (fini te character) for every a, A and E , a .J..A E if a nd only if, for all finite tuples bin E , a .J..A b.

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234

(14) (ex tension) for eve ry ii, A a nd B , t here is a t uple 5' having the same length a nd t he same ty pe over A as 5 a nd satisfying ii' 4-A B ; (15) (sy mmetry) for eve ry

s, ba nd

(16) (tran siti vity ) for eve ry and ii 4-B C .

n a nd

A,

n4-A bif a nd only if b4-A 5;

A ~ B ~ C , ii 4-A C if a nd onl y if

s 4-A B

4-A B ' , b, iI be tu ples in n havin g t he sa me type p over A and satisfying b 4-A B , b' 4-A B ' resp ectively; then there is some s in n realizing t he sa me type as b over B a nd as iI over B ' , a nd sat isfying s 4-A B UB' .

(17) (amalgamation ) let A be a model ofT, B , B ' "2 A satisfying B

The read er may directly reali ze what t hese properties imply when t uples ii are repl aced by arbitrary small subsets, so wh en on e deal s with B 4-A B ' , a nd rewrit e them in this enla rge d setting. Rem arkably sy m metry - na mely (I5) - is a key proper ty of the forking ind ep endence within simple t heories : ind eed , sim plicity is eq uivalent to t he sy m me t ry of forkin g. Also t ransit ivity (16) a nd local character (12) have t he same crucial role. This was ob served by Kim a nd Pillay as t o local cha racter, a nd has been recen tl y show n by Kim in the other two cases . Theorem 7 .2 .8 A theory T is simple if and only if, one of the follo wing proposit ion s holds:

(i) the fo rking in dependenc e satisfies symmetry (15); (ii) the fo rking in depen dence sa tisfies transit ivity (16) ;

(iii) the forking indepen dence satisfie s local characte r (12). In conclusion, sim ple t heories a re those wh er e independ enc e ca n be reasonabl y introduced and developed in the axiomatic way we said before. Of course, this is a pos itive property towards classifiability, But one can wond er which is the reverse of t he med al , in other words what happens in non sim ple t heories . Well , Shelah proved a qui te negative resul t a bout them: in fact , t hey are not classifiabl e, a nd so must be rejected in our classification progr am. Theorem 7. 2.9 (Shela h) If T is not simple, then I (T , A) = 2'\ for every unc oun table cardinal A. Sim ple t heories exclude o-rn inimal t heories, as ob ser ved before. More ge nerally one can show t hat no com plete t heory of linear order s can be simple.

7.3. STABLE THEORIES

235

Theorem 7 .2.10 No lin early ordered infinite structure has a simple theory. In particular, no complete theory of linearly ordered infinite structures is classifiable.

It is perhaps worth repeating t hat, in sp ite of th is result , infinit e linea rly ord ered st ructures are not so bad . On t he contrary, some of them , like dense or discr ete linear ord ers, real closed fields and , more generally, o-rninimal mod els satisfy some nice and powerful model theor eti c properties; in all th ese frameworks a suit a ble independ ence notion , sat isfying the axioms (11 )- (16) can be introduced . So things are not so sick as t hey look. We will come back to t his point at th e end of Chapter 9. To conclude t his sect ion, let us sket ch a br ief history of sim ple t heori es. Simplicity was first int rod uced by Shela h in 1980, but its relevan ce was not complet ely rea lized at that t ime; indeed Shelah laid a majo r emphasis on a st ro nger notion - st a bilit y, the matter of the next section - a nd regarded simplicity as a generalization of st a bility, and not directly as a key dichotomy. The role of simplicity in introducing independ ence and so towards classifiability was neglect ed until t he nineties , when one obser ved t hat severa l int erest ing algebraic st ruct ures - most notably, existent ially closed fields with an a ut omorphism - have a simple unstable theory. Then Kim in 1996 showed that the forking independence sat isfies symmet ry (15) wit hin simple theories a nd, together wit h P illay, prove d the close connection bet ween indep end ence and simplicity introduced before (in Theorem 7.2.7). Kim again realized in 2001 t he key role of sy mmet ry for the for king independence in the simple framework . Now sim plicity t heory has deser ved its own room in Mod el Theory, as an autonomous a nd relevan t part of t he classification program .

7.3

Stable theories

As said in t he last sect ion, Shelah's original analysis put its emphasis on st a bility rat her than simplicity. St abi lity looked t he key property towards independenc e. But, after Kim and P illay, one realized that it is simplicity th at plays t he cent ral role here , while st a bility strengths simplicity in t he se nse we a re going t o explain. There are several ways to int rod uce stability , and their equivalence is not so im mediate as one wou ld like. We adopt th e following definition, which underlines t he conn ection wit h simplicity : T st ill denotes a complete first ord er theory with no finite mod els in a countable la nguage L, and n is a big saturated model of T .

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CHA P TER 7. CLA SSIF YING

Definition 7.3.1 T is stable if and only if T has an independence syste m I satisf ying the follo wing additional assumption:

(18) (stationarity over models) fo r ever y model A of 1', tupl es a, ;; in f2 and small subset B ;;2 A in f2, if a, a have the same type over A an d are independent from B over A (with' respect to 1), then they have the same type also over B. T is called unstable when it is not stable. A struct ure is said to be stable or unstable when its theory is. The content of (18) is clear: t here is a unique way t o ext end a ty pe p over a model A of T t o a ty pe over B ;;2 A of tuples ind ep endent fro m B over A. Let us propose an easy example to illust rat e this condit ion (18), and so stability.

Example 7.3.2 Just for a change (?!) , we deal with a strongly minimal T a nd I-types over a model A ofT. The algebraic l-types are realized in A a nd so extend uniquely over a ny B ;;2 A. But th e only non- algebraic p E 51 (A) may have several extensions over B; however onl y one of th em do es not fork over A , an d this is t he unique non- algebraic I-type over B , in other words the ty pe of any eleme nt ou t of acl(B). What happens if we enl arge our a na lysis from mo dels of T to arbitrary small sets A ? Now even an algebraic l -type p over A may have several extensions over a B ;;2 A , an d all of t hem do not fork over A; however th eir nu mbe r is finit e and cannot exce ed 1
Theorem 7 .3.3 1f T is stable, then T is simple. Proof. Let I be a n independ ence system of T sat isfying (18). We hav e to check amalgamation (17). So t ake a model A of 1', t wo sets Band B ' ind ependent over A (with res pect to I) , and t wo tu ples b, iJ having t he same type over A and satisfying (b, B , A) E I , (iJ , B ' , A ) E I resp ectively. Using extension (13) , one find s two tuples d~ J! having the same typ e as b over B a nd as b over B ' res pect ively, a nd both inde pe nde nt over B U B ' over A. '

7.3. STABLE THEORIES

237

In particular, J, Jt have the same type over A. By (18), they have the same type also over B UB'. Hence lhas the same type as Jt, and consequently as ii, over B '; furthermore l has the same type as b over B. So l satisfies • (17) as c. Then any ind ependence system I satisfying (18) in a stable T is good, and must equal the forking independence: for a, A and B as usual,

(a,B,A) E I

~

atA B.

Accordingly (18) can be restated as follows for a simple T.

(18) Let A be a model of 1' , B be a small subset of n including A. Then every type over A has a unique non-forking extension over B. Once again recall that the existence of a non-forking extension is just stated in (14). Uniqueness may fail for a stable T over arbitrary subsets A of n; but also in this extended framework stability bounds the number of the possible non-forking extensions over B of a type over A, in the following sense. Theorem 7.3.4 T is stable if and only if T is simple and, for every small A ~ nand pES (A) , there is a cardinal K, (less than 1 n I) such that, for any small B :2 A, p has at most K, non-forking extensions in S (B). Another equivalent way of defining stability relies upon a counting type characterization (resembling w-stability). Theorem 7.3.5 T is stable if and only if there is some infinite cardinal A such that, for every small A ~ n of power A, IS(A)I = A. We wish to underline another equivalent characterization , saying that stability just excludes definable infinite linear orders, in the following sense. Theorem 7.3.6 T is unstable if and only if it satisfies the order property: there is a formula
CHAPTER 7. CLASSIFYING

238

Hence instability is a negativ e condit ion in the classifica tion perspective. On the other hand , which are t he benefits of the stability assumption? Some of them are already described by t he definition a nd t he equivalent (po sitiv e) ch aracterizations listed before. In particular we emphasize once again t ha t , for a stable T a nd for a mod el A of T , ever y t ype p over A enlarges uniquely to a non-forking extension over a ny set B 2 A. Let p lB denote this extension in S(B) ; plB can be characterized as follows

* for every

L-formula
** for every formula

Th e last theorem impli es in its turn C o r o lla ry 7 .3 .10 Let A -< B be models of a stable th eory T . If

We will check t hese results in det ail in 7.5 in t he restrict ed fram ework of wstable t heories. Now, to conclude t his sect ion, let us propose some examples of stable or uns table t heories.

7.4. SUPERSTABLE THEORIES

239

Examples 7.3.11 1. Every w-stable theory T is st able. T his will be dedu ced by the definition of st a bility in 7.5 below; it follows even more directly from other charact eri zations (for instance, from T heorem 7.3 .5). Indeed , the part icular case of strongly minimal t heories was alread y dealt with a few lines ago. So a ny infinit e vectorspace, as well as a ny algebraically closed field , or a ny differentially closed field of characteristic 0, has a st able t heory.

2. Any module (over a countable ring R) has a stable theory: this ext ends what we have just observed a bout vectorsp aces. The proof requires very basic preli min a ries a bout the model theory of modules, an d ca n be found in the refere nces quoted at the end of the cha pter. 3. Real closed fields do not have a stable theor y. Ind eed RC F is not even simple, and a nyhow it houses infinite linear orders. 4. The theory of random gr ap hs is unstable, although simple. In fact, let A = (A, R) be a random gr aph , B ~ A be a small subset of n. Use com pact ness and find a and at in n such that n 1= R( a, x) for every x E Band n 1= R(a , b) for a unique b E B , with b 1- A . Then a and at have t he same t ype over A a nd eac h of them is independent from B over A. But they do not have th e same typ e over B. 5. Similarly, existentially closed fields with an a ut omorphism are simp le and unstable. 6. On t he cont ra ry, sepa ra bly closed fields and differenti ally closed fields in pr ime characteristics are stable.

7.4

Superstable theories

Also in t his section T is a complet e first order t heory with infinite models in a countable language L , a nd n denotes a big saturated mod el of T . Let us underline once again a particular feature of the st rongly minimal case: this ass umption implies, among other things , that T is stabl e and , over any small subset A of n, th ere is a unique non-algebraic I-typ e p , which is realized by all t he elements outside acl(A); moreover, for a and B in p(n), so out of acl(A) ,

a.J!A B mean s a E acl(B) .

240

CHA P TER 7. CLASSIF Y IN G

In particular YA satisfies wit hin p(Q ) t he properties (0 1) - (D4) st ated in C hapter 5. Now conside r an ar bit rary sta ble T. In t his enla rged setting t here may be seve ral non- algeb raic I-typ es over a small A; fix one of th em , call it p a nd look at t he set p(Q) of its rea lizations in Q. We wond er whet her YA still sat isfies (Dl) - (D4). Assume mom entarily t he following condition.

(*) For every small B 2 A, p has a unique non-forking ext ension over B. Recall t hat p has at least one non-for king extension over an y B: t his is ju st what is stated in (1 4). (*) req uires t hat all t he eleme nts a E p(Q) satisfying a t A B realize t he same ty pe also over B. Not ice t hat, as T is stable, every p over a mod el of T sat isfies (*). Call p stationary when (*) holds. So take a non- algeb raic stationa ry t yp e p over a small A. It is easy to check t hat YA satisfies (D'l), (D2) a nd (D4) in p(Q). In fact , let a , b E p(Q) a nd

B

~

p(Q).

(D1) If a E B , t hen a YA B . As p in not algebraic we can find infinitely many eleme nts bo = a, bl, ... , bn , ... realizing p in Q . T he formul a v = bo is in tp(a / B) bu t no other bn can satisfy it. Hence tp( a/ B ) fork s over A acco rding to Shela h's definition , and so a YA B. (D2) If a YA B , t hen t here is some finite B o ~ B for which a YA B o. This follows directly from (13). (D4) If a YA B U {b} but a tA B , t hen b YA B U {a}.

Oth er wise transit ivity (16) implies b tB a , whence a tB b by symmet ry (15); as a tA B , (16) a pplies again a nd gives a tA B U {b}. On t he oth er side, t here do exist non algebraic stationary p's for which YA do es not obey th e last condi tion: (D3) if a YA B U {b} and bYA B , t hen a YA B. Here is a counterexam ple.

Example 7.4.1 Consider t he t heory of t he usu al Cartesian pla ne R 2 wit h the two projections

(x, y) 1--7 (x, 0),

(x, y) 1--7 (0, y),

Vx, yE R

onto t he z-axis and t he y-axis respectively. T here are t hree non-a lgebraic I-types over t he empty set; t hey cor respond in R 2 to:

241

7.4. SUPERSTABLE THEORIES (i) any element (x , 0) with x

=1= 0,

(ii) any elemen t (0, y ) wit h y

=1=

0,

(iii) a ny eleme nt (x, y ) wit h x, y

=1=

0

respectively (the only algebraic l-type concerns (0,0) , that is the unique point lying in both the projection sets). Let p deno te t he third ty pe in t he previous list. Now let us conside r t he algebra ic l-ty pes over R 2 ; t hey can be parti tioned int o t hree classes: (a) a new eleme nt (00, y ) with a n old ord inate y E Rand 00 1. R ; (b) a new element (x , (0) with an old ab sciss x E R and 00 1. R; (c) a ent irely new (00, (0) wit h 00 1. R . Act ua lly the same analysis works over every mod el of T . T he ty pes in (a) and (b) have Mo rley rank 1, while th e t hird ty pe has Morley rank 2. A ccordingly T is w-stable with Morl ey ran k 2 and Morley degr ee 1. T he thi rd type is t he only non-forking extension of p ; actua lly, even t he ty pes in (a), (b) wit h x , y =1= 0 enlarge p, bu t t hey fork over 0 (when y = 0 and x = 0 t he ty pes in (a) a nd (b) extend t hose in (i), (ii) resp ecti vely) . It is also straight forward to check t hat p is stationa ry. Now take

A =

0, B

= R 2, a = (00, (0), b = (x, (0)

with x =1= 0 in R . Clearly a a nd b realize p, in particular a -l- R 2 . Bu t a ¥ R 2 U {b} and b ¥ R 2 • T his cont ra dicts (D3) . Definition 7.4.2 A (n on -algebraic stati on ar y) i-type p over a small A is called regular when (D3) holds in p(f2). Our classification purposes suggest t o handle stable t heories T adm itting s uitably many regul ar ty pes . In fact , roughly speaking, t he mo re t hey are, t he bet t er one can expect to a pproach T . Now pu t: Definition 7.4.3 A stable theory T is called superstable if and only if, for every small Band pE S (B ), there is a fi nite subset A of B over which p does not fork. A structure A is said to be superstable when its theory is . Notice t hat superstability generalizes what happens in t he strongly minimal case . But what is more rem arkabl e for our purposes is t he following resul t.

CHA PTER 7. CLASSIFYING

242

Theorem 70404 Let T be a superstable theory, A and B be two models of T such that A is an elementary su bstructure of B and A =f:. B. T hen there is some b E B - A such that the typ e of b over A is regular. So reg ular types are not rare within superstable theories. O n the other ha nd , if superst a bility fails , then no classification can be expected .

Theorem 704.5 Let T be sta ble unsuperstable. T hen , for every uncountable power A, I (T , A) = 2,\. In particular T is not classifiable. Accordingly supe rstability provides a new dichot omy in Shela h's approach to classification. For, it ens ures sufficient ly many regu lar types, while its negation excl udes a ny classification . Supe rstability can be introd uced in other equivalent ways. Let us quote a cha racterization based on a count ing types cr iterion (just as we did in t he stable setting) .

Theorem 704.6 T is superstable if an d on ly if, for every A ~ 2No and every set A of power A, there are at most A i -t ypes over A . Let us concl ude t his section by list ing some examples and counterexamples of superstable theories.

Examples 704.7 1. Separably closed fields , or even differentially closed fields in a prime characteristic are not superstable (although t hey ar e stable). 2. On t he other ha nd , all w-stable t heories are superst a ble. T his can be eas ily deduced from t he previous definiti on, but we will postpone t he details to 7.5 below. 3. There do exist superstable non w-stable theory. Fo r instance t he t heory of the additi ve gro up of int egers is so.

7.5

w-st able theories

Just to have a short break in our outline of She la h's classificatio n, let us com e back in t his section to a more fam ilia r setting , t he one of w-stable theories . W hat we plan to do is to examine closerly within th is part icular framework t he main notions int rodu ced so fa r in our report . Acco rdingly we will check that w-stable theories are simple, stable and superstable; we will characterize forking , independence and so on in the w-stable setting; we

7.5. w-STABLE TH EORIES

243

will provide proofs a nd details of t he main t heo rems stated t ill now. So, all t hroughout t his sect ion, T will deno t e an w-stable t heory in a coun t abl e language L. The first fact we wan t t o check is th at w-sta ble t heories are st a ble as well. Ind eed wit hin Shelah 's dicho to mies, and in our outline, simplicity pr eceded stability, a nd consequ entl y one might reason abl y ex pect t o meet simplicity before stability at this point ; we will explain later why we a re reversing t his ord er a nd postpon ing simplicity in t his sect ion. We provided severa l equivalent characterizations of stability in Section 7.3; in particular we said that stable theories are ju st t hose failing to have t he ord er property. And act ually it is easy enough to show t hat w-stability impli es stability if we refer to t his characterization. Theorem 7 .5 .1 N o w-s table th eory T satisfi es the order property. Equivalently, any w-s table T is stable.

Proof. Let T be w-stable. Suppose t owa rds a cont radict ion t hat t here are a n L-for mul a cp (v, 'Ill) an d t uples ai (i E N ) in n (say of length n) such t hat, for every choice of i a nd j in N ,

n F cp (ai , aj)

{::}

i::; j.

A straightforward a pplicati on of th e Compact ness Theorem shows th at we can replace N by Q a bove; in other word s, t here are tu pies ai (i E Q) in n such t hat, for every choice of i and j in Q ,

n F cp(ai, aj)

{::}

i::; j .

n

Notice t hat t here do exist definabl e subsets X in such t hat t he ind exes i E Q such t hat ai E X form an inter val of positiv e length I(X ) in Q ; for instan ce, when X = n n, I (n n) = Q is so. C hoose X as before with minimal Morley rank and degree. Fix a poin t j in the interior of th e corres ponding interval I(X) and look at

X o = X n cp(n n, aj) ,

Xl = X - X o.

Both X o an d Xl are definable subsets of X . Furthemore I (X o) = {i E I (X ) : i ::; j} an d I (X I ) = {i E I (X ) : i > j} a re intervals of posit ive leng th . So X o and Xl have t he same Morley rank and degree as X ; but t his • cont radict s t he fact th at t hey parti tion X. Another equivalent way of introducing st a ble th eories conc ern s definability of types: let us check that w-stability implie s stability by referr ing to t his

CHAPTER 7. CLASSIFYING

244

characterization. Recall t hat a typ e p over a small subset A of the uni vers e n of T is said to be definable over A if and only if, for every L-formula <.p(V, w), the set of t he t up les a in n for which <.p (v, a) E p is A-definable. T is st able exactl y when a ny ty pe over any mod el M of T is definable over M: t his is what we want to check for an w-stabl e T. We begin by t wo technical lemm as.

nn

n

Lemma 7.5.2 Let X ~ be a definable se t, A be a small su bset of such that eve ry automorphism f of n fi xing A poin twise fixes X setwise. T hen X is A-definable. Proof. Let <.p (v) be a formu la defining X, a nd let P denote the set of the ty pes p over A of elements a in X . Notice t hat, in this cas e, every realization b of p is in X ; in fact there is a n A-au tomorphism f of n sending a to b; f fixes X setwise, and so , as a is in X, b is in X , too. In conclusion, for every pEP a nd a 1= p, n 1= <.p (a). Now repeat the sa me a rgument as in Theorem 6.5.5: by compact ness, for every pEP t here is a finite conjunction of formulas in p, and hence a single for mul a <.pp (v) of p such that

Furt hermore, for every a EX , P = tp( a/A) is in P, a nd a E <.pp (n n). In conclusion X = U PEP <.pp (n n), in other words {Urpp : pEP} is an op en covering of Urp. By compactness again, there is a finite subset Po of P such that {Urpp : p E Po} cover s Urp , hence X = U PEPo <.pp (nn). Accordingly X is defined by t he L(A)-formula V PEPo <.pp(v). • Lemma 7.5 .3 Let T be an w-stable theory, M be an ~o -saturated m odel of ~ be a non-empty M -definable set, Y be any definable (poss ibly non M -definable) subse t of n such that Y ~ X and Y has the same Morley rank as X . T he n Y n u» i= 0.

T, X

nn

°

Proof. Put 0' = RM(X) , d = G M (X ). Recall S; 0' < 00 . T hen proceed by induction on t he pair (0' , d) (with resp ect to t he lexicogr aphic order) . If 0' = 0, then X is finit e, and so X ~ u». This implie s Y ~ u» . Further more y i= 0 because has t he same Morl ey rank as X . Now suppose cv > 0, d = 1. Then RM(X - Y) < cv. Put (3 = RM(X - V) . Owing to Proposition 5.7.9, there are infinitely man y pairwise disjoint M definable subsets X i (i E N) of Morley rank (3 inside X . In part icula r there exists some natu ral i for which RM((X - Y) n X i) < (3 . Accordingly RM (Y n Xi ) = (3, and by induction Y n X i n u» i= 0. Hence Y n M n i= 0.

7.5. w-STABLE THEORIES

245

At last assume a > 0 and d » 1. By Proposition 5.7.9 once again , X can be decomposed as the disjoin t union of d i\1-defina ble sets X o, ... , X d-l having Morl ey rank a and degre e 1. For some i < d, Y n X i has Morley rank a and hence Morley degree 1. By induction Y nXinMn i= 0, whence YnMn i= 0.

.

The previou s lemm as have a prevalent technical flavour. The next lemma is more substa nt ial and will be used several t imes lat er in this section; it ensures t hat, in an w-stable t heory T , for every definabl e X c n n and formu la
respect ively are A -defi nable.

For every a E nm , RM (
Proof.

Hence it suffices t o show t hat, Vi < d, t he set of th e tuples a E nm sat isfying RM (

246

CHAPTER 7. CLASSIFYING

The second step will even prove th at , for every ii E nm sat isfying RM (
X)<

Q' .

Ass ume moment arily t hese pr elimin ar y steps show n, a nd consider all t he finit e sets D ~ X such t hat, Vii E nm ,

Let ~ denote the class of t hese sets (~ may hav e t he same power as n) . Not ice that, for every D E ~ , one ca n writ e a s uitable formu la Q'D (w) (po ssibly with param eter s) ju st st ating "D n
In fact , if ii E n m satisfies Q'D (w) for some D E ~ , t hen D n
s E n n,

D n
n X) <

Q' .

Oth erwise, for every natural i , one can find tw o t uples ai E s uch t hat , for an y i , j E N ,

RM (
n F .
Q',

~ j < i.

nm ,

b~ E

nn

7.5. w-STA BLE T HEORIES

247

Let us see how t o int roduce ai, b~ for a given natural i. Suppose aj , bj given for all j < i . Then D = {bj : j < i} is a finit e s ubset of 'l9 (Mn ), and conseq ue nt ly there is ai E nm for which

bu t

RM (
n F n 1=

j


for eve ry i a nd j in N , a nd so ob eys t he order pr op er t y. In t he same way on e shows the next st ep. Second step: for ever y ii E nm for which RM (
a; E n

m

,

It suffices to a pply t he sa me procedu re as in the first step to 'l9(M n)
.6. has power <

1nl, a nd

aD( n m ) (whe re, for every D E .6. , aD is introd uced as befor e, and hen ce is M-de finable because D ~ M n ) eq uals the set of t he t uples ii E nm for which RM (
Third ste p: UDE~ aD( n m ) is defin abl e. To get t his, first no tice t ha t what we ha ve proved with resp ect t o
248

CHAPT ER 7. CLA SSIF YING

RM (X -
0', t hat is RM (
Va Enm ,

RM(
n X) < 0'

~

aE

U O'D (n

m

).

DEt::.. o

As UD Et::.. o O'D(n m ) is definabl e, we a re done.

•

At last we are in a position to show: Theorem 7.5.5 Let T be an w- stable theor·y. Th en eve ry typ e p over a sm all A ~ n is A-definable.

Proof. Let '19 (v) be a for mula of p having t he sa me Morley rank and degr ee as p. In particular , let 0' de note t he rank of p. For every formula

~

'l9(V) A
Mo reove r, if t his is t he case, then '19 ( v) A

By Lemma 7.5 .4, th e t uples

a E nm

satisfying

(th at is RM('19(nn) n
Proof. Let '19 (v) be a formul a of p havin g t he same Morley rank 0' and t he same Morley degree d as p. Accordingly t here are d formulas 'l9 i (v, Xi ) (i < d) wit h parameter s xi in n such t hat

7.5. w-STABLE THEORIES

(ii) 'l9i(nn ,xj) n 'l9 j(nn, xi) =

249

0 for

(iii) R M ('l9 (n n) n 'l9 i( n n, xi )) =

Q'

j

< i < d,

for ever y i < d.

The class of the seq uences xi in n sat isfying (iii) for a given i < d is Mdefinable becaus e 'l9 (iJ) has got its own paramet ers in M. Accordingly t he existence of some seq uences i'o, . . . , Xd- l satisfying (i) , (ii) and (iii) ca n be ex pressed by a su itable first ord er sente nce wit h pa ram eters in M . Consequently t hese t uples xo ,... , Xd - l can be chosen in M . Correspo nding ly we ca n ass ume t hat, for every i < d, 'l9( iJ, xi ) is a formul a with param eters in M; so there is i < d such t hat 'l9(iJ, xi) E p, as V i 1, t hen (iii) implies RM('l9(iJ) A'l9 j(iJ, xj )) = Q' for every j < d, j =1= i, a nd hence GM('l9 (nn) n 'l9 i (n n, xi )) < d. But t his cont radicts 'l9(iJ) A 'l9i(iJ, xi) E p. So d = 1, as claimed . ... Now we t urn ou r attent ion t o simplicity. Our aim is t o pro ve t hat w-stable th eorie s are simple. More specifically, we will show that , for an w-st ab le theor y T , th e t riples (ii, E , A) such that ii is a t uple in n , B 2 A a re small s ubsets of nand R M(tp(ii/A )) = R M (tp(ii/B )) form a good ind ependence system of T . This confirms t hat T is simple, and pro ves also t hat, for ii, A an d B as before, ii is ind ep endent of B over A ii -!-A B if a nd only if tp (ii]A) has the same Morley rank as its extension tp(ii/ B). Incident ally recall t hat, for an y ii, A and B , tp(ii/B) always includes tp(ii/A ), a nd consequent ly

RM(tp(ii/B))

< RM (tp(ii/A)) .

Theorem 7 .5.7 Let T be an cc-stable, and let I be the set of the triples (ii, B , A) where ii is a tuple in n, A ~ B are small subsets of n and R M (tp (ii/B )) = R M (tp(ii/A )). Then I is a good independence system of

T. Proof. We have to check t ha t I sat isfies th e condit ions (11)-(17) listed in Section 7.2. (11) is trivia lly t rue, as automorphisms preserve both Mo rley rank and Morley degree. (12) Fix ii, B , and pick a formula
250

CHA PTER 7. CLASSIFYING

(16) Fix A ~ B ~ C and a. Recall RM (tp(ajA) ) ~ RM (tp(ajB)) ~ RM(tp(ajC)) . Co nsequent ly RM (tp(aj A)) and RM(tp(ajC)) coincide if and only if both t hese ranks eq ual RM (tp(aj B )). (13) Take A ~ B e a. If R1vf(tp(ajA)) = RM (tp(ajB )), t hen we ca n apply wh at we have ju st checked in (16) and dedu ce RM (tp(aj A U b) ) = RM(tp(ajA)) for every bE B . Conversely suppose RM(tp(ifjA)) > RM (tp(ajB )). Choose a formula
(15) Let A be a small subset of n, if, b be two sequences in n. We have to pro ve t hat RM (tp(aj A U b)) = RM (tp(ifj A)) implies RM (tp(bj A U a)) =

7.5. w-STABLE THEORIES

251

RM(tp(bjA)). Let n, m denote the length of ii, b respectively. First assume that A = M is the domain of an ~o-saturated model of T. Let
n F ViNw(X(v, w) -7
a'

•

a'

CHAPTER 7. CLASSIFYING

252

At t his point we ca n also check (18) and confirm once again in t his way t hat a n w-st a ble theory T is stable. Indeed , given any type P over a small A ~ n and B 2 A , t he number of non-forking extensions of p in S( B ) ca nnot exceed the Morley degree of p , and so is an yhow finit e. So, when A is t he do main of some mod el of T , (18) is ju st a rephras ing of Theor em 7.5.6: in fact , the Morley degree of p is 1 in this case . At t his point , it is imm ediate to deduce also t hat w-st a bility impli es superstability. C oro llary 7.5.8 An w-stable theory T is superstable.

Proof.

T is stable, and the set A in (12) is finite .

'"

Now we want to discuss in our w-stable setting the concepts of heir and coheir introduced in Section 7.3 , and to show their equivalence with the notion of non-forking exte nsion. Let us recall briefly how heir and coheir are defined . F irst we fix our fram ework: T is a( n w-stable) theory, M is a mod el of T, B is a small subset of n including M , n is a positiv e int eger , p is a n n-type over M and q is a n extension of p in Sn (B) . In t his setting

• q is a n heir of p if and on ly if, for every formul a iJ(v, b) E q, t here exists a t uple a in M s uch t hat iJ( ii, a) E p, • q is a coheir of p if a nd only if, for every form ula iJ( V, b) E q, iJ(M

0.

n

,

b) =f:.

'rVe are going t o show t he equalit ies heir = coheir = non-forking extension a mong t he ty pes over a sma ll set B extending a given ty pe p over a mod el M of T . F irst let us recall what we showed before in t his section, namely t hat a ty pe p over an y sma ll set A is A-definable and so, for every L-formula iJ(v, w), finds an L (A )-formu la diJ(w) such th at a t uple ain A sat isfies diJ(w) if and only if iJ( v, a) E Pi indeed , if
253

7.5. w-STABLE THEORIES Theorem 7.5.9 The following propositions are equivalent:

(i) q does not fork over A; (ii) q is an heir of p; (iii) q is a coheir of p. In particular the heir (and coheir) of p over B is unique. Proof. Let q E S( B ) be a non-forking extension of p. As already underlined , q is unique as p has Morley degr ee 1, and, for every L-formula 'O(v, ill), there is a n L (M)-form ula d'O( ill) such that, for every b in B ,

'O(v, b) E q {::}

n F d'O(b)

(and, for every a in M , 'O(v, a) E p if and only if M F d'O(a)) . So suppose th a t , for some bin n, 'O(v,b) E q. Then n F :JilldO(ill) , and consequent ly M F :Jill dO(w) because dO(w) is an L (M )-formula . Let a E M sat isfy M F dO (a) , then '0(v, a) is in p and in q. So q is an heir of p. Now suppose that q' E S(B ) is a not her heir of p. Th ere are a form ula '0(v, ill) in L a nd a tuple bin B such that oio, b) E q - q'; in particular n F d'O(b) , a nd hence

-,'0 (v, 01)

1\

d'O(a!) E p.

