annalisa marcja, carlo toffalori a guide to classical and modern model theory trends in logic 2003

8/12/2019 Annalisa Marcja, Carlo Toffalori a Guide to Classical and Modern Model Theory Trends in Logic 2003

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A GUIDE TO CLASSICAL AND MODERN MODEL THEORY


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TRENDS IN LOGICStudia Logica Library

VOLUME 19Manag ing EditorRyszard W6jcicki, Institute ofPhilosophy and Sociology,

Polish Academy ofSciences, Warsaw, PolandEditorsDaniele Mundici, Department ofMathematics Ulisse Dini ,

University of Florence, ItalyEwa Orlowska, National Institute of Telecommunications,Warsaw, PolandGraham Priest, Department ofPhilosophy, University of Queensland,

Brisbane, AustraliaKrister Seger berg, Department ofPhilosophy, Uppsala University,SwedenAlasdair Urquhart, Department of Philosophy, University of Toronto, Canada

Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,Germany

SCOPE OF THE SERIESTrends in Logic is a book seri e s co ver ing essen tia lly th e sa me area as th e j ourn alStudia Logica - that is, co nte mporary formal logic and its applica tio ns andre la tions to othe r di sciplines. The se i nclude artifi ci al int elli genc e, in formatics,cogni tive sci e nc e, phi losophy of science , and the ph ilosoph y of langu age.How ever , thi s li st is not ex ha us tive, mor eo ver , the ran ge of applications, comparisons and sources of inspirati on is open a nd e volves ove r ti me .

Volume Edito rRyszard Woj:k k i

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


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A GUIDE TO CLASSICALANDMODERN MODEL THEORY

byANNALISA MARCJAUniversity of Florence, Italy

and

CARLO TOFFALORIUniversity of Camerino , Italy

KLUWER ACADEMIC PUBLISHERSDORDRECHTI BOSTON I LONDON


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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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Printed in the Netherlands.


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PrefaceThis book deals with Mod el Theory. So the first que s tion t hat a possible,recalcitrant reader might ask is ju st : What is Mod el Theory? Which are it sintents a nd applications? Wh y should one try to learn it? Another , mor epa rticular que stion migh t be th e following on e. Let us ass ume, if you like ,t h at Mo de l Theor y d eser ves so me a tte nt io n. Wh y should one use thi s bookas a guide t o it ?T he an swer t o t he former qu esti on may sound problema ti c, bu t it is q uit esimple, at leas t in ou r opinion. For , Mode l Theor y has been dev elopin g ,since it s bir th , a numb er of methods a nd co ncept s t hat do ha ve t heir int r insic relevan ce, but also provide fruitful an d notable applications in variousfields of Mathematics. We could mention her e it s role in Algeb ra an d Algebraic Geom etry, for instance the analysis of differentially closed fields an dth e resul t s on t he differ ential closur e of a differ en ti al field ) , or p-adic field s and th e asy m pt otic solut io n of Art in s Conjecture), as well as th e recentHrushovski s mod el t heoretic approach to classical problems, like MordellLang s Co njectu re or Manin-Mumford s Conject ure.So Model Theor y is to da y a lively, sprightly and fertile research a rea, w hic hsurely deserves t he atte nt ion of t he mathem atical world and, consequent ly,its own references. This recall s t he latter question above. Act ually ther e doex ist so me excellent t extbooks ex plaining Mod el Th eory, suc h as [56] a nd[57]. Also Poizat s book [131] should b e menti on ed; it wa swritt en mo re th anten yea rs ago, bu t it is st ill up-to-date, an d it has been recen tl y t r a nsla te din En glish. In add ition more specialistic references t reat adequat ely somepa rticular fields in Model Theory, s uch as stability t heory, simplicity t heory,o-minimality, classification th eory an d so on.Nevert heless, we believe t hat t his book has it s own role a nd its own originality in t his setting . Ind eed we wish t o address t his work not onl y to t heexperts of th e area, bu t also , and mainly, to youn g people having a basicknowledge of mod el t he or y a nd wishing to pr oceed toward s a deeper a na lysis , as well as to mathematicians which are not directly involved in Mod el

v


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VI PREFACETheory but work in relat ed a nd overlapping fields , s uch as Algebra and G eometry. Accordingly we will emphasize t he frequ ent an d fruitful connectionsbetween Model Theor y a nd t hese branch es of M ath em a tic s d iffere nt ia llyclosed fields, Artin s Conj ecture, Mordell-Lang s Conjecture an d so on ). Ineach case, we aim at giving a detailed report or , a t leas t , at sketching th emain ideas an d techniqu es of th e model t heore tic approach.Our book wishes also to follow a historical perspective in introducing ModelTheory. Of course, t his do es not mean to provid e a full history of ModelTheory although s uch a project could be interesting and wor thy of so meat tention), but ju s t to inser t any ba sic concept in th e historical fram eworkwh ere it was bor n , and so to better clarify the reasons why it was introduced.Hence, after shortly recalling in Chapter 1 basic Mod el Theory structuresan d theories, compactness and definability), we deal in Ch apter 2 withquantifier elimination, in particular with the work of Alfred Tarski on algebraically closed fields and real closed fields. We will discuss the rol e ofquantifier 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, an d so on).C ha pt er 3 will be concerne d with Abr ah am Robinson s ideas: mod el co mpleteness, model companions, existent ially closed structures . We will conside r again algebraically closed fields an d real closed field s, bu t we will illustrate also other crucial classes, like differentially closed fields, sep arablyclosed fi elds, p-adically closed fields and, finally, existent ially closed difference fi elds a rather recen t ma t t er , with som e rem ark abl e a pplicat io ns t oAlgebraic Geometry).C ha pt er 4 deal s with imagin ary eleme nt s . They a re esse nt ia lly classe s ofdefinable equivalence relations in a structure A , so elements in some quotien tstructure. We describe Shelah s construct ion of Aeq , englobing these classesas new element s in th e whol e st r uct u re , and we show that th ese imagin ar yelements can be sometimes eliminated, because the corresponding quotien tsca n be sim ula t ed by som e s uit a b le definable su bsets of A .Chapters 5 and 6 are devoted to Morley s Theorem on un countable categoricity . Actually it s proof will be given only in C ha pt er 7, but here wedescribe Morley s ideas -algebraic closure, totally transcendental th eories,prim e models , an so on- and we illustrate t heir richness an d th eir applications.We will be led in th is way t o on e of t he main topics in Mod el Theory, namelyt he C lass ifica t io n Pr ob lem. We will ex pla in in C ha pter 7 t he more relevan tideas in t h e formid abl e wo rk of Sh elah on t h is m at t er sim plicity, stability,superstability , mod ula rity) , a nd we will di scu ss t heir significa nce in so me


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PREFACE vuimportant algebraic classes, like differential fields, difference fields, an d soon. We wish also to deal with the Zilber program of classifying structuresup to bii nterpretabil ity, in particular with Zilber s Conjecture on stronglyminimal sets, an d its brilliant solution du e to Hrushovski.Also Chapter 8 largely owes to Hrushovski. In fact, after illustrating inmore det ai l t he n at ur al connection between Model Theory an d AlgebraicGeometry, we will describe th e Hrushovski proof of Mordell-Lang conjecture;we will refer very quickly also to the Hrushovski solution of the relatedManin-Mumford conjecture. In particular we will realize how deeply ModelTheory, actually both pu re Model Theory an d Model Theory applied toalgebra are involved in these proofs.The final Chapter is devoted to a comparatively) recent and fertile areain Model Theory: o-minimality. We will expound th e basic results on -minimal theories, an d we will discuss some intriguing developments, including Wilkie s solution of a classical problem of Tarski on the exponentiationin the real field.We as su me some famili ari ty wi th the basic not ions of Algebra, Se t Theoryand Recursion Theory. [65], [66] or [78], an d [121] respectively are goodreferences for these areas. Incidentally, let us point out that we ar e workingwithin the usual Zermelo - Fraenkel axiomatic system, including th e Axiomof Choice. We also assume some acquaintance with basic Model Theory,such as it is usually proposed in an y introductory course. However, Chapter1 is devoted, as already said, to a short an d somewhat informal sketch ofthese matters.As it s title states, this book aims at being only a guide. We do not claimto provide an exhaustive treatment of Model Theory; indeed ou r omissionsare likely to be much more numerous an d larger than the topics we dealwith. But we have aimed at giving an al most complete report of at leasttwo crucial subjects w-stability an d o-minimality), and at providing th ebasic hints towards some conspicuous generalizations such as superstability,stability, an d so o n) .In a similar way, we have treated in detail some key algebraic examples algebraically closed fields, real closed fields, differentially closed fields incharacteristic 0), but we have provided at least some basic i nf or ma tio n onother relevant structures like p-adic fields, existentially closed fields withan automorphism, differentially and separably closed fields in a prime characteristic). In conclusion, we do hope that the ou tc om e of ou r work is asufficiently clear an d terse picture of what Model Theory is, an d providesa report as homogeneous an d general as possible. Incidentally, let 11S say


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viii PREFACEt hat this book is not a lit er al t ranslatio n of t he form er it ali an ve rs io n [108];all t he material was revised a nd rewr it ten ; ou r t reat ment of so me t o pics ,like qu an tifier eliminat ion a nd mo del com pleteness , a re entire ly new ; an dwe ha ve a dde d some relevan t matters , such as prime mode ls a nd Morley sT heo re m on un cou nt a ble categorical t heories.