So M F d'O(al ), while 'O( V, a!) 1. p: t his is a cont radict ion. It follows th at q is the only heir of pin S (B ). This im plies t hat (i) and (ii) are equ ivalen t . Now we check (ii) {::} (iii) . F irst suppose B - M finite; let blist the elements of B - M , and let a realize q. Observe that

tp(a/M U b) is a n heir of tp(a/M) if and only if for every L(M)-formula t.p (v, ill) satisfying such that n F t.p(a, m)

n F t.p (a, b) , t here exist s mE M

a nd hence if and only if

tp(b/M U a) is a coheir of tp(b/M) . So t he eq uivalence (i) {::} (ii) a nd (15) imply that tp(a/ M U b) is an heir of tp(a/M) if and only if tp( b/ M Ua) is a coheir of tp( b/ M ). This est ablishes t he equivalence of (ii) and (iii) when B - M is finite. To ext end t his conclusion to the gener al setting, it suffices to obser ve that q is a n heir (a coheir) of p

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if a nd only if, for every finit e subse t Bi, of B , t he rest rict ion of p to MU Bi, is an heir (a coheir respectively) of p. ... At this poin t we can complete what we observed in 6.2 a bout the connected component of an w-stable group.

9 be an eo-stable group. Then the connected component GO of 9 has Morley degree 1.

Theorem 7 .5.10 Let

Proof. Take two ty pes p , q over G hav ing th e same Morley rank as G an d containing th e formu la
As said before , w-stability impli es superstability. Recall th at , roughly speaking , t he supe rstable theories T a re t hose ensuring that every proper elementary extension M < N , M =f N between mod els of T realizes some regular type over M ; a nd t he regul ar typ es a re just the non- algebraic types p over M such t hat ¥A sat isfies (D3), a nd consequ ently (Dl) - (D4), in p(D). Ou r aim is to examine close rly these matters in t he w-st abl e framewor k, where an elegant approach proposed by Lascar allows a sim pler t rea tm ent . Indeed stronger tools are availa ble in the restricted w-st a ble setting; for instance, we will use over and over again (exist ence a nd uniqueness of) prime models over subset s below: this is a featu re of w-stable th eories which may fail in arbitrary superst abl e theories. Accordingly fix a n w-s table T a nd a mod el M of T ; we wish t o examine types over M. By the way, let us introduce t he following notation: for X a small subset of D, let M (X) denote the mod el of T prime over M U X; when X is t he do main of some finit e tuple a, we write M (a) instead of A1(X) ; notice th at , if a a nd a' have t he same ty pe over M , then M (a) a nd M (a') ar e isomorphic by a fu nction fixing M poin twise and mapping a into a'; in ot her words, the isomorphism type of M(a) over A1 do es not depend directly on a, bu t on t he typ e of a over M . Wh at we fi rst need in our analysis is a crit erion com pa ring types p and q over M , a nd measuring whether realizing p is easier or harder than realizing q. This is provided by t he so called Rudin-Keisler relation ?RK (although it

7.5. w-S TA BLE T HEOR IES

255

was Lascar who introduced t his noti on, fain tl y inspi red by a Rudin-Keisler order relation conc erning ultrafilt ers). Here is its definition . Definition 7.5.11 Let p an d q in S(M) . W e put p '2RK q if and only if, f or eve ry tupl e ii satisfying p, the m odel M (ii) prim e over M U ii con tains some realizati on b of q, an d we put p r- tuc q if and on ly if p '2 RK q and q '2RK p.

Notice t hat, owing t o what we said ab ou t M (ii) before, we could eq uivalent ly write "for some if' inst ead of "for every if' in t his definition. '2 RK is a preorder relati on : it is reflexive and t ra nsitive, bu t it is not anti symmetric. So r-ruc is a n eq uivalence relation a nd '2R K determines an o rde r relation in the quotient set S (M )/ r-ruc among rvRK-classes of t ypes. Rou ghl y speaking, p '2RK q mean s t hat, wherever p is realized , q is reali zed as well; an d consequent ly p
Notice t hat every RK-mini mal type p is rvRK-eq uivalent to some I- typ e. In fact t ake ii 1= p ; as p is not algebra ic, t here exists some a in ii out of M; t hen tp(a/M) is not algebraic and tp( a/M) S,RK p , whence tp(a/M ) r-tuc P by RK- minimality. There is anot her relevan t relation betw een typ es a rising in t his set t ing, and closely connected with rvRK ; it is called orth ogonality a nd is defin ed as follows. Definition 7.5.13 Let p and q in S(M) . W e say that p is orthogonal to q and we writ e p 1. q if and only if, for every ii 1= p and b 1= q, {.M b.

s

Notice t hat 1. is sy mmetric. Roughl y speaking , p 1. q mean s t hat realizing p a nd realizing q are "independent" facts. Here a re so me simple exam ples concern ing '2RK , rvRK and 1.. Examples 7.5.14 1. Let K be an algebraically closed field . Recall t hat t here is a unique non-algebra ic I- typ e over K , that of t he t ra nsce ndental eleme nts over K. Mo re genera lly look at a rb itrary ty pes p and

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q over K. Let n F= p a nd b F= q (where s, bmay have differen t length s). Then p ~RK q if a nd only if t he t ranscende nce degr ee of K (il) over K is not smaller t ha n t he t ra nscende nce degree of K (b) over K , and p rv RK q if a nd only if t hese t ra nscende nce degrees coincide. In par ticular t he Rudin-Keisler relation det ermines a total ord er in t he quo tient set S (K) I
Notice that , owing t o what we observed before about A1 (a ) , we could say "for some a F= p" instead of "for every a F= p" in t his definition . Now let us propose some exam ples. Examples 7.5.16 1. The only non- algebraic I- t yp e over a n algebraically closed field , or also over a n a rbit rary stro ngly minim al structure, is strongly reg ular: v = v works as
7.5. w-S TA B LE THEORIES

257

2. Tak e a structure (M, E ) where E is an equivalence relation with infinitel y man y infinit e classes and no finite class, as in Example 7.5.13 , 2. Each ty pe list ed in 7.5.13 ,2 is st rongly reg ula r , Pa beca use of E( v , a) and q because of v = v . The basic fact a bout st rongly regula r types is: Theorem 7.5.17 Let T be w-s table, M and N be models of T such that i= N . T hen there is some a E N - M for which tp( a/ M ) is strongly regular.

M is an elemen tary su bstructure of N an d M

P roof. Let a E N - M be such t hat p = tp (a/M) has a minim al Morley rank, a nd let th e formula
Proof. Let b F q. As q is not algebraic, M (b) properly extend s M. So apply th e pr eviou s t heorem and gets some stron gly regular p reali zed in M (b) : p ~R[{ q. .. O n the oth er side Theorem 7.5 .19 Eve ry strongly regular p E S I(M ) is RI( -m inimal.

Proof. This needs a more laborious a pproac h, and some preliminary ste ps which perh ap s hav e t heir own intrinsic int erest. F irst let us fix our setting. There is some L (M)-formula
~

M, b ~M a. T hen b ~M M(a) .

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CHAPTER 7. CLA SSIF YING

Notice t hat we do not use here th e hypothesis th at p is st rongly regul ar. What we want to show is t hat tp(b/ M( a)) do es not fork over M , mor e precisely is t he heir of tp(b/M ). Accordingly take a formul a X(v, C) in tp(b/M (a)) with parameters c in M( a). The ty pe of c over M U {a} is isola ted , say by "7 (w, a). We claim t hat "7(w, a) isolates tp(CjMU {a, b}) as well. Otherwise t here is some t uple 2 such t hat 1= "7(2, a) bu t tp(c/M U{a , b}) =1= tp(2/M U {a , b}), and so t here exists some formul a 'I/; (w, a, b) (possibly with furth er param et ers from M ) for which 1= 'I/; (c, a, b) A -,'1/; (2 , a, b) . Consequent ly 3w3W" ("7 (w, v) A "7 (W" , v) A 'I/; (w, v, b) A -,'1/; (W" , v, b)) is in tp(a/M U {b}). As a tM b a nd so tp(a/M U {b}) is t he heir of tp(a/M) , th ere is some m E M sat isfying

bu t this gives a contradiction because "7 (w, a) isolates a single type over M U {a} . Accordingly "7( w, a) isolates t he typ e of c also over M U {a, b}. This t yp e contains X(b, w) , so

'Vw("7 (w, a) -7 X(v, w)) A 3w"7 (w, a) E tp(b/M U {a} ), whence, as before, x(v , m) E tp(b/M ) for some tp(b/ M (a)) is the heir of tp(b/M ), as claim ed .

m in M .

In conclu sion ,

The second preliminary step prov es:

S tep 2.
p r"VRK q.

'"

Using mor e or less t he same technica l premises one shows also t he following not abl e characterization of r"VR!{ within RI<-minimal ty pes . T heorem 7.5.20 Two RI< -minimal types over M are r"VRK-equivalent if

and only if they are orthogonal.

7.5. w-STABLE THEORIES

259

We omi t the de tails. There a re other relevan t t hings t o underline abou t strongly regular typ es. First of all , strongly regular typ es ar e regul ar as well: checking t his requires t he sa me ingred ients as in t he las t t heorem and some equivalent charact erizations of both t he involved notions. Again , we omi t a full t reatment of t his point (t his section is going t o be quite long , and actually we already prov ed a bout strongly regul ar ty pes wh at we will need in t he rest of the cha pter) . However , it is worthy poin tin g ou t that t here do exist even in t he w-stable setting some regular ty pes which a re not st ro ngly regul ar (and inde ed a re not "'RK-equivalent t o a ny strongly regular typ e). There is a not her point in t his framework deserving a few word s. In fact , ~ RK (and consequently '" RK) as well 1.. are pre serv ed und er passing from types P, q over a mod el M of an w-stable theory T to their non-forking exte nsions over some elementary extension M': P ~RK q, P "'RK q, P 1.. q if an d only if plM' ~RK qlM' , plM' "'RK qlM' , plM' 1.. qlM' resp ectively. The same ca n be said about RK-minimality. With resp ect t o strongly regul ar ty pes , one sees that , if M < M' , P is a l-typ e over M a nd cp(v ) is a n L( M)-formula making P st rongly regular , th en th e same formul a makes plM' strongly regul ar as well. The las t matter we wa nt t o deal wit h in t his section st ill concerns nonforki ng extensions and w-stable t heories. It is a t heorem (actua lly valid even for stable t heories) which will t urn ou t t o be a useful tool in t he proof of Shelah's Uniqueness Theor em in Sect ion 7.7. It is generally ca lled t he Open Mapping Th eorem, a nam e clearly evo king t opology ; of course, t he topological spaces we a re concerned with are t he Boolean spaces of typ es over small s ubsets of n. In particular , we conside r two small subsets A ~ B in a n w-stable t heory T, and we look at t he spaces S(A) and S (B) . It is easy to realize that the restriction map r from S(B) in S( A) , sending every typ e q over B into its restriction to A (so into th e set of t he L(A)-formulas of q) is cont inuous and surject ive: cont inuit y follows from th e trivial rem ark that L(A) ~ L(B) , hence L(A)-formulas are L(B)-formulas as well, a nd any open set in S(A) can be regarded as the image und er r of some open set in S (B); surject ivit y is obvious. Now consider the set of t he types q over B t hat do not fork over A. Let N (B , A) denote t his set. First notic e Proposition 7.5.21 N( B , A) is a closed sub set of S( B ).

Proof. T he clai m is obv ious when A is t he dom ain of some mod el A of T . In fact we know t hat , for every q E S (B ), q E N( B, A) if and only if q is t he coheir of its restricti on t o A , a nd it is easy t o realize t hat t he las t

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CHAPTER 7. CLASSIF YING

cond it ion just ens ures that q is in t he closure of t he set of t he types over B of t he t uples in A . Accordingly N( B, A) equ als t his closed set . Now let A be any subset of B. Use extension (13) and com pactness, and build a model M of T ind ep endent of B over A . Notice t hat every type in N( B, A) has a non -forking extension over B U M , and so by t ransit ivity enlarges to some ty pe in N( B U M, A) , which , again by t ransit ivity, lies also in N (B U M , M ). On t he other hand , every type q E N (B U M , M ) restricts t o a type in N( B, A ). In fact, let c realize q, henc e let c -!-M B . Use sy mmet ry and dedu ce B -!-M M U c. As B -!-A M, t ransit ivit y yield s B -!-A MU and con sequ en tl y B -!-A A U c. Use sy mme t ry and deduce c -!-A B , in ot her word s t hat the type of c do es not fork over A , as claimed . Then the restriction map from S (B U M) onto 8(B) - a cont inuos function between compact spaces - se nds N (B U M , M) just onto N(B, A). As the for mer set is closed , it s im age is, t oo. ..

s

Now we are in a position to prove t he Op en Mapping T heorem . T h e orem 7. 5.22 Let A ~ B be small se ts, r denote the restriction m ap fro m N (B, A) onto 8(A) . T hen r is open . In oth er words, for eve ry L(B )fo rm ula O(V) there is an L (A )-form ula O'(V) suc h tha t a type q over B non forking over A contains O(v) if an d only its restrict ion to A in cludes 0' (V) . Proof. T here is no loss of ge nera lity in repl acing B by a mod el A1 of T elementarily incl ud ing B and saturated in some power > IAI. In fact, let r', r'' denote t he restricti on map s from N (NI, A) onto 8(A) and from N(M, A) in 8 (B) resp ecti vely ; it is an easy exe rcise t o check that the lat t er fun cti on maps N(M, A) onto N( B, A) and t hat r' is just t he com posit ion of r" a nd r . Conseq uent ly, if r' is open, then r is, as r" is cont inuous . Henc e we ca n replace B by M , as claimed; accordingly, let r denote from now on t he restriction map from N (M, A ) onto 8(A). Let U be an op en set of N(M , A) . We can assume t hat U is t he set of the types over M which do not fork over A and include a given formula
7.6. CLASSIFIA B LE T HE OR IES

261

«: (q) E U, and so, as q a nd

g-1 (q) have t he same rest riction t o A , q E U* . Now let us deal wit h t he other implication. Let q E U*, hence t here is some type p E U such that p and q have t he same restrict ion t o A. For every for mula X(1o) of tp(bjA), ep(v, b) A X(b) is in p; as p does not fork over A, t here is some t uple if in A for which ep(iJ, if) A X(if) occurs in t he restricti on of p t o A a nd consequent ly in q. Recall t hat q E S(M) is definabl e, whence t here is some L (M)-formula "ep(iJ, 10) E q" defining t he t upies c such t hat ep (iJ, C) E q. Accordingly t he set of L (M )-formulas

{ "ep( e. 10) E q"} u tp(bjA) det ermines a (possibly incomplet e) typ e over M. As M is sat ur at ed in some power > IAI, we ca n even find a t uple c in M realizing t hese for mulas; in par t icular ep(v, C) E q. Enla rge t he identity map of A by b t----7 c, and get an '" a utomo rphism 9 of M such t hat ep(iJ, g(b)) E q, as requi red.

7.6

Classifiable theories

We cont inue and conclude in t his section our outline of Shela h's classification progr am. T still deno tes a com plete first ord er theory with infinite models in a co untable lan gu age L , n a big sat ur ated mod el of T . As unsuperst abl e t heories have t oo man y models and are not classifia ble, we should ass ume T s uperstable. However , to sim plify our exposit ion, we will even assume T wstable, and we will t reat in det ail t his restrict ed fram ework. In fact w-stable t heories are supe rstable as well (but not conversely) a nd enjoy some st ronger properti es (such as existence and uniqu eness of prim e mod els) making t hem more t ract able. To introduce our next ste ps, let us refer once again t o st rongly minimal t heories, and even to the more particular case when T = AC Fp for a given p = 0 or prime. Recall St einitz's an alysis of an algebraically closed field IC of cha racterist ic p: given • t he prime subfi eld IC <> of IC , • a t ranscende nce basis B of IC (over IC<» , IC is fully det ermined up to isomorphism as t he algebraic closure of t he extension of IC<> by B , which in its t urn depend s only on t he cardinality of B (t he t ransce nde nce degree of IC ). This pro vides a qui te sat isfact ory answer to our classification purposes. But , in view of a possible generalization, we

262

CHA P TER 7. CLASSIFYING

have to be very car eful in weighing t he specific advantages of th e particular algebraically closed (or also st rongly minim al) setting a nd in checking which of t hem ca n be extended to a broade r fra mework. Ab ove all, we have to recall t hat over K <> (as well as over any subfield of K ) t here is a unique non- algeb raic l-type, t hat of t ranscende ntal eleme nts; by t he way, t his ty pe is strongly regular. Now t ake a ny w-stable T a nd a mod el M of T . Keepin g t he previous exam pie in mind, we examine t he struct ure of M. M (element arily) includes t he model M <> of T prime over 0. To construct A1 upon )\,-1 <>, look at the nonalgeb raic I- types over M <> rea lized in M . Unlike t he example of fields, in this general sett ing we s hould expect to meet qu it e a lot of ty pes; however we ca n restrict our attention to t he RK-minim al ones, partition th em according to t he eq uivalence relation r-tuc and choose a st rongly regular represent ative in each rv RK- class. Incident ally recall t hat t he resu lt ing ty pes are pairwise ort hogo na l. At t his poin t take a maximal independ ent set of realizations for eac h of t hese types , glue t hese sets, form t heir union X a nd t he mod el M <> (X ) prime over M<> U X in A1. If t his mod el equals M (just as in algebraically closed fi elds) , t hen we a re don e: t he isomorphism type of M sho uld be given by t he sizes of t he inde pende nt sets of realizations of t he involved strongly regular ty pes. Bu t it may happen t hat, given a strong ly regul ar type over M <> realized in M , say by a, after for ming t he model A1<> (a) ofT prime over lVf<>U{ a} , one meets some new non-algebraic ty pes over M<>(a) which are orthogon al to a ny type over M<> an d are realized in M (as t he next Exam ple 7.6 .1 will show) . One sees t hat t his excludes M = M <>(X ). Accordingly we have to repeat t he previous machinery over a nd over again until M is reached (prov ided t hat A1 ca n be event ually reac hed) . Example 7.6.1 Let T be t he t heory of a l -ary function 1 such th at eac h eleme nt has infinitely man y pr eimages via 1, t here is an element 0 for which 1 (0) = 0 a nd no a =I- 0 satisfies r (a) = a for any positive integer n. The reader may check t hat T is complete : we do not wish t o linger over t his point and we pr efer to descri be in detail t he non-algeb ra ic I-ty pes over a model M of T . In t his setti ng, first we have to consider, for a ny posit ive integer nand a E M, t he type PM (n , a) of an element satisfying r(v) = a, -,(fn-l( v) = b) for every b E M. In addition t here is a t ype qM determined by t he for mul as -,(r( v) = a) for any a E M and positive integer n. No fur th er I- typ e a rises. In pa rt icula r SI (M) is countable when M is, a nd so T is w-stable. Incident ally notice t hat t he set t heoretic union of two mod els of T is again a mod el of T .

7.6. CLASSIFIABLE THEORIES

263

Now observe that all t he I-types mentioned above are strongly regular. Let us check t his. First con sider PM(n , a) for a E M , n a positive int eger. Let x F PM(n ,a). To form A1(x) , one t akes M , a nd one adds first z , f (x), ... , (x), an d th en ~o pr eimages for each of th em , and for ever y pr eimage, and so on. Con sequently th e on ly elements of M( x) - M real izing PM(n, a) are t hose sat isfying t he formu la r(v) = a. T his confirms t hat PM(n , a) is st ro ngly regu lar. Now look at x F qM. This time, to form A1( x) we t ake M a nd we add f n(x) for every posi ti ve int eger n , and t hen ~o preimages for each of them , and for every preimage, and so on. All t hese elements sat isfy as« , and so th e poin ts of M (x) - M realizing qM ar e exactly those satisfying, for instance, v = v . So also qM is st rongly reg ular. Now take an ~l -sat u rated model M of T. M <> (the model pri me over 0) is an element a ry subst ruct ur e of M up to isomorphism . As M <> is count able, M con tains a realization x of PM<> (1, a) for every a E M, Also A1' = M <>(x) is an eleme ntary substructure of M . Now consider PM' (I , x), which is a st rongly regular type, and again can be realized in M because M is ~l- saturated and M' is countable . Furth ermore PM'(I , x ) is orthogonal (equivalently is not f"VRK-equivalent) to any non-algebraic I-type P over M <>; in other word s PM,(I , x ) 1.. P/M" In fact , if P = PM<>(n, a') for some a' E M <> and positive integer n , t hen one eas ily observes PIM' = PM,(n , a') , while, if P = qA1 <>, t hen pIM' = qM' , In any case, piM' is strongly regular. To conclude PM,(I , x ) 1.. P!M' , we can eq uivalent ly check pIM' rfRK PM,(I ,x), which is eas ily proved, as, for ever y y F PM,(I , x) , M'(y) contains no realization of piM' (and conversely). In conclusion t here exist some st ro ngly regu lar types over M' realized in A1 but ort hogonal t o t he nonalgebraic types over M <>. Moreover t his pro cedure can be repeated as man y t imes as you like (afte r realizin g PM' (I , x ) by y in lVI , one can form PM'( y) (1, y), a nd so on) .

r:'

Let us com e back to our general analysis. We wonder und er wh ich conditions any model M of our arbitrary w-stable t heory T can be reached at th e top of th e construction descr ibed before. To clarify this point we hav e to make our framework mor e preci se. So let us first recall what follows. Definition 7.6.2 For any non-zero ordinal A, let A<w denote the set of the fini te ordered sequen ces of elements of A. A<w is partially ordered by the relation ::::; according to which, for s , tE A<W, s ::::; t if and only if s is an initial segment of t . A<w has a least element in ::::; (the empty sequence < » ,o every S EA <w different from <> has a (unique) predecessor (a greatest elem ent < s}, which will be denoted by s: ; s is called a succe ssor of S- .

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CHA PTER 7. CLASSIFYING

A tree of A<w is a downward closed subs et C : if SEC, tE A<w an d t ~ s, th en t E C as well. The following definition describes in detail the models M which can be built in the way sketched before: th ey are exactly those having a presentation of the form we a re going to explain .

Definition 7.6.3 Let M be a model of T of power A > ~o. A presentation of M is a pai r (C , ((M s, as) : s EC)) such that C is a tree of A<W, fo r every sE C M s is a mod el ofT and as E Ms, and:

(i) M <> is prime over 0; (ii) if s, t E C and s = t'>, th en )\I{t = M s(at} and Pt = tp(at/M s) strongly regular;

(iii) if s, t , t f

E

is

C and s = i " = i':' , then ei ther Pt = Pt' or Pt .L Pt' ;

(iv) for every SE C , (at : t E C , t " = s) is indepen dent over M s, namely, for eve ry t E C su ch that C = s , at tMs {at ' : t f E C, t f =1= t , t': = s};

(v) if s , t

E

C and s = t " >, then Pt is orthogonal to all the non-algebraic

types over M s; (vi) )\1{ is prime over UsEc

u. .

Of course we are int erest ed in those t heories T such that every model gets such a presentation .

Definition 7.6.4 An to-sta ble theory T is called presentable when, fo r all models )\I{ o, M 1 , M 2 and )\I{ oj T such that • M o is an elementa ry substruct ure of both M 1 an d M

2,

• M 1 tMo M 2 (in other words ai t Mo M 2 f or every ai E M 1 , or, equzvalently, a2 tMo M 1 f or every a2 in M 2),

• M is prime over M 1 U M 2 , for eve ry non-algebraic type pES (M) there ex ists some type over M over M 2 that is not orthogonal to p .

1

or

Presentability is a new dichotomy within t he classification problem. In fact , on the one side, one shows :

7.6. CLASSIFIABLE THEORIES

265

Theorem 7.6.5 (Shelah) If T is an eo-stable theory and T is not presentable, then I(T, A) = 2-\ for every uncountable cardinal A. In particular, T is not classificable. On the other hand Theorem 7.6.6 (Shelah) If T is ea-stable and presentable, then every uncountable model M of T has a presentation. A model M may admit several presentations. Actually there is a "quasi uniqueness" theorem stating under which conditions two "different" presentations yield isomorphic models; but it is impossible to discuss it here shortly, so we omit its treatment, and we conclude our report about presentability and presentations by proposing some examples and, in particular, an w-stable theory which is not presentable. Examples 7.6.7 1. Let T be the theory of two equivalence relations El, E 2 such that any El-class and any E 2-class share infinitely many common elements. One checks that T is complete. Here are the nonalgebraic I-types over a model M of T: • for a E M, the formulas Et{v, a), E 2(v , a) and ,(v = b) for all b « M determine a type PM(a); • for a E M and i = 1,2, the formulas Ei(V , a) and ,E3-i(V, b) for all s « M give a new type PM(a, i); • finally there is a type qM determined by the formulas ,El (v, b) and ,E2 (v , b) for all s « M. In particular, if A1 is countable, then Sl (M) is. Hence T is w-stable. Moreover it is straightforward to check that all the types listed above are strongly regular and, for all a E M, PM(a, 1) ..1 PM(a, 2). Now fix a model Mo of T and a E 1110 • For i = 1, 2, let Xi realize PMo (a, i) and Mi denote MO(Xi): so Mi is built by taking Mo and adding a new E3_i-class (the class of Xi) having countably many common elements with any Ei-class in M o- One can see that both M 1 and M 2 are elementary extensions of Mo ; furthermore PM(a , 1) ..1 PM(a, 2) implies Xl -!-Mo X2, whence M 1 -!-Mo M 2 (see the proof of Theorem 7.5.19), Now form the model M of T prime over M 1 U M 2 ; M is obtained just by adding countably many elements to M 1 U M 2 in the intersection between the E 2-class of Xl and the El-class of X2. Let X be an element in this intersection , and consider p = PM(X) : P is

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266

a non-algebraic type over M , and one can check t hat P is or thogon al (equivalent ly, is not "'RK-equivalent) to any ty pe over M 1 or M 2 • So T is not present abl e. Let us also check t hat, for every uncount abl e cardin al >. , J(T, >') j ust eq uals 2'\. In fac t take t he disjoint union I of tw o sets h and J2 of power >.; let R be an irreflexive sy mmet ric bin ary relation in I such t hat, if 81, 82 E I and ( 81 , 8 2 ) E R , th en 8 1 E h and 8 2 E h or conve rsely (so (1, R ) is a bipartite graph). Now build a model MR of T where the E i-classes correspond to th e elem ents of I; for ever y i = 1, 2 and , for every El -class Xl and E 2 -class X 2 ,

a nd

IX 1 n X 2 ! = ~o

otherwi se.

It is clear th at MR has power>. for ever y R. Furt hermore, for R =1= R' , A1 R i:- A1R' . So I (T , >') ~ 2.\ and conseq uent ly I (T , >. ) = 2.\: in fact , t here exist 2.\ man y relations R as befor e. 2. On t he contrary t he t heory T in Example 7.6.1 is present able. In fact take t hree models M o, A11> M 2 of T such t hat JV1 0 is a n elemen t ary substructu re of both M 1 a nd JV1 2 and M; -!-Mo M 2 . We observed t hat M 1 U M 2 is t he dom ain of a mod el M of T extending M 1 and M 2 ; of course, M is prime over M 1 U M 2 • Let p be a non-algebraic ty pe over j1;[ (and keep th e same notat ion as in 7.6.1 ). If P = PM( n , a) for some a E M and so me positiv e integer n , t hen P "' RK PMi (n , a) where i = 1, 2 sat isfies a E Ms, If P = qM , t he n P is '"RK-eq uivalent to both qM l and qM 2' So T is pr esentable, as claim ed . Nevertheless T is not classifiable, as it has 2.\ many pairwise non isomorphic models in every uncountable power >.. Let us see why. Ind eed , for every X, we ca n build 2.\ non isomorphic mod els sat isfying the furt her assum pt ion for every a E M a nd for some natural n f n(a) = O. Notice t hat such a mod el M ca n be viewed as a t ree of >. <w with resp ect t o t he relati on ~ defined as follows: \la , b E M , a ~ b if a nd only if r(b) = a for some natural n ; in parti cular a = b: if and only if f( b) = a , and < >= O. The isomorphism class of M clearl y det erm ines t he iso mo rphism ty pe of (M, ~) as a t ree . Moreover every

7.6. CLASSIFIABLE THEORIES

267

point in (M,~) has infinit ely many successors, a nd IMI = A. So it suffices to show t hat t here exist 2-\ pairwise non isomorphic trees of A<w sat isfying t he las t addit iona l proper tie s. Associate a t ree C (v) of A<w with any ordinal v < A as follows. (a ) F irst let v = 0: in C (O) th e roo t <> has A success ors, while an y fur t her vertex has ~o successors. (b) Now let v = J-l + 1: in C (v) <> has ~ o successors, and each of them is t he root of a t ree isomorphic t o C (J-l) . (c) Finally let v be a limit ordinal: in C (v), < > has a successor s/' for every J-l < v , a nd s/' is the root of a tree isomorphic to C (J-l) . At this point , let us bui ld for ever y S <;;; A a tree C (5 ) of A<was follows: < > has a success or s., for every v < A and , for every t/ < A, S V is in it s turn th e root of a t ree isomorphic t o C( v) if v E 5 and t o w<w otherwise. It is clear t hat jC(5) I = A for every 5 ~ A and th at different subsets 5 I: 5 ' yield non isomorphic trees C (5 ) 'f!. C( 5' ). At t his poin t one may wonde r what is so wrong in t he las t example to exclude a ny classification of models. More generally one may as k why a present abl e t heory may be non-classifiabl e. Recall th at , if T is presentabl e, t hen any uncount abl e model A1 of T has a present ation , whose "skelet on" is a t ree C of IMI< w. Of course t his is a good feature t owa rds a genera l class ificat ion of mod els . Bu t th e poin t is t hat some involved t ree might be non well found ed . Let us recall what th is mean s. Let C be a ny t ree (say in A<W ). On e ass ociates with a ny poin t s in C a rank r (s) (an ordinal, or 00) in t he following way. First we define r (s) ~ a for any ordinal a . We pro ceed by induction on a :

1. r (s)

~

a if a = 0;

2. if a is limit , then r(s) ~ a means r (s) ~ {3 for every ordinal {3 < a; 3. if a = (3 + 1, th en r(s) ~ a means t hat , for some t E S with s = t r ; r(t) ~ (3 . If t here is an ordinal a for which r(s) 'l a, t hen t he least ordin al with t his property is necessarily a s uccessor ao + 1, and we pu t r( s) = ao. Otherwise (when r(s) ~ a for every ordinal a) we put r (s) = 00. We say t hat C is well found ed if r( <"> is an ordinal. Now let us arrange t hese definitions in our setting, when T is any w-st abl e present abl e t heory. So every uncoun t abl e mod el of T has a presen t at ion ,

CHAPTER 7. CLASSIF YING

268

where th e points of th e involved t ree corr espond to strongly reg ular types. Take any model M of T an d an y st rongly regu lar type p over M. We want to associate wit h p a depth D p (p ) (an ordinal or 00) in a way inspired by th e above assignment of a rank to the points of a tree. First we defin e Dp(p) ~ a for every ordinal a . As usual, we pro ceed by induction on a . Definit ion 7 .6.8

1. If a = 0, then Dp(p ) ~ a .

2. If a is limit, then we put Dp(p ) ~ a when Dp(p ) ~ f3 fo r eve ry ordinal f3 < a . 3. If a = f3 + 1 is a successor f3, then we put Dp(p) ~ a ex ac tly when, for every x realizing p , there is som e strongly regular type q over M (x) such that Dp(q) ~ f3 and q is orthogonal to all th e types over M .

Now we ca n introdu ce Dp(p) . Defi nitio n 7 .6. 9 If there is an ordinal a such that Dp(p ) 'i. a , then the least ordinal with this property is a succes sor a o + 1, and we put Dp(p) = ao . Otherwise (wh en Dp(p ) ~ a for ev ery ord inal a ) we put Dp(p ) = 00 .

Now we define the depth of the whole theory T Dp(T) . As usua l, we agree that 00 is great er than any ordin al. Definition 7 .6.10 T he depth of T Dp(T ) is the least upper bound of the depths of the strongly regular types over models of T. T is called de ep if Dp(T ) = 00 , and s hallow if Dp(T ) is an ordinal.

Actu ally t he original definition of depth requires some rearrangements du e to a technical conv enience in th e proofs. Bu t thi s does not concern our sket ched treatment here. At t his point it is easy to realiz e that the (pr esentable) th eor y T in 7.6.1 (a nd lat er in 7.6.7.2) is deep . In fact , let M be a model of T , a E Mo , p be the stro ngly regul ar ty pe PM(l , a) (accor ding t o t he not ation introduced in 7.6.1) . We hav e see n t hat , for every x 1= p , S(M(x)) cont ain s a st ro ngly regul ar ty pe q = PM( x )(l , x) having depth ~ 0 a nd or thogon al t o all t he non- algebraic ty pes over M . Conseq uently Dp(p) ~ 1. Bu t th en eve n q has depth ~ 1, and so Dp(p) ~ 2. Iterating this pro cedure, one eventu ally gets Dp(p ) = 00. Recall t hat T is not clas sifiabl e, because it gets too many models in any uncountable power. But this is a general fact, as th e followin g theorem clarifies.

7.6. CLA SS IFIA BLE TH EORIES

269

Theorem 7.6.11 (Shel ah ) 1fT is an cc -stable, presentable and deep th eory , th en I (T , A) = 2>' fo r eve ry uncountable cardin al A. I n particular T is not classifiable.

This is not the case for a sha llow T. In fact , first one shows t hat Dp(T ) < Wl (and inde ed , for every ordinal et < Wl , ther e is som e w-stable presen t abl e sha llow th eor y TOt having depth et). Moreover , for every un coun t abl e A, one ca n upp erly an d und erly bound I (T , A) with resp ect to Dp(T) in an effecti ve way, impl ying, among oth er t hings, I (T , A) < 2>' for some A. In t his sense, we ca n conclude Theorem 7.6.12 Let T be an w- st able th eory. Th en T is classifiabl e if and only if T is presentable and shallow.

Of course, t his stateme nt makes sense provided t hat we ag ree wit h Shela h's prop osal t hat T is not classifiabl e if a nd on ly if I (T , A) = 2>' for every unco untable Aj should we think t hat a classifiabili ty pr oof requires t o show t he existence of an invari an t system of some ordinal rank, we migh t reasonabl y d oubt t hat Theor em 7.6.12 is t he las t word a bout classification , a nd wonder if Shela h' s classification pro gr am is fully reached in t his t heorem . T his is a su btle an d delicate matter, a nd may be discussed as long as one likes. So here we limit ou rselves t o a few comments which ju st aim at introducing a possible deba te and do not cla im t o suggest a ny conclusion. F irst one should und oubt edly acknowledge how for midable Shela h's work was; in some sense t he "de pt h" itself of t he new notions and tec hniques pro posed by Shela h wit ness its validity a nd a ut horitat iveness. Not surp risingly, Shela h himself celeb rated it s conclusion even in t he t it le of a prelimin ary version of t he final pa per, which ju st stated "W hy am I so hap py". Actually Shelah's happiness is quite easy to und erst and a nd t o sha re. It should be also mentioned t hat Melles proposed some years ago an alternative approach to classifia bility of a mor e recurs ion t heoretic flavour, so lookin g for effective class ificat ion invaria nts; however Melles himself pro ved t hat his persp ecti ve event ually agrees wit h Shela h's point of view. It should be also said t hat Appenzeller (a nd others) observed some intriguin g connections bet ween t he main notions ar ising in Shela h' s classification , a nd some concepts coming from Stat iona ry Logic a nd Descrip tiv e Set T heory. This is quite interesting, and provides a fur th er corroboratio n of Shelah 's persp ective. Fi nally let us discuss what happens wit hin superstable (an d possibly non w-stable) t heories T. T he main t roub le in t his enla rged framework is t hat

CHA P TER 7. CLASSIFYING

270

prime mod els may fail. However prim e models (and existence and uniqu eness properties) st ill make sense provided we restrict our attent ion to a suitable subclass of M od(T), form ed by sufficientl y sat ur ate d models (called a-models) . This allows to define wh at T presentable, or T deep, or T shallow mean s, and t o dedu ce t hat , when T is not pr esent abl e, or is presentable a nd deep , T is not classifia ble: in fact , in t hese cases T has too many a- models , a nd consequent ly t oo many mod els, t o get a classification . But t he point is whether a present abl e shallow supersta ble T is classifiable. According to Shelah 's persp ecti ve, t his requires to upperly bound t he nu mber of mod els of a given uncountable power (wa rn ing: we have said models, and not a-models , bounding th e latter ones do es not impl y a priori restricting the number of t he former ones). Again the main difficulty in handling this sett ing is the possible lack of prim e models , especially in t he framework proposed in t he definition of pr esent abl e theor y. And actually Shelah singled out t he following key property, called t he existence property. Definition 7.6.13 A superstable T is said to satisfy the existence property if and only if, fo r every choice of m odels A10 , A1 1 , ) \.1 2 of T such that

• Mo is an elem entary substructure of both M 1 and M

2,

• M 1 tMo M 2 ,

there is a model M of T pr im e an d ato mic over M 1 U M 2 • This is t he las t dichot omy in t he superstable case , a nd provides t he final dividin g line between class ifiab le a nd non- classificabl e t heories. In fact t he following resul ts hold . Theorem 7.6.14 (Shelah ) 1f T is superstable but fa ils to have the existence property, then l(T , A) = 2'\ for every un countable cardin al A. Theorem 7.6.15 (Shelah) 1fT is a superstable, presentable, shallow theory and has the existence property, then T is classifiable. Of course, t he comments following T heore m 7.6.12 a re still valid and do concern also Theo rem 7.6.15.

7.7

Shelah's Uniqueness Theorem

In the forth coming sections we discharge t wo debts we con tract ed in Ch apter 6: the proofs of Shelah 's Uniqueness T heorem for prime mod els in w-stabl e

7.7. SHEL AH'S UNI QUENESS T HE OREM

271

t heo ries and Morley's T heorem on uncount ably categorical t heories . In fact bot h require, in addit ion to t he techniq ues develop ed in C hapters 5 and 6, t he mor e sophisticated tools introduced in t his C hapter , in par ticular t he material of Section 7.5 on w-stability. First we deal wit h Shela h's Uniqueness T heorem . vVe recall its fram ework: we are con cern ed wit h an w-stable t heory T and a small s ubset X of it s uni verse n. We have t o show t hat , if A o a nd A 1 a re two models of T both elementarily including X a nd prime over X , t hen t here is a n isomorphism betw een A o and Al fixin g X pointwise. As we saw in 6.4 , Ressayr e's Uniq ueness T heorem for co nst ructible sets reduces th e whole problem to prove: Theor e m 7.7.1 (Shelah) Let T be an w- stable theory , X be a small subs et of n, A be a model of T eleme ntarily including X . If A is prime over X , then A is constructible over X.

Proof. According to Morl ey's Existence Theorem (6 .4.17) t here is some mod el B of T con structible over X . Fix su ch a mod el B. We can freely assu me t hat X is a subset of B and t hat t he co rres pond ing inclu sion is elementary. As A is prime over X , t here is an eleme ntary embedding of A into B fixin g X point wise. Again , wit hout loss of ge nerality we can ass ume t hat A is an eleme ntary su bst ructure of B. In par t icular X ~ A ~ B , and all t hese inclusion s are elementary. We claim that t hese ass um ptions X

~

A

~

B , B const ructible over X

(so for get ting t he addition al hyp othesis t hat both A a nd B a re models of T ) are s ufficient t o impl y, for T w-stable, t hat A is const ructible over X , as ex pected . We proceed by induction on the power IB - XI of B - X. If IB - XI is countable, then lA - XI is count a ble, too; furt hermore B is at omic over X becau se B is con structible over X ; hence we can apply Lemma 6.4 .6 and conclude that A is const ructible over X, as claimed. So ass ume IB - XI = A un count able. Fix an y const ruction (bv : t/ < A) of B over X : her e A is viewed as an initi al ordinal. Correspondingly we build a n increasing sequence of subse ts C; (v < A) of B satisfying 1. B=XUUv<'\ C v

a nd, for eve ry v 2.

< A,

ICvl = Ivl + ~o ,

3 . Cv is closed (in t he se nse of Section 6.4) ,

272

CHAPTER 7. CLA SSIF YING

4. for ever y t uple

c in

C v, the type of as it s extension tp(cjA U Cv).

c over

A has t he same Morley rank

To for m this sequence, start by putting Co = 0. Moreover , for a limit u, let C; be t he unio n of the preceding sets in the sequence . It is straightfo rwa rd t o check t hat 2, 3, and 4 are satisfied in these cases . Now take a successor ordinal v + 1. If b; E Cv, then put C v+1 = Cv. Otherwise form Cv U {by} a nd ca ll it Cv(O). For every tuple bin Cv(O) th ere is a finite subset A (b) of A for which

RM(tp(bjA)) = RM(tp(bjA(b))) . Take Cv (O) U Ub'A(b) and enlarge it to a minimal closed ext ension C v (1 ) in B, in t he way we saw in Section 6.4. Apply t he sa me procedure used to enla rge Cv(O) to C v(l), and then to Cv(n) for every natural n, so building for every n a closed set Cv(n + 1) 2 Cv(n) in B such t ha t, for each bin Cv (n ), there is a finite subset A(b) of A n Cv(n + 1) for which

RM (tp(bjA )) = RM(tp(bjA(b))). Finally pu t C v+1 = UnCv(n) . Again , a st ra ightfo rward check prove s 2, 3 and 4 for Cv+!, It is also clear that B = X U Uv<"' C v a nd hence 1 holds. Now notice t hat, for a given t/ < A, 5. Cv+! is constructi ble over X U Cv,

The latter claim follows directly from 2, as IB-XI = A > Ivl+~o. 5 requires some more work . First form a (possibl y transfinite) list of the elements in

C v+1

-

(X U Cv ) (dJ1- :

J-L

< a)

extracting t hem from the given constructi on of B over X . Notice th at, for ever y /3 < a , C; U {dJ1- : J-L < /3} is closed becau se both C; a nd Cv+! are closed. So use Lemma 6.4.9 a nd deduce that the type of d(3 over XUCvU{dJ1- : J-L < /3} is isolated. Hence {dJ1- : J-L < a} is a construction of C v+1 over XU C v , a nd 5 is proved . Owing t o 5 and 6, t he ind uct ion hypothesis a pplies to

X U c,

~

X U o, U (C v+1

n A) ~

X U C v+1

7.8. MORLEY'S THEOREM

273

a nd shows that xU C v U (Cv+! n A) is con structible over x U C v. For every v < x, fix a const r uction of X U U (Cv+! n A) over X U and glue all t hese constructions t o buil d a (t ra nsfinit e) list of t he eleme nts of A

c,

c.,

where, for every v , (all : fl v :S fl < flv+d constructs Cv+ 1 n A over X U Cv. We claim that (*) is act ually a construct ion of A over X: in oth er word s, for eve ry fl , tp(a ll / X U{aT] : 1] < fl}) is isolat ed. In fact , suppose fl v :S fl < fl v+l , it suffices t o show

In fact , the former type is isolated and conseq uent ly th e Open Mapping Theor em ensur es t hat t he latter is isolated , too. Now, for every tuple c in

c.,

RM(tp(CjA )) = RM(tp(CjA n Cv))

by 4. Now we use wh at we saw in Section 7.5 a bout ind ep end ence in w-sta ble th eori es. F irst noti ce t hat t ra nsit ivity (16) implies

Then sy mmetry yields

As t his hold s for every t uple c in Cv, (13) implies th at t he ty pe of all over X U C; U {aT] : 1] < fl} do es not fork over X U {a T] : 1] < fl} , and this is ju st wh at we hav e claimed . So A is const ructible over X , a nd we are done. ..

7.8

Morley's Theorem

At last here are to show Morley 's Theorem on uncountabl y categorical t heories. Let us recall it s fram ework: for a complete t heory T wit hout finit e mod els in a count a ble lan gu age L , t he theorem says t ha t T is categorical eit her in all uncount ab le ca rdina ls, or in non e of t he m. In oth er words Theor e m 7. 8 .1 (Morley) 1f T is categorical in some uncoun table cardin al, th e T is categorical in ev ery uncoun table cardin al.

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Just to prepare the proof of Morl ey 's Theorem , let us premit a particular case as a warm-up . Proposition 7.8 .2 If T is stmngly minimal, th en T is categorica l in every uncountable power .

P roof. For eve ry mod el B of T , t he isomor phism class of B is given by its dimension with resp ect to the ad dep endence relation . Wh en B is uncountable , t his dim ension ju st equals th e power of B becau se B = ad(X) forces IXI = IBI for every X ~ B . ..

So the point is: how far is a t heo ry T categorica l in some uncount able power f-l from being st ro ngly min imal? Can we recover inside T a possibly "weaker"

for m of "st ro ng" minimality making t he previous argument work ? Not ice that there do exist som e t heor ies T which are categorical in any infinit e cardinal but are not strongly minimal: for inst a nce, consider t he theory of t wo infinite disjoint sets A and A' with a bijection f between them . Anyhow take a f-l-cat egorical T where f-l is a fixed uncountable cardinal. We keep thi s ass umpt ion all t hroughout t his section, unless ex plicit ly st at ed . First we observe: Lemma 7.8.3 T is w-sta ble. P roo]. The strate gy is sim ple. We deny w-st ability a nd consequently we succeed in building t wo models of T of power f-l which cannot be isomorphic, so contradicting the hypothesis that T is f-l-categorical. To do th is, first we recall that , if T is not w-stable, th en there is a countable X ~ n over which there are unco untably many I-types. So th ere is some model of T extending X a nd realizing un count a bly many types, and using t he Lowen heim-Sko lem T heorem we ca n get a model B of T sat isfying these prop ert ies and having power f-l. The second part of t he proof uses a method due to E hrenfeucht and Mostowski and valid for every th eory T and every uncount a ble cardinal f-l : it yields in t his gen eral framework a model C of T of power f-l realizing at most ~o I-typ es over an y count a ble subset. Ehrenfeucht-Mostowski 's method is q uite sophisticated , and its int erest for our purposes in this book is confin ed to this lem ma ; so we omit its detailed t reat ment, and we limit ourselves to a pply it to our par ti cular set t ing . In fact, for a f-l-categorica l T, t he model C just built in t his way cannot be isomorphic to the structure B obtained before. ..

275

7.8. MORLE Y'S T HEOREM

Of course , Lemma 7.8.3 ca nnot im ply t hat T is st ro ngly minim al. But , as a consequence of t he w-stability of T , ind eed as a co nsequence of this only hyp othesis (we do not need fl-categoricity here) , we show t hat t here is some strongly minimal formula -possibly involving parameters- in T . Lemma 7.8.4 Th ere is som e strongly minimal form ula cp(v ) in T.

Just t o avoid any misunderst anding lat er , let us und erlin e on ce again t hat here formul a mean s for mula with param eters. P roof. Oth erwise, for every formula cp( v) for which cp(n) is infinit e, one ca n find a formul a cp' (v) s uch t hat cp' (n) ~ cp(n ) and both cp' (n) an d cp(n )_ cp' (n) are infinite. Using t his fact a nd start ing from CPO (v) : v = v, one defines , for every finite ord ered sequence s of 0 a nd 1, a formul a CPs (v) in such a way t hat , for a ny s , cps(n) is infinite a nd cpso(n ) and cps1(n ) pa rti tion cps (n ). T he para meters involved in these formulas a re as ma ny as t he sequences s , a nd so form a countable set X . On t he other side, every bra nch t of t he t ree {a, 1}w determines a( n incomplet e) l-Lyp e Pt over X given by t he for mulas CPt ln (v) wher e n ranges over naturals and t in denotes t he restriction of t to t he first n naturals; furth ermore differen t br an ches yield different ty pes . This gives at leas t 2 No I- typ es over X and contradicts w-st ability . •

Bu t fl-cat egorici t y im plies even more. In fact , for a fl-categor ical T , t he formula cp(v) ca n be chose n wit h parameters in t he model A of T pri me over (owing to w-stability this mod els exists and is unique up to isomorphism ). This result requires the following premis e, which is now ju st a techn ical lem ma, bu t will play a key role lat er in t he proof of Morley's T heorem .

o

Lemma 7.8.5 Let a( v, il) be a fo rmula (wi th param eters ii in n) suc h that a( n , ii) is infini te, and let B o an d B l be two models of T suc h that B o is an elementary substruct ure of B l , B o =1= B, and both Bo and B 1 elem entarily include ii. Th en a(Bo, ii) is properly included in a(B 1 , ii).

P roof. Suppose not. Accordingly t here a re two model s B o a nd B 1 of T elementarily includin g ii suc h t hat Bo is an eleme ntary substr ucture of B 1, B o =1= B 1 but a( B 1, il) equa ls a(B o, il) (a n infinite set ) . Apply t he Lowenh eim-Skolem Theor em t o t he t heory of (B 1 , B o) in a lan guage ext ending L by a l-ary relat ion sy mbol for B o and new const an ts for t he eleme nts in ii, and get a countable model (B'1 , Ba) of t his t heory: so a nd Ba a re still mod els of T elementarily inclu din g ii, Bo is a n elementary s ubstruct ure of Bi , B =1= Bi , a(Bi , il) = a(B ii), B e; and consequent ly a(Bi, il ) = a(B o, ii) are countable. Hence, unless replacing B o and B 1 by B o

Br

o

o,

o,

CHA P TER 7. CLASSIFYING

276

a nd Br resp ecti vely, we can ass ume t hat a (B l , if) = a (Bo, if) is count able. Now we define for every or dinal t/ > 0 a model B v of T eleme ntarily extending Ba (and if) and even t he Bp's for p < u, properly including all of them a nd stilI satisfying a (Bv , if) = a( Bo, if) . Notice th at this is sufficient for our purposes becau se, pr oceedi ng in t his way, we eve nt ua lly build a mod el B of T of power f-l eleme ntarily extending Ba bu t satisfying a( B, if) = a( Bo, if), hence admitting a countable a( B, if); on t he other side , a sim ple use of Co m pactness Theorem yields another model C of T of power f-l such t hat , for every t u ple if in C ad mitting t he sa me type as if over t he em pty set , la (C , 01= f-l . Co nseq uently B and C cannot be isomorphic, a nd t his co ntradicts t he f-l-cat egoricity of T . So let us build the Bv's. vVe alr eady introduced B l . For a limit v, put B; = Up O. For simplicity we limi t ourselves t o p = 1, v = 2; indeed what we are going to say in t his case applies to any p > 0 as well, a nd so is ge nerally valid . First use ind ep endence t heory, mo re precisely (13) an d (14), and find an isomorph ic copy B~ of B l inside n, correspond ing t o B l by a n isom orphism fixin g Ba poin t wise, and satisfying B~ ../.-Bo Bi . To build B~ , consider a language L * enlargin g L by a constant b* for eve ry b E B l , and in L * t he t heo ry T * saying th at , for eve ry bin B l ,

b* satisfies t he non -forking extension of tp(bj Ba) over Bi . Any finit e por tion To of T* has a mode l; in fact , let bglue all t he t upIes from B l a rising in t he sent ences of To, use (14) and obtain a t upIe b* realizing t he non-forking extension of tp(bj Ba) over Bi ; recall that every subseq uence of b* has t he sa me prop erty. By com pactness , T* has a mod el. T he elements b' em bo dy ing t he constants b* in t his mod el form a st r uct ure B~ isomo rphic t o B l over Ba (as, for every L (Bo)-formul a
n F x ( b~, b)

<=?

n F dx(b).

7.8. MORLEY'S THEOREM

277

This remains true if we enlarge Ba to B~ U {d} : for every also for b= (iJi, d) with iJi in B~ ,

n F= X(b~ , Let us see why. Oth erwise n

iJi , d)

{:?

F= -' (X ( b~ ,

b in

B~ U {d} , so

n F= dX(iJi, d).

iJi , d) H dX(iJi , d)) a nd consequentl y

n F= 3z(a( z, ii) 1\ -'(x (b~ ,

b', z)

H

dx(b' , z)));

in other word s,

3z(a (z, ii)

1\

-' (x (v, iJi , z) H dx(iJi , z))) E tp(b~ / BD.

As B, an d B~ a re ind epend en t over Ba , there is som e b in Ba for which " n F= :3z(a (z, ii) 1\ -' (x (b~ , b , z) H dx(b , z))).

"

"

Then t here is some d" E B l (and indeed in a (Bl , ii) ) such t hat

81

F= -' (X (b~ ,

b , d") "

H

dX(b , d")). "

Recall a( Bl , ii) = a (Bo, ii), so d" E B a. Bu t t his cont radicts t he choice of dX over B a. Now apply what we have j ust observed to t he formul a 1/;(v, b~ , d). As n F= 1/; ( b~, b~ , d), it follows n F= d1/; (b~ , d), where d1/; is t he defining formula. So 8~ F= 3zd1/; (b1 , z), and we find d' E B~ such t hat B~ F= d1/; (b1, d'), a nd hence n F= 1/; (b~ , b1 , d'). T his means that d' has t he same ty pe as d over B l U B~ , a nd so d = d' E B~ ; bu t a (B~, ii) = a (Bo , ii), hence d E a (Bo , ii) . This accomplishes our proof. .. Now we are in a position to show, as promised Lemma 7.8.6 Let A be a model ofT p rim e ov er 0. Th en th ere is a strongly m in imal f orm ula ep (v) in T with parameters from A . Proof. Proceed as in Lemma 7.8.5 , but work in A instead of n. Using w-stability, find again a formul a ep (v) with pa ramet ers in A such t hat ep(A) is infinite but has no par ti t ion into 2 A-definable infinite s ubsets. This do es not mean a priori t hat ep ( v) is strongly mini mal , as in general we cannot imagin e what happens if we allow parameters out of A . However we claim t hat in our parti cul ar setting , for a JL-categorical T, ep( v) is ju st strongly minimal. Let us see why. Suppose not , so, enla rging ou r perspective to n, we ca n find a n L-for mula 1/; (v, Z) and par a meters b in n such that both

278

CHA P T ER 7. CLASSIF YING

1\ 'IjJ (v, b) and

1
However , ju st owing t he choice of
..

Bu t we ca n even ass ume t hat our strongly mini mal
The read er may t ry to check t his as a n exercise. In pa rticular , if T is J.Lcategorical, t hen T' is; a nd, if we succeed in proving t hat T ' is categorica l in every uncountable power A, th en we can say t he same of T . So, wit h no loss of genera lity, we ca n replace T by T', in other words to assu me t hat T is a J.L-categorical t heory wit h a strongly mini mal L-formula
7.9. BIIN TERPRE TABILITY AND ZILBER CONJECTURE

279

to ad , a nd so (at least in uncount a ble powers) t he cardi nality of

Lemma 7.8.8 For eve ry model B of T , B is prim e over
P roof. As T is w-stabl e, t he re ex ists a mo de l B' of T pr ime over

P roof. (Morley 's Theor em) . Let B o a nd B 1 be t wo models of T having the same unco unt abl e power ).. . Owing to Lemma 7.8.8, each B, (i = 0, 1) is prime over

~o. T hen X o a nd X l correspond to each other by some elementary biject ion , which enlarges t o a n eleme ntary bijection h between

7.9

Biinterpretability and Zilber Conjecture

We have devoted several sections t o t he problem of classifying struct ures up to isomorphism an d to Shelah 's analysis of this qu estion. But , needl ess to say, iso mo rphism is not the onl y pos sible classifying eq uivalence relation , even within structures. Anot he r possible criterion , deeply related to Model Theory, is interpret abili ty. The following t wo examples illustrate t his alternative persp ective a nd it s underl ying idea.

280

CHA PTER 7. CLA SSIF YING

Examples 7 .9.1 1. Natur al numbers (viewed as non-negati ve int egers) form a definable set in t he ring (Z, +, .) of integers: as recall ed in Chapter 1, a celebrated T heorem of Lagr an ge says t hat they are exactly t he sums of four squa res in (Z, +, .). So t he whole struct ure (N, +, .) is definabl e in (Z, +, '), because t he addit ion and multipli cation in N ar e ju st t he rest rict ions of t he corres ponding op erations of Z . This is a fund am en t al result: in fact , as (N , +, .) lives in (Z , +, .) as a definable struct ure , (Z, +, .) inh erits its und ecid abili ty phenomena related t o God el Incompleteness Theor ems , and in this sense is a "wild" st ructure.

2. In t he same way (N, +, .) lives in the ration al field (Q, +, .) as a definable structure. This is a deep theor em of Julia Robinson . So even the rational field inherits th e complexity of (N , +, .) and its und ecidability. So, genera lly speaking, when we meet a st ruct ure A in a lan guage L and we realize t hat A defines, or also interprets anoth er structure A' (possibly of a different lan guage L'), t hen we ca n reason abl y ag ree t hat A inh erits th e full complexity of A' , and conseq uent ly is at leas t as difficult to dominate as A' is. Of course t his ca n be extended t o classes of st ruct ures . In t his enla rged fram ework , we com pa re two classes of struct ures, K in a lan gu age L and K' in a lan guage L' resp ecti vely. For simplicity, we can ag ree t hat both K and K' are elementary. We ass ume t hat t here ar e suitable L-formulas defining , or also int erpreting, in a ny structure A E K a struct ure A' E K' and t hat, conversely, every A' E K' ca n be recovered by some A E K in t his way; we ass ume also that t hese formul as do not depend on t he choice of A in K . Then we say t hat K interprets K' and in t his case we ca n agree that K inh erits t he complexity of K' . Here are some furth er exam ples illustratin g this point. Examples 7.9 .2 1. Recall that a graph is a st ruct ure (G , R) where R is a sy mmet ric irr eflexive binary relation , usu ally ca lled adja cency . T he (elemen tary) class of graphs interprets a ny class of structures, and so inherits in t his way t he full complexity of math em at ics . The proof of thi s fact requires patience rath er t ha n ingenu ousness . To avoid t oo many te dious det ails , let us illustrate its idea in a par t icular case, and see how graphs int erpret arbitrar y binar y rela tion s. So take any struct ure (A, R ) where R is a binary relation on A , and form a graph (A', R' ) as follows. Let A' include A. Moreover, for every a E A, add t wo new vertices ao and al in A', both adjacent t o a (so

7.9. BIINTERPRETABILITY A ND ZILBER CONJECTURE

281

(a, ao ), (a, ad E R' ). Finally, for every pair e = (a, b) E R, add in A' t wo new ver tic es eo, el sat isfying (a, eo), (eo, el ), (el , b) E R', t hree more ver tic es adjacent to eo, and four mor e vertices adj acent t o el . For inst an ce, here is a pictur e of (A', R' ) when A = {a , b, c} and R = { (a, b)} .

bo

A

Co

It is an easy exercise to realize that a ny (A, R) ca n be definably recovered inside t he cor res ponding (A' , R' ) in a way ind ep end ent of t he particul ar choice of (A, R). In fact , A is ju st t he set of t he ver tic es in A' having eit her 2 adjacent nod es , or 3 adjacent nod es such t hat 2 of t hem have no fur th er adjacent nod e, while R is t he set of t he pairs (a, b) E R such t hat t he re a re eo, el E A' for which (a, eo), (eo, el) (el, b) E R' , eo has t hree adjacent nod es in addit ion to a and el has four adj acent nod es besides eo and b. 2. T he same can be said of t he class of groups, a nd even of the class of nilpoten t groups of class 2 (a compa rat ively slight generalizat ion of a belia n groups) . This is a beautiful result of A . Mekler , s howing that nilpotent groups of class 2 int epret graphs a nd so, t hrough them , any class of structures. The proof uses brilliantly some non -trivial notions and tools from group th eory. T he conclusion is clear: groups, and even nilpotent groups of class 2, are a class as bad as possible, and inherits t he full complexity of mathematics. 3. We said in 7.1 that , for a given (count a ble) field K , K(x , y)-modules (i. e. K-v ectorspaces with two distinguished endomo rphism s x and y) ar e a n intractable class: no classification ca n be ex pected, even for finit e dim ension al objects , otherwise the word problem for grou ps would be solvable. Of course, every class interpretin g K (x , y)-modules is at least as complicat ed as t hey a re, a nd hence definiti vely a bad class. Several not abl e clas ses of modules share t his negative fea ture. For instance,

282

CHAP TER 7. CLASSIF YING

t his is t he case of K[x , y]-m odules (wit h t wo commuting unknowns x an d y), or Z[x]-modules, an d so on. The book of Prest quoted in t he references at t he end of t he cha pter includ es a discussion of t his point and a great deal of notewort hy examples. So a possible way of classifying st ructures, or even classes of st ructur es, is up t o mutual interpretability (biinterpretability) . Accordingly, one could t ry t o characterize struct ures by looking at what is definabl e, or also int erpretable, in t hem : gro ups , fields, a nd so on. Incident ally notice t hat a relevan t emphas is on t he role of definabilit y already a rises within Shelah 's classification analysis (for instan ce, t hink of th e ord er property, lookin g at t he orders definable in a given struct ure ). However the st udy of mu tual int erpretability in mathem atics did pr ecede Mod el Theory , or , at least , mod ern Model Th eory. For exam ple, let us mention t he celebrated Malcev corres ponde nce between groups a nd rings , essent ially showing t hat t he class of uni t ary rings is biint erpret a ble with a suitable class of nilpotent groups of class 2, a nd confirming in this way how complicated t hese groups are. Bu t who mainly developed t he biint erpretabilit y program in Mo del T heory was Boris Zilber. Zilber 's orig inal project concerned t he class ification of uncou nt abl y categorical t heories (t hose where Morley 's Theorem a pplies) by lookin g at which gro ups, or fields , an d so on , are definabl e in t hem. As already observed, uncoun tabl y categorica l t heories includ e the stro ngly minim al ones, a nd t he lat ter t heories a re t he simplest possible (if we excludes finite structures) . Acco rdingly t heir exam is a reason abl e first step t owa rds a genera l approach. So t he qu estion is: what stro ngly minimal struct ures look like? Ca n we reason abl y classify t hem by lookin g at what t hey int erpret ? Let us outline Zilber an alysis in t he strongly minim al setting. Recall t hat a st ruct ure A is said to be st rongly minimal if it s complete theory is: so, for every B element arily equivalent to A , the only definable subsets of B a re t hose finit e or cofinite . We introdu ced in Cha pte r 5 several examples of strongly minim al struct ures . To summa rize them , let us work inside a fixed algebraically closed fi eld A. Examples 7.9.3 1. Firstl y view A as a struct ure in t he lan gu age 0, so as a mere infinite set . In t his case , t he t heory of A equa ls t hat of infinite sets , and is strongly minimal. T he st ruct ure of A is very poor , and no (infinite) gro up is definable here. Mo reover • for every subset X of A , acl (X ) = X ,

7.9. BIINTERPRETABJLITY AND ZJLBER CONJECTURE

283

• for every positive int eger n , the defina ble s ubset s of An are the finit e Boolean com binations of

with i, j = 1, .. . , n, bE M . 2. Let A a be t he pr ime subfield of A, and look at A as a vectorspace over Aa in the appropriate language. Now t he complete theory of A is t hat of infinite vectorspaces over Aa, and is again strongly minimal. But t his time • for every subset X of A , acl(X) is the subspace of A spa nned by X, • for every positive integer n, the definable subsets of A n a re t he finite Boolean combinat ions of cosets of pp-definable subgroups of A , and for n > 1 their class is larger t han in Example 1. Notice that an infinite group is trivially definable in A, but no field ca n be interpreted inside A . 3. At last , view A just as a n algebraically closed field. If p is its characteristic, then the complet e theory of A is ACFp and is strongly minimal. Moreover • for every subset X of A, acl(X) is t he algebraic closure of A in the field theoretic sense, • for every positive integer n, t he definabl e su bsets of A n a re the construct ible ones , in other words t he finite Boolean combinations of algebraic varieties of A n . Now take any strongly minimal st ruct ure A.