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Contents

Structures1.1 St ru ctures1.2 Sentences1.3 Embeddings1.4 The Compactness Theorem1.5 Elemen tary classes and t heories1.6 Com plete theories1.7 Definable sets1.8 Reference

2 Quantifier Elimination2.1 Elimin ation sets . . .2.2 Discrete linea r ord er s .2.3 Den se linea r orders . .2.4 Algeb raically closed fields an d Tarski)2.5 Tarski aga in: Real closed fields . . . .2.6 pp- elimination of quantifiers and modules2.7 Strongly mini mal th eorie s .2.8 o-minimal theories .2.9 Computational aspects of q. e.2.10 References .

3 Model Completeness3.1 An introduc tion .3.2 Abraha m Robinson s t est .3 .3 Mo del co mplet e ness an d Algebra3.4 p-adic fields a nd A rt in s Conjecture .3.5 Existentially closed st r uct u res .3.6 DCFa .

IX

1159

1820303542434752546168767879828585889196

103109


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x3.7 SCFp and DCFp3.8 ACFA . . .3.9 ReferenceEliminat io n of imaginaries4.1 Interpretabilit y .4.2 Imaginary elements . . . .4.3 Algebraically closed fields4.4 Rea l closed fields . . . . .4.5 The elimin ation of imaginaries sometim es fails .4 .6 Reference . . . . . . . . . . . . . . . .

5 Morley rank5. 1 A tale of two cha pt ers5.2 Definable set s 5.3 Types .5.4 Saturated mod els . . .5.5 A parenthesis: pure injective modules5.6 Omi t ting ty pes . . . . . .5.7 The Morley rank at last .5.8 Strongly minimal sets .5.9 Algebraic closure and definable closure5.10 References .

w stab ility6.1 Tot ally transcend ent al t heories6.2 w-st able g ro ups6.3 w-stable elds .6.4 Prime models .6.5 D C F revisi ted6.6 Ryll-Nardzewski s Theorem and other t hi ngs6.7 References .

7 Class ify ing7.1 Sh elah s C lass ification Theory7.2 Simple t heories .7.3 St able theori es .7.4 Superstable t heories7.5 w-stab le th eories 7.6 lassifiable th eories .

CON TENTS1121151192121123126129131132 33133133136143150156158168172180 818118419219620921722022221227235239242261


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ONT NTS

7.7 Shela h s Uniqueness Th eor em .7.8 Mo rIey s Theorem .7.9 Biin t er pr et abi li ty a nd Zilber C onj ec tu re7.10 Two algebraic examples7.11 References .

8 Model Theory and Algebraic Geometry8.1 Int roduction .8.2 Algebraic varieties ideals types . .8.3 Di mensio n and MorIey rank . . . .8.4 Mor phisms and definable functions8.5 Manifolds .8.6 Algebraic gro ups .8.7 The MordelI-Lang Conjecture8.8 References .

9 O minimality9.1 Int rod uct ion .9.2 The Monotonicity Theorem9.3 Cells .9.4 Ce ll decomposition a nd other t he or em s .9.5 Their proofs .9.6 Defina ble groups in o-rninimal structures9.7 O-minim ali ty and Rea l Analysis .9.8 Variants on th e o-minimal t h eme9.9 No rose wit hout thorn9.10 References .

BibliographyIndex

xi270273279286289292912922942972993013043103 331331832032432933934134634734835363


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Chapter Structures1.1 StructuresThe aim of t his chapter is to sket ch ou t basic mo del t heory. We wish tos um ma rize some key facts for people already acquainted with them , butalso, at the sa me t ime, t o introduce t he m to people unfamili ar to logic, an dperh ap s disliking too m an y logica l d eta ils. Accordin gly we will use a rathercolloquial tone . The fundamental question to be answered is: what is ModelTheor y? As we will see in mor e d etail in Section 1.2 , Mod el Theory is -or,mor e pr ecis ely, was a t it s beginnin g- the study of t he rela tionship betw eenmath ematical formulas and st r uct u res s at isfying or rejecting th em. Bu t , inord er t o fully appreciate t his matter , it is ad visa ble for us pr eliminarily torecall what a st r uct ure is , a nd which kind of formul as we ar e dealing with.This section is devoted to th e form er concept .Structures a re an algebraic notion . Actually, since G alois, Algeb ra is notonly th e s olving of e qu at io ns , or literal calcu lus , but becomes th e science ofst ructures gro ups, rings, fields, an d so o n) . This new direction gets clearerat the beginning of t he la st ce nt ur y, with S te in it z s work on fields a nd, la t er ,t he pub lication of th e Van der Wa erden book . What is a st r uct ur e? Basically, it is a non-em pty se t A , with a collect ion of di st ingu is hed element s,op er a tion s , and rela tio ns . For instance , t he set Z of int egers with th e usualop erations of addition + a nd multiplication . is a st r uct ure, as well as th esame set Z with th e ord er rela tion Note that , in th ese ex am ples, th e underlying set is t he sa me th e int eger s) , but , of course , t he st r ucture changes:in t he form er ca se we hav e t he ring of integers , in t he la t t er t he int eger s asa n ord ered set . To make t his kind of difference a mo ng struct ures clearer, weh ave t o choo se a language, in oth er words to specify how many distinguished

1


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2 CHAPTER 1. STRUCTURESelements, 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 (foraddition and multiplication), while, in the latter case, a binary relation (forthe order) is enough. Notice that the language of the ring case works as wellfor all the structures admitting two binary operations, and hence possiblyfor structures which are not rings; for instance, the reals with the functions

f(x, y) = sin(x - y) , g(x , y) = eX 'Yfor all x and y in R provide a new structure for our language, but, ofcourse, the algebraic features of this structure are very far from the basicproperties of integral domains. Accordingly it is advisable , from a generalpoint of view, to distinguish the constant, operation and relation symbols ofa language L and the elements, operations and relations embodying thesesymbols in a given structure for L. Symbols are something like the charactersin a tragedy (like Hamlet) , while their interpretations in a structure are theactors playing on the stage (Laurence Olivier, or Kenneth Branagh, or yourfavourite 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, henceeither finit e or denumerable (but most of what we shall say can be extendedwithout problems to uncountable languages).Definition 1.1.1 A structure A for L is a pair consisting of a non emptyset 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 ofA (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.


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1.1. STRUCT URES 3x am ples 1 1 2 1. A graph is a non empty set A with a binary relation

P both irreflexive and symmet ric. Hence a graph can be viewed as astruct ure A in the language L consisting of a uniqu e binary relationsymbol R , with RA = P. Also a non empty set A partially ordered bysome relation ca n be regarded as a structure in t he same languageL ; this time , RA = ~

2. A (multiplicative) groupy is a structureof th elanguageL = {I , ., I} ,where 1 is a constant , . and I are opera tion symbols of a rity 2 and1 respectively. 19 represents the identi ty element in y, while .9 and 1

9 denote the product and th e inverse operation in y . Actually onemight enrich L with some additional symbols; for instance, one mightintroduce a new binary operation symbol [ ] corresponding to th ecommutator operation in y . But, for a and bin G, [a , b] is jus t a . ba - I . : , so [ ] is not really new, and is implicitly defined by L .Actually we will prefer L later ; bu t it is notewor thy that L can captureand express some fur th er operation s (and rela tions and constants) ofy besides t hose literally interpreting its symbols.