Definition 7.9 .4 A is called trivial if, for every X

~

A,

acl (X) = UxEXacl(X) .

Every (pu re) infinit e set is trivial. But, of course, vectorspaces and algebraically closed fields are not. Moreover no trivial st rongly minimal structure ca n interpret an infinite group.

Definition 7.9.5 A is called locally modular if, for eve ry choice of X , Y ~ A suc h that X n Y i= acl (0),

(*)

dim (X U Y)

+ dim (X n Y) =

dimX

+ dim Y.

CHA PTER 7. CLASSIFYING

284

Every t rivial structure A is locally modular (in fact , for every X and Y , acl (X U Y ) = acl (X ) U acl (Y ), so a basis of X U Y ca n be form ed by taking a basis a X n Y , extending it t o a basis of X a nd a basis of Y , and gluin g t hese bases together). Bu t also vect orspaces a re modular: in fact , in t his case , (*) is ju st t he Grassm an formula (and do es not need t he ass umpt ion X n Y :f: acl (0)) . On t he cont ra ry, no algebraically closed field A (of t ra nscende nce degree 2': 4) is locally modular. In fact, choose ao, aI , a2, a3 E A alge braically ind epend ent over the prim e subfield Aa , and form th e extensions

Then dim X = dim Y = 3 but dim(X n Y) = dim Ao(aa) = 1, whence X n Y :f: acl(0) , and dim(X U Y) = dim(Aa(ao , aI , a2, a3)) = 4. So (*) fails. Wha t is t he significa nce of local modularity? Basically a locally modular A eit her is triv ial or ca n define an infinite group. Furth ermore one obser ves th at a ny group 9 definabl e in a locally modular A is a belia n-by-finite (in other wor ds , it has a normal abelia n subgroup of finit e index); and every subset of any ca rtesia n power definabl e in A is a finite Boolean combination of cosets of definable subgroups of Accordingly, no infinite field is definable in A. In t his setting Zilber raised in 1984 t he following problem , generally called Zilb er Tricho tomy Conject ure.

en

en.

Conjecture 7.9.6 (Zilber ) Let A be a st rongly m in imal non locally modular struct ure. T hen A interprets an infi ni te fi eld K . Furthermore, for eve ry posit ive integer n , the subsets of K " definable in A are ju st thos e definable in K (and hence coin cide with the construciible ones). Recall that , owing to Macintyr e's Theor em , any infinite field interpretable in a n w-stabl e st ruct ure must be algebraically closed. Hence the importance of this conj ecture is clear: according to it any st rongly minim al st ruct ure A eit her is t rivial, and so looks like an infinite set (as in Exa mple 7.9.3.1) , or is locally modular and not t rivial, and t hen resembl es a module (as in Example 7.9.3.2), or looks like a n algebraically closed field , becau se it interpret s s uch a field (Example 7.9. 3.3). Hence t he conj ecture would pro vide a qui te sat isfactory classification of strongly minim al struct ures (and t heories) up to biinterpret abili ty. Bu t in 1993 Hru shovski showed t ha t Zilber 's Conjecture is false.

7.9. BIINTERPRETABILITY AND ZILBER CONJECTURE

285

Theorem 7 .9.7 (Hrushovski) Th ere do exist strongly minimal structures A which are not trivial but cannot int erpret any infinite group.

Clearly such a structure A is not locally modular and does not interpret any infinite field. However Zilber Conjecture (more precisely, a suitable restatem ent) do es hold in certain topological structures deeply related to st rongly minimal mod els: the so called Zariski geometries . To introduce them, let us come back to Example 7.9.3.3 , so dealing with algebraically closed fields A. We know that , for every positive integer n, the algebraic varieties of A n a re preserved und er finite union and arbit rary intersection , and form the closed sets in the Zariski topology on A n. These topologies a re Noeth eri an: none of them admits a ny infinite strictly decre asing sequence of closed sets. Moreover they sat isfy th e following properties (m and n denote below positive integer s) .

Ut, ... , f m) be a function from A n in A m . Assume that eac h component f i (1 ~ i ~ m), as a function from An in A , either projects A n onto A or is constant . Then f is continuous.

(Zl ) Let f =

( Z2 ) Every set {iI EA : a; = aj} with 1

~

i, j ~ n is closed.

(Z3) The projection of a closed set of A n+! onto A n is a constructible set in A n. (Z4) A , as a closed set, is irr educible.

(Z5) Let X be a closed irreducible subset of A n. For every ii E An-I , let X (iI) denote th e set of the elements b E A such that (iI, b) EX . T hen there is a natural N such t hat, for every ii E An -I, either IX (iI) I ~ N or XCiI) = A. In particular, when n = 1, every closed proper subset of A must be finite. (Z6 ) Let X be a closed irreducible subset of A n, d denote the (topological) dimension of X. Then, for every i, j a mong 1, . . . , n, X n {iI E A n : ai = aj} has dimension 2 d - 1. (Zl) - (Z6) restate in a topological style some properties which are well know n, or easy to check. For inst a nce (Z5) follows directly from some simple algebraic facts and the st rong minimality of A by a compactness argument: the reader may check this in detail as an exercise. Now it is easy to realize that even infinite sets a nd vectorspaces over a countable field satisfy (Zl) - (Z6) provided one takes as closed subsets the finite Boolea n com binations of t he following sets:

286

CHAPTER 7. CLASSIFYING

• the sets of the tuples ad mit t ing a fixed coordinate in a given plac e, or equal coordinates in two different places, when dealing with pur e infinite sets; • the cosets of pp-definabl e subgroups when dealing with vectorspaces. It is easy to control t hat in both cases t hese sets ar e actually t he closed sets in a su it a ble topology.

D efinition 7.9.8 A Za riski st ructure (or geometry ) is a collection (A, {Tn : n positive integer}) where A is a non-empty se t, for eve ry n T n is a No eth erian topology on A n and (Zl) - (Z6) hold. Hence the examples 7.9 .3 produce Zariski structures. Conversely, let (A, {Tn n positive integer} ) be any Zari ski structure. Assum e t hat A is t he domain of some st ruct ure A (in a language L) such that, for every positive integer n , the subsets of A n defin abl e in A are j ust the finit e Boolean combina t ions of closed sets in T'; (and so coincide with th e construct ible sets in T n ) . Then it is easy to check that A is st rongly minimal; mor eover t he possible triviality, or local modulari ty of A do es not depend on L , or on t he L-structure of A , bu t only relies upon th e charac terizat ion of th e definabl e sets of A and so, afte r all, upon (Zl ) - (Z6) . More notably, in t he restricted fram ework of Zari ski struct ures, t he Zilber Trichotomy Conjecture is t rue, as shown by Hru sho ski and Zilber himself.

T heor em 7.9.9 (Hr ushovski-Zilber) Let (A, {T n : n posit ive intege r}) be a Zarisk i struct ure, A be a strongly m inimal st ru cture with domain A such that, for eve ry posit ive in teger n , th e su bsets of A n definable in A are ju st th e cons iructible sets in T n . If A is not locally modular, th en A interprets an algebraically clos ed fi eld J( , and J( is unique up to definable isomorphism. Moreover, for eve ry positive integer n , th e subsets of K " definable in A coinc ide with the on es definable in J(.

7.10

Two algebraic examples

Let us s ummarize briefly some of the main notions introdu ced in t his chapter by examining t wo relevant classes of algebraic struct ures, and their first order th eori es: differentially closed fields of characteristic 0, and existent ially closed fields with an automorphism (again in charact eristic 0). Both these examples play an important role in the mod el t heoretic solution of some not abl e qu estions of Algebraic Geom et ry: we will describe t hese probl ems and t heir solut ion in t he next cha pter.

:

7.10. T WO A LGEBRA IC EXA M PLES

287

1. D C Fa . F irst let us deal wit h differenti ally closed fields of characterist ic O. Let us recall once again t hat t heir t heory D C Fo is complete and quantifier elimina ble in its natur al lan gu age L , containing the sy m0, 1, D and nothing mor e. So definabl e sets are easy t o bols classify: as we saw in Chapter 3, t hey includ e t he zero sets of (finite) systems of differenti al polyn omials - in oth er words, t he closed sets in t he Kolchin t opology - as well as t heir finite Boolean combinat ions t he construct ible sets in t his topology -, bu t nothin g else. As a ty pica l Kolchin closed set in a different ially closed field (K, D ) let us mention t he field of const an t s C (K ) = {a E K : D a = O}. T his is an algebraically closed field - ju st as K -, and is strongly minim al even in L; in fact , D is identically 0 on C (K ) a nd so adds no fur ther definabl e objects to th e field st ruct ure on C (K ).

+, " - ,

DC Fa eliminates the imagin aries, t oo. Mo reove r DC Fa is w-st able with Morley rank w, so ind ependence makes sense in D C Fa , and ind eed it is ruled by Morley rank: for ii , A and B as in Section 7.2,

s +A B

{:::=;>

R M (tp (il/ B )) = R M (tp (il/A )).

Of course t his raises t he question of characterizing algebraic ally Mo rley ra nk wit hin differenti ally closed fields of cha racterist ic O. Bu t t here are also oth er ways of describing forking an d inde pendence in DC Fa, having a pret ty algeb ra ic fl avour. For inst an ce, one can pr elimin arily obser ve t hat , for every small A , acl (A) - in t he model t heo ret ic sense - is ju st t he (field t heo retic) algebra ic closure of t he different ial subfield generated by A Q(Dia : a E A , i EN); at t his point , B ju st means one ca n realize t hat, for ii, A and B as usu al , ii t hat acl(AUil) and acl(B) are (algebraically) ind epend ent over acl( A). Among other things, this cha racterization sugges ts a n alternative rank notion, specifically concerning th e differential fr am ework: this is called differential degree or D-degree and denoted D-dg: for H a differential field a nd ii a tuple of elements in n, D-dg(il/H) is th e t ranscende nce degre of the differenti al field generated by H U ii over H . So, for ii, A an d B as before, and A and B differential subfields for simplicity,

+A

il

+AB

{:::=;>

D-dg(il/B ) = D-dg (il/A ).

However we have t o be ca reful here: t he last equivalence does not mean t hat R M a nd D -dg coincide . Their relationshi p, an algebra ic cha racterization of R M in D C Fa and t he con nection amo ng differenti al

288

CHAPTER 7. CLASSIFYING

degr ee, Morley rank and other possib le ranks in D C Fo a re described in the refer ences ment ioned at the end of th e chapte r. Now let us deal with biinterpret ability, in particula r let us consider st ro ngly minimal sets in differ entially closed fields IC of charact eristic O. They include the cons tant s ubfield C (IC ) (which also has different ial degree 1, as it is easy to check). C (IC ) is not locally modular, in fact the a rgume nt prop osed in t he last sect ion for algebra ically closed fields applies to C (IC ) as well. But what is most remarkabl e in this sett ing is a theorem of Hru shovski and Sokolovic sayi ng that Zilber Trichot omy Conjecture holds withi n strongly minimal sets in DCFo. In fact all t hese sets are Zariski structures, and so obey the Hru shov skiZilber Th eor em . We can say even more: any strongly minimal set S which is not locally modular, and henc e interprets an infinite field, doe s interpret t he field of constants C(IC) up to a definable isomorphism . We will provide more details about t hese mat te rs in Section 8.7 .

2. AC F A. Now we deal with existe nt ially closed fields with a n autom orphism . For simplicit y, we still work in cha racterist ic o. Let AD F A o denote the corresponding theory in the natural language L = {+ , ., - , 0, 1, O" } where 0" is the sy mbol representing the a utomorphism . Recall that, t his ti me , fixing t he characteristic is not sufficient to ensure com pleteness: in order to characterize a mod el of AC FAo up to element a ry equivalence, one has also to describe the act ion of t he a ut om orphism on th e prime subfield Q. Moreover ACF A o does not eliminate t he qu an tifiers in L, alt hough it is obviously model complete (as a mod el companion) . Accordingly definabl e sets exhibit some more complications than in the differential case. In fact , t hey include the zero sets of (finite) syst ems of difference polynomials , as well as their finite Boolean combinat ions; the former are the closed set s, a nd t he lat ter t he constructible ones in a suitable topology. But now , as qu an tifier eliminat ion fails, we have to conside r also the projections of const ruct ible sets - a nd nothing else, owing to model completeness - to ca pt ure the whole class of definabl e sets. An exam ple of a closed set in a mod el (IC , 0") of AC F A o is it s fixed su bfield Fix (O" ) = {a E I< : er(a) = a}. This is not algebraically closed (in particular it is not st rongly minimal); but one can see t hat it is a pseudofinite field , so an infinite model of the theory of finite fields. ACF A o elimi nat es the imaginaries. T his time no existentially closed field (IC, 0") with an au t omor phism is w-st able , or even st able. How-

289

7.11. REFERENCES

ever (K, a) is sim ple (as well as its fixed su bfield , a nd a ny pseudofinite field) . So ind ep end ence makes sense in ACFAa, and comes directl y from forkin g, bu t ca nnot be ruled by Morley ran k. Any how an explicit algebraic characterization ca n be don e as follows. We work for simplicity in a big saturated mod el (K , a) of AC F Aa. F irst one obser ves t hat, for every small A, acl(A) coincides wit h t he algebra ic closure in t he field t heoret ic sense - of t he differ ence subfield generated by A Q (a i (a ) : a E A, i E Z )

(here we use t he cha racterist ic 0 assu mp t ion; prim e cha racterist ics cau se some major trouble). At thi s point one shows th at, for ii, A a nd B as usu al , ii t A B ju st means that acl(A U il) and acl(B) are (algebraically) ind epend ent over acl (A). This yields an appropriate notion of rank , of a pret t y algebraic flavour , called difference degree or a -d egree and deno ted o-dq: for H a difference fi eld a nd ii a t uple of elements in n, a -dg (il/ H ) is t he t ra nsce nde nce degree of t he difference fi eld generated by H U ii over H . When finite, t he difference degree can reason abl y repl ace Mo rley rank and provides a good notion of dimension in t his unstable setting; on t he oth er side , when a-dg (il/ H ) is infinite, t hen clearly t he a i (a) 's (when i ra nges over integers) are algeb raic ally inde pe ndent over H. So, for d, A an d B as before, a nd A and B difference s ubfields for simp licity, ii

tA B

{:::::::?

a - dg (il/B ) = a - dg (il/A )

at least when t he la t ter degrees are finite. Wh en X is a ny definabl e set in f{ n , the difference degree of X over H a - dg (X/H) is t he maximal difference degree of a t uple ii in X over H . In particul ar the fixed subfield of K get s difference degr ee 1. In this sense, Fi x (a) is a "minimal" definabl e infinite set of K. Notably, an adapte d version of Zilber Trichotomy Conj ecture holds for these "minimal" sets in ACFAa, a nd even ensur es in part icular t hat, very roughly speaking, Fix (a) is t he only non "locally modul ar" exam ple among t hese structures.

7.11

References

T he classification issue fro m t he point of view of Descripti ve Set T heory is treated in [68]; t he par ticul ar and int riguin g case of torsionfree a belian groups of finit e rank is dealt wit h in [163]. Finite dim ension al vect orspaces

290

CHAPTER 7. CLASSIFYING

wit h two distinguished endomorphisms over a fixed field, and the wildness of t heir classification problem are described in [136] or, more specifically, in [137]. The mai n references on Shelah's classificat ion theory are just t he Shelah book [149] and its revised and up dated version [151]. Most of t he topics of this chapter are treated there in detail. Another good an d per haps mor e accessible source on classification and stability t heory is [8]. See also Makkai's paper [101] . Vaught's Conjecture is proposed in [174], and its solution in the w-stable fram ework ca n be foun d in [152]. Lascar's pap er [84] provides an enjoyable disc ussion of t his matter. The (uncountable) spectrum problem for complet e countable first order theor ies is fully solved in [53]. Simple theor ies were introduced in [149], but it was Kim who showed, together with Pill ay, t heir relevance within the classification program: see [71] and [73]. Kim again observed the key role of symmetry, transitivity a nd local character [72]. Wagner's book [175] provides a general a nd exhaustive report on this theme. As already said , stability, superst ability and the further dichotomies ari sing in t he classification program are treated in Shelah books [149, 151], in [8] or also in [101]. [85] provides a nice and terse introd uction to stability, using and emphasizing t he notion of heir and coheir. [83] pursues t his approach and deals in particular wit h Rudin-Keisler order and strong regu larity. The effectiveness aspects of Shelah 's classification progr am is discussed in [114], while [2] examines its connect ions with Stationary Logic. Turning our attention to t he algebraic examples, let us mention [136] or also [56] for modules, a nd [19] for pseudofinite fields . [110] treats differentially closed and separably closed fields, and includes a wide bibliographical list on them . [62] a nd [155] provide ot her key references on DCFo; see also [134]. Existentially closed fields wit h an automorphism are j ust t he su bject of [20]. An explicit exa mple of such a field can be found in [58] and [88]; see also [95]. Shelah's uniqueness theorem is in [148], Morley's theorem in [117]. Anot her proof of Morley 's Theorem is given in [9]; see also Sack's book [146], or [57]. T he Ehrenfeucht - Mostowski models quoted in Sect ion 7.8 are int roduced in [38]. Finally, let us deal with biinterpretability. Malcev 's correspondence is in [102] while Mekler's th eorem on nilpotent groups of class 2 is in [113]. Zilber 's program is developed in [182], where Zilber's Conjecture is also proposed . The negative solution of this conjecture is in [59], and the HrushovskiZilber theorem on Zar iski structures in [64].

Chapter 8

Model Theory and Algebraic Geometry 8 .1

Int roduction

We have often emphasized in t he pas t chapters t he deep relati onship bet ween Model Theor y and Algebra ic Geometry : we have see n, a nd we are going t o see also in t his cha pter t hat seve ra l relevan t notions a rising in Algebra ic Geom etry (like variet ies, morphisms, manifolds, algebraic gro ups over a field K ) are definabl e object s a nd are conseq uent ly concerne d wit h t he mod el t heor etic machinery develop ed in t he previous pages. For inst an ce, when K is algebraicall y closed , t hey a re w-stable st ruct ures.T his connection ca n yield, a nd is act ually yielding, significa nt frui ts in both Mod el T heory and Algebr aic Geom etry. On th e one hand , seve ral t echniques a nd ideas originated a nd employed within t he specific sett ing of Algebraic Geome t ry ca n inspire a more abstract model th eor etic treatment , applying to a rbit ra ry classes of st ruct ures. In this sense Algebraic Geometry over algebra ically closed fields can suggest new directions in th e study of w-stability: we will describe this connection in many sect ions of t his chapter. However a parallel analysis ca n be developed inside other relevant areas, like differentia lly closed fi elds (and Differential Algebraic Geom etry) , or existe nt ially closed fields with a n a ut omor phism , and so on . On t he oth er hand , it is right t o observe that t he benefits of t his relationship regard not only Model Theor y, bu t also, and releva nt ly, Algebraic Geomet ry. In particular, we will propose some prominent problems in Algebra ic Geometry, whos e solution do es profit by Model Theor y and its t echniques. This will be the aim of t he final section of this cha pte r.

291

292 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY

8.2

Algebraic varieties, ideals, types

Let J( be a field , n be a positive integer. We already introduced in th e past chapters the (algebraic) varieties of K" : They a re t he zero sets of finite systems of polynomials of J([X] (where x abbreviates, as usual, (Xl , .. . , x n)), a nd so are definabl e in J(. Moreover they form the closed sets in the Zari ski t opology of K"; accordingly even the Zariski open or const ructible sets ar e defina ble in J( . But the varieties of K" are also closely related to the ideal s of J([X]. In fact one can define a function I from vari eties to ideals mapping a ny variety V of K" into the ideal I(V) of t he polynomials f(x) in J([X] such that f(a) = 0 for every a in V . Checking that I(V) is ind eed a n ideal is straight for war d; I(V) is even a radical ideal , in other words it coincides with its radical rad I (V ): if f(x) E [([X] and , for some positive int eger k, fk( x) E I(V) , t hen f (x) already occurs in I (V). In particular I is not onto. Bu t there is also a nother function V in the other direction , from ideals to varieties, mapping any ideal 1 of J([X] (in pa rt icular a ny ra dical ideal) into t he set V(I) of t hose elements a E K" a nnihilating all the polynomials of I

V (1) = {a E J(n : f (a)

= 0 vf (x)

El}.

Due to t he Hilbert Basis T heorem, I is finit ely generated , and so V(I) is a variety. Ind eed , any vari ety V can be ob t ained in this way by definition; in oth er word s V is onto. Not ice also t ha t V(I) = V(rad I) for every ideal I . The definition of I an d V trivially impli es t ha t, for every ideal I of J([X] , I(V(1)) "2 I. As I(V(I)) is a radical ideal , I(V(I)) "2 rad T, Hilbert 's Nullstellensatz (see Ch apt er 3) ensur es that , when J( is an algebraically closed field, eq uality hold s: fo r every ideal I of J([X] ,

I(V(1)) = rad I , It is easy t o deduce that , if J( is algebraically closed , t hen I a nd V deter mine t wo bijections, the one inverse of t he other, betw een varieties of K" and radical ideals of J([X] . We will st ill den ote these restricted bijections by I , V respectively. Notice t hat both reverse inclusion : for instan ce if V , W a re two varieties of K" , then

I(V) "2 I(W)

~

V

~

W.

We assume from now on that J( is a n algebraically closed field. By t he way recall that, under t his condit ion , what is definable in J( is w-stable of finite Morley rank , becau se J( is st rongly minimal.

8.2. ALGEBRAIC VARIETIES, IDEALS, TYPES

293

Let us restrict our at tention from radical ideal s of K[X] t o prime ideals (t hose ideals I in K[X] such that , if a product of two polynomials of K[XJ lies in I , th en at least one factor polynomial is in I as well) . Parallely we con sider, among vari eties of K" , the irreducible ones , so t he non-empty vari eties V t hat ca nnot decompose as a union of two proper subvariet ies . It is easy to check that the previous bijections I and V betwe en vari etie s of K" and rad ical ideals of K[X] link irreducible varieties of K" and prime ideals of K[XJ: for every vari ety V of K"; V is irreducible if and only if I(V) is prime. Notice also that I(0) = K[XJ . A closer relationship links irreducible varieties and prime ideals. For inst a nce, it is known that every non-empty variety V of K " ca n be expressed as a finite irredundant union of irreducible varieties, and that this decomposit ion is unique up to permuting the involved irreducible varieties (which ar e con sequ ently called the irreducible components of V) ; th e irredundancy of the decomposit ion just means that no irreducible component of V is included in t he union of the other components. Sp ecularly, every proper radi cal ideal I of K[XJ can be expressed as a finit e int ersection of prime ideals minimal with resp ect t o inclusion ; even t his representation is uniqu e up to permuting the involved minim al prime ideals. On e ca n also realiz e th at , und er t his poi nt of view , for every non- empty vari ety V in K " , t he irredu cible components of V correspond to t he minimal prime ideals occurring in th e decomposit ion of I (V ). So far we have summa rized -some very famili ar t opics of basic Algebraic Geom etry. Now let Model Theor y intervene. As we saw in Section 5.3, prime ideals of K[XJ - a nd hence, through t hem , irreducible variet ies of K" - directly and na turally corres pond to n-types over I< . In fact , for a n algebraically closed field K , th er e a re t wo bijections i and p , one inverse of the other, between n-types over I< and prime ideals of K[XJ. Basically, for every n-type p over I<,

i(p) is the ideal of the polynomials f(x)

E

I<[XJ such that the formula

f(11) = 0 is in p,

a nd, conversely, for every prime ideal I of K[XJ ,

{J(11) = 0 : f( x) E I} U {.(g(iJ) = 0) : g( x) E I<[XJ - I} enl arges t o a unique n-typ e p(I) over I<. Accordingly, for every irred ucible variety V of K " and polynomial f (x) E I<[ xl, f (ii) = 0, Vii E V {::} f (x) E I(V) {::} " f(11) = 0" E p (I (V) );

294 CHAPTER 8. MODEL THEORY A ND ALGEBRAIC GEOME Tny p(I(V)) is called a generic type of the variety V . More gen erally a generic type of an arbit ra ry non- empty variety V of K" is a ge ne ric type of an irr educible compon ent of V . T his model t heoretic notion of generic type just corresponds to the idea of a ge neric poin t of a n irreducible variety V int roduced in Algebraic Geometry: the latter is a po int just a nnihilat ing t he polynomials in I(V) a nd not hing more , and so can be eq uivalent ly defined as a realizati on of the generic type p(I(V)). It can be obtained as follows. As V is irreducible, I( V) is prime an d conseq uent ly t he quotient ring /C[X] /I(V) is an integral domain ; /C[X]/I( V) contains a subring {k + I(V) : k E I<} isomorphic to /C and a t uple + I( V) a nn ihilat ing just the polynomials of I(V) ; so t he field of fractions /C(V) of /C[X]/I(V) extends /C - up to isomorphism - a nd includes a generic point a(V) = x + I(V) of V . To conclude this section let us observe what follows.

x

Proposition 8.2 .1 Let n be a positive integer, /C be an algebraically closed fie ld, V be an irreducible variety of K" , 0 be a Za riski open set of K " satisf ying V nO ::j:. 0. T hen th e generic typ e p of V con tains th e formula "v EO".

Proof. In fact V - 0 is a variety properly included in V ; conse quently I( V - 0) =:> I(V). If p does not con t ain "v E 0 " , then p has t o include "v E V - 0 " , a nd hen ce ever y formula "f (v) = 0" when f (x) ranges over I(V -0) ; accor dingly, for some polynomial f( x) (j. J(V) , "J(v) = 0" belo ngs to p: a contrad iction . .,.

8.3

Dimension and Morley rank

We main t ain throughou t t his section our assumption t hat /C is a n algebr aically closed field . Let V be an irreducible variet y of K" : Algebraic Geometry equ ips V with a dimension in the followig way. As we saw in the last section , there is a minimal field /C (V) extend ing /C by a realization a(V) of the generic type of V , in other words by a generic point of V . The dim ension of V (dim( V)) is just t he transcendence deg ree of /C CV) over /C . T his m ak es sense for a n irred ucible V , bu t can be eas ily extended t o an y non-em pty variet y V of K", In this en larged setting the dim ension of V (dim(V)) is t he maximal di mension of a n irr educible component of V. F inally, t he dim ension of a con structible no n-empty set X dim (X) is the dime nsion of t he closure of X in t he Zariski t opology (a non-em pt y variety) .

295

8.3. DIMENSION AND MORLEY RANK

On the other side every vari ety V of K" (irreducible or not) and, more generally, every constructible set X ~ K " is definable in K; K is algebraically closed, hence w-stable; accordingly V and X have th eir Morley rank . We want t o compare the dimension and the Morley rank of V , or X , and to show that they coincide. We start by examining an irreducible var iety V. Lemma 8 .3.1 Let V be an irreducible variety of K" , p be its generic type.

Then dim (V) = RM(p) .

Proof.

p is realized by a(V) in K (V) . So the MorIey rank of p equals the transcendence degree of K(V) over K (see Section 6.1), and hence the dim ension of V.

•

Lemma 8.3.2 Let V , W be two irreducible varieties of

their generic types. If V C W , then RM(p) > RM(q).

x»,

p, q denote

Proof. V C W impl ies I(W) "2 I (V) and hence (*) for every f(x) E K[X] , if f (a(V))

= 0, t hen

f(a(W))

= 0 as well.

In particular t he t ranscende nce degree of K(V ) over K is not smaller t ha n that of K (W ) over K , and so RM (p) 2: RM(q) . Now ass ume RM(p) = RM(q), t hen K (V) , K(W) have the same transcend ence degr ee over K. Recall t hat K (V) = K (a(V )), and similarly for W . (*) impli es that , if i }, .. . , i m ~ n and ail (W ), , aim(W) form a t ransce ndence basis of K (W ) over K , t hen ail (V ), , aim(V) are algebraically independent over K and so form a transcendence basis of K(V) over K. Accordingly one can define an isomor phism of Kw = K (ail (W ), .. . , aim(W)) onto Kv = K(a il (V) , . .. , aim (V )) fixing K pointwise and mapping ai(W) into ai(V) for every i = i}, .. . , i m . This isomo rphism can be enlarged in t he usu al way to an isomorphism between Kw [x] and Kv [X]. By using (*) once again, one sees t hat , for every j = 1, ... , n with j =I- i} , ... , i m , the mini mum polynom ial of aj(W) over Kw must correspond to the minimum polynomial of aj(V) over Kv in this isomorphism. Acco rd ingly we obtain an isomorphism of K (W) onto K (V) fixing tc pointwise and mapping a j (W) into a j (V) for every j = 1, . . . , n. Bu t a(V) realizes p and a(W) realiz es q, when ce p = q and , in conclusion , V = W. • Now we can show t hat dim ension and Morley rank coincide for a n irreducible variety V.

296 CHA P TER 8. M ODEL THEORY A ND A L GEBRA IC GEOMETRY Theor e m 8.3.3 Let V be an irreduci ble varie ty of K", dim( V ), GM(V) = 1.

T hen RM(V ) =

Proof. Let p be t he generic ty pe of V . p is t he only n-type over /C containing t he formul a "5 E V " th at define s V, and satisfying V( i(p)) = V. If q is another n-ty pe over /C cont aining "5 E V" , t hen i(q) :J i(p), whence V (i (q)) C V (i (p)) = V. By Lemma 8.3.2, RM (q) < R M (p). T hen R M (V ) = R M (p) and GM(V) = 1. By Lem ma 8.3.1, R M (V ) = R M (p ) = dim (V ). • C o r o llary 8.3 .4 Let V be an irreducible varie ty of K " (with the relat ive topology of th e Zariski topology) . If 0 is a no n-empty open set of V , then RM (O ) = RM(V ) . IfW C V is a closed se t of V, th en RM(W) < R M (V).

Proof. T he forme r claim follows from Proposition 8.2.1 and t he fact t hat, if p is t he generic point of V , t hen R M (p) = R M (V ). At t his poi nt t he lat ter clai m is a consequ ence of t he fact t hat V has Morley degree 1. ... C oro llary 8. 3.5 Le t V be a no n- empty variet y of K", Th en R M (V) = dim(V), furth ermore G M (V ) equals the number of th e irredu cible com ponents of V having th e same dim en sion as V.