3. A field K is a structure of t he language L = {O, 1, +,-, .} where 0and 1 are constan t , and +, - and are operation symbols (each havingan obvious interpretation in K). Alte rnatively, K ca n be viewed as ast ruct ure in th e language L' = L U { I } with a new operation symbolI ; obviously, I has to be interpreted wit hin K in the inverse oper at ion for nonzero elements of K . However , according to the generaldefinition of struct ure given before, - I JC should denote a 1-ary oper at ion with dom ain K . So we run into the problem of defining 0- 1 ; thiscan be overcome by agreeing, for instance, 0- 1 = 0, bu t this solut ionmay sound slightly artificial. So we will pr efer t o adopt below t helanguage L when dealing wit h fields. Indeed, when a and b are twoelements in a field K, then a = b- I can be equivalent ly expressed bysaying a . b = 1.

4. An ordered field is a st ruct ure in the language L = {+ , - , 0, 1, ~obtained by adding a new binary rela tion symbol Its in terpretationin a given ordered field is clear: the order rela tion in t he field.

5. Let N denote the set of natural numbers . 0 is an eleme nt of N ; t hesuccessor s (mapping each natural n into n + 1) is a 1-ary functionfrom N t o N. Giuseppe Peano poin ted out that t he Induction Principle(together with t he auxiliary conditions that s is 1 - 1 but 0 is not in


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4 CHAPTER 1. STRUCTURESits image) fully characterizes (N , 0, s). A suitable language to discussthis st ructure should include a constant symbol and a I-ary operationssymbol.

6. Let IC be a (countable) field . A vectorspace V over IC can be regardedas a st ructu re in the language LK = {O , +,-, r (r E K)}, where isa con stant , + and - are operation symbols with arity 2 and 1 respectively, and , for every r E K , r denotes in LK a l-ary op eration symbol ,to be interpreted inside V in th e sca la r multiplication by r . The othersymbols in LK are interpreted in the obvious way. The assumpt ion onthe cardinality of K has the only role of ensuring LK count able. Moreover, what we have said so far easily extends to right or left modulesover a (countable) ring R with identity; the corresponding language isobviously denoted by L R .

As already said, we should distinguish symbols and interpretations , for inst ance, a binary relat ion symbol R and t he relation RA embodying it in as t ructure A (sometimes an order rela tion in a partially ordered set, but elsewhere possibly the adjacency relation in a graph). But , to avoid too manycomplicat ions, we will often confuse (and actually we already confused) thelanguage symbols and their mos t natural interpretations. For instance, inExample 1.1.2,6, we denoted in the same way the addition symbol + of LRand its obvious interpretation in a given R-module M , namely the additionin M .We will be interested in several algebraic notions concerning structures. Inparticular embeddings play a crucial role in Model Theory. So let s recalltheir definition.Definition 1.1.3 Let A and B two struct ures in a language L . A homomorphism of A into B is a function f from A into B such that

(i) for every constant c of L, f(cA) = cB;(ii) for every positive integer n , for every n-ary operation symbol P in Land for eve ry sequence i i = (aI , . . . , an) in An , f(pA(ii)) = pB(J(ii))(hereafter f( ii) abridges (J(al) , . . f( an)) );

(iii) for eve ry positive integer n , for eve ry n-ary relat ion symbol R of Land for every sequence ii in A n, if ii E RA , th en f (ii) E R B.r is called an em bed ding of A into B if f is injective and, in (iii), f(ii) E R Bimplies ii E RA for every ii in An. When there is some embedding of A


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1.2. SENTENCES 5into E, we write A E. An isomorphism of A onto E is a surjectiveembedding. When there is some isomorphism of A onto E, we say thatA and E are isomorphic and we write A E. Finally, an endomorphism(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. Ifthe inclusion of A in to B defines an embedding of A into E, A is called asubstructure 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 . Wewonder if A is the domain of a subst r uct ure of E. One promptly realizesthat th is may be false. Indeed(i) if c is a constant of L , it may happen that cB is not in A;(ii) if F is an n-ary operation symbol in L , it may happen that the restric

tion of F B to An is not an n-ary operation in A, in other words that Ais not closed under FB ;

(iii) on the cont rary, if R is an n-a ry relation symbol in L, then R B n An isan n-a ry relat ion in A.So A is not necessa rily t he domain of a substructure of E. However theclosure 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 ainof a substructu re of L , usually denoted (A) , and called the substructuregenerated by A : in this case A is said to be a set of generators of (A) . Noticethat these notions can be introduced even in the case A = 0, provided thatL contains at least a constant symbol. E is called finitely generat ed if thereexists a finite subset A of B such that E = A).F inally, let L ~ L' be two languages , A be an L-structure, A' be an L'struct ure such that A = A' and the interpretations of the symbols in L arethe same in A and in A'. In this case, we say that A' expands A , or alsothat A' is a n expansion of A to L' ; A is called a res triction of A to L.

1 2 SentencesGiven a language L, after forming the st r uct ures of L, one builds, in acomplementary way, the formulas of L, in particular th e sentences of L , andone defines when a formula (a sente nce) is true in a given structure. T his ist he realm of Logic rather th an of Algebra.


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6 CHAPTER 1. STRUCTURESActually there are several possible ways of introducing formulas and truth ,according 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 howformulas and tru th are usually introduced in t he first order logic. For simplicity let us work in the par t icular sett ing of na tural numbers (full generaldetails and sha rp definitions can be found in any handbook of basic Math ema tical Logic, such as [153]).Consider the natural numbers and t he corresponding st ruct ure (N , 0, s),where s denotes the successor function . The corresponding language L includes a constant (for 0) and a 1-ary operation symbol (to be interpretedin s). As an nounced at t he end of the previous section, we denote thesesymbols by still using 0 and s : this is not completely correct , but simplifi esour life. In the first ord er set t ing, formulas can be built by using additionalsymbols

count ably many element vari abl es VD, V i , Vn , (just to respectour countable framework ; oth erwise we can use as many variables aswe need) ,

t he basic connectives /\ (a nd) , V (or), --, (not) (and even (if . . . ,then) , H (if and only if) if you like) , t he quantifier s V (for all) and :J (t here exists) , pa renth eses (, )

a nd a symbol = to be int erp reted everywhere by t he equality relat ion. Atthis point one form s the te rms of L . Essent ia lly they are polynomials; inour case t hey are built sta rt ing from the constan t 0 and t he variables Vn nna tural) and using the operation symbols (so s in our setting). The secondstep is to construct the atomic formulas of L: basically they are equationsbetween terms, but , when the language includes a k-ary relation symbol R ,we have to include every statement saying that a k-uple of terms satisfies R .At this poin t t he formul as of L a re built from the atomic ones inductivelyin th e following way:

1. one can negate , or conjunct, or disjunct some given formulas 0: , (3, . . .and get new formulas -- 0: , 0: / \ (3, 0: V 3 ;

2. one ca n t ake a formul a 0: and a va riable Vn and form new formulasVvno: , :Jvno: ;3. nothing else is a formul a.


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1.2. SENTENCES 7For a and /3 formulas, a -- /3, a H /3 ju st abridge - a V /3, a -- /3 )1\ (/3 -- a )respectively. Let us propose some examples in ou r framework of naturalnumbers. The injectivity of s can be expressed by the following formul a inour language L

while the formulaVvo-,(O = s vo))

says that 0 is not in t he image of s. Actually these formulas are sentences(each occurring variable is under the influence of a corresponding quantifier).In general , an occurrence of a variable v in a form ula a is bounded if it isunder th e influence of a quantifier Vv, ::lv, and free otherwise; a is ca lled asen tence if , as already said, each occurrence of a variable in a is bounded.When writing a(v), we want to emphasize th at th e variables freely occu rringin th e formula a are in the tuple v.2. and 3. are very restrictive conditions, and are the distinctive peculiarityof first ord er logic. Actually, in Mathematics, one sometimes uses V and ::l onsubsets (rather than on elements ) of a st ruct ure. This is ju st what happensin our sett ing concerning (N , 0, s) with respect to t he Indu ction Principle.In fact, Induction says

for every subset X of N , if X cont ains 0 and is dosed under s, thenX=N .

This statement uses V on subsets, and this is not a llowed in fi rst order logic.Accordingly, the Induction P rinciple cannot be writ ten (a t least literally inthe form proposed some lines ago) in the firs t order framework. T his mightlook very disappointing: consequently, one may search more powerful andexpressive ways of constructing formulas, for instance by allowing quantificat ion on set variables (t his it the so-called second order logic). But actuallyfirst order logic enjoys several important and reasonable technical theorems,th a t get lost and do not hold any more in these alternative worlds. Wewill discuss these results lat er , but it may be useful to quote already now atheorem of Lindstrom say ing (very roughly speaking) that first ord er logicis the best possible one (see [11] for a det ailed expos ition of Lindstromtheorem).However, formulas and sentences are not sufficient to form a logic. Wh atwe need now to accomplish a complete descrip tion of our setting is a not iona truth. We want to define when a sentence of a language L is true ina structure A of L , and , more generallly, when a sequence ii in A makes


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8 CHAPTERl . STRUCTURESa formula a v) true in A. This ca n be done in a very nat ur al way, sayingexactly what one expects to hear. For instance t he sentence 3v(v2+ 1 = 0) ist rue in the complex field just because C contains some elements i satisfyingt he equation v2 + 1 = 0; a nd in t he ord ered field of reals yI2 makes theformul a v2 = 2A v 2: 0 t rue because satisfies both its conditions, while -yI2 ,or 1, or other elements cannot satisfy t he same formula. See again [153], orany handbook of Mathematica l Logic for more details on the definition offirst order t r ut h . We omit t hem her e.Incid ent ally we note that, according this not ion of t rut h, a V {3 jus t means- { .a A ....,{3) , and Vvna says the same thing as ....,3vn( a) . So we could avoidthe connect ive V and the quantifier V in our alphabet and, consequent ly,in our inductive definition of formula, and to introduce a V {3 and Vvna asabbreviat ions, just as we did for a ---+ {3 and a H {3. In this perspective,formul as a re obtained from the atomic ones by using A, .... 3 and nothingelse.Moreover one can see t hat, accord ing to th is definition of t rut h, up to suitable manipulations, each formula ep w) can be wri t ten as

where QI , . . . , Qn are quantifiers, v = V I , .. . , vn) and a(v, w) is a quan ti fier free form ula, and even a disjunction of conjunctions of atomic formulasand negations. (*) is ca lled t he normal form of a formula . When ep w) is inits normal form and every quantifier Qi (1 i n) is universal V (existential3), we say that ep(w) is universal (ex iste nt ia l, respecti vely).Before concluding this section, we would like to emphasize t hat t he study oft his t ruth relation between structures and sentences is jus t Model Theory,at least according to the feeling in t he fifties. In fact , one says t hat astructure A is a model of a sentence a , or of a set T of sentences in thelanguage L of A , and one wri tes A 1= a, A 1= T respectively, whenever a ,or every sente nce in T , is true in A . Model Theor y is ju st t he st udy of thisrelationship between struct ures and (sets of) sentences. Tarski provides anauthor itative corroboration of this claim , when he wr ites in 1954 [158J

Whithin the last years, a new branch of metamathematicshas been developing. It is called theory of models and can beregarded as a part of the semantics of formalized theories. T heproblems studied in the theory of models concern mut ual relations between sentences of formalized theories and mathematicalsystems in which these sentences hold.


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1.3. EMBEDDINGS 9It is notable that this Tarski quotation is likely to propose officially for thefirst time the expression theory of models. Accordingly, one might fix 1954as the birthyear -or perhaps the baptism year- of Model Theory (if onelikes this kind of matters). Actually, several themes related to the theoryof models predate the fifties; but one can reasonably agree that just inthat period Model Theory took its first steps as an autonomous subject inMathematical Logic and in general mathematics.

1 3 EmbeddingsWe already defined in 1.1 embeddings and isomorphisms among structuresof the same language L. We followed the usual algebraic approach. However there are alternative and equivalent ways, of more logical flavour, tointroduce 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 Ainto B. Th en the following propositions are equivalent :

(i) f is an embedding of A into B;(ii) for every quantifier free formula If (if) in L and for every sequence iiin 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 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 awinning and 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 functionfrom A onto B. Then 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 iiin 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 i fB F 1f J ii));

(iv) for every 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)).


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10 CHA PTER 1. STRUCTURESI t can be obser ved that , when f is any embedding of A into B, for everyqu antifier 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 i i), w) , then A F= Vw a ii , w) .Definition 1.3.3 T wo s truc tures A and B of L are e lementa r i ly equivalent A == B) if they satisfy th e same sen ten ces of L .As an easy corolla ry of Theorem 1.3 .2, we have :Theorem 1.3.4 Isomorphic structures are elementarily equivalent.Converse ly, it may happen that elementarily equivalent structures A and Ba re not isomorphic. We will see count erexamples below. However it is aneasy exercise to show t hat , for fini te structures, elementary equivalence andisomorphism ar e j ust t he same t hing.Now let us int rod uce a related notion: partial isomorphism.Definition 1.3.5 Let A an d B be structures of L. A part ia l i somorphismbetween A and B is an isomorphism between a subs tructure of A and asubstructure of B. A and B are said to be partially ieomorphic A Bif there is a non empty set J of part ial isomorphisms between A and Bsatisfying the back-and-forth property: for all f E I ,

(i) fo r eve ry a (: A, t here is some gE l such that f 9 and a is in thedomain of g

(ii) f or eve ry b E B , there is some g E l such that f ~ 9 and b is in theimage of g.

Example 1.3.6 Two dense linear ord erings wit hout endpoints A = A, s;and B = B , S;) are parti ally isomorphic.In fact , le t I include all t he possible isomorphisms between a fini te substruct ure of A and a finit e subst ructure of B. I is not empty, because, for everya E A and b E B , a t-+ b defines a par ti al isomorphism in I. Now take anyf E I ; let ao < a l < . . . < an list t he elemen ts in t he dom ain of f andbo < bl < .. . < bn t hose in t he image of f ; so f (ai ) = b, for every i S; n .


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1.3. EMBEDDINGS 11Pick a E A, and notice that th ere exists some b E B such that, for everyi ~ n ,

a; a b, ~ b.This is trivial when a is in th e domain of f . Otherwise, one uses the factsthat B has no minimum when a < aa, that B has no maximum when a > an,and , finally, th at t he order of B is dense in the remaining cases. Define g E lby put ting

Domg = Domf U {a}, Img = Im f U {b} ,g ~ I, g(a) = b.

Clearly g satisfies (i). (ii) is p roved in the same way.Remark 1 3 7 If A B, then A 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 structures that admit a different cardinality, andyet a re pa rtially isomorphic. For instan ce, t his is t he case of two denselinear ord erings without endpoints. We have just seen that th ey are always partially isomorphic, indipendently of their cardinalities; in particular

R ~ ~ p (Q, ~But one can also find partially isomorphic non isomorphic struct ures withthe same cardinality. For instance, still consider dense linear orderings without endpoints , and notice that (R, ~ (R + Q , ~ ((R + Q, ~ denoteshere the disjoint union of a copy of (R, ~ and a copy of (Q, ~ where(R, ~ precedes (Q, ~ . Both (R , ~ and (R + Q , ~ have t he continuum power. But they cannot be isomorphic, because (R + Q , ~ unlike(R , ~ contains some countable intervals, and any order isomorphism mapscountable int ervals onto countable intervals.However, with in countable models, par tially isomorphic struct ures are alsoisomorphic.

Theorem 1 3 8 Let A and B be countable partially isomorphic structures.T hen A B.


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12 CHAPTER 1. STRUCTURESThe pr oof is obtained as follows. First one list A and B in some way

A = {an : n E N}, B = {bn : n E N}.Let J be a set of parti al isomorphisms between A and B ensuring A c::=.p B.Due to (i) and (ii) one enlarges a given 10 E J by defining, for every naturaln, a function In E I such that , for any n,

2. an is in the domain of [z,3. bn is in the image of fz n+l.

Pu t 1= UnENln. Owing to 1. , I is a function; 2. impli es t ha t its dom ainis A, and 3. ensures that it s image is B. ord er to conclude that I is anisomorphism , we have to check that , for every atomic formula cp(v) of L andevery sequence ii in A, A 1= cp(ii) if a nd only if B 1= cp(J (ii)). Bu t t his iseasily done , as t here is some n such tha t ii is in t he domain of In' and Inrestricts I and is an isomorphism between it s domain and it s image.A noteworthy consequence of t he theorem isCorollary 1.3.9 (Canto r) Two countable dense linear orderings withoutendpoints are isomorphic .Hence linearity, density and lack of endpoints cha racterize t he order of rat ionals up to isomorphism . I t should be underlined t hat Cant or s originalproof used a different argument; but a subsequent approach of I-Iausdorffand Huntington inaugurated the back-and-forth method. In fact, what theydid was just firstly to observe that two dense linea r order s without endpoints are partially isomorphic (according to our modern terminology) , andconsequently to deduce that , if one adds the countability assumption, th enisomorphism follows; t he latter point can be easily generalized to arbitrarystructures (and actua lly th is is what Theorem 1.3.8 says) . Now let us com-pare c::=.p and =.Theorem 1 3 10 Partially isomorphic structu res are elementarily equivalent.In fact let A and B be partially isomorphic struct ures in a language L , andlet I be a se t of parti al isomorphisms between A and B witnessing A c::=.p B.