P roof. V is t he (finite) union of its irreducible components. Then t he Morley ra nk of V coincides wit h t he maximal ra nk of its com po nents. So by T heore m 8.3.3 and the definition of dim (V) R M (V ) = di m (V ). Moreover, if Vo a nd VI are two different irr edu cible components of maximal rank of V, t hen Vo n VI is a closed subset of Vo properly includ ed in Vo. By Corollary 8 .3.4, R M(VonVI) < R M(Vo). This implies t hat G M (V) equals the number ... of the irr educible components of maximal rank in V.

Dim ension and Morl ey rank coincide even for constr uct ible sets X ~ K" : Recall t hat, owing to Tarski's qu an tifier elimination Theor em, constructible just mean s defina ble in /C . Furth ermore a const ruct ible X ~ K " ca n be re presented as a union of finit ely many sets, which are in t heir t urn t he intersection of a variety W - so a Zariski closed set, t he zero set of a finite system of eq uations - a nd a Zari ski open set 0 - t he set of t he elements of K " satisfying finit ely many inequ ations

8.4. MORPHISMS A ND DEFINABLE F UNCTIONS

297

with 90(£), " " 9s(£) E K[Xj, or also, equivalent ly, a single inequa tion

II9j(£)

#0

- .

j ss

An open set defined by a unique inequation is called principal. Corollary 8.3.6 Let X ~ tc- be construct ible (hence definable). Th en RM(X) = dim (X) . Moreover, if X denotes th e Z aris ki closure of X , th en R M(X) = R M(X) and RM(X - X) < RM(X). Proof. We know dim(X) = dim(X) , and dim(X) = RM(X) (by Co rollary 8.3. 5) . So the form er claim is don e if RM(X) = RM(X). As observed befo re, X a finite union of sets of t he form W nOwher e W is a nonempty variety of JCn and 0 is op en in JC n. As a ny non- empt y variety of JCn decompo ses in its t urn as the union of its irredu cible components, we ca n assume th at every variety W occurring in t he above representation of X is irreducible. By Corolla ry 8.3.4, RM(W n 0) = RM(W) = dim(W). Con sequ ently the Morl ey rank of X (that equa ls the maxim al rank of the sets W n 0) coincides with the Morley rank of X . This proves t he former claim. Bu t , by Corollary 8.3.4 once again, RM(WnO) < RM(W) for Wand 0 as before and W irreducible. So, in conclusion, RM(X - X) < RM(X) . ...

8.4

Morphisms and definable functions

JC still denotes an algeb raically closed field. Let n , m be positive integers , V , W be two algebr aic vari et ies in JCn, JCm respectively. Algebraic Geomet ry defines what a morphism from V to W is: it is a function f from V int o W such t hat, for every i = 1, . .. , m, t he com posit ion f i of f a nd t he ith projection of K m onto K is a polynomial map. One easil y sees that a mo rphism is a cont inuous fun ction with resp ect to the to pology ind uced on V a nd W by the Za riski topology. But what is rem arkabl e for our purposes is t hat a variety morphism is always defin abl e in JC. For inst an ce, if f is, as above, a morphism from V to W , t hen "f (v) = W" is defined by the formula

"v E V"

1\ " $ E

W"

1\

1\

" f i(V) = uu",

l
Conversely what ca n we say about an arbitrary defin able fun ction f in JC? Certainly both t he domain and the image of f a re definab le sets . Furthermo re, if the image of f is a subset of K m, t hen, for every i = 1, . .. , m,

298 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY

the composit ion f i of f and t he i-th projection of K ": onto J( is definable. So we are restricted to characterize definable functions from a subset of K " into J( , for some positive integer n. In this framework one ca n observe what follows . Theorem 8.4.1 Let f be a definable fun ction from K " into K.. . Th en th ere are a non-empty open subse t 0 of K.. n, a rational fun ction r and (wh en K.. has a prime characteristic) a positive integer h suc h that • if car K.. = 0, th en f equals r on 0; • if car K.. is a prime p and Fr denotes th e Frobenius map from K.. to K.. (that mapping any a E J( into Fr(a) = a/'], th en f coincides with r -:>. r on O .

Just to underline th e power of this result, let us recall that , owing to Corollary 8.3.4, every non- empty op en subset 0 of K" has th e sa me Morley rank n as K" ; while th e Morley rank of K.. n-o is smaller (for , K" is an irreducible vari ety of rank n). Secondly, it is worth recalling th e general fact that , if f is a fun ction from a vari ety V of K " into K.. and , for every a E V , there is an op en neighbourhood 0 of a such t hat f equa ls some ration al funct ion in O n V , t hen f ca n be globa lly expressed as a polynomial fun ction . Now let us show Theor em 8.4. l. Proof. Let Q be the uni verse of the theory of K.. , t} , .. . , t« E Q be algebraically independent over K.. . As f is J(-definabl e , an y a utomorphism of n fixing K.. and t l , • • • , t n pointwise acts identically also on f(i) (f denote s here th e tuple (t}, ... , tn)). Whence j(i) is in dclt K n i) . So, if K.. has cha racterist ic 0, then f(i) = r(i)

for some suitable rational function r with coefficients in prime characteristic p , then

J(,

while, if K.. has

where r is as before and hp is a positive integer. Put s = r when car K.. = 0, s = r otherwise. The eleme nts in K " where f coincides with s form a set X definable in K.. (just as f and s) , and th e form ula "s (v) = f (v)" defining it is in th e type of f over J(. As t} , ... , t n are algebraically independent over K.. , t his t ype has Morley rank n . Hence RM(X ) 2: n . As X ~ K" ; t he Morl ey rank of X must equa l n , a nd RM (J( n - X ) < n. It follows th at t he

r -:» .

299

8.5. MA NIFOLDS Zari ski closure of K " - X has Morley ra nk an open set 0 in X of Morley rank n where

8.5

< n , and so its complement is f = s. •

Manifolds

Throughout this section IC still denotes an algebra ically closed field and n a positive integer. We deal here with (ab stract) manifolds in IC n a nd we show that th ey are defin abl e objects. F irst let us recall their definition . Definition 8 .5.1 A manifold of IC n is a structure V = (V, (Vi, f i) i5: m ) where m is a natural and • V is a subset of K " (called atlas); • V o, ... , V m are subsets of V , and V is th e union Ui<m Vi ; • fo r every i ~ m , f i is a bijection of Vi onto a Zariski closed set Vi (a coordinate chart of th e atlas V) ; • for i ,j ~ m and i

i- i , f i(Vi n Vj)

=

o.,

is an open sub set of Vi;

• for i, j ~ m and i i- i . f ij = fi . f T 1 (a biject ion between Vji and V ij ) can be locally exp ressed as a tu ple of rat ional fu n ct ions.

Manifolds include several familiar exa mples. Examples 8.5.2 1. Every algebraic vari ety (so every Zari ski closed set ) V of IC n is a manifold , provid ed we set m = 0 a nd choose Ui, = Vo = V as the only coordinat e cha rt of t he atlas; the res ulting man ifold is called affine . 2. Let 0 be an op en principal set of IC n; 0 is defined by a single inequ at ion -' ''g(v) = 0" ; notice that the formula "g(v) . V n +l = 1" defines a closed V in IC n+l , a nd it is easy to control t hat the projection of K n +l onto K " by t he first n coordina te s determines a bijection f of V onto O . Accordingly (V , (V , f -l )) is a manifold with t he only chart O . Such a manifold is called sem ia ffine . 3. Also the projective space p n(IC) can be equipped with a manifold structure. In fact, view p n(IC) as t he quo tient set of K n+l - {a} with resp ect to the equivalence relation rv linking two non zero tu ples x = (xo , Xl, . . . , x n) a nd fl = (Yo, YI , . .. , Yn) in K n+l if and only

300 CHAPTER 8. MODEL T HEORY AND ALGEBRAIC GEOMETRY

if t here is some k E J( such that Xi = kYi for every i ~ n. For x E J( n+l - {O} , let (xo : X l : •• . : x n ) be t he class of x with resp ect t o t his relation . Moreover , for i ~ n , let

• A i de note th e set of th e elements (x o : such that Xi i= 0,

Xl : • • • :

x n ) in p n(!C )

• f i be the function of A i int o K " mapping any (xo : X l

.In (3Zll. Xi '

Xi _l

· · · , ~,

5.±.!. xi

~)

, •• . , xi

: ... :

xn )

•

It is st ra ight forward to check that (pn(!C), (A i, Ji)i5:. n ) is a manifold .

Notice t hat affine and semiaffine var iet ies are definable -even as structuresin /C . Moreover p n(!C ) is interpretable in !C both as a set (since t he relation rv is 0-definable) and as a manifo ld . But a lgebraically closed fields uniformly eliminate th e imaginaries, so we can view p n(!C) even as a definable st ructur e in !C. Mor e gen er ally one can show T heo rem 8.5.3 Let V = (V, (Vi , Ji) i5:. m ) be a manifold of a structure defina ble in !C.

«» .

T he n V is

Proof. As algebraically closed fields have t he uniform eliminat ion of imaginari es , it is sufficient t o show t hat V is a struct ure interpretable in !C . In fact, every map U, of t he atlas V (i ~ m) is definable in !C . V ca n be rega rded as the quotient set of t he disjoint union of t he cha rts U, (wit h i ~ m) with respect to the equivalence relat ion ident ifying Uij and Uji via f ij for every i < j ~ m; moreover , for every i ~ m , Vi can be definably recovered as t he image of U, by th e projection into the quotient set V , and f i is given by the invers e funct ion of this projection (restrict ed to Ui) . So our claim is proved if we show t hat, for every i, j ~ m with i i= i , f ij is definable. But f ij can be locally expressed as a rat ional function, and its domain Uij is an open subset of U, and accordingly can be written as a finit e union of principal open sets. So th e th eor em is a direct con sequence of the next resu lt.

Lemma 8.5.4 Let 0 be a prin cipal open of !C n, and let q(x ) E [([X] be a polynomial satisfying 0 = {ii E K " : q(ii) i= O}. Let f be a funct ion of 0 in to K" which can be locally expressed as a ratio nal fun ct ion. T hen there are a polynomial r( x) E [([X] and a positive integer m such that f(ii) = r(ii)/qm(ii) for eve ry ii EO . In particular f is defi nabl e.

8.6. ALGEBRAIC GROUPS

301

Proof. We know that 0 is canonically homeomorphic to the closed subset V of K n+l defin ed by q(v) . Vn+l = 1. Under this per spective f can be replaced by the functi on 1* of V into 1< mapping any tuple (ii, an+d E V in f(ii) . Eve n 1* ca n be locally expressed as a rational function , and hence as a polynomial function in (x, xn+d (by th e general fact we recalled before the proof of Theorem 8.4.1). So th ere is some polynomial s(x , xn+I} E 1<[x, xn+l ] such t hat

J*(ii, (q(ii))-l) = s(ii, (q(ii))- l)

vs En.

Let m deno te t he degree of s(x, Xn+l ) with resp ect t o Xn+l' T hen there is some polynomial r (x) E 1<[X] such that , for every ii EO ,

f(ii)

= J*(ii, (q(ii))-l) = qm r (a(-a~) .

..

Notice th at a manifold , when regarded as a definable structur e, may lose part of it s geometric features. For instan ce the Za riski closed subset of K 2 defined by Xl . (X2 - 1) = X2 . (X2 - 1) = 0 is an affine manifold , form ed by the line X2 = 1 and t he poin t (0,0) , a nd hence is th e disjoint union of tw o closed sets. However , consider the manifold given by t he proj ective line pl(K) (as seen in Exam ple 8.5.2, 3). Fro m the definable point of view, its atl as has two charts, each of them is a line and th ese lines coincide excep t a single point . So the resulting manifold is again the union of a line {(1 : x I) : X l E 1<} and a poin t (0 : 1), a nd as a definabl e object is qui te similar to the previous one. Bu t p l (K) is not the union of two distin ct closed sets. On t he oth er side, it is noteworthy t hat every defin able subset X ~ K " can be easily equipped with a manifold st ruct ure. In fact X decomposes as a union Ui<m (Vi n Oi) where m is a natural a nd, for every i ::; m , Vi is a Za riski closed of K" and O, is a pr incipa l open set , so th at Vi n O, is canonica lly hom eomorphic to a closed U, of K n+l , as observ ed before. On this basis, it is easy t o build a manifold structure on X (with Uo, ... , Um as atl as maps) . Furt her more a man ifold V , viewed as a definable structure, is w-stable.

8.6

Algebraic groups

A basic example of algebr aic group over a field K is the linear group of degree n over K GL(n, K) , where n is a positive integer. Observe:

302 CHAPTER 8. MODEL THEORY AND ALGEBRA IC GE OMETRY 2

• G L(n , /C) is a prin cipal op en of /C n , because it is defined by the diseq uat ion -, (det ( if) = 0) ; hence G L(n , /C) is canonically homeomorphic 2 t o t he Zariski closed of /C n +1 given by t he equat ion det( if) . vn2+ 1 = 1;

• t he product op er ation· in GL (n , /C) is a morphism of variet ies and so is definable. Accordingly the group G L (n, /C) is a st ructure definable in /C . Moreover , if we ass ume /C algebraically closed , GL(n, /C) is an w-st a ble group of finit e Morley rank . Also linea r algebraic groups are examples of algebraic groups. Recall t hat a linear algebraic group over a field /C is a closed subgroup 9 of some linea r group G L (n , /C ). In particular a linear algebraic group 9 is a variety over /C, hence is defina ble (as a grou p) in /C a nd is w-stable of finite Morley rank when /C is alg ebraically closed . Under t he last ass um ption, we can say even more: ind eed , for an algebraic ally closed /C , t he linear algebra ic gro ups are ju st t hose subgro ups of t he linear groups GL(n , /C ) which a re definable in /C . Let us see why. Theorem 8 .6 .1 Let /C be an algebraically closed fi eld, n be a posit ive integer, 9 be a subgroup of G L( n, /C ). Th en 9 is closed if and only if 9 is defi nable (in /C) .

Proof. Clea rly, if 9 is closed -in other word s 9 is a variety- , then 9 is definabl e. Con versely suppose t hat 9 is a definabl e group; let C be t he closure of G with resp ect to t he Zari ski topology, and let a be an elemen t of 2 C. Every open set of /C n containing a overlaps G. Conseq uent ly, for bE G , every open 0 including ba overlaps G; in fact , if ba E 0 , then a E b-10 ; b-10 is open because th e left multiplication by b is a morphism of varieties, and so is cont inuous; whence (b- 10) n G i= 0 and so 0 n bG i= 0; as bEG, bG is ju st G, hence 0 n G i= 0. In conclu sion , for every a E C, Ga < C. If th ere is any a E C - G, then Ga ~ C is disjoint from G and has the same Morl ey rank as G . So RM (G - G) ~ RM(G) , which contradicts Corollary 8.3.6. Hence C ~ G and G is closed . ..

As alread y said, linear groups a nd linear algebraic gro ups exe mplify algebraic gro ups. In fact an algebraic group over a field /C is defined as a manifold over /C ca rrying a group str uct ure whose product a nd inverse operations a re (manifold) morphism s . Of course, understanding t his definition preliminarily requires to stat e what a prod uct of t wo manifolds is, a nd t o realize th at t his produ ct is a manifold

8.6. ALGEBRAIC GRO UPS

303

as well; mo reover we should specify what a manifold mo rph ism is. We omit here the det ails. The former point is comparat ively simple and natural to cla rify, while the concept of morphism is more complex t o introduce; but , once this is done, one ca n easily show th at manifold morphism s a re defin able (as it is reason abl e to ex pect) . Hence, if K is algebraically closed , t hen every algebra ic group g over K is a st r ucture defin able in K an d so w-stable of finite Morley rank. As already said, linear algebraic groups are algebraic gr oups; in fact they just correspond to affine (or also semiaffine) manifolds. And indeed Theorem 8. 6.1 ca n be regarded as a parti cular cas e of a general result ensuring that every group definabl e in an algebraically closed field K is an algebraic group over K up t o definabl e isomorphism. This fact was shown by Hrushovski (and Van den Dries) , who observ ed t hat it is impli citl y contained in some results of A. Weil. So it is commonly quoted as the Hrushovski-Weil Theorem. Theorem 8 .6 .2 (Hrushovski-Weil) Let K be an algebraically closed field and g be a group definable in K. Th en g is isomorphic to an algebrai c group over K by a fun ction definable in K

Let us spe nd some more words about the connection between algebraic groups a nd w-st a ble groups. We have seen that every algebraic group over a n algeb raically closed field K is w-st abl e of finit e Morley ra nk . Of course w-st able groups include furt her relevant examp les: for ist an ce, a ny divisible torsionfree abelian group - so basically an y vectorsp ace over Q - is w-stable, a nd even stro ngly min imal. More generally, it was shown by Angus Macintyre that the w-stable (pure) abelian groups are just the direct sums of a divisible abelia n group and a n abelian group of finite exponent . However the techniques used in t he investiga t ion of a belia n groups do not seem appropriat e to handle w-stable groups. On the cont ra ry, algebraic groups and t heir t heory fit very well for w-stable groups. This is not surprising. In fact , even neglecting the sim ila rit ies we emphasized in t he last sect ions between va rieties an d definabl e sets, or dimension and Morl ey rank , and so on , we can recall t ha t several notions introduced in Chapter 6 for st udy ing w-st abl e groups clearly come from Algebraic Geometry. In this sett ing , it is worth me nt ioning t he following conjecture, proposed by Ch erlin in 1979. Conjecture 8 .6 .3 (Cherlin) Let g be an w- slable group of finit e Morley rank. If g is si mple, then g is an algebraic group over an algebraically clos ed field.

304 CHA PTER 8. M ODEL T HE ORY AND A LGEBRA IC GEOMETRY

Ch erlin 's Conje cture is still an op en qu esti on. We should also point ou t t hat some remarka ble pro gress in st udying w-stable groups of finit e Morley rank has been obtained by using ideas and tec hniques coming from finite groups, in particular from t he classification program of finite simpl e groups.

8.7

The Mordell-Lang Conjecture

Every promise must be honoured . And consequently, after em phasizing so man y t imes t hat Mo del Theor y does significant ly a pply t o Alge bra ic Geo met ry, here we are t o propose one a pplication (indeed a beau tiful and deep application, in our opinion ): Hru shov ski 's proof of a qu estion of Lan g usually called Mordell-Lan g Conjecture. Wh y is this solut ion notewor thy? Basically because it is t he very first proo f of t his conject ure, at least in t he genera l for m we will state in 8.7.6 below; bu t also, an d mainly, becau se it la rgely involves Model Theor y (strongly min imal sets, Zari ski str uct ures , differentially closed a nd separably closed fields , as well as the materi al of t his chapter). So let us introd uce t he Mordell-Lang Conject ure, and briefly sket ch its hist ory. We assume some acqua intance with Algeb raic Geometry. T he qu estion originally rose within Diophantine Geo met ry, which deals wit h the roo t s of sys te ms of polynomials over th e ra tion al field Q or also over a number field F (that is a n extension F of Q of finite degree). T his was t he setting where Mordell ra ised in 1922 t he following problem. Conjecture 8.7.1 (Mo rdell) Let F be a number field, X be a curve of genus > l over F. T hen X has only finit ely m any F -rationals points. Incident ally recall t hat a curve X of genus 1 is an ellipt ic curve, a nd so is naturally equipped with a gr oup structure . In a mor e abstract persp ective, on ca n obser ve what follows . Remarks 8.7.2 1. A curve X of genus 2': lover F is a Za riski closed subset of it s J acob ian J (X ). 2. A t heorem of Riema nn says t hat t he J acobian J (X ) is an ab elian variety (t hat is a connected complete algeb ra ic group); by t he way, every a belia n variety is actua lly a n abelian group. 3. A t heorem of Mordell and Weil ensures t hat, if A is an abelia n vari ety in t he complex fi eld defined over our number field F , t hen t he set 9 of t he F-rational points in A is a finit ely generated subgroup.

8.7. THE MORDELL-LA NG CONJECT URE

305

So Mordell's Conj ecture ca n be restated mor e genera lly as follows. C onjecture 8 .7 .3 Let A be an abelian varie ty of C , X be a curve of C em bedded in A , 9 be a fi nit ely generated subgroup of A . T hen either X is an elliptic curve , or X n G is fi nite. A simila r qu estion was raised by Manin a nd M umford in 1963. C onject ure 8. 7 .4 (Manin-M umford) Let A be an abelian variety ofC , X be a curve of C em bedded in A , 9 = Tor A be the torsion subgroup of A. T hen either X is an elliptic curve, or X n G is fi nite. Actually the orig inal form of Manin-M umford Conjecture said that, if X is a cur ve of genus > 1 and A = J( X) is its J acobian , th en X n Tor A is finit e. Bu t t he more general stateme nt given in 8.7.4 is eas ily obt aine d as in th e Mordell case . A possible unifying approac h covering both the Mordell a nd t he ManinMumford problem uses t he notion of gro up of finite type: in our characte ristic 0 fr am ework, t his ca n be introduced as an abelia n gro up 9 wit h a finitely generated subgroup S such t hat, for every g E G , t here is some posit ive int eger m for which mg is in S . In fact , every finite ly gene rated abelian gro up 9 is of finit e type (just take S = G) , a nd every to rsio n group is of finite ty pe as well (via S = {O}) . Accord ingly the conjectures of Mordell an d Manin-Mu mford can be rega rded as two parti cular cases of t he following mo re ge nera l qu estion. C onject ure 8 .7 .5 Let A be an abelian varie ty over C , X be a Zari ski closed s ubset of A , 9 be a su bgroup of finit e type of A. T hen X n G is a (possibly empt y) fin it e uni on of cosets of su bgroups of g. T his statement was formul at ed by Lang in t he sixt ies, and is usually called th e Mordell-Lang Conjecture. As underlined before, it impli es a positi ve a nswer t o Mordell's Conjec t ure: to see this, ju st take a curve X o of genu s > lover a number field F , embed X = Xo(C) int o its J acobian A = J(X ) a nd apply t he Mordell-Weil T heore m ensuring th at t he group 9 of F-ration al points in A is finitely generate d . Accordingly decomp ose X o = X n G as a finite union of cosets a + H where a E G and H is a subgroup of g. Take a coset a + H . Its closure a + H is includ ed in X o - an irr edu cible set of di mension 1 -. Conse quent ly, if a + H is infinit e, then X o ju st equa ls a + H an d so inherit s a group st ructure , and genus S; 1. T his means t hat, if t he gen us of X o is > 1, t hen every coset a + H must be finite, whence X o itself is finite .

306 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY

These are the questions we wish to deal with. Now let us report about their solution. Mordell's Conjecture was proved by Faltings in 1983. The echoes of this result spread far and wide, also because it implied an asymptotic solution of Fermat's Last Theorem: in fact, by applying Mordell's Conjecture (or , more precisely, Falting's Theorem) to the projective curve over Q

for n ;::: 3, one gets only finitely many zeros for every n. Also the Manin-Mumford Conjecture was positively answered by Raynaud in 1983. The Mordell-Lang Conjecture (as stated before) was solved just a few years ago: first Faltings handled the particular case when the group 9 is finitely generated , and then McQuillan provided a general positive solution , using Falting's work and other contributions of Hindry. So far we have limited our analysis essentially to the characteristic 0, and to number fields. What can we say when passing to function fields, or prime characteristics? First let us deal with function fields . Still working in characteristic 0, Manin had proved in the sixties the following analogue of Mordell's Conjecture in this setting: if K is a function field over an algebraically closed field Ko (of characteristic 0) and X is a curve of genus> 1 over K, then either X does not descend to Ko (in which case X(K) is finite), or X is isomorphic to a curve X o defined over Ko (and all but finitely many points of X(K) come from elements of X(K o)). When considering prime characteristics p, even the notion of group 9 of finite rank must be rearranged. In fact, what we have to require now is that 9 has some finitely generated subgroup 5 such that, for every g E G, there exists a positive integer m prime to p satisfying mg E S . However 8.7.5 - as it was stated before - does not hold any more. In fact A itself is a torsion group without elements of period p; but there may exist some curves of A which are not finite unions of cosets of subgroups of A. A reasonable restatement of 8.7.5 in the general setting, for an arbitrary characteristic (0 or prime), is the following. Conjecture 8.7.6 Let Ko -< K be algebraically closed fields, A be an abelian variety over K having trace 0 over Ko (this means that A has no non-zero abelian subvarieties isomorphic to abelian varieties over Ko). If X is a Zariski closed subset of A and 9 is a subgroup of A of finite rank, then X n G is a (possibly empty) finite union of cosets of subgroups of g.

In 1994 A. Buium proved this form of the Mordell-Lang Conjecture in characteristic O.

8.7. T HE M ORDELL-LANG CONJECTURE

307

Theorem 8.7.7 (Buium) 8. 7.6 is a true statement when /Co -< /C are algebraically closed fields of characteristic o. What is not eworth y for our purposes in Buium 's line of proof is his use of Differenti al Algebraic Geometry; ind eed Differenti al Alge bra promptl y recalls Mo de l T heory a nd its treatment of differenti ally closed fields. So it is righ t t o spe nd a few words to desc ri be Buium 's strategy : one equips /C wit h a deri vation D whose constant fi eld is ju st /Co, one embeds 9 in a differenti al algebra ic gro up 91 and, finally, one shows by ana lytic arguments th at X n C l is a finite union of cosets of 91 a nd one t ra nsfers this proper t y to 9. But it was Ehud Hru shovski who first proved t he Mo rdell-Lang Conjecture in it s more general form , in any cha racterist ic, following the initial Buium a pproac h and then using mod el th eor etic methods an d , above all, Zari ski geom etries , differentially closed fields in cha rac te rist ic 0 and separably closed fields in prime cha racteristic, in addit ion t o Morley ra nk, elimin ation of im aginari es and t he defina bilit y resul ts of t his chapte r. It should be emphasized t hat no alte rnative genera l proof of the conjecture is known ; and ind eed Hrushovski proposed , so me time lat er , a new mod el t heoretic proof of t he Manin-M um ford Co njecture, based on a cruc ial use of existentially closed fields with an aut om orphism (in pa rticula r Zilber 's Trichotomy in ACFAa) , a nd getting in t his way nice effective bo und s of the number of involved cosets in a decomposition of X n C. Coming back t o t he Mordell-La ng conjecture , we ca n say

Theorem 8 .7 .8 (Hr ushovs ki) 8. 7.6 is a true statem ent in any charact eristic. T his concludes our short and lacu nose history of Mo rdell-Lan g, Morde ll and Manin-Mumford Conjectures. Which is our purpose now? Ce rtainly we do not aim at providing a complete report of Hru shovski 's proof: t his would requ ire man y pages and serious efforts; moreover th ere do exist sever al nice ex pos itory pap ers a nd books wholly devoted to a det ailed ex posit ion (some of t hem are mentioned a mong the references at t he end of th is chapter) . On t he oth er hand , we would like t o spe nd a few words abo ut Hrushovski 's a pproach, ju st to explain where Mo del T heory int ervenes an d why it plays a decisive role. W ith t his in mind , we will sketch Hru shovski 's proof in t he characterist ic 0 case, where so me old frie nds of ours - differenti ally closed fi elds - a re involved. Then we will shortly comment the prime characteristic case, where differenti ally closed fields a re profitably replaced by the separa bly closed ones.

308 CHAPTER 8. MOD EL T HE ORY AND ALGEBRAIC GEOMETRY

So take t wo alge braically closed fields 1(0 -< I( of charact eristic o. Let A be an ab elian variety over I( wit h trace 0 over 1(0 , X be a Zari ski closed set in A , y be a su bgroup of A of finit e rank. Our claim is that X n G is a (possibly empt y) finit e union of cosets of subgroups of y. (a) W ithout loss of generality one ass umes that I( has infinite tra nscende nce degree over 1(0. Then one equips I( with a derivation D making I( a differential field , and even a differenti ally closed field, whose constant subfield C (I( ) coincides with 1(0. J ust to fix our symbols, let L deno te from now on th e usua l language for fields, and L' = L U {D} that of differential fields. So 1(0 is strongly minimal both as a str uctu re of Land L': in fact D is identically 0 on 1(0 and so adds no definable objects to the pure field 1(0 . On t he cont rary, I( is a st rongly minimal struct ure in L , as an algebraically closed field, but it is not any more as a differenti ally closed field; indeed 1( , although w-stable, has Morley rank w in L'. Not ice also t hat, owing to what we saw in t he past sect ions, the a belian variety A is definable (even in L) in 1(. (b) At this poin t one recalls a general resu lt of Manin on differenti al fields: the derivation D yields a group homomorph ism p (definabl e in L') from A onto (I(+) d , where 1(+ is the addit ive group of I( and d is t he dim ension of A. The kernel of p is definabl e in L' a nd has a finite Morley ra nk. Now we deal with y . As y has finite rank and 1(+ has no non zero torsion elements, t he grou p p( G) is finitely generated and t here are go, . . . , gm E K such that

p(G) ~

L

i<m -

Q . s. ~

L

](0 ·

gi·

i<m -

Let II denote L i<m ](0 . gi. H is definabl e (in L' ) and has finite Morley ra nk. Hence p-l (H) is a subgroup of A exte nding y; moreover p-l(H) is definabl e (in L') and has finite Morley rank because both H a nd t he kern el of p are definabl e of finite Morley rank. Without loss of gener alit y for ou r purposes , we ca n replace Y by p-l(H). In fact , if X n p-l (H) is a finit e union of cosets of subgroups of p-l (H) , t hen t he same can be said about X n G a nd G . So we can ass ume t hat y itself is definabl e a nd has a finit e Morley rank. Now, ju st to explain Hr ushov ski's ap proac h in a mor e accessible way, let us restrict a little mo re ou r framework to the particular case when X is an irreducible curve (the setting of the original Mordell Conjecture). If X n G is finite , then we are don e. Otherwise X n G - as a definabl e

8.7. T HE M ORDEL L-LANG CONJECTURE

309

set of finit e Mo rley rank - contains a defina ble strongly minimal subset 5 . As usual, 5 can be viewed as a strong ly minimal str ucture. (c) Now we use t he result of Hr ushovski a nd Sokolovic saying that Zilber's Tri chot om y Conjecture holds for strongly min imal sets definabl e in differenti ally closed fields of characteristic O. In fact , t hese strongly minim al sets a re Zariski struct ures , a nd so obey t he Hru sho vski-Zilber Theor em. T his a pplies to 5, of course . Bu t what Hru sho vski also poin ts out is t hat 5, as a Zariski st ruct ure , is locally modular. In fact, as 5 is st rongly minimal, it suffices to exclude finitely many points fro m 5 to get an ind ecomposable set . So t here is no loss of genera lity in ass uming th at 5 itself is indecomposable, a nd conseque nt ly each t ranslate bS with bE G is also ind ecomposable. Up to replacing 5 by b- 1 5 for a suitable bE G we ca n even assum e t hat t he identi ty element 1G of G is in 5 . Hence we are ju st in a positi on to a pply Zilber 's Ind ecomposability T heorem; accordingly, one ded uces t hat the subgrou p generated by 5 in g is definab le, and ind eed every element in t his subgrou p ca n be ex presse d as a ·c- 1 wit h a and c in 5. Hence 5 interprets an infinite gro up an d so, as a Za riski struct ure , it ca nnot be t rivial. T his means t hat eit her 5 is locally mod ula r or 5 interprets an infinite (algebraically closed) field. We have to exclude t he lat t er opt ion . To obtain t his, one uses a result of Sokolovic already ment ioned in 7.10 and saying what follows. (d) An infinit e field definable in a differenti ally closed field of characte rist ic 0 a nd having finit e Morley rank is isomorphi c to t he constant subfield by a definable fu nction . Recall t hat, owing to t he elimination of imagin aries , t here is no difference between definabl e or interpretable wit hin different ially closed fields. So, if 5 int erpret s a ny infinite field, t hen it defines even C (K ) = Ko up to an £I-definabl e isomorphism. Consequent ly the subgroup t hat 5 generates is isomorphi c t o some group go £I-definable in Ko by a function I also definabl e in L'. As D = 0 in [(0 , go is definable even in L (just as I and 1-1 ). T hen we can a pply t he Hru shov ski-Weil T heorem and dedu ce t hat go is an algebraic gro up over Ko up t o a n L-definabl e isomorphi sm. At t his point one checks t hat go defines a n abelia n subva riety of A in K , which cont radict s the hyp oth esis t hat A has t race 0 over Ko. In concl usion, 5 must be locally modu la r. Now 5 is of t he form (a + L ) - {bo, .. . , btl for some strongly minimal subgroup L of g and suitable a, bo, . . . , b, E G . X is Za riski

310 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY

closed , a nd so contains t he closure a + L of S as well. Hence X equals a + L and consequently inherits a group structure, as Mordell 's Conjecture requires. This concludes our outline of Hru shovski 's proof in cha ract erist ic 0 (when X is a curve). Bu t , as already said , the real novelty of Hru shov ski 's T heorem concerns prime characterist ics p. So it is worth spending some word s also on this case. The pla n of the proof is simila r, but requires some necessary rearrangements. In particular, one can avoid to refer to differentially closed fields and directly handle sepa rably closed fields (with no derivation) . (a) F irst one replaces with no loss of gene rality K by a separably closed non algeb raically closed extension having finit e degree over KP. O bserve that now the theory of K is not w-stable, although it is stable. (b) T he role of the kernel of J-L is now played by n npn A , which is not a definable set, but is the intersection of infinitely many definable sets. The other crucial points in th e proof a re: (c) any strongly minimal struct ure definabl e in K is st ill a Zari ski st ructure (so Zilber's Trichotomy holds);

(cl) a field definable in K and having Morl ey rank 1 is isomorphic to Ko by a definabl e function (a result of M. Messmer). For more details, look at the referen ces quoted below.