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1.3. EMBEDDINGS 13Then one can show that , for every choice of a formula


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14 CHAPTER 1. STRUCTURESWe say that A is elementarily embeddable in E, and we write A


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1.3. EMBEDDINGS 15When A Band f is the inclusion of A into B, we say that A is anexistential subst ructure of B. When t here is some existent ial embeddingof A in B, we say that A is ex istentially embedded in B and we wri teA


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16 CHAPTER 1. STRUCTURESA itself becomes a structure of L A) provided we interpret quite naturally,for every a E A , t he constant cor responding to a in a. The result ing structurewill be denoted AA.Which a re the st ruct ures elemen tarily equivalent to AA in L A )? They areob tained as follows. Take an L-struct ure B where A embeds elementa rily,say by f ; for every a E A , let f (a ) int erpret t he constant of a. One get s int his way a struct ure BJ A) of L A), and it is easy to check that BJ A) == AAConversely, every st ructure elementarily equivalent to AA can be obtainedin this way.More generally, for every struct ure A of L and for every subset X of A , onecan introduce a new language L(X) by adding to L a new constant for everyelement x in X. Ax is the L(X) -structure expanding A and interpreting,for every x EX , the constan t symbol corresponding to x in x it self. Ofcourse, there do exist other structures elementarily equivalent to Ax. Letus see how to construct t hem. Let B be a struct ure of L.Definition 1.3.16 A funct ion f fro m X in to B is called elementary if ,for eve ry formula cp if) in L and for every sequence i in X ,

A F cp(i ) {:} B F cp (J (i )).Not ice that , when X = A, an element ary function from X in B is j ustan elementary embedding of A in B. Moreover, for any X , an elementaryfunction f from X in B enjoys t he following properties.(i) f is 1 - 1 (use CP (VI' V2) : VI = V2) .(ii) f - I it self is an elementary function (from f (X) in A).(iii) A and B are elementa rily equivalent (apply t he definition of elementary

function to the sente nces cp of L).Now take an L-structure B. Let f be an elementary function from X intoB , and, for every x E X , let f( x) int erpret the constant of x in L(X). Onegets in thi s way a st r uctu re BJ X ) of L(X) , and even a model elementarilyequivalent to Ax. Conversely, every st ruct ure == A x ca n be obtained in thisway.Remark 1.3.17 Let cp V, w) be a formula of L , d be a sequence in A , i bea sequence in X. Fussy people will like to distin guish

(a) A F cp ii, i) (in t he sense t hat A satisfies t he L-formula cp(v, w) ifV, wa re embodied by i i, i respectively);


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1.3. EMBEDDINGS 17(b) Ax F i.p ii , x) (in the sense that Ax satisfies the L(X) -formula

i.p if, x) if if is embodied by il);(c) AA F i.p(il, x) (in th e sense that AA satisfies th e L(A)-sentencei.p(il, x)).

However one easily shows that (a), (b) and (c) are equivalent. So we willuse t he simplest notation (that of (a)) to mean any of these conditions.We conclude this section by mentioning withou t proof two theorems onelementary emb edd ings. We will state them for simplicity in the case whenthe involved embeddings are just inclusions but they extend to arbitrary(elementary) embeddings.The former theorem provides a criterion to check whether a given subset ofa struct ure B is the domain of an elementary substructure of B (rememberthe discussion at the end of 1.1) .Theorem 1 3 18 (Tarski-Vaught) Let B be a structure of L , A be a subsetof B. Th en the following proposition s are equivalent:

(i) A is the domain of an eleme ntary substructure A of B;(ii) for every fo rmula a( if , w) of L and for every sequence ii in A , if B F:3wa(ii, w), then there exis ts 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 arelation ~ . For every i E I , let A i be a struct ure of L. Suppose t hat , forevery choice of i j in I , A i is a substruct ure of A j: this means th atA- C A and- J

for every constant c of L , cA ; = cA j , for every n-ary operation symbol F of L , FA; is the 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 iEIAi(and hence including Ai for all i E 1), and interpreting the symbols of L asfollows:

for every constant c of L, cA = cA ; where i is any element in J;


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18 CHAPTER l . STRUCTURES for every n-ary operation symbol F of L an d for every choice of

aI , . . . , an in A,

where i E I satisfies aI, . . . , an E Ai; for every n-ary relation symbol R of L and for every choice of aI , . . . , an

in A,

where i E I satisfies aI, . . . , an E Ai.I t is clear that A is well defined and extends Ai for every i E I. Straightfoward techniques show:T heorem 1 3 19 (Elem entary Cha in Theor em) Suppose that , fo r eve rychoice of i ::; j in I , Ai is an elemen tary substructure of Aj. Then , fo revery i E I , Ai is an elemen tary substructure of A .

1 4 The Compactness TheoremT he Compactness Theorem is t he most powerful to ol -a nd inde ed a keyfea ture- in classical Model T heory. It statesTheorem 1 4 1 Let S be an infinite set of sentences in a language L . Suppose that every fin ite subset of S has a model. Then S has a model.Notice t hat the converse is obvious , because a model of S is a model ofevery (finite or infinite) subset of S . But t he t heorem ensures that , if everyfinite subset of S has it s own model (hence different subset s may admitdifferent models), then there is a global model sa t isfying all the sentencesin S . fact , there are several sit uations where the Compactness Theoremapplies and guarantees satisfiability for sets S of sentences for which it isvery difficult to imagine a gene ra l model directly, bu t it is quite simple toequip every finite subset with a suitable private model: we will see some ofthem la ter in 1.5. fact th is sect ion and (implicitly) the next one will bedevoted to discussing t his fundamenta l theorem and, in det ail:

it s proof; its name;


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1.4. THE COMPACTNESS THEOREM 19 its role in producing nonstandard and , in some sense , unexpected

models, and , as th e reverse side of t he same medal , in bound ing theexpressiveness of first order logic and in excluding that some fam iliarprinciples, like induction on na turals, may be writ ten in any way int he first order framework ;

in spite of this, it s plausibili ty, supported by metamathematical considerat ions on t he nature of math ematical proofs;

finally, some words about t he already mentioned t heorem of Lindstromsaying t hat the Compactness Theorem, together with a related result(th e downward Lowenheirn Skolem Th eorem) fully cha racterizes firstorder logic.

As said, let us begin by discussing the proof. Ther e a re several possibleways to show the Compactn ess Theorem. For instance, the re is an approachbased on t he algebraic notion of ultraprodu ct and due to Keisler (see [39]) .Another classical proof was proposed by Henkin. Let us outline very quicklyits idea .Wha t we have at t he beginning is a( n infinite) set of sentences S such thatevery fi nite subset has a model. Some technical -and non t rivial- prelimin arywork shows that t here is no loss of gene ra lity in assuming t hat S satisfiestwo fur th er cond itions:

1. S is complete, in oth er words , for every sente nce rp in L , either rp or-' rp is in S;

2. S is rich: if S contains a sentence of the form :Jva:(v), t hen t here is aconstant symbol c in L such that a:(c) is in S .

At this point a qui te art ificia l construct ion produces t he model we are looking for. Basically, th e domain is ju st the set of terms without variables in L(the so-called Herbrand universe of L) ; th is is non-empty owing to 2. TheL-structure arises in a rather reasona ble way. 1 and 2 play a key role inshowing that what we build is a model of S .It is worth emphasizing that the model we get in our proof is countable (fora countable L ; when the language has a larger cardinality , it is easy tocheck t hat our argument s t ill works and produces a model of power s: -X).So, as a byproduct of t he Henkin proof, we have that , when S has a model ,then S has a countable model: this is the so ca lled Downward Lowenli eimSkolem Theorem, and is a notable result . Vve shall discuss its relevance in1.5.


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20 CHAPTER 1. STRUCTURESNow let us t reat t he reason s of t he t heorem name. Act ually compactnessrecalls topology. In fact , we will see later in Chapter 5 that th e th eorem hasa topological content and implicitly says that a certain topological space iscompact .We shall see with in a few lines in 1.5 t hat the Com pactness Theorem produces some strong and severe expressiveness restrictions in first order logic.For ins tance , we will show that , ju st owing to Compactness, condit ions likefiniteness, or popular statements such as the Minimum Principle, cannot beexpressed in a first order way. On t he other hand, one should agr ee thatwhat t he Compactness T heorem says is a qui te reasonable statement, especially if one considers t he following corollary. Let 5 be a set of sente nces ofL and a be a sentence of L ; we say that a is a logical consequ ence of 5 , andwe write 5 F a , when a is true in all the models of 5 .Corollary 1 4 2 If 5 F a , then there is a fin ite subse t So of 5 such thatSo Fa.Proof. Clearly 5 F a if and only if 5 U {-,a} has no mode ls. Bu t , owingto Compactness, t his is equ ivalent t o say that t here is a finit e subset of5 U {-,a} withou t a ny model s. Wi th no loss of generality we can assumethat th is finite set is of the form So U {-,a} where So 5 . But , aga in,stat ing that So U {-,a} has no models is equivalent to say that So F .p ...Now, anot her fundamental result in firs t order logic , deeply related to compactness -the Completeness Theorem- says that one can explicitly providea notion of provabili ty accompanying and supporting th is concept of consequence in such a way t hat the logical consequences of a given 5 are jus twhat is proved by 5 at the end of a sequence of rigorous deductions. Sowhat compact ness in conclusion emphasizes is the finit ary nature of mathemat ical proofs; t his feature can be regarded as an aut horitat ive witness init s favour and, t rough it , as a support to first orde r logic.

1 5 Elementary classes and theoriesWhen considering, for a given language L , structures, formulas and t ruth,two problems arise quite naturally:

(a) given a set T of L-sentences, classify th e models of T (their classwill be denoted Mod(T));


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1.5. ELEMENTARY CLASSES AND THEORIES 21(b) given a class K of Le-structu re , characterize t he set of t he L-sentences

true in all the structures of K (this set will be called the theory of Kand denoted T h(K )).

According to its declared intents, Model T heory should be mainly concernedwith P roblem (a) . However , also (b) arises quite naturally in the modelt heoretic framework. For instance , consider a class K formed by a singlestructure, like t he complex field , or the real field. We will see lat er that Kcannot be represented as M od(T ) for any T. But it may be qui te interestingto realize in an explicit way which sentences are true in t he only structurein K.However we have to admit that the previous statements of (a) an d (b) aresomewhat vague and unprecise. First of all, what do classifying or characterizing mean? This is not a minor question ; on the contrary, it is a verydelicate and central matter. For instance , t he classificat ion problem for aclass of structures touches and overlaps several basic ope n questions in Algebra . So we should be more detailed about this crucial point . Of course,one can reasonably agree that a classification should identify isomorphicstructures. But t his is still a partial and indefinite answer; we should fixmore precisely which criteria, tools and invariants we want to use in ourclassification problem. We shall try to clarify these fundamental questionsin the next chapters. Here we limit ourse lves to discuss other points, mainlyconcerning (a). T is a set of sentences in a language L .

1. Let a be a sentence of L, and suppose that every model of T is also amodel of a (so a a logical consequence of TT 1= a). Hence Mod(T) =Mod (T U{a }). Consequent ly we can assume wit h no loss of generalityfor our purposes that

for every sentence a of L , if T 1= a , then a E T.A set of sentences in L with this property is ca lled a theory of L. Itis a simple exercise to show that, given a set T of sentences of L, T isa t heory if and only if t here is a class K of struct ures of L such thatT = 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 thefollowing (equivalent) conditions:(i) for every sentence a of L , either a rf. T or -,a rf. T;(ii) there is some sentence of L which is not in T;


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22(iii) M od(T) - 0.

CHAPTERl. STRUCTURES

To show the equivalence among (i) , (ii) and (iii) is an easy exercise . (i) saysthat the consistent theories are just t hose excluding any con tradiction. (iii)ensures that t hese th eories are exact ly those admitting at least one model. Itis clear t hat , within Problem (a), we are exclusively interested in consis tenttheori es. So we can assume in (a) th at T is a consistent th eory; accordinglyhereafter theory will always abbreviate consistent theory.A rigid model theoretic perspective might limit the classification analysis tothe classes of models of (consistent) theories . But open minds could prefera more general study, providing an abstract t reatment of th e classificationproblem for arbitrary classes of structures. Hence it is worth underliningthat there do exist classes K of L-struct ures which are not of th e formK = M od(T) for any theory T of L. We propose here some examples; theCompactness Th eorem is a fundamental tool in th is se t t ing .Definition 1.5.1 A (non-empty) class K of structures of L is said to beelementa ry (or also ax iomat izable) if there is a set T of sentences of L(without loss of generality , a theory T of L) such that K = M od(T ).Now let us propose a ser ies of examples, as promised. P ar t of t hem aimat poin ting ou t t hat several classes of st ructures are explicitly element arybecause their defini tions can be naturally wri t t en in a first order way. Bu toth er cases are not elementary : it is here that the Compactness Theoremplays it s role and first ord er logic shows its expressiveness bounds.Examples 1.5.2 1. Let L = 0 (so the struct ures of L a re the non -empty

sets), K be the class of infinite sets . Infin ite means admitting a tlea st n + 1 elements for every natural n . Given n, the property there are at leas t n + 1 elements can be expresse d in a first ord erway by the following sentence of L

o; :Jvo . . . :Jvn 1\ - '(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-empty)sets . Finite means having a t most n + 1 elements for some naturaln . Given n , the proposition there are at most n + 1 elements can


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1.5. ELEMENTARY CLASSES AND THEORIES 23be expressed in a first order way by the sentence -,an+l . But nowM od( {-,an+ : n E N}) is not K, indeed it equals the class of thesets having only one element. So the approach in 1 does not work anylonger. However, assume that K is elementary, hence K = M od(T)for a sui table 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: itsuffices 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 amodel. This is a set both finite (as a model of T') and infinite (as amodel 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 finitearbitrarily large structures (in the sense that, for every positive integern, there is a structure in K whose size is larger than n). A class Kof this kind cannot be elementary; in other words, the theory of Kdoes 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 . Soone can wonder which are the infinite models of the theory of thesefinite structures. We will consider the particular case of fields later inExample 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-). InL we consider the class K of linear orders. I t is easily seen that Kis elementary, because the properties defining linear orders are firs torder sentences (and indeed universal first order sentences) of L. Forinstance, linearity can be expressed by

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


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24 CHAPTER 1. STR UCTURESIn the same way the class of dense linear orders without endpoints iselementa ry; in fact, density is st a ted by

VvoVVI :JV2 (vo < VI -+ Vo < V21\ V2 < vd ,and lackness of endpoints by

VVO:JVI (VI < Vo),VVO:JVI (vo < VI )

(vo < VI abbreviates here Vo:::; VII\ --, (VO= vd ).The set of the logical consequences of the sentences quoted so far is thetheory of dense linear orders without endpoints; we will denote it byDLO-. I t is formed by t he se nt ences tru e in every den se linear orderwith no endpoint s. Recall t h at (Q, :::; , (R, :::; ) are dense linear orderswithout endoints , and consequently their theories include DLO- (andone may wonder if they actually equal DLO-).The reader can check directly t hat t he following classes of L-structuresare elementary:

dense linear orders with a leas t but no last element, or a last butno least element, or both a lea st and a last element,

infinite discrete linear orders with or without endpoints (an orderis discrete when ever y eleme nt, bu t t he leas t on e -if any- , has apredecessor and eve ry eleme nt, but t he last on e -if any- , has asuccessor) .

4 . We still work in L = {:::;} (wh ere j is a binary relation symbol), bu tthis t ime we deal with t he clas s K of well ordered sets (so ordered setswhere every non-empty subset has a least element ) . Hence (N , :::;) E Kowing to t he Minimum Principle, while any dense linear order A, :::; ,even with a minimum, does not lie in K (in fact, given b > a in A ,which is t he least element > a in A?). So the situation is , in somesense, opposite to the last (elementary) ex ample 3 .Suppose that K is elementary, so K = M od(T) for a suitable set T ofL-sentences. Pu t

L' = L U {cn : n E N}where , for every natural n , Cn is a co nst a nt symbol, a nd in L' look atthe following set of sentences

T = TU {Cn+1 < Cn : nE N }.