8.8

References

The con nection between Mod el T heory a nd Algebraic Geometry is clearl y explained by Poiz at's book [131] (recentl y translated in English [135]): this is a very rich reference on t his matter. In particular it includes a proof of Hru shovski-Weil Theorem 8.6.2 . Cherlin's Conjectu re was raised in [24]. The classification of w-stable groups was given by Macintyr e in [89]. Grou ps of finit e Morl ey rank a re exam ined in [15]. Now let us deal with Mordell-Lang Conjecture and Manin-Mumford Conjecture. A geom etrical int roduction ca n be found in Lang's book [80]; a short but reson abl y ex ha ust ive history of these two questions is also in the recent Pillay paper [128]. Pillay's book [126] explains the main model theoretic techniques involved in Hru shovski's approach. Hrushovski 's original proof

8.8. REFERENCES

311

of t he Mo rdell-Lang Conjecture is in [60]. A det ailed exposit ion is in th e book [16]. [50] and [127] a re shorter surveys , both very read abl e. As already said , Hrushovski 's proof in cha racteristic 0 case is based on some pr elimin ari es concern ing differenti ally closed fields: t hey ca n be found in [155] and [62]. For a prime characterist ic, separa bly closed fields are enough: th e basic prelimin ari es are desc ribed in [115]. Finally, let us deal with Manin-M umford Conjecture: Hru sho vski 's proof is now in [61] . It is based on t he mod el t heory of ACFA, as develop ed in [20] a nd [21] .

Chapter 9

O-minimality 9.1

Introduction

The last part of this book is devoted to o-minimal st ructures . As we saw in t he past chapt ers , t hey a re the infini t e expansion s M = (M , :S; , . . .) of linear orderings such that th e subsets of M definabl e in M are as t rivia l as possibl e, and restrict to th e finit e unions of singlet ons and open int ervals, possibly with infinite end points ± oo (eq uivalent ly to t he finit e unions of op en , closed , ... int ervals in th e broad sen se including half-lin es and the whole M ). a-minimal str uct ures are not sim ple according to t he definition provided in Ch ap ter 7, ju st because t hey define and even expand infinite linear orders; conseq uent ly no good ind ep endence system can be develop ed inside th em , and t hey are not clas sifiable in She lah 's sense. Despi t e thi s, and just owing th e relative triviality of th eir 1-ary definable sets , one ca n see that t hey enjoy several relevant model th eor etic properties and , among th em , a sat isfact or y notion of independence with a related dimension partly resembling Morley rank . Furthermore, they include a lot of noteworthy algebraic exampies. Ind eed we hav e already seen that th e ordered field of real s (R , +, " - , 0, 1, :S;), as well as any real closed field , is o-rninimal; discrete or dense infinit e linear orders, like (N , :S; ), or (Z , :S;), or (Q, :S;), or (R , :S; ), ar e o-minimal as well; and one ca n show t ha t even divi sib le ordered abelian groups, such as (Q, +, 0, :S; ) and (R , +, 0, :S; ), are o-rninimal. In particular , consid ering t he order of th e reals, or expa nding it by addition , or addit ion and multiplication together , yield o-rninimal st ruct ures. On the oth er side, t here do exist some ex pansion of (R, :S;) which are not o-rninimal.

313

314

CHA PTER 9. O-MINIM ALITY

For inst ance, ext end the ord er of reals by t he sinus fun ction si n; in t his enlarged structure, Z -a denumer abl e set of isolated points- get s defin abl e (by sin 1r V = 0) , and so o-mi nimality get s lost. Of course the same argument applies to cos . T his cha pter has a two fold aim. On t he one hand , we will provide a n a bstract structure t heory for o-minimal models M. The sta rt ing poi nt will be ju st the definit ion of o-mi nimality, and t he consequent class ification of t he defina ble subsets of M . But we will see t hat, on t hese seemingly poor grounds , one ca n develop a sign ificant t heory, including a nice characterization of definable subsets of M " for every positi ve int eger n, as well as of definabl e function s from M" into M . This will also lead t o the alr ead y recalled notion of dim ension, satisfying several rem arkable properties resembling those of Morley rank in w-st abl e t heories. Act ually th ere is a good deal of similarity bet ween t he o-mini mal framewor k and t he w-stable set t ing, ju st in t hese di men sions, but also in definabl e groups and in other matters. We will em phasize t hese connections in Sections 9.2-9.6. In part icular Section 9.4 and Section 9.5 will prove, among other th ings, t he already ment ioned a nd not ewor th y fact t hat o-minimalit y, unlik e minimality, is preserved by elementary eq uivalence : if M is a n o-minimal st ructure, t hen every mod el of the t heory of M is o-minimal as well. Accordingly a complete th eory T is said to be o-minimal whe n some (equivalently every) model of T is o-minimal. T he subseq uent section 9.6 will treat definabl e groups, definabl e manifolds , and so on, in o-minim al st ruct ures. On t he other side , we will prop ose ot her relevan t exam ples of o-minim al struct ures (t o which t he previous gene ral t heorems apply) . T his will be t he t heme of Section 9.7, where we will see t hat certain ex pa nsions of t he rea l field by fam ilia r functions , such as ex ponent iation, or suitably restrict ed a nalytic functions, are o-minimal. Here o-minimality largely overla ps real algebraic geomet ry a nd real a nalytic geomet ry, both in acquiring tec hniques and constructi ons fro m t he geomet ric framework towards a general and lar ger spect re of applications, and in providing a new light and in op ening new per sp ecti ves wit hin t he geometrical setting itself. The subseq uent section 9.8 will deal with some variations on t he o-minimal t heme, most not ably wit h a notion of wea k o-m inimality , enla rging t he 0 minimal setting an d intensively st udied in t he latest years. At las t , t he final section 9.9 will int rodu ce very shortly t he qui te recent and attracti ve work of A. On shuus a bo ut a notion of ind ependence enla rging both forkin g ind epend ence in simple t heor ies and algebra ic ind ependence in o-minim al th eories t owards a common general fram ework. Now a few historical not es. O-minimality began its life in th e eight ies; its

9.1. INTRODUCTION

315

origin refers to a classical problem of Tarski, asking whether the real field, expanded by the exponential function x t--+ eX, still has a decidable theory. As we know, decidability closely overlaps definability, so a deeply related question is what is definable in (R, +, " -, 0, 1, :S, exp): is the theory of this structure quantifier eliminable, or model complete? This was the scenery where 1. Van Den Dries introduced o-minimality at the beginning of the eighties. But who gave a considerable impulse to this notion were A. Pillay and C. Steinhorn, who proved the basic structure theorems on o-rninimal structures and greatly developed their abstract theory. In 1991 A. Wilkie partly solved Tarski's Conjecture, showing that the theory of (R, +, " -, 0, 1, :S, exp) is model complete, and even o-minimal (as we will see in 9.7, decidability is still an open question, involving a deep number theoretic problem, usually known as Schanuel's Conjecture, while quantifier elimination fails). This emphasized t he connection with analytic geometry, mentioned some lines ago. And indeed o-minimality became, and still is, a matter of interest not only to model theorists, but to geometers and analysts as well. To conclude this section, we give the proof that any o-rninimal ordered field is real closed. This is the converse of a result we already know, ensuring that every real closed field is o-minimal, and can be viewed as the o-rninimal analogue of Macintyre's theorem saying that any w-stable field must be algebraically closed. The proof requires a very basic machinery from 0minimality -just the definition itself- in addition to the necessary algebraic grounds. Let us preliminarily examine o-minimal groups. Here (and later in this chapter) intervals possibly admit infinite endpoints, and so include half-lines, and the whole line, in case. Lemma 9.1.1 Let A = (A, 0, +, -, :S , . ..) be an o-minimal structure expanding an ordered group (A, 0, +, -,:S) and let H be a subgroup of (A, 0, :S) definable in A. Then H = {U} oppure H = A.

+, -,

Proof. Suppose towards a contradiction that there exists some subgroup H i- {O}, A of (A, 0, +, -, :S) definable in A. Owing to o-minimality, H decomposes as a finite union of pairwise disjoint intervals (possibly closed, or with infinite endpoints). Accordingly write

(0)

H =

U t, j :::; s

where 10, ... , Is are intervals and s is minimal. Notice that H is infinite, because it must contain all the multiples nh of any nonzero element h E H

CHAPTER 9. O-MINIMALITY

316

when n ranges over int egers. Hence t here is j ::; s for which I j is infinite. Without loss of generality pu t j = O. Moreover we ca n even ass ume th at l a contains 0 a nd is sy mmetric with respect to 0 (in t he sense t hat -c E l a for every c E la). This is becau se H is a subgro up, and so includes 0 and is preser ved under + a nd -. As H =f:. A, la is [-a , a] or] - a, a[ for some a > 0 in A. In t he form er case , eve ry b in A satisfying a < b ::; 2a decomposes as b = a + (b - a) where a E la ~ Hand 0 < b - a ::; a , so even b - a is in 10 a nd conseq uently in H ; hence b E H. Therefor e [-2a , 2a] is an interval in H properly including 10' T his impli es t hat [-2a , 2a] shares at least one element with some interval Ij where 0 < j ::; s , say with I s. Put I~ = [-2a , 2a]Uls .

So

Jb

is an interval including 10 U I s and contain ed in H . Co nseq uent ly

H = I~ U

U

O<j<s

i.,

a nd t his cont ra dicts t he choice of s in (0). In t he lat t er case, fix b E A such t hat 0 < b < a, t he n 0 < a - b < a , and consequent ly even a - b is in l a. It follows a = b + (a - b) E H. We get in t his way an interval [- a, a] properly including l a and contained in H . But t his contradicts as before t he minim alit y of s in (0). ., As a conseq uence, one ca n give a full characterization of o-minima l ordered groups. They ar e exa ctl y t hose listed before, namely t he ord er ed div isibl e a belia n groups. Theor em 9 .1.2 An o-m in imal ordered group A = (A , 0, and divi sibl e.

+, - , ::;) is abelian

Proof. For every a EA , th e cent ralizer C (a) of a is a definable subgrou p of A , a nd consequently equ als eit her {O} or A. If a = 0, then clearly C(a) = A. On t he other side, when a =f:. 0, C (a) = A as well, becau se a E C (a) and t his excludes C (a) = {a}. Hence C (a) = A for every a E A, a nd A is abelian . Now take a positive int eger n : n A is a definabl e subgro up of A , clearl y eq ualling A when A = {a}; on t he ot her side, if A =f:. {a}, t hen nA =f:. {a} an d consequent ly nA = A . Then n A = A for every positi ve int eger n , III other words A is divisibl e. .,

Coming back t o ordered fields, we ca n eventua lly pro ve

9.1. INTRODUCTION

317

Theorem 9.1.3 Let /C = (K, 0, 1, +, " -,::;) be an a-minimal ordered ring with identity 1 (in particular let /C be an a-minimal ordered field). Then /C is a real closed field.

Proof. First we claim that /C is an ordered field, in other words the set /C* of the nonzero elements in K is an abelian group with respect to -, It suffices to show that the set «> of the elements > 0 of K is so. In fact, for every a E «>, aK is a definable subgroup of (K , 0, +, -, ::;), and aK -I {O} because a > O. So, owing to Lemma 9.1.1, aK = K, and in particular there exists some b E K satisfying ab = 1. As a > 0, b is positive as well. This proves that x> is a(n ordered) group with respect to -, It remains to check that «> is also abelian. To show this, it suffi ces to observe that «> is o-minimal , and then to use Theorem 9.1.2. In fact «> is definable (as a group) in /C, and consequently every subset X of «> definable in «> is also definable in /C, and hence is a finite union of non-empty intervals of K. All the end points of these intervals lie in «> U {+oo}, with the only possible exception of the left most end point in the first interval, that might equal 0, but can be replaced in this case by -00 in «>. In conclusion, X is actually a finite union of intervals in «>. Hence «> is o-minimal and consequent ly abelian, as claimed. Now let us prove that /C is real closed. Accordingly take a polynomial f(x) E K[x] and two elements a < b in K satisfying f(a) . f(b) < 0, for example f(a) < 0 < f(b). We have to show t hat there is some c E K such th at a < c < band f(c) = O. Recall that /C is an ordered field, and hence ::; is dense in K and in la , b[. The polynomial function that f defines is continuous (with respect to the order topology) , so both la, b[+= {d E K : a < d < b, f(d) > O} and

la, b[-= {d E K : a < d < b, f(d) < O} are open sets. If la, b[-= 0, then the continuity of f is contradicted in a. Hence la, b[- and similarly la, b[+ are not empty. Moreover both la, b[+ and la, b[- are definable, and accordingly decompose as finite unions of intervals, indeed of open intervals. As la, b[+ and la, b[- are disjoint , t here is some c E]a, b[ out of both la, b[+ and la, b[-: so c is a root of [, f( c) = O. '"

318

9.2

CHA PTER 9. O-MINIMALITY

The Monotonicity Theorem

Let M = (M, ~ , . ..) be an a-minimal struct ure . First obser ve t hat M is a t o pological space with resp ect to t he ord er t opology a nd so, for every positi ve int eger n , M " is a topological space as well, with resp ect t o th e product t opology. As already recalled , t he o-minimalit y of M ju st requires t hat t he only definabl e subsets are th e finit e union of singlet ons a nd op en inter vals (possibly including half-lin es and t he whole M) . Bu t what ca n we say abou t th e defina ble subsets of M " when n > I ? We give here a very partial and pr eliminary an swer , dealing with l-ary definable functions f. We show t hat, if the domain of f is an open interval la , b[ with a < b in M U {±oo} , then one can partition la , b[ into finitely many intervals such that, in each of t hem, f is either const ant , or strict ly incr easing , or strict ly decreasing, and a nyhow cont inuous according to t he orde r topology. T his is t he so-called Monotonicity Theorem , say ing in detail what follows. Theorem 9.2.1 Let A1 be an o- m inimal structure, X ~ M, a, b E M U {±oo}, a < b, a an d b be X -defi nabl e when belonging to M . Let f be an X -defi nable f unction of la , b[ in to M . Th en there are a posit ive in teger n and aD, al , .. . , an E MU {±oo} suc h that

1. a = aD < al < ... < an = b, and aI , ... , an- l are X -defi nable;

2. fo r eve ry i < n , f is eithe r constant or strictly m on otonic in ]ai , ai+d; 3. fo r every i < n , if f is strictlu mono to nic in uu, ai+l[, then f( ]ai, ai+l [) is also an in terval, and f con tains a biject ion preserving or reversing ~ bet ween ]ai , ai+d and f (]ai' ai+d ) · In particular f is conti n uous in every inter val ]ai , ai+d for i

< n.

Notic e t ha t, when mor e genera lly f is an arbitrary definable function with both domain an d image in M , t hen the domain of f is definable as well, a nd conseq uent ly is a finit e union of singletons and op en int erval s. Each of t hese intervals satisfies t he ass umpt ions of Theorem 9.1.2 , and hence inh erits its conclusions. Notice also t hat Theor em 9.1.2 implies, as a simple conseq uence, t hat, if M is a n a-minimal structure, a, b E M U {±oo} , a < b a nd f is a definabl e function from la, b[ into M , t hen f (x) has a limit in MU {±oo} when x -T a+ and x -T b>. A full proof of Theorem 9.1.2, as stated before, would require seve ral tec hnical det ails a nd would be quite long. We prefer to propose here a simpler

9.2. THE MONOTONICITY THEOREM

319

argument showing only the continuity result when M expands (R, :S;) (actually this is more or less what we will need later). Proposition 9.2.2 Let M be an o-minimal structure expanding (R, :S;) , X ~ M, f be an X -definable function from ]a, b[ into R . Then th ere are aI, ... , an E]a, b[ such that al < ... < an, aI, ... , an are X -definable and f is continuous in ]a, al [, ]an, b[ and each interval ]ai, ai+l [ for 1 :s; i < n. Proof. Let S denote the set of the points in ]a, b[ where f is not continuous. It is an easy exercise to prove that S is X -definable. If S is finite , then we are done: for, aI, .. . , an are just the elements of S. Hence suppose S infinite. So S contains an infinite open interval I. For every natural n, build two infinite open intervals In and I n such that, for every n,

(i) In

~

I,

(ii) the (topological) closure I n+l of I n+l is included in In'

(iv) the length of I n is smaller than

nil'

Let us see how to define these intervals. First put ID = I. Then take any natural n , suppose In given and form I n and I n+l as follows. If f(In) is finite , then for some d E f(1n) the preimage {c E In : f(c) = d} is infinite. But {c E In : f( c) = d} is definable, and hence includes some infinite interval; f is const a nt , hence continuous, on this interval, which contradicts In ~ I ~ S. Accordingly f(In) must be infinite. But f(In) is definable, too, and hence includes in its turn an infinite interval; let I n be such an interval, notice that we can assume with no loss of generality that the length of I n is < The preimage of I n in In is also definable, and contains some infinite interval. Let I n+l denote such an interval; we can assume In+! ~ In. This determines the In's and the In's for every n. Now put I' = nnEN In. Clearly I' = nnEN In' whence I' =1= 0 because R is compact. Pick d E 1', we claim that f is continuous in d (this contradicts dES and so accomplishes our proof). Take any interval U containing f(d). Owing to (iv), there is some natural n for which U ;2 I n. But this implies U;2 f(In+l) where I n+l is an open neighbourhood of d. ...

nil'

A final remark. When M expands the field of reals, we can say even more, and state a smooth version of the theorem: in fact, one can partition ]a, b[ into finitely many intervals where f is of class cm for every positive integer m.

320

9 .3

CHAPTER 9. O-MINIMALITY

Cells

Now we move to characterize in an o-minimal struct ure M = (M, :S , . . .) th e definable subsets of M": in particular t he definable n-ary function s, for every positive int eger n . To make our life easier , we assume all throughou t t his sect ion (and also in th e following ones) t hat :S is dense without end points: in particular every interval ]a, b[ with - 00 < a < b < + 00 in M must be infinite. As alr ead y said , the reason of t his restriction is ju st to make our treatment and our proofs sim pler; in fact , all th e results we will show below ca n be extend ed -by the appropriate a rr ange ments- to any o-rninimal st ruct ure M. On the other side , t he ord er of reals is j ust dense without end point s, so our fram ework include all the exp an sions of (R , :S ), in particular all the st ruct ures enlarging t he real field; as we said befo re, the notab le o-minimal exam ples we will propose in Section 9.7 lie in t his set t ing. We know that the basic definable subsets of M are t he int ervals and t he singlet ons. More gener ally, th e basic definable subsets of M " are t he cells. So let us define what a cell is, mor e precisely what a k-cell in M " is.

Definition 9 .3 .1 First suppose n = 1. A sub set C of M is a O-cell if an d only if C is a si ngleton, and a 1-cell if and only if C is a non- empty open in ter val, possibly with infi nite endpoints . N ow let n > 1, and let k a natu ral number :S n. A subse t C of M " is called a k -cell if and only on e of the follo wing conditions hold: 1. there are a k- cell D of M n - 1 and a contin uous and defi nable fun ct ion f of D into M such that C is the graph of I, namely

C = {(11, b) E M " : 11 E D , b = f(I1)} ; 2. k ~ 1 and th ere are a (k - I )-ce ll D of M n-l and two fun ctions f and g with domain D such that

(i) eithe r the image of f is a subset of M and f is both continuous and definable, or f(l1) =

-00

VI1 E D ,

(ii) either th e im age of g is a subset of M and g is both continuous and definable, or g(l1) = + 00 VI1 E D , (iii) f(l1) < g (l1) VI1 E D , (iv) C

=

{(11, b) E M n : 11 E D , f (l1)

< b < g (I1)}.

On e eas ily sees t hat every k-cell of Mn is defin able, and t hat a k-cell of jU n is op en in M " if and only if k = n . One can also observe

9.3. CELLS

321

Proposition 9.3.2 Let M be an o-minimal str ucture, k S; n be posit ive integers. For every k-cell C of M" , there is a definable homeomorphism of C onto a k- cell of M k • Proof. It suffices to see , for n > k a nd n > 1, how to det ermine, for every k-cell C of M n , a definable homeomorphism 1re onto some k-cell C' of Mr:" , and then t o iterate th is procedure as long as one needs. First ass ume that C is t he gr aph of some cont inuos definable func tio n fro m a k-cell D of M n-l into M. In this case it suffices to put C' = D , and choose as 1re th e proj ection of C onto C' . Now ass ume

C = {(a , b) E M n

:

a E D , f (a)

< b < 9 (a)}

where D is a (k - I)-cell of M n-l , and I, 9 sat isfy the cond it ions in 9.3.1 , 2. We pro ceed by ind uction on n , If n = 2, then k - 1 = 0, and D red uces t o a single point a of M. P ut C' =JJ(a) , g(a)[ , 1rc(a , b) = b for every b E C '. Now let n > 2. We know th at there is some definabl e homeomorphism 1rD of D onto a (k - 1)-cell D' of M n-2. Consider the two fu nctions one gets by composing l , 9 respectively a nd the inver se of 1rD . T hey have dom ain D' a nd sat isfy t he ass um pt ions in 9.3.1,2. Accordingly they define a k-cell C ' of M n - l ; further mo re 1rc(a , b) = (1rD(a) , b)

V(a , b) E C

det ermines a definab le homeomorphism of C onto C'.

...

When M expands a real closed field (for instance, when M is the real field , or even a n expansion of it) , the cells in M ca n be charact erized as follows. Proposition 9.3.3 Let M be an o-minimal expansion of a real closed fiel d, k , n be two natural nu mbers satisf ying n 2:: k , 1. If C is a k -cell of M n, th en th ere exis ts a definable homeomorphism of C onto ]0, I [k. Proof. When k = 0, our claim is trivial, becau se a O-cell reduces t o a singlet on. So t ake k > 0. Owing to the previous proposition, we can assume n = k. We proceed by induction on n . If n = 1, t hen C = ]a, b[ where a , b EMU {±oo} a nd a < b. So the required homeomorphism between C and ]0, I[ is easily obtain ed: for instance, if both a an d b are in M , th en it suffices to map every x in C = ]a, b[ into x - a b- a'

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in t he remaining cases one proceeds in a simila r way. Now s uppose n > 1 (a nd our claim true for n - 1). Let C be an n-cell of M n. There do exist an (n -I )-cell D of M n-l and two functions f and 9 as in 9.3.1 ,2 such t hat C is t he set of t hose t uples (ii, b) in M " for which ii E D a nd f (ii) < b < g(ii). By induction , t here is a defina ble homeomorphism h of D ont o ]0, l[n-l. If both f and 9 take th eir values in 1\1, t hen

( a,~ b) ~

(h ( ~)

b - f (ii) ) V(ii, b) E C

a , g(ii) _ f(ii)

is th e required hom eomorphism. All t he other cases ca n be handled simila r way, ju st as when for n = 1. •

III

a

In particular, when M expands th e real field Rand k > 0, every k-cell of M is connecte d in th e ord er topology ; for, ]0, l[k is. This do es not hold any longer when R is replaced by any real closed field. For instance, if Ra is t he orde red field of real algebraic numbers, th en Ra is real closed , and hence o-minimal; Ra is a Lcell of itself, but is not connected becau se, for every real t rascende ntal t , Ra pa rti tion s as R a = {r E R a : r < t } U {r E R a : r > t}. Hence connect ion gets lost. However ever y cell in an o-minimal st ruct ure M sa t isfies a wea k form of con nection, wit h resp ect to ope n definable sets . In fact , conside r t he following notion. Definition 9.3.4 Let M be an o-m inimal structure, n be a posit ive integer. A defi nable set X ~ M " is said to be definably connected if and only if X cannot part it ion as the disjo int un ion of two non-empty open definable subsets . Wh at we will see now is t hat every cell in an o-minimal struct ure M is definably connected . First we give an equivalent charact erization of defin abl y connecte d sets. An open box of M n is t he ca rtes ia n product of n op en int erval s of M . Hence op en boxes form a basis of op en neighbourhoods of t he product topology of M n. Furthermore, for Y ~ X ~ u», an element s of X is called a boundary point of Y in X if a nd only if every op en box of M n containing ii overlaps both Y a nd X - Y. Lemma 9.3.5 Let A1 be an o-m inimal structure, n be a posit ive in teger, X be a definable su bset of M": T hen X is defi nably connected if and only if, fo r every proper non-em pty defi nable subset Y di X , X contains at least one bounda ry point of Y in X.

9.3. CE LLS

323

Proof. X is definably connected if a nd only if, for every definable Y ~ X such t hat Y =1= 0, X , eit he r Y or X - Y is not op en , in other words ther e is eit her a point a E Y such that every open box cont aining a overl aps X - Y , or a point E X - Y such that every open box cont aining overl aps Y . Accordingly X is defin ably connected if a nd only if, for every Y as a bove, t here is a E X which is a bound ary poin t of Y in X . ..

a

a

Now we can show, as promised, T h e orem 9.3.6 Let M be an a-minimal structure, n be a posit ive integer. T hen every cell C of M n is defi nably connec ted.

Proof. We proceed by induction on n . If n = 1, t he n t he claim is trivial becau se the cells of M red uce to singletons a nd open intervals. Hen ce assume n > 1. Let C be a cell of J\;f n . If C is a O-cell, so a singleton, t he n C is definably connected . If C is a k-ce ll for some positi ve int eger k < n, t hen, owin g t o Corollary 9.3.3 , C is defin ably hom eomor phi c t o a cell C' of M k ; by t he induction hypothesis , C' is defin ably co nnected, wh en ce C is defin ably connected , t oo . Fin ally suppose t hat C is a n n-cell of M n. Then ther e exist a cell D of M n-l a nd two fun ctions 1 and 9 as in 9.3.1, 2 such th a t C = {( a, b) E M n : a E D , 1(a) < b < 9 (a) } . D is definably connected by t he induction hyp oth esis. To deduce that even C is defin ably connected we use Lem ma 9.3.5 . Accordingly take a definabl e subset Y of C such t hat Y =1= 0, C . F irst sup pose t hat, for some a E D , t here a re t wo eleme nts b an d b' in M suc h t hat (a, b) E Y a nd (a, b') E C - Y. Then t he int erval ]1(a), g(a)[ contains at leas t one boundary poin t bo of t he definable set {b E M : (a, b) E Y}; t his implies t hat (a, bo) is a bo und ary po int of Y in C . Now suppose t hat , for every a E D , either

{ (a, b) E C : bE M} or

{ (a, b) E C : s « M }

~

~

Y

C - y.

Let Z den ot e t he set of t he eleme nts a of D satisfying t he for mer cond it ion. Y =1= 0, C implies Z =1= 0, D . As D is defin abl y connected, t here is some boundary point ao of Z in D. Let b E M sat isfy (aa , b) E C, we claim th at (aa, b) is a boundary point of Y in C . Let B a n op en box of M n containing (aa, b). As C is open , we can suppose B ~ C . Let B ' denote the projection

CHAPTER 9. O-MINIMALITY

324

of B in M n - 1. Then ao E B ' a nd B ' ~ D. Since ao is a boundary point of Z in D, there are a1 and a2 in B ' such t hat a1 E Z a nd a2 tj Z . Con sequently, if b1, b2 E M satisfy (a1 ' b1), (a2 ' b2) E C respectively (in particular (a1 ' b1), (a2 ' b2) E B) , then (a1' b1) E Y, (a2' b2) tj Y . Hence (ao, b) is a bou ndary poin t of Y in C . In conclusi on , C is defina bly connect ed . ..

9.4

Cell decomposition and other theorems

The aim of this section is to introduce and to state t he basic general theorems for o-rninimal struct ures M = (M , ~ , . . .): we want to characterize all the defin a ble sets and funct ions in M , a nd we wan t also to emphas ize some relevant consequences a nd, am ong t hem, the already ment ioned fact t hat 0minimality is preserved und er elementary equivalence. As th e proofs of t hese fundamental results a re qui te long and intricate, we will defer part of t hem, a nd t he cor responding de tails, to t he next section ; here we provide ju st a basic outline, illustrating these cent ral core s of the theory of o-minimalit y. So a reader simply int erested in a general view may limit her , or his attent ion t o t his section, and to skip the next one. Let us remind once again that , for simplicity 's sake, we a re assuming that (M , ~) is dense without endpoints: this is tacitly acce pted all throughout these sections, unless otherwise stated. The first result we prop ose j ust describes definabl e sets (and functi ons) in o- rnini mal struct ur es. It is a beau ti ful a nd powerful cha racterization, called Cell Decomposition Theorem. In fact , it says t hat every definabl e set decomposes as a fi nite union of cells. Theorem 9.4.1 Let A1 be an o-m inimal structure, n be a positive integer. 1. Eve ry definable set X uni on of cells in M n.

~

M " can be expressed as a fi nit e (disjo int)

2. Furthermore, if X is the domain of a definable fun ction f with values in M , then on e can decompose X as a fin it e disjo int union of cells, such that f is contin uous on each of them . Not ice that t his generalizes what we know when n = 1; in fact, in that case every definable X ~ M is a finit e union of point s and open intervals (in other words, of O-cells a nd I-cells respectively), a nd, when X is th e do main of some definable fu nction I, one can also suppose that f is continuous on eac h of these pieces, owing to the Monotonicity Theorem. But now we can ext end t hese results t o any n .