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1.5. ELEMEN TA RY CLASSES AND THEORIES 25Let Tbbe a finite subset of 1 , t hen there is some natural N such that

Tb ~ T U { Cn+ l < Cn : n EN, n ~ N}.Then Tb has a model because T U {C n+ l < Cn : n EN , n ~ N}has : it suffices to t ake the well ordered set (N , ~ to interpret Co,Cl , . . . , CN+ in N +1, N, . .. ,respectively, and any furth er constantCn (with n > N ) ar bit ra rily. By the Compactness Theorem, 1 doesadmit a model

A = A, ~ (C;; ) nEN) .Let A = A , ~ then A is a model of T, a nd hence is a well orderedset; however it contains t he non-empty subset

X={C;;: nEN}admit t ing no minimum, because , for every natural n, ~ < c;; . Sowe get a contradiction. Consequent ly K is not elementary. In otherwords th ere are linearly ordered sets which are not well ordered butsatisfy the same first order sentence as well ordered sets.

5. Let now L = {O , 1, +,-,.} be our language for fields. We consider inL the class K of fields. K is elementary. In fact t he definition itself offield can be wri t t en as a series of first order sentences (in most cases,of universal first order sentences) in L . For instance

says that any non zero element has an inverse.Also t he class of algebraically closed fields is elementary, although thecorresponding check is a lit tle subtler. fact what we have to say nowis that , for every natural n , any (monic) polynomial of degree n + 1has at least one roo t. So the point is how to quantify over polynomialsof degree n + 1. However recall that such a polynomi al is ju st anordered sequence of length n + 2 of element s in the field : the first isthe coefficient of degree 0, the last is t he coefficient of degree n + 1(and equals 1 if we deal with monic polynomials) ; so what we have towrite is just , for every n ,

(where vi has the obvious meaning, for every i ~ n + 1).


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26 CHAPTER 1. STRUCTURESThe logical consequences of t he sentences listed so far form the theoryof algebraically closed fields, usually denoted ACF . Now let p be aprime, or p = o. Also the class of (algebraically closed) fields of characteristic p is elementary; for, it suffices to add to the previous sentencesthe 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 these sentences. In conclusion, forevery p prime or equal to 0, we can introduce the theory ACFp ofalgebraically closed fi elds of characteristic p. Among the algebraicallyclosed fields in characteristic 0 recall the complex field C , as well asthe (countable) field Co of complex algebraic numbers; th eir th eoriescontain ACFo, and one may wonder if they equal ACFo. Recall alsothat every field K has a (minimal) algebraically closed extension K;in particular, for p prime, Z/pZ is an example of algebraically closedfield in characteristic p .Since we are t reating fields , let us consider agai n finite fields, and,mor e exactly, the infinite models of their theory we met in example2: th e so ca lled pseudofinite fields. As observed before , one can askwhich is the structure of these fields. J. Ax equipped them with a veryelegant axiomatization, explaining the essential nature of finite fieldsin t he first order setting: in fact , pseudofinite fields are just the fieldsK such that:* K is perfect,* K has exactly one algebraic extension of every degree,* every absolutely irreducible variet y over K has a point in K.All these conditions can be written in a fi rst order way, although thisis not immediate 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 theset of the following sentences in L:(i) the field axioms (see Example 5);(ii) those characterizing the linear orders (see Example 3);(iii) the sentence saying that sums and products of nonnegative ele

ments are nonnegative


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1.5. ELEMENTA RY CLASSES AND THEORIES 27Also the class of real closed ordered fields (t hose satisfying t he Intermediat e Value Property for polynomials of degree ~ 1) is elementary,it suffices to add t he new sentences:(iv) for eve ry natural n ,

A 0 < Vo + vI .W+ ... + Vn . u i + w n+1 A u < w -+-+ u < v A v < w A Vo+ vI . V+ ... + Vn . vn + vn +1 = 0).

T he logical consequences of (i), (ii), (iii) , (iv) form the theory of realclosed ordered fields, usually denoted RCF . Examples of real closedord ered fields are t he ord ered field of the real numbers as well as the(coun t able) ord ered field R o of real algebraic numbers. T heir t heoriesinclude RCF , and one may wonder if actually t hey equal RCF .

7. Let R be a (countable) ring with identi t y. Conside r t he languageLR = {O , +,-, r (r E Rn of (left ) R -modules . The class of left R -modules is elementary becau se it equ als t he class of mod els of t hefollowing sentences in LR:(i) t hose ax iomatiz ing the abe lian groups in the language with 0, +

and - ;(ii) for every r , s E R , if r+s and r s denote the sum and the product

(respecti vely) of r a nd 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 ,\IVo(1Vo = vo).

The logical consequences of the previous sentences form the theoryn T of left R-modules. Of course , t he re is no reason to pr efer theleft to the righ t , at leas t in this case ; indeed, one can check that event he class of right R -modules is element ary, and consequent ly one canintroduce t he theory Tn of right R-modules.


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28 CHAPTER 1. STRUCTURESLet us come back to our classification problem for elementary, or also nonelementa ry classes. The following fundamental theorem can suggest t hat ,even in the elementa ry case, t his problem is not simple, as the class of modelsof a theory T can include many pairwise non-isomorphic structures.Theorem 1.5.3 (Lowenheim-Skolem] Let T be a theory in a (countable)language L . Suppose that T has some infinite model. Then, for every infinitecardinal A, T admits some model of power A.The proof just uses Compactness in the extended framework of languagesof arbitrary cardinali ties. In fact one enlarges L by A many new constantsymbols c; (i E l , III = A) and one gets in t his way an 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 model; in fact it tu rns out that , for somefinite subset 10 of I ,

so, in order t o obtain a mod el of T ~ it is sufficient to refer to an infinitemodel A di T , as ensured by t he hypothesis, and to interpret the finitelymany const ants c; (i E 10 ) in pairwise different elements of A. At th is point ,Compact ness applies and gives a model of T ' (hence of T) of power :S: A.But this model has to include the A many distint interpret ations of the c/s,and so it s power is exactly A.T herefore, if a theory T of L has at least an infinite model then T has a modelin each infinite power (and two models with different cardinalities cannotbe isomorphic). Of course , one may wonder how strong is the assumptionthat T has some infini t e model. Not so much , if one recall s that a theoryT admitt ing finite models of arbitrarily la rge size must admit also someinfinite models. Another reasonable question may concern how many modelsT adm its in any fixed infinite cardinal A. One can check t hat t heir numbercannot exceed 2\ bu t this upper bound can be reached, for every A, by somesuitable T s. T he opposite case, when T has just one mod el in power A (upto isomorphism) , will be of some interest in the next chapters; we fix it int he following definition.Definition 1.5.4 Let T be a theory with some infinite model , A be an infinite cardinal. T is said to be A-categorical if and only it any two modelsof T of power A are isomorphic.


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1.5. ELEMENTARY CLASSES AND THEORIES 29We wish to devote some more lines t o t he Lowenhcim-Skol em theorem .Among other things , it confirms th at elementa ry equivalence is a weakerrelation than isomorphism. In fact, take an infinite struct ure A , and usethe Lowenheim-Skolem to build a model A' satisfying the same first ordersentences as A but having a differen t cardinality, I t is easily checked thatA ,A' are elementary equivalent; but , of course, t hey cannot be isomorphic.Now recall what we pointed out in 1.2 : the Induction P rinciple (in its usualform) canno t be written in the fi rst order style in the language for (N , 0, s)because first order logic forbids quantification on set variables . However,as far as we know, one might find an equivalent statement that can be expressed in t he first order set t ing ; in this sense , Induction might become afirst orde r statement. Well, the Lowenheim-Skolem theorem excludes thisextreme possibility. For, the Induction P rinciple characterizes (N , 0, s) upto isomorphism, while the Lowenheim-Skolem theorem ensures us that anytentat ive first order equivalent translation (even involving infinitely manysentences) has some uncountable models. So th is translation cannot exist.

The Lowenheim-Skol em t heorem emphasizes other similar expressivenessrestrictions in first ord er logic. For instance, it is well known that t he ord eredfield of teals is, up to isomorphism , t he only complete ordered field (herecompleteness means t hat every non-empty upperly, lowerly bounded set ofreals has a least upper bound , a greatest lower bound respectively). Socompleteness cannot be expressed in a first order way, because any tentativefirst order translation should be true in some real closed field wit h a noncontinuum power.