9.4. CELL DECOMPOSITION AND OTHER THEOREMS

325

As already said , t he proof of the Cell Decomposition Theorem will be given in detail in t he next sect ion. But Cell Decomposition , and it s a nalysis of definable sets, is also t he key t ool in showing another basic fact in 0minimality, that is it s preserving under elementary equivalence. Theorem 9.4.2 If M is an a-minimal structure, then the theory of M a-minimal.

is

In fact , which is t he trouble in this claim ? Actually we do know th at, for a n o-minimal M, for every formula O( v, w) in the lan gu age L of M a nd t uple d in M , O(M , il) is a finite union of (possibly closed) intervals . But suppose th at , when ii ranges over M, the minimal number of t he intervals involved in these decompositions of O(M , il) cannot be upperly bounded , and so, for every natural N, on e find s some tuple il( N) in M such that any decomposition of O(M , il( N )) requires at least N intervals. If this is t he case , then it suffices a st raightforward application of Compactness Theorem to provide some M' == M and some t uple a' in M' for which O(M' , a') can not be expressed as a finite union of int ervals , and con sequently to conclude t hat the t heory of M is not o-rninimal. Hence the cr ucial point in showing that the o-rninimality of A1 is preserved by element ary equivalence is to uniformly bound , for every form ula O( v, w) as befor e, the minimal number of intervals necessary to decompose O(M , il) wh en il ranges over M. T his is a definability que stion concerning formulas in arbitrarily many free vari ables, and so directly refers to Cell Decomposition. On the other hand, boundi ng t he number of the invol ved int ervals in O(M , il) is the same as bounding the total number of their end points (forming a defina ble, and finite set). So the key ste p towards the proof of Theor em 9.4 .2 is Theorem 9.4.3 Let M be an a-minimal structure in L , Ifl(V, w) be an Lformula such tha t, for all il in M , Ifl(A1, il) is finit e. Th en there is a positive integer N such that, for eve ry il in M , IIfl(M , il) I ::; N.

The proof of Theorem 9.4.3 will be deferr ed until the next section. But , as we have just pointed ou t , T heorem 9.4.2 is an almost immedi ate conseq uence of Theorem 9.4.3. Let us see in detail why.

Proof. (Theorem 9.4 .2) Let L be the language of M , and let 7)( v , w) be an L-formula ; in pa rticular, let n deno te the lengt h of w. For every d E M"; O(M, il) is a finit e union of intervals. Let Ifl(v, w) be t he L-formula saying v is an end point of 0(... , w).

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CHAPTER 9. O-MINIMALITY

T hen, for every ii E M."; cp(M, ii) is finite; hence, by Theor em 9.4.3 , there exists a positive integer N such that , for every ii E M" , cp(M, ii) cont ains at most N elements, whence B(M , ii) has at most N end poin ts. Bu t t he sentence ViiEj ~ N v cp( v, w) remain s t rue in every struct ure M' == M. Accordingly, for every A1' == M and bE B(M' , b) has at most N end poin ts, and hence is a finit e unio n of intervals. In conclusion t he t heory of M is o-minimal. ..

u»,

The bounds given by Theor em 9.4.3 on formul as B( v, w) ca n be extended to formulas B(V, w) with an a r bit rarily long v. In fact the following result holds . Theore m 9 .4.4 Let M be an o-minimal structure in L , B( V, w) an L formula (n be the length of v and m be that of w) . Th en there exists a posit ive integer N such that , for every M' == M and tuple ii in M ' , B(M, n, ii) is the un ion of at most N cells in M' . Before beginning t he proof, a nd ju st for pr eparing it , let us premi t a sim ple exa mple. Ass ume n = 2. Let B(V I, V2, w) be a n L-formul a , ii be a t uple in M. Suppose t hat t he definabl e set B(M 2 , ii) decomposes as a disjoint union of 2 cells in NI : the form er is a singlet on, so a O-cell, while t he lat ter is a I-cell, a nd more precisely is t he graph of a cont inuous definable func tion f whose dom ain is t he open int erval ]a, b[ wit h a < bin M . Notice t hat t here a re a n L-formula 1]( V I, V2 , Z) a nd a sequence e in J\1 such t hat, for every Cl and C2 in M with a < Cl < b,

Now consider the L-formula

1\

"1](', " Z) defines a cont inuous function of dom ain ]u, w[" 1\ 1]( V l, V2,

Z))) 1\ -, (u <

UI

<W

1\ 1]( UI ' U2,

Z) ))).

b in M , or even in a mod el M' of t he t heory of M , satisfying N F B(i (b) are ju st t hose for which B(M ,2, b) has a cell decomposit ion as 'l9(M 2, ii) (t he disjoint union of a singlet on and a gra ph of

It is clear t hat t he t uples

9.4. CELL DECOMPOSITION AND OTHER THEOREMS

327

a definabl e continuous function) . It is also obvious t hat t his is quite gener al: given an L-formula B(v, w) as in t he statement of 9.4.4, a t uple 5 of th e same length as w in t he univ erse n of t he t heory of M and a cell decomposition of 'l9(nn , 5), one can build a n L-formula Bii ( w) such t hat a t uple bin n satisfies Bii if and only 'l9(nn , b) has a cell decomposition just as 'l9(n n, 5). Proof. (T heorem 9.4.4). Owing to Theorem 9.4.2 , a ny structure M' element aril y equivalent to M is o-m inima l, and hence, by Theorem 9.4.1, a ny definable subset of (in part icular B(M m , 5) for 5 E M' m) is a finite union of cell. We have seen t ha t t here is an L-formula Bii ( w) (without paramet ers) describ ing t he form of a given cell decomposition of B(M,n, 5). Let denote the (countabl e) set of all these formul as Bii (1V) when 5 ranges over M' n a nd M' is a model of th e th eory of M. Use Com pactness Theorem a nd get finitely many formulas

u»

such t hat

Vw

V 'l9i(W) E Th (M ). i «:s

Co nseq uent ly t here is a positi ve int eger N s uch th at , for every M' == M and 5 E M'?" , B(M,n, 5) decomposes as the union of at most N cells: N is just t he maxim al num ber of involved cells in t he decom posit ions described by 'l9 o(w), ... , 'l9 s (w). • Recall t hat , when A1 ex pan ds t he real field R , every cell of M is also connected . Hence in t his case, for every formul a B(V, w), t here is a positiv e integer N such that , for a t uple 5 in M m, B(M n , 5) is t he union of ~ N connected components . vVe will see in Section 9.7 several relevant examples of o-rninimal exp ansions of t he real field R. In this fr am ework it is worth stat ing the following result of Wilki e' s. Theorem 9.4.5 (Wilkie) Let M expand the ordered field of reals by C oo fu nctions from some cartesi an powers R t of R into R. Assume that, for every quantifier free fo rmula B(v, w) in the languag e L of M , there is a posit ive integer N such that, for any tuple 5 E Mm, B(M n, 5) decomposes as the union of ~ N connected compone nts . T hen the same is true fo r every fo rm ula of L. In part icular, M is o-m inimal. Cell Decomposition is an import an t tool also in developin g a dim ension t heory inside o-minimal structures M , a nd in equ ipping every definable X

328

CHAPTER 9. O-MINIMALITY

in M with a natural number (its dim ension). Recall t hat no o-minimal M is simple, and hence w-stable; accordingly t he Morley ra nk of X might be 00 . More generally, as simplicity fails, other possible rank notions ari sing wit hin simple, or stable, or supersta ble settings and replacing R M in t hese enla rged fr am eworks lose t heir interest in o-minimal mod els. Bu t Cell Decomposition do es assign a dim ension to X in a qui te reasonabl e way. Let us see how. Definition 9.4.6 Th e dimension of a k- cell of M n is k. Th e dimension of a definabl e X in M is the maximal dim en sion of a cell arising in a cell decomposition of X. Of course, a cell decomposition of X is not unique; but th e maximal dimension of an involved cell is, and do es depend only on X. So the previous definition makes sense for every X. There is another alte rn at ive way t o introd uce a dimension notion in M. In fact , as we saw in C hapt er 5, t he alge braic closure acl determines a dependence relation ~ in M (and in every mod el M' of t he t heory of M ). T his relation generates in its t urn an ind epend ence system as ax iomatized in sect ion 7.2 , so satisfying t he condit ions (11)-(16) (but not t he fur th er requirement (17) )). Wi th resp ect to this indep end ence notion , we ca n define the dim ension of a t upie ii = (al' . . . , am) in M'": as t he size of a minimal subsequence b suc h t hat ii lies in acl(b) . T hen we ca n introduce, ju st as in t he st rongly mini mal case , t he dim ension of a definable set
9.5. THEIR PROOFS

329

The uncountable sp ectrum of an o-m inimal T th eory takes everywhere t he maximal value I(T, >.) = 2" V>. > ~o . With respect to th e count a ble framework , it is wort h emphasizing t ha t Vaught 's Conject ure holds, in t he following st ro ng form. Theorem 9 .4.8 (Mayer) Let T be a (com plete) o-m in imal theory. Th en, up to isomorphism, either T has conti nuum many countable mod els, or there are two naturals n and m such that T has 3n . 6m countable models. Of course, one might get curious in reading t he st ate ment of this th eorem: why, and where, do n and m arise? Basically, they depend on a careful a nalysis of types in o-minimal structures. The interested read er may directly consult Laura Mayer 's work , quoted below .

9.5

Their proofs

This section provides t he proofs of Theorems 9.4.1 and 9.4.3 , stat ed in Section 9.4. As said, th ey are long and intrica t e. In spit e of this, we think it right to propose th em for at least two reason s. Th e form er (and th e principal) is t hat we believe t hat T heorem 9.4 .1 (t he Cell Decomposition Theorem) and T heor em 9.4.2 (the one saying t ha t o-rnin imalit y is pres erv ed under element a ry equivalence) a re two beau tiful and fundamental results and deserv e a full report , including a t echni cal preliminary like Theorem 9.4 .3. The la tter reason just concerns t he intricacy of t he proof; actually this is du e t o it s length and ingenui ty, bu t do es not depend on a relevan t a nd deep t heory, ind eed t he premises it needs a re rather element ary and accessible (they ju st include the definition of o-minimality, t he Monotonicity T heor em, some properties of cells and an induction argument). So our ex position should require no pa rticular efforts but a lit tl e attention and pat ience . And anyhow the reader who is not intere sted in th ese details may neglect this section and proceed directly to th e next ones, t hat will not use these proofs. So conside r a n o-minimal st ructu re A1 in a lan gu age L ; for simplicity, we keep our ass umption t hat t he order of A1 is dense withou t end poin ts. W hat we said in the last section in introducing Theorem 9.4.3 s uggests t he followin g definition . Definition 9.5.1 Let
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330

• tp(v, w ) is finite in X if an d only if, for every ii E X, tp(M, ii) is fin it e; • tp(v, w) is uniformly finite in X if and only if there is a pos it ive in teger h such that , fo r every ii EX , the size of tp(M, ii) is at mo st h.

When tp(v, w) is finite in X , we can introduce two parti al functions tp_ a nd tp+ mapping any ii E X into t he minimal and the maximal element in tp(M, ii) resp ectively -provided t ha t tp(M, ii) is not empty, of course-. If X is definab le, then tp_ a nd tp+ a re definable as well.

Definition 9.5.2 Let n be a positive integer, X be a definable su bse t of M": A decomposition of X is a part it ion of X into fin itely many cells. If y ~ X is defi nable, we say that a decomposition P of X partitions Y when no cell in P overlaps both Y and X - Y. Here is a not her t echnical preliminary notion.

Definition 9 .5 .3 Let C be an open cell of M n (so an n-cell} , tp(v, w) be a fo rm ula fin ite in C . Call a point ii E C good for' tp(v, w) if and only if the following conditions hold : 1. for every b E tp(M , ii), there are an open box B ~ C containing ii and an open interval I containing b such that tp(A1 n+l ) n (I X B) is the graph of some continuous funct ion of B in I ; 2. fo r every b E M - tp(M, ii) , there is an open neighbourhood of (ii, b) in lU n+1 disjo int f rom tp(Mn+l ).

A poin t ii E C which is not good for tp(v, w) is called (with no particular imaginat ion) nasty for tp(v, w). Not ice t hat both good and nasty poin ts for tp(v, w) form definabl e subsets of t he cell C . At this point we ca n begin our proof. The followin g lem ma is its crucial step .

Lemma 9 .5.4 Let M be an o-minim al structure, n be a positive integer, C be a cell of M n.

(l) n For eve ry eleme nt i in a fin ite set I of indexes, let X i denote a definable subset of C . T hen there exists a decomposition of C partitioning each

X i.

9.5. THEIR PROOFS

331

(2)n Let f be a definable function of C into M . Then there is a decomposition P of C such that f is continuous on any cell of P. (3)n A formula
Lemma 9.5.4 immediately implies both Theorem 9.4.1 and Theorem 9.4.3.

Proof. (9.4 .1.1) M" is an n-cell. If X is a definable subset of M", then (l)n in Lemma 9.5.4 provides a decomposition P of M" partitioning X. Hence X is the (finite) union of the cells of P it contains. .. Proof. (9.4.1.2) Just apply (2)n to the cells of the decomposition of X given by 9.4.1 , 1. .. Proof. (9.4.3) This is just (3)n when C = M":

..

Now let us show Lemma 9.5.4. We proceed by induction on n .

(1 h This just rephrases o-rninimality. (2h This is the Monotonicity Theorem (in the weak form we saw in 9.2).

(4h If C reduces to a singleton , then the claim is trivial. Hence assume that C is an open interval ]a, b[ where -00 ::; a < b ::; +00. Suppose that, for some positive integer h, the set Y of the points c in ]a, b[ such that 1 1. Let c be an end point of Y in C, and put L + 1, 'Vc' El'. Assume for simplicity that c is a left endpoint of 1'. Define the following function 9 in 1': for every c' E 1',

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CHAPTER 9. O-MINIMALITY

g(c') is the least element d'

E

tp(M, c') such that d' =F gi(C') for every i

~

L.

9 is definable and its domain coincides with the whole interval I'. So, as observed in § 2, 9 has a limit in MU {±oo} when x -+ c+ . d

Moreover

= lim x --+ c+ g(x).

tp_(c') < g(c') < tp+(c')

Vc' El' ,

and so d cannot equal +00 or -00, in other words d E M. If then d = d; for some i ~ L , and consequently

1=

tp(d, c),

on the other side , we know that, for every c' E I' ,

g(c') =F gi(C') ,

g(c'), gi(C') E tp(M, c').

But we contradict in this way the fact that c is good for tp (recall Definition 9.5.3 , 1). Accordingly 1= -.tp(d, c), which again excludes that c is good for tp (this time by 9.5.3 ,2) . This yields the required contradiction. Hence (4h holds.

(3h The claim is trivial when C is a singlet on. Accordingly suppose C =

la,

b[ where -00 ~ a < b ~ +00. The set of th e elements c of C for which f/J is definable, and consequently is a finite union of singletons and open intervals in C. Of course, it is enough to show that tp(v, w) is uniformly finite on every interval in this decomposition. Accordingly we can even assume tp(M , c) =F f/J Vc E]a, b[;

tp(M, c) =F

in particular tp_ and tp+ are defined throughout ]a, b[. Owing to (2h , we can even suppose (up to replacing again la, b[ with a suitable subinterval) that both ip: and tp+ are continuous in ]a, b[. Now let Y denote the set of those points in C =la, b[ that are nasty for tp. If Y is finite , say Y = {co , ... , Ct } with a < Co < .. . < Ct < b, then ip: a nd tp+ are cont inuous in each interval la, co[, ]Ci, ci+d for i < t, ]Ct, b[, and every point in these int ervals is good for tp. So we are just in a position to apply (4h, and accordingly the size of tp(M, c) is constant throughout every interval, and, in conclusion, tp(v , w) is uniformly finite in la, b[. Hence it suffices to show that Y is finite. Suppose not. Anyhow Y is definable, and hence, by o-rninimality, it contains some infinite int erval.

9.5. THEIR PROOFS

333

Without loss of generality, we can assume that this interval just equals

C =]a, b[, and so that every point e of ]a, b[ is nasty for <.p. Then, for every e E]a, b[, there exists d E M satisfying one of the following conditions:

(i) d E <.p(M, e) but, for no pair of open intervals I and J containing e and d respectively, <.p(M 2) n (J X 1) is the graph of a function of I in J; (ii) d rJ. <.p(M, e) but every open box B containing (d, e) overlaps <.p(M)2 as well.

In both cases we say that (d, e) is a black sheep of <.p (of type (i) or (ii) according to whether (i), or (ii), holds). Observe that, for every e E]a, b[, there is a minimal dE M for which (d, e) is a black sheep of <.p. In fact , as <.p( v , w) is finite in ]a, b[, there are at most finitely many d E M such that (d, e) is a black sheep of type (i). Consequently it suffices to prove that , if dl, d2 E M, dl < d2 and, for every d E]d l , d2 [, (d, e) is a black sheep of type (ii) of ip, then also (d l , e) is a black sheep of <.p. This is certainly true when d l does not belong to <.p(M, e) (in this case (dl , e) is a black sheep of type (ii)). Accordingly assume d l E <.p(A1, e) and fix two open intervals I and J containing e and d l respectively. If d E In]d l, d2[, then d rJ. <.p(M, e) and any open box including (d, e) -in particular, any open box in J X I including (d, e)- intersects <.p(M 2). Then <.p(M 2) n (J X 1) cannot be the graph of a continuous function of J in J , and (dl, e) is a black sheep of type (i) of <.p. Therefore we can consider the function g mapping any e E]a, b[ into the minimal d E M for which (d, e) is a black sheep of <.p. g is definable, and so, owing to (2h, we can find an open interval I in ]a, b[ such that g is continuous in I and one of the following conditions holds:

(iii) for every e E 1, (g(e), e) is a black sheep of type (i) of sp; (iv) for every eEl, (g(e), e) is a black sheep of type (ii) of <.p. As before, we can suppose that I is just ]a, b[. Assume (iii). Introduce two functions gl and g2 as follows. For every e E]a, b[,

• glee) is the maximal element dE <.p(M, e) satisfying d < gee), if such an element exists, and is

-00

otherwise;

• g2(e) is the minimal element dE <.p(M, e) satisfying d > gee), if such an elements exists, and is

+00

otherwise.

CHAPTER 9. O-MINIMALITY

334

Both gl a nd g2 are definable (in the obvious sen se) ; moreover we can suppose that gl is cont inuous or const a nt ly -00 in ]a, b[ and , simila rly, g2 is continuous or constant ly +00 in ]a, b[. C hoose c E]a, b[, d1 , d2 E M such that By the definit ion of gl , g2 and the continuity of gl, g2, g, t here exists a n int erval I ~ ]a, b[ containing c s uch t hat, for every c' E J,

Hence again t he definition of gl and g2 implies that
• gl (c) is the maximal elem ent d E g(c). Both gl a nd g2 have do mai n ]a, b[ becau se, for every c E]a , b[, .p: (c) < g(c) <
9 (c') E .I,

gl (c'} , g2 (c') rJ. J.

Hen ce
ti

>

1. Ass ume

(l) j , (2) j , (3) j and (4)j for 1 ::; j < n .

(l) n Let C be a cell of M n. For every i E I let X i be a definable subset of C. We a re looking for a decomposition of C part it ion ing each X i. If C is a k-cell for k < n , then there is some defin ab le hom eomorphism ITa of C onto a cell C' of M n-1 . By (1)n- 1 t here is a decomposition p I of C' part ition ing

9.5. T HEIR PROOFS

335

eac h 7!'C (Xi ). Through 71'01 , p' can be lifted to a decomposition P ofC partitioning every X i. So ass ume t hat C is ju st an n-cell . Consequ ently t here are a n (n - I )-cell C' of A1n - 1 and two funct ions f and 9 satisfying 9.3.1, 2; in part icular

C = Ha , b) E M " : a E C', f(a) < b < g(a)}. Let 7!'C be the projection of C onto C'. By (1)n-1 there exists a decomposition P' of C ' partitioning each 7!'c(Xi)' For every Y' E P' , let Y = {(a,

b) E C : a E Y'} .

Clearl y these sets Y partition C when Y' ra nges over P'. So it suffices to show that , for every Y ,

(v) t here is a decomposition of Y partitioning eac h set X i for which 7!'c(Xi)n

Y'-# 0.

To simplify t he notation , assume without loss of generality that 7!'c(Xi) Y' -# 0 for every i E I. Fi x i E I and , for ever y a E Y' , conside r

n

X i(a) = {b E M: (a, b) E Xi} ' Th ere is a formu la
ft , ... ,

n,

mapping any a E Y' int o t he first , . .., t he hi-th eleme nt of Bi (a). All t hese functi ons a re definable; by (2) n-1, we ca n assume that t hey are cont inuous on Y'. Unless partitioning agai n each X i, we ca n also s uppose t hat, for every i and j in I , h = 1, .. . , hi and h' = 1, ... , hj , exactly one of t he following cases holds: Va E Y' , f i(a) = f~/(a) ,

336

CHAPTER 9. O-MINIMALITY

vs E Y' , vs E Y' ,

f~(a) < f~,(a) , f~(a) > f~/(ii) .

Accordingly we can rearrange t he fun ctions f~ (i E I , 1 ~ h ~ hi ) and form a new sequence

go , . .. , gt s uch t hat, for r ~ 8 ~ t, gr(a) < gs(a) for every a E Y'. But t his implies th at t he sets of t he tuples (a, b) E Y s uch that a E Y' a nd

f(a) < b < go(a), gs(a) < b < gs+t{a) (8 < t) , gt(a) < b < g(a) , b=gs(a)

( 8 ~t)

resp ecti vely, form a decomposition of Y parti tioning each X i , as claim ed. This concludes t he proof of (l)n . Acco rdingly ass ume fro m now on also (l)n .

(2)n Let C be a cell of A1n a nd let f be a definabl e function from C into

M . Wh at we have t o find is a decomposition P of C in cells wh er e f is continuous . If C is a k-cell for so me k < n , t hen t here exists a defin able hom eom orphism KC of C on to a cell C ' of J\I1 n- l . By (2)n-I ' ther e is a decomposition P' of C ' in cells such that f K CI is continuous on each of them , and K CI lift s P' to a decomposition P as required . So assume that C is an n- cell, in other words an op en cell in M n . Let

Cl be the set of tuples (a, b) E C such th at

defines a continuous fun ction f b on some op en box B of M n-l cont aining a and satisfying B X {b} ~ C, C 2 be t he set of tu pies (a, b) such t hat X n J----7 f (a, x n ) defines a function fa eit her constant or strictly monotonic on some op en int er val I con t aining b and satisfying {a} x I ~ X , a nd, in t he lat t er case, also f(l ) is an op en interval and f is a bijection bet ween I a nd f(l ).

9.5. THEIR PROOFS

337

Use (l)n and get a decomposition P of X partitioning both Cl and C 2 • So it is enough to show that f is continuous on every cell of P, and even on every open cell D of P, owing to what we observed at the beginning of this point. Then there are a cell D' of Mn-l and two functions ft, h satisfying 9.3.1,2 and

D = {(a, b) E l'vr

a E D' , ft(a)

< b < h(a)}.

Notice:

(vi) D < Cl . In fact let b E M satisfy (a, b) E D for some suitable a E u»:', As the domain of 9 includes D, fb is defined in some open subset of D'. By (l)n-l and (2)n-l, there is some open cell in D' where fb is continuous. For a in this cell, (a, b) E Cl. Hence D n Cl =I- 0. As P partitions Cl, D ~ Cl. Now we claim

(vii) D

~ C 2 and, for every a E D' , fit is either constant or strictly monotonic in]ft (a), h(a)[ (and, in the latter case, the image of]ft (a), h(a)[ is an open interval and fit is a bijection between these intervals).

D ~ C 2 can be shown by proceeding as for Cl. Now take the Monotonicity Theorem, there are a natural m and bl , that fl(a) < bl < .. . < b-: < ft(a),

a E D'. ... ,

Owing to bm E M such

fit is either constant or strictly monotonic in each interval J among jjj (a), bl [, ]bj, bj+l[ (for 1 ::; j < m) and ]bm , h(a)[, and, in the latter case, even fit(J) is an interval and fit induces a bijection between J and fit (J). Choose m minimal. If m = 0, then the only involved interval J is just ]fl(a), h(a)[, and (vii) is trivial. On the other side, if m > 0, then fit is neither constant nor strictly monotonic in any open interval containing bl ; as (a, bd E D and D ~ C 2 , (a, bt) E C 2 , which contradicts the definition of C 2 • So m = 0, and we are done. At this point, we are in a position to conclude the proof of (2)n' In fact, let (a, b) E D, J be an open interval containing f(a, b); we are looking for an open box B of Mn including (a, b) and satisfying f(B) ~ J. Owing to (vii), there is a closed interval I = [bl , b2 ] of M such that b is in the interior of I, I ~]ft(a), h(a)[ and fit(I) ~ J. By (vi) there exist two open boxes n, and B 2 of M n - l both containing a, and satisfying the additional conditions

n. X

{bi} ~ D,

f(Bi ' bi) ~ J

Vi = 1, 2.

CHAP TER 9. O-MINIM ALITY

338

Let B' be an open box of M n- 1 such that B' ~ B 1 n B 2 , ii E B'. Then f (B'x 1) ~ J . In fact, let a E B' , b' E I , so f( al ) < b1 becau se B' x{bd ~ D a nd b2 < 12 (;' ) because B'' X {b 2 } ~ D . Consequent ly

it (a < b1 < b' < b2 < 12 (al ) . ') Furthermore f (;' , bt) E J , f(;' , b2 ) E J. As fa! is eit her con stant or st rictly monotonic in ]f1(;') , 12(;')[, f (;' , b') E J. Accordingly f(B' X 1) ~ J. T his accom plishes our proof of (2)n. So we ca n ass ume even (2) n tr ue from now on. At t his point let us deal with

(3)n. (3)n Let i.p (v, W1 , ... , w n ) be a formula, C be a cell of M n such t hat i.p (v, W1, . . . , w n ) is finite in C . The case when the dim ension of C is strictly

smaller t ha n n ca n be handled as befor e. So we ca n limit our a na lysis to t he case when C is a n n-cell, in other words is op en. Let

Cl be t he set of those poin ts (ii, b) E C suc h t hat ii is good for i.p (v , W1 , ... , W n-1 , b) , C 2 be t he set ot the point s (ii, b) E C such t hat b is good for i.p (v, ii, w n ) resp ectively (ii a bbreviat es here (a1 ' . . . , an-1)). By (l) n' there is a decomposition P of C parti tioning both Cl an d C 2 • We claim

(viii) for every open Y in P, Y

~

Cl and Y

~

C2 •

In fact , let (ii, b) E Y , and let B be an open box in Y including (ii, b). Th e projecti on B' of B onto t he first n - 1 coordinates is an open box of M n- 1. By (3)n-1 , i.p (v, W1, . . . , Wn-1 , b) is un iforml y finite in B' . By (1)n-1 and (2) n-1, there are a n op en cell C ' ~ B' an d a posi t ive integer h such t hat, for everya E C', i.p (M, ai, b) has size h a nd t he functions mapping an y;' E C ' into ' the first , ..., the h-th elem ent of i.p(M, ai, b) resp ectively are continuous. Let;' E C'. It is easy to see t ha t ;' is good for i.p(v, W1 , . . . , W n-1 ,

b).

T hen (;', b) E Cl, and Y n Cl =1= 0. Hence Y ~ Cl . Y < C 2 is show n in a similar way. In conclusion, if Y is an open set of P, then for ever y (ii, b) E Y , d is good for i.p (v, W l , . . . , Wn-l , b), b is good for i.p(v, ii, w n). So it suffices to show wh at follows.

9.6. DEFINABLE GRO UPS IN O-MINIMAL S T RUCTURES

339

(ix) IfY is an op en cell of M n,
Simila rly (4)1 implies

In co nclusion

1
a nd t his accomplishes t he proof of (3)n' T he last claim t o be examined is (4)n' Bu t now t he proof is a direct con sequ ence of what we have just obse rved . In fact, recall t hat, if Y is an op en box of M n and t he poin ts (a, b) of Y a re good for

9.6

Definable groups in a-minimal structures

Which st ruct ures are definable in o-minimal models? The aim of this sect ion is ju st t o measure how complicated o-mini mal struct ures are up t o biin t erpret a bility, a nd so t o answer t he previou s q uestion , and to reali ze which groups , or rin gs, or ma nifold s are definabl e in t hem . In par t icul ar t he int er est in definable manifolds arises qui t e naturally from t he connection bet ween o-minimality and (a nalytic) geometry und erli ned at t he beginning

340

CHAPTER 9. O-MINIMALlTY

of this cha pt er. We saw what a manifold is (inside a n algebraically closed field K) in Section 8.5 , where we observed that every manifold in K is definable in K. In an ar bitrary o-rnin im al st ruct ure M a manifold may not be definable. Accordingly, first we fix wh at definable manifold means. It is just a finite family (V, (Vi, f;)i~ m) where m is a natural number a nd

* *

*

V =

Ui ~ m Vi

is t he at las,

each f i is a bijection from Vi onto a definable open subset U, = f i(Vi) of M" for some natural n independent of i , for i, j ~ m and i =Iin Ui;

* for

i , Ui,j

= f i( Vi n\!j) is, again, definable and op en

i, j ~ m and i =I- i , f i,j = f i between Ui,j and Uj,i'

i;'

is a definable hom eomorphism

After fixing this definition, let us look for groups and manifolds definable in o-rninim al st ructur es. Even at a first superficial sight on e can meet some non-trivial exam ples: for instance, it is quite obvious t hat , for a real closed field K, the linear groups GL(n , K) a re definable in K. Indeed , a sharp analysis displays some notable similarities with the w-stable framework. In particular, by ad apting the Hru shovski-Weil Theorem 8.6.2, Pillay showed Theorem 9.6.1 Let M be an o-m inimal st ructure, and g be a group definable in M. Th en g can be equipped with a (uniqu e) definable manifold structure mak ing it into a topological group. When M expands th e real field , t he manifold topology makes g into a locally E uclidean t opological group an d in conclusion, owing to the Montgomery, Zippen an d Gleason solut ion of Hilb er t 's Fifth Problem , into a Lie group. Definable groups hav e been intensively studied in o-rninimal st ructures . In particular we would like to mention an o-rninimal analogue of Cherlin 's Conjecture, proved by Peterzil , Pillay and Starchenko. Theorem 9.6.2 Let M be an o-m inimal structure, g be a connected group definable in M and having no definable non-trivial normal abelian subgroup. Then there is a definable isomorphism of g onto the conn ected component of an algebraic group over a real closed field. Not ably, a local version of Zilb er 's Conjecture is t rue in t he o-rninimal sett ing, as shown by Peter zil and Starchenko. Let us discuss briefly t his matter.

9.7. O-MINIMALITY AND REAL ANALYSIS

341

Giv en an o-rninimal M, call an element a E M trivial when there are no open interval/including a and no definable f from / 2 into I which is st rictly monotone in each variable. T heore m 9.6.3 (Tri chotomy Theorem ) Let M be an 'No-saturated o-minimal structure, a E M . T hen exactly one of the follow ing conditi ons holds :

(i) a is tri vial, (ii) there is some convex neighbourhood of a where M induces a structure of an ordered vectorspace over an ordered divi sion ring,

(iii) there is some open interval including a where M induces a structure of a real closed field .

9.7

0-minimality and Real Analysis

In t his section, we introduce some new examples of o-minimal st r uctures . They concern some expansions of t he real field closely related t o Real An alysis and Geometry. Ind eed Model Theory meets t hese areas within o-rninimali ty, and provide s new ideas , new t ools and, definit ively, new pers pectives in studying t he involv ed st ructures . 1. R ex p

The first exam ple we wish to deal with is the most famous as well. It concern s the exponent iat ion in t he real field . We have seen in Ch apt er 2 Tarski 's Theorem showing that the t heory of th e real field R has the elimination of quantifiers in the lan guage L of ordered fields: accordingly

definable = sem ialgebraic in this set t ing. Tarski also gav e an effective procedure reducing an y formula

342

CHA PT ER 9. O-MINIMALITY

Tarski also proposed t he following que stion . Expan d R to a st ruct ure R ex p = (R, 0, 1,

+, " - , ~ ,

exp)

where ex p is the l-ary function mapping any real x into eX. Accordingly add a l-ary operation symbol (for exp ) t o L and denote by L ex p t he enla rged lan gu age: hence L ex p = L U {exp} = {O, 1,

+, " - , ~ ,

ex p }.

Conjecture 9 .7.1 (Ta rski) T he theory of R ex p is decidable.

On e ca n observ e t ha t the th eor y of R ex p is not qu an t ifier elimina ble in L ex p : this was shown by Van den Dries in 1982. However in 1991 Wilkie proved it s model completeness. Theorem 9.7.2 (Wilkie) T he th eory of R ex p is model complet e.

In particular , the definable sets in R ex p ca n be obt ained as follows. For n any positive integer , call a subset E of R " expon ential when it has t he form

for a suit able real polynomial f wit h 2n unknowns. Notice t hat expon enti al sets are closed und er finite un ion and intersect ion (as the points a nnihilating at least one of finit ely man y polynomials a re ju st the zeros of their product , and the poin t s annihilating a finite system of real polynomials are just the zeros of the sum of their squa res). But , according to the Van den Dries remark on (the failure of) qu antifier eliminat ion, the definable sets in R ex p are something larger than the finit e Boolean combinat ions of exp onent ial sets. So let us introduce subexpon ential sets. A subexponent ial set in R " is just t he image of an exponential set of R n +m (for som e m) und er t he proj ection map of R n +m onto the first n coordinat es in R " . Clea rly exponential an d subexponent ial sets are definable. What Wilkie showed is that s ubexponent ial sets are closed under complement. This implies model com plete ness, a nd prov es that in R ex p

definable = s ubexponential.

9.7. O-MINIMALJTY AND REAL ANALYSIS

343

At this point, one can use a theorem of Khovan skii saying that every exponential set, and consequently every subexponent ial set , has only finitely many connected components. Just apply this result to definable (equivalently subexponential) subsets of R and get Corollary 9 .7 .3 R exp (and its theory) are o-minimal. Later Ressayre gave a nice axiomatization of T h( R exp), showing t hat its mode l theory requires very simple global infor mat ion a bout exponentiation. But now let us come back to Tarski's Conjecture. W hat can we say about it ? Well, there is a famous conjecture in transcendental number theory, due to Schanu el a nd saying: Conjecture 9 .7.4 (Schanuel) Let n be a positive integer', al, .. . , an be complex numbers lin early independent over th e rational fi eld Q . Then the transcendence degree of Q (al ' .. . , an, e a j , •• • , e a n ) over Q is at least n. Remarks 9 .7 .5 (a) Schanuel's Conjecture has been proved in some particular cases, for exa mple when n = 1, or al , . .. , an are algebraic (Lindemann) . (b) 1, e are linearly independent over Q . Hence Schanuel's Conjecturewould imply t hat Q (l , e, el, ee), in other words Q (e, ee),has transcendence degree 2 over Q , and hence e, e" are algebraically independent. Nevertheless, as far as one presently knows, it is still an open question whet her e" is irrational. (c) 1, in are linearly independent over Q . Hence Schanuel's Conject ure would imply that Q (l , in , el , ei 7l" ) , hence Q (e, i1r) (as e i 7l" + 1 = 0), has transcend ence degree 2 over Q , and conseque nt ly that e and n are algebraically independent: but this is still an open question, as well know n. It is generally felt that a solution of Scha nuel's Conjecture is vary far , and should go beyo nd the present knowledge in Mathematics. However a positive answer to the qu estion of Schanuel would imply a solution of Tarski's Conjecture as well.

Theorem 9 .7.6 (Macintyre-Wilkie) If Schanuel's Conjecture holds, then Th( R exp) is decidable.

CHAPTER 9. O-MINIMALITY

344

2. R an

Now we deal with real an alytic fun ctions f. Her e we have t o be very ca reful in fixing our setting. In fact , we hav e t o recall what happens when we expa nd the reals by sin (or cos ): o-minim ality gets lost. However one sees t hat o-rninimality is preserved if we restrict t he dom ain of t he sinu s function t o a su itable interval] - ~ , ~ [ . Accordingly one takes a language L an enlarging the language L of ord ered fields by a l-ar y op eration sy mbol j for every function f a nalytic on som e op en subset U of R " containing th e cube [0, l]n (n ran ges , as usual , over positiv e integers , and th e only reason to choose [0, 1] instead of another interval is ju st to fi x and normalize our set t ing); th en one t akes the Lan-structure R an expanding the real fi eld R a nd int erpreting any symbol j in t he funct ion equalling f in [0, I]" a nd ass uming th e constant value 0 elsewhere . Notic e that L an is uncoun t abl e. R an is called t he real field with restri cted analytic function s. Theorem 9.7.7 (Van den Dri es) Th(R an ) is o-min imal.

model complete and

Van den Dri es ' a nalysis also determines what is definabl e in R an . In fact , t he definable sets a re exactly t he so called globally s ubanalytic sets. They are ob t ained as follows. Call a subset A of an a na lyt ic manifold X semian aly tic in X if t here is a n op en covering U of X s uch t ha t , for ever y U E U , A n U is a finite union of sets {a E U : f (a ) = 0, 9o(a), ... , 9k(a )

> O}

where f and t he 9 'S ar e ana lyt ic function s on U . At t his point call a subset B of X suban aly tic in X if there is a n op en covering U of X such that, for every U E U , B n U is th e image of some A semia nalyt ic in U X R'" by t he projection map from U X R '" onto U (here m may depend on Band U ). F ina lly call S ~ R " globally subanaly tic if it is suba nalyt ic in th e analyt ic manifold (P 1 (R) )n (where P 1 (R) is t he real projective line). Gabrielov showed a "theo rem of t he complement" for subanalyt ic sets in an analyt ic manifold X, ensuring th at t hey are ju st closed und er com plement . This is t he key result in showing t he mod el completeness and t he o-minimality of the t heory of R an' a nd also in proving th at

definabl e = globally s ubanalytic

9.7. O-MINIM ALIT Y AND REA L ANALYSIS

345

in R an. Does Th( R an) admit quantifier elimination? Denef and Van de n Dri es showed t hat the answer is positive, provided one extends the language L an by a symbol - I for the inverse function (with the usual convention 0- 1 = 0). Theorem 9.7.8 (Denef-Van den Dries) Th( R an) elim inates the quantifiers in t.. ; U {-I} . 3. R an,exp

F inally let us examine what happ ens when we expand t he reals bot h by t he exponent ial fun ct ion a nd t he rest rict ed analytic functions . Let L an ,exp = L an U { exp} the corresponding language, R an,exp t he result ing struct ure in Lan ,exp. First Van den Dri es and Miller, adapting Wilkie's wor k on exponentiation, pr oved Theorem 9.7.9 (Van den Dri es-Miller ) Th e theory oj R an,exp is model complete and o-minimal. Subsequently, Van den Dries, Macintyre and Marker fou nd a different proof providing a nice axiomatization of the theory of R an,exp in Ressayre's style. They got also quantifier elimination in a language extending L an ,exp by the logari th m function log . Theorem 9.7.10 (Van den Dries-Macintyre-Ma rker ) Th e theory o] R an,exp eliminates the quantifiers in the language L an,exp U {log} . Notably, t he logari t hm function cannot be ignored to obtain qu an tifier elimina t ion . T he Van de n Dri es-M acint yr e-Marker approac h also provides an explicit desc ript ion of t he definabl e sets in R an, ex p , following t he sa me lines as in t he cases before . Of course t hese examples are very far from exhausting a general displ ay of t he a-m inimal ex pansions of the real field (a wider information ca n be foun d in the references quoted at the end of t he chapter) . But t hey ca n illust rate how rich and int eresti ng t his research field is. Let us conclu de t his section with some final remarks partly exceeding the a-minimal limits. In fact, it is noteworthy that , although Model Theory and Real Analysis closely interact via o-minimality, Complex Analysis has

346

CHAPTER 9. O-MINIM A LIT Y

raised a lot of difficulties to a model t heoret ic t reatment . For inst a nce, whi le expanding the reals by exponentiation gives an o-mi nimal structure (by Wilkie's Th eo rem), (C , +, " - , 0, 1, ex p) defines the integers by the formula exp(27riv) = 1, so t hat there is a very lit tle hope to dominat e its first order t heory, its definable sets, a nd so on. Indeed , the zero sets of com plex an alytic fun ctions can be quite pathological. Some years ago , Boris Zilber proposed a satisfactory st rategy to develop th e mod el theory of the complex exponentiation, but his program needs som e very strong conjectures on t ranscendental numbers (even beyond Schanuel's P roblem). Zilber also followed a more successful approach, looking at analytic compact manifolds X rather than at an alytic function s: in fact , t hese manifolds can be viewed as first order structures in a language with a relat ion for any su banalytic subset of every power of X. In this setting one shows Theorem 9 .7.11 (Zilber) T he theory of a compact complex manifold eliminates the quantifiers and has a finite Morley rank.

9 .8

Variants on the a-minimal theme

Strongly min imal theories have a natural en largement to totally transcenden t al (i. e. w-stable) theories via Morley ra nk. In the ordered setting nothing is known ext ending sistematically o-minimality in a parallel way. However just the ord ered framework suggests several notions widen ing 0 minimality : they have been intensively studied in th e latest years. In pa rticular we want to discuss here briefly weak o-minimality. As said, we still work wit hin linearly ordered structures M = (M , ::; , ...). Recall that M is o-minimal when every definable subset D of M is a finite union of interval s (possibly with infinite endoints); not ice t hat intervals ar e convex. Definition 9 .8 .1 M is called weakly a-minimal when every definable subs et D of M is a finit e union of con vex (definable) sets. Remark 9.8.2 Of course, o-rnin imality implies weak o-minim ality. Moreover , among exp ansions of the real line (R, ::;), the converse is also true, and weak ly o-minimal just means o-m inimal. This is because (R , ::;) is Ded ekind comp lete, and every bo unded set has its own least upper bound and its own

9.9. N O ROSE WITHO UT THORNS

347

greatest und er bound ; in particular every convex set is an interval (in t he broader sense recalled before). However t here do exist weakly a-minimal structures which are not a-mini mal. Example 9.8.3 Take t he ordered field of real algebra ic numbers Ra. This is a real closed field , an d so an a-minimal st ruct ure. Add a 1-ar y relation select ing t he elements of Ra lying between - IT and IT (or, if you like, bet ween a ny two reals a < b wit h a or b t ranscende ntal) . The resulting st ruct ure is not a-minimal an y more, because D = {r E R a : -IT < r < IT} is convex and definable, but ca nnot be expressed as a finit e union of intervals with real algebraic end points. But act ua lly D is convex, and ind eed one can see that t he new structure is weakly a-minimal. Notably, every expa nsion of an a- minimal struct ure by convex subsets is wea kly o-minimal. This is a beau tiful result of Baisalov-Poizat , generalizing t he las t exa mple and answering in t his way a question of Cherlin. Oth er releva nt exam ples, ar ising from several frameworks in Algebra , ca n be proposed. By t he way, weak o-rninimality was first introduced by M. Dickmann in 1985, dealing wit h certain ord ered rings extending real closed fields. Not surprisingl y, weakl y a-minimal st ruct ures do not behave so well as 0 minimal do. In particular weak o-minimality is not preserved by eleme ntary equivalence (so t here are wea kly a-minimal struct ures whose t heo ry has some non wea kly a-minimal models) . Furt hermore Monotonicity and Cell Decomposition fail as well as existence and uniqueness of prim e mode ls. However some " wea ker" versions of these results ca n be recovered , and a relevant, alt hough not so fluent , th eory has been developed.

9.9

No rose without thorns

We have seen t hat a- minimal t heo ries admit an ind ep endence noti on related to algebraic closure and satisfying t he same basic proper ties (11 )-(16) forkin g ind epend ence has in sim ple t heor ies. However t hese ind ependence notions -forking indipend ence in sim ple t heories and algebraic ind epend ence in a-minimal th eories- were introduced in a different way and were develop ed ind epend ently. So a natural qu estion arises in Mod el Theory, Le. to

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348

find a new conce pt of ind ep end ence so con vincing t o satisfy (11)-(16) in most t heo ries and so general t o enla rge bo th t he pr eviou s cases. This is the cont ent of a recent work of Alf On shuus wh o, following suggest ions from Thomas Sca nlon, int ro du ced • a new noti on of indep endence (ca lled thorn-independence) a nd • a related class of t heor ies (named rosy theories) whe re t horn-i nde pe nde nce enjoys all t he bas ic ass umptions (11 )-(16) , so local character, sym met ry a nd so on. Rosy t heories include both sim ple and 0 minim al t heories, as well as fur ther relevant examples. Thorn-indep endence agrees wit h algebraic indep endence in t he o-minimal case a nd with forking ind ep endence in stable t heo ries: in fact , wh en t hese pages are written (at t he end of 2002 ) , it is not clear whether t horn- indepe nde nce eq ua ls forkin gindep endence even in the simple setting, although t his has been checked to be t rue in all t he known key examples of sim ple t heor ies. Notably symmetry, or also local character, is a key property towards ros iness. In fact , a t heo ry T is rosy if a nd onl y if t ho rn- inde pe nde nce satisfies sy mmet ry or local character .

9.10

References

O- minim al t heories where introdu ced by Van den Dries [166] a nd extensively st ud ied by Pill ay and Steinhorn in [129] and (together with J. Knight ) in [74]. Van den D ries' book [169] provides a nice a nd stim ulating treatment of o-rnini mality, also describing it s genesis a nd moti vat io ns, a nd em phas izing it s con nect ions wit h real analysis and real alge braic geometry. T hese int er acti ons ar e illust rat ed in t he mor e recen t survey [170], where t he 0 minimal ex pa nsions of t he real field are examined. Also [109] gives a sho rt, but captivating introd uction to o-rninim ality, A ge neral proof of Monotonicity Theorem 9.2.1 can be found in [129]. Wilki e 's Complement T heorem 9.4 .5 is show n in [178], a nd t he Pill ay-St einhorn T heorem 9.4.7 on prime models in o-rnin imal t heories is in [129] again . Laura Mayer 's sol ution of Vaug ht Conjectu re in t he o-rn inimal setting is in [111]. Pill ay 's a na lysis of t he groups defina ble in o-minimal struct ures (T heorem 9.6.1) is in [125], while t he o-rninimal analogue of C herlin Conjecture shown by P et erzil , Pill ay and Starchenko (9.6.2) is in [123] a nd the P et erzilStarchenko Theorem 9.6.3 is in [124].

9.10. REFERENCES

349

As already said [170] provides a general sur vey of the main o-minimal expansions of the real field, and a rich and detailed bibliography on this matter. In particular Wilkie's Theorem 9.7.2 on R ex p is in [177], and the Khovanskii's results on exponential sets in [70]; Ressayre's approach to the theory of R ex p can be found in [139]; the Macintyre-Wilki e Theorem on the decidability of the theory of R ex p and its relationship to Schanuel's Conjecture is in [99]. The o-minimality of the theory of Ran is proved in [167], using [49], while the Denef- Van de n Dries treatment -including the quantifier elimination result in a language with the inverse operation- is in [30]. The o-minimality of the theory of Ran,exp is already shown in [171], but the subsequent analysis of Macintyre, Marker and Van den Dries is in [97]. Zilber 's Theorem 9.7.11 on compact complex manifolds can be found in [183]; see also [128] . Now let us deal with weak o-minimality. This was introduced in [32], and extensively examined by Macpherson, Marker and Steinhorn in [100]. The nice theorem of Baisalov-Poizat (mentioned at the end of Section 9.8) is in

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Index Cherlin 's conject ur e, 303, 310 Ch evalley 's theorem , 59 Church-Turing thesis , 39, 80 class eleme nt ary, 22, 105, 221 of fields, 25, 103 of finite sets, 22 of graphs , 280 of infini te sets , 22 of mo d ules, 27 of nilpote nt groups of class 2, 281 of ordered fields , 26 classifi cation problem, 20, 221-227 , 289 closure algebraic, 133, 172, 287, 328 definable, 133, 172 differential , 209 real, 175 cohe~, 238, 252, 290 com pact ness, 18 theorem of, 18, 243, 325, 327 connecte d com ponent , 189 constructible set , 38, 58, 118, 134, 198, 199, 220, 271, 292, 295 construc t ion , 199 coordinate chart , 299 cylindrical algebra ic decomposition, 80

adiacency relation , 280 algebraic geometry, 291 complex , 37 real , 38, 341 algebra ic numbers com plex, 87 real, 87 algebraica lly ind ep endent set , 170 alg orithm , 39, 80 amalgamation, 231, 234 analysis complex, 345 annihilator , 42 Artin 's conj ect ure , 86, 96-103 Art in 's theorem , 95 atl as , 299 atomi c set , 198 automorphism , 5 Ax 's theorem , 60 Ax-K ochen-Ershov theorem , 100 back-an d-forth property, 10 basis , 169 Baur-Monk theorem , 70, 74 biinterpretability, 279-286 , 288, 290, 339 Boolean space, 136, 180 boundary point, 322

definable manifold , 340 definable sets, 35-42 , 45, 59, 78, 112, 121, 126, 129, 133-136 , 222, 280, 320, 344, 345 Boolean algebra of, 36, 134, 143, 219 conv ex , 346 definably connected , 322

Canto r 's t heorem , 12 Cant or-Bendixson rank , 155, 159 cell, 320 cell decompositi on th eorem , 324, 329 charact er finit e, 228, 233 local , 228, 233

363

364

INDEX

ind ecomposable, 190 dep end ence, 227 depend ence relation, 180, 184, 328 algebraic, 170 linear , 169 derivation , 109, 307 descriptive set th eory , 222, 289 difference degree, 289 differential algebra, 209 differential degree , 287 differentially algebraic, 210 differentially transcendental , 210 dimension, 294-297 of a definable set, 328 of a tuple, 328 of a vectorspace, 169 dimension theory , 327 effect ive pro cedure, 40 , 44 , 79 Ehrenfeucht-Mosto wski mod el, 274, 290 eleme nt ary cha in t heorem , 18, 144 eleme nt ary equivalence, 324, 347 eliminat ion of im agin ari es, 125, 194, 307 uniform , 125, 300 elim inat ion sets, 43 embedding, 4,9-18 , 85, 88, 103 elementary, 13, 85, 103 existe nt ial, 14, 89, 105 endomorphism, 5 exist ence property, 270 expo nent ial set , 342 extension , 5, 228, 234 eleme nt ary, 14 non forking, 237, 238, 240 field w-stable, 220 algebra ic closure of, 33, 86, 93, 103, 117 algebraically closed , 26, 32, 45 , 80- 82, 86, 103,105, 117, 134, 160, 184, 230, 239, 255, 284, 287, 291, 294

elim ination of im aginaries for , 126 mod el com pleteness of, 91 qu an tifier elim inat ion for , 5461 with an automo rphism, 117 com plex elim inat ion of quantifiers for , 82 const ant subfi eld of, 110, 116 difference, 115 existe ntially closed , 117 inversive, 116 differential , 109, 209 existe nt ially closed , 110, 115, 209 differential closure of, 111, 115 differenti ally closed , 106, 109-11 5, 119, 134,211 ,220 ,239 ,242 , 290, 291, 304, 307 axioms for, 111 existe ntially closed , 239 ,290 , 307 with an automo rphism , 291 fixed subfield of, 116 formally real , 95 Henselian , 99 imperfecti on degree of, 114 locall y com pact, 98 of complex numbers, 33 of meromorphic fun cti ons , 110 of ra ti on al fun ct ions , 109 ord ered , 129, 316 , 327 o-minimal ,317 real closure of, 180 p-adic, 86 perfect , 113 pseud ofinite, 26, 102, 118, 288, 290 real , 321 elim ination of quantifiers for , 82 real closed , 43 , 78- 82 , 87, 95,104, 105, 129, 134, 239, 313, 317 elim ination of im aginaries of, 129-131

365

INDEX model completeness of, 93 quantifier elim inat ion for , 6168 theory of, 27 residue, 101, 102 separably closed , 112-115 , 119, 239, 242, 290, 307 structure of, 3 superstable, 220 transcenden ce basis of, 33 transcendence degree of, 33, 86, 184 valued , 99, 100, 102 of p-adi c numbers, 102 field of definition , 194 finite in uniformly, 330 Fi scher-Rabin theorem , 80 forking, 233, 237, 287 formal Laurent ser ies, 98 formul a , 5 positive primitive, 41 atomic , 6 existential, 8, 88 finite, 330 normal form of, 8 quantifier free , 8 T -equivalent , 43 true in a structure, 5, 7 universal , 8, 88 Fr aisse's th eorem, ] 3 Frobenius morphism, 113, 116, 118, 174 function definable, 297-299 elementary, 16, ]35 , 196 function field , 306 generic element , 190 geometry, 286 graph , 280 random , 231 structure of, 3 group pp-definable , 283

w-st abl e, 184-192 , 220, 302, 303, 310 abelian- by-finite , 284 algebraic, 301-304 cent re of, 121 definable, 302, 317 existe nt ially closed, 106, 119 lin ear, 121,301 linear alg ebraic, 302 o-minimal, 315 of finit e Morl ey rank , 303, 310 of finit e type, 305 ord ered abelian , 101 divisible , 313 quotient , 122 sp ecial, 122 structure of, 3 torsionfree abelian, 222, 289 groups elementary class of, 105 Godel Incomplet eness theorem , 40, 280 heir , 238, 252, 290 Hensel's lemma , 98 Herbrand universe , 19 Hilbert 's Basis t heorem , 38, 292 Hilbert 's Nullstellens atz , 82, 86, 93, 119, 292 Hilbert's Seventeent h problem , 86, 95, 119 homomorphism, 4 pure, 151 Hrushovski- Weil theorem , 303, 310, 340 ideal differ ential , 210 prime , 210 prime , 293 radica l, 292 ideal eleme nt , 139 indep endence, 287 independence system , 236 good , 231, 249 ind ep end ent set , 177

366 Ind uct ion Principle, 7, 29 infinite sets theory of , 22 inj ectivity-implies-surj ectivity th eorem , 60 interpret ability, 279 invariance, 228, 233 invariant statement , 70 , 73 invariant system , 225 irr edu cible components , 293 isomorphism , 5 partial , 10 Knight - P illay - St einhorn th eorem, 78 Kolchin constructible, 112 , 134 Kolchin topology, 112 Lagrange's th eorem , 43 , 280 language, 1 Lindstrorri's theorem , 7, 29 linear order expansion of, 313 linear orders , 78 class of, 23 dense, 52-54 , 93 elim inat ion of qu an tifiers for , 52 dense wit hout end points, 24, 32 d~ c ret e , 24, 47-52 , 93 elimination of qu antifiers for , 48 th eory of , 23, 218 linearl y independe nt set, 169 locall y modular , 288 Lowen heim-Skolem theorem , 28, 85, 182, 226, 274, 275 downward , 19, 29 Macintyre's theorem , 192, 284 manifold , 299-302 affine, 299 sem iaffine, 299 Manin-Mumford conj ecture, 305 ,310 model, 8

INDEX A-satu rated , 144 A-universal, 145 homogeneous, 147 minimal , 87 pri me, 87, 133, 196-209 ,216 ,220 , 254, 270, 328 saturated , 133, 143-150 , 180 weakly A-hom ogeneous, 145 model com panion , Ill , 115, 117, 215 module, 239 algebraically com pact, 152 ind ecomposable, 153 pure inj ective, 152 pure inj ect ive hull of, 152 modules, 27, 41 , 290 theory of, 68-76 monotonicity th eorem , 348 Mord ell's conjecture , 304 Mord ell-Lang conj ect ure , 304-310 Morl ey degree, 161, 189, 220, 241 of a type, 165 Morl ey rank , 158-168 , 180, 220, 230 , 241, 288, 294, 298, 307 , 313 , 346 of a ty pe, 165 Morley 's existe nce t heorem , 271 Morl ey 's th eorem , 133, 181,271 ,273279, 290 mor ph ism , 297, 302 n- typ e , 293

com plet e, 137 Neumann's lem m a , 70,72 nilradical, 108 non -for king extension , 252, 259 number field , 304 om itting types th eorem , 157, 198, 217 op en box , 322 open mapping th eorem , 259, 273 ord er property, 237, 243 ordered field , 104, 313 of rea ls, 65, 95 real closur e of, 93 st ructur e of, 3

INDEX

367

orthogonality, 255 p-adic topology, 96 p-b asis , 114 P=NP problem , 80 parameters , 35 partition , 330 polish space, 222 polynomial , 110 difference, 116 differential, 110, 115 sep arable, 113 pp-elimination of quantifiers , pp-formula, 41, 69, 151 pp-type, 152 pr edecessor , 47, 263 presentation , 264 proj ective space, 299 Priifer group, 155 pure injectivity, 150

70~ 76

quantifier elimination , 43-82 , 87, 88, 345 random graph t heory of, 239 rank , 158 real analysis , 341 recursive sets, 39 recursively enumerable set , 40 residue field , 99 Ressayre's Uniqueness theorem , 202,

271

ring commutative, 107 different ial , 109 existe nt ially closed , 106 ordered , 317 reduced , 109 rmgs eleme ntary class of, 105 Robinson 's test , 88-91 Rudin-Keisler relation, 254, 290 Ryll-Nardzewski 's theorem , 217, 220 Schanuel's Conjecture, 343

semialgebraic set , 38, 67, 134 sem idefinite positive, 95 sentence, 5, 7 separant , 211 Shelah's uniqueness theorem , 220, 270273,290 sign change prope rty, 98 small subset of n, 148 smooth equivalence relation , 223 spectrum function , 226 stationarity over models , 236 stationary logic, 290 strong homogeneity th eorem, 147 strongly minimal set , 163, 168-172, 288, 304 structure X-definable, 121 X-interpr etable, 123 w-stable , 185, 220, 291, 308 R an,etcp, 345 Ran , 344 R exp , 341 basis of, 177 definable, 40, 121, 302 dimension of, 177 existent ially closed , 105, 110, 119 expansion of, 5 exte nsion of, 85 interpretable, 123 locally modular, 283 minimal , 77, 168, 176, 179 o-minimal, 78, 178, 179, 313, 318 restriction of, 5 simple, 232 stable, 236 strongly minimal, 168, 256, 282, 287, 292, 308 superstable, 241 trivial , 283 two-sorted , 100 universe of, 2 unstable, 236 structures, 2 elementarily equivalent , 10 subanalytic set, 344

368 globally, 344 subexpon ential set , 342 subgroup definable connecte d, 188 pp-d efinabl e, 41, 69 submodule pure, 151 subst ructure , 5, 85 element ary, 14 existe nt ial, 15 finitely generated , 5 generated, 5 successor , 47, 263 symmet ry, 228, 234 Tarski 's t heorem, 54, 296, 341 Tarski-Seidenb erg th eorem , 38, 67 Tarski-Vaught th eorem , 17, 158 Terj ani an 's counte rexa mple, 100, 102 terms, 6 t heory, 21,22 A-categorical , 28 of vecto rspaces, 34 ACF, 26,45 , 54, 58, 88, 91, 111 ACFA, 117, 118, 131,231,288 ACFo, 33, 57,87 ACF p , 26, 32, 33, 45, 57,77 , 88, 149, 169, 184, 197,21 9, 261 D C F o, Ill , 112, 197, 209-217 , 287 D CFp ,115 dLO ,52 dLO+, 48, 50, 51, 77, 87 DLO- , 24, 32, 52, 53, 160, 174, 181, 182, 217, 218, 226 RC F , 27, 33, 61, 65, 66, 80, 87, 88,9 1, 93, 95, 104, 111, 129, 175, 197, 341 SCFp, 114 T p , 102 Tn, 27 w-stable, 181-184,220 ,230,242261, 270, 346 K. T ' , 169

INDEX n T , 68, 70, 73 Booleanl y A-categorical , 219 categorical, 133, 274 classifiabl e, 225, 227, 261-270 complete , 19, 30, 102 complet ions of, 31, 45 consistent, 21 decidable, 40, 44, 342 deep , 268 depth of, 268 ind epend ence syst em of, 228 model companion of, 105, 117 mod el complete, 34, 46, 85-96, 102, 103, 117, 119, 212, 288, 342, 344, 345 not classifiable, 237 o-minima l, 78-79,178 ,226 ,234 , 313, 344, 345, 348 of a 1-ar y function , 262 of a class of models, 21 of a model, 31 of an equivalence rela tion , 256 of infinite sets , 32, 282 of two equivalence relations, 265 present abl e, 264 rich, 19 rosy, 348 shallow, 268 simp le, 227- 235, 249, 289, 290, 328 st abl e, 235-239 , 243 strongly minimal , 76-77,163 ,168 , 182, 184,220 ,221 ,225 ,227 , 236, 239, 261, 274, 346 sup erstable, 239-242 , 252 totally t ranscendent al, 133, 181184, 346 un stable, 236, 237 weakly o-minimal, 346 t horn-i ndependence, 348 top ological space com pact , 140 Hausdorff, 140 totally disconn ected , 140 t ra nscendence bas is, 170, 295

369

IND EX t ra nscendence degree, 33, 92, 149, 170, 294 transitivity, 228, 234 tree, 264 rank of, 267 well founded , 267 trichotomy t heorem , 341 Turing machine , 39, 79, 80 ty pe, 133, 136-1 43 RK-m inim al , 255 algebraic, 141, 166 complet e, 137 consiste nt , 137 definabl e, 238, 244 dep th of, 268 gen eric, 190, 294 isolated , 141, 156, 198 reali zati on of, 139 regular , 241 stabilizer of, 188 stationary , 240 st rongly regul ar , 256 type of, 138 ultrafilter , 136 uniqueness theorem , 146 un iversal domain , 86 un iversal ity t heorem , 145 valuation map , 99, 101 valuation rin g, 99 variety abelian , 304 algebra ic, 37, 285, 292, 299 irredu cibl e, 293, 294, 298 Vau gh t 's conject ure , 290, 329 Vau ght 's t heorem , 32 vectorspace, 230, 239 st ructure of, 4 theory of quant ifier elim inat ion for , 75 weak homogeneity th eorem , 145 well ordered sets class of, 24

word probl em , 107 Zar iski geome t ries, 285 Zariski st ruct ure, 286 Zari ski to pology, 38, 117, 292, 296, 297, 302 Ziegler sp ect ru m , 154 Zilber 's conject ur e, 279-286 ,289 ,309 , 340 Zilber 's Indecomposability theorem , 190, 309 Zorn 's lemma, 94

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