On t he ot her side, we will see that the Lowenheim-Skolcm t heorem is avery useful and powerful technical tool in first order model theory (ju st asthe Com pactness Theorem ). And actually the expressiveness restrictions remarked 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 quoted before in 1.2.Indeed, what Lindstrorn shows is t hat, if you have a logic (namely a reasonable system offormulas and tru th) and you demand t hat your logic satisfiesthe Compactness Theorem and the weaker form of the Lowenheim-SkolemTheorem , ca lled Downward Lowenheim-Skolem Theorem, introduced in 1.4and requiring -for countable languages- th at any set of sentences adm ittinga mod el does have a countable model , then your logic is t he first order logic.In t his sense the first order framework is (Leibni zianly) the best possibleone.


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30 CHAPTER 1. STRUCTURES1 6 Complete theoriesLet 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, welimit our analysis to elementary classes K = Mod(T), where T is a firstorder theory. This choice is not so partial and na rrow as it may appear . Infact, it certain ly includes the cases when T is explicit ly given and equipsK with an effect ive lis t of firs t order axioms, as in the positive examples ofthe last section; but it is also conce rned with other , and worse situations.For instance, think of the theory T of finite sets , or groups , or fields, or,in general, of a class of finite arbitrarily large structures , so t hat T hasalso infinite models. Alternatively, think of the theory T of a single infinitestructure A: due to the Lowenheim-Skolem Th eorem, T has some modelsnon-isomorphic to A. In t hese cases, T is introduced by specifying somecrucial models, but this does not determine in an explicit way a priori whichfirst order sentences belong to T , and which are excluded; indeed we couldjust be interested in finding an effect ive axiomatization as in the previousexamples, and we could aim both at describing T and also -a s a rela tedmatter- at classifying its models.These are th e set tings we wish to consider. Actually we should also admitthat we have not clearly explained up to now which kind of classificationwe pursue; however we have agreed that this classification should identifyisomor phic models but distinguish non isomorphic structures. Also, we haveseen that isomorphic models sat isfy the sam e order orde r sentences. So apreliminary classification is just up to elementary equivalence, and aims atdistin guishing non elementarily equivalent structures. Once this is don e,we could restrict our analysis to structures satisfying t he same first orderconditions; Le. fix a structure A and classify up to isomorphism t he modelsof its theory T = Th({A}) (by the way, let us abbrevia te for simplicity1 h { A} ) by Th(A)).Which is an intrinsic syntactical characterization of such a theory 1 ? Basically it is complete according to the following definit ion.Definit ion 1 6 1 A (consistent) theory T of L is said to be complete if,for every sentence


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1.6. COMPLETE THEORIES 31On t he other hand , every complete theory T can be represented in this way.In fact , fix any model A of T . Clearly T ~ Th A). Conversely, let ep bea sentence of Th A ) , then -' ep j. Th A) and so -' ep j. T; as T is complete,ep E T .Notice that t he same arg ument shows t hat, if T T ' are consistent t heoriesand T is cornplete, t hen T = 'I ,Now notice what follows.Rem a r k 1.6.2 Every (consistent) t heory T of L can be enlarged in at leastone way to a complete t heory in L. In fact , it suffices to consider T h A )where A is any mod el of T . A complete theory exte nding T is called acompletion of T .So our classification project can be organized as follows.

First , determine struct ures up to elementary equivalence, I I I otherwords find all the complet ions of a given theory T;

then, class ify up to isomorphism the models of a complete T .We deal in this section with the former problem, hence with complet ionsand , definitively, wit h complete t heories. Incomplete t heories are easy tomeet.Example 1.6 .3 For inst ance, t he t heory of g roups it is not complete (asthere are both abelian and nonabelian groups , and commutativity can bewritten in a universal first ord er sent ence) . In th e same way, the theoryof fields is not complete (why?) . Also the theory of linear orders is notcomplete (as t here are both dense and non dense total orde rs, as well asord ers with or without a minimum or a maximum).On the other hand , the previous remark pointing out t hat a complet e Tis the t heory of any model of T seems to provide a gr eat deal of completet heories; but these examples a re not satisfactory. In fact , as already said,wha t we reasonably expect is to have complete theories T equipped withan explicit list of basic axioms, ensuring that t he sentences in T are jus tt he consequences of these axioms. Now, when we look at Th A) for somestruct ure A (t he field of complex 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 ob tainsuch a bas ic ax iomatization in t he mentioned cases. A possible strategy t osolve these problems might be t he following. Given A , prepa re a t entativeexplicit axiornat ization and the corres ponding theory T. Of course, A should


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32 CHAPTER 1. STRUCTURESbe a mod el of T . At this point , check if T is complete, by some suitableprocedures. If yes , T = T h(A ).Unfortunately, checkin g completeness for a theory T as before is not simple. We mention here a celebrated sufficient (but non-necessary) condition,found ed on t he notion of A-categor icity.Theorem 1.6.4 (Vaught) Let T be a theory of L . Suppose that every modelof T is infinite and T is A-categorical for some infinite cardinal A. Then Tis complete.Proof. Suppose towards a con tradiction that T is not complete; let


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1.6. COMPLETE THEORIES 33Proof. An algebraically closed field lC is always infinite; in fact , if ao, .. . ,anare 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, and so is a new element . At thispoin t , in ord er to apply Vaught s Theorem , we have to prove \-categoricityfor some infinite \. But t his is ju st a consequence of Steinitz s analysis ofalgebraically closed fields . For, this analysis essentially impli es (in our terminology) that , fixed p = 0 or prime, t he t heory AC Fp of algebraically closedfields of characteristic p is \-categorical for every uncountable cardinal ,\ (soVaught s Theorem applies and yields cornpleteness) . Let us recall brieflywhy (we will provide an a lte rnat ive, detailed proof of the complete ness ofAC Fp in Chapter 2). Any algebraically closed field in characteristic p canbe obtained as

lC = lCo S)where lCo is the prime subfield of lC (hence lCo is isomorphic to the rationalfield 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 independent subset ), and- denotes the algebraic closure in lC. Furt hermore the isomorphism ty peof lC is fully determined by the cardinality of S (the transcendence degreeof lC). Accordingly, one ca n realize that ACFp has

~ o pairwise non isomorphic counta ble models (correspondingly to thet ranscendence degrees 0, 1, . . . , ~ o

for every uncountable cardinal \ , exactly one isomorphism class ofmodels of power \ , because a ll these models share the same t ranscendence degree \.

Hence ACFp is \-categorical in every cardinal ,\ > ~ o and consequent lycomplete. . .In particular , two algebraically closed fields lC l and lCz having the samecharacteristic p but different transcendence degrees d l -I dz are elementarilyequivalent, but cannot be isomorphic. Hence, when lC l and lC z are countable,they are not even partially isomorphic.Not ice also that t he field of complex numbers is a model of ACFo, and soAC 1 0 equa ls it s t heory : we find in this way an explicit list of axioms (thatof AC1 0) for the theory of the complex field.Let us propose a further application of Vaught s Theorem to deduce completeness . Perhaps at t his poin t someone may expect t o meet RC1 andthe theory of the real field in ou r list of examples. But we have to delaythis appoin tment . Indeed , RCF is complete (and hence equals the theory


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34 CHAPTER 1. STRUCTURESof t he ordered field of reals) , bu t RCF is no t A-categorical for any infinitecardinal A. So a different approach is necessary : we sha ll follow this newstrategy in t he nex t chapter. On t he contrary, Vaught s Theorem applies tovectorspaces over a countable field . Let us see why.

orolla ry 1 6 8 Let JC be a countable field. Then the theory KT' of infinitevectorspaces over JC is complete.

Proof. Clearly KT' has no fini te models. Moreover we know that two (infinite) vectorspaces with the same dim ension over JC are isomorphic. Consequ ently, for every cardinal A bigger than ~ o there is a unique isomorphismclass for all the JC-vectorspace of power A (in fact , each of them has dimension A). In other words, KT' is A-categorical for every cardinal A > ~ o . ByVaught s Theorem, KT' is complete .

On t he contrary, KT' may not be categorical in ~ In fact , when JC is infinite,JC , JC2, .. . , JC{ No) are count able JC-vectorsp aces wit h distin ct dimensions, andso cannot be isomorphic, hence they are not even par tially isomorphic . So eleme ntary equivalence cannot imply partial isomorphism (and isomorphism ).The reader may check d irectly what happens when JC is finite .We conclude this section by introducing another notion related to completeness. It will be used in Chapter 3 to show th at RCF is complete. Recallthat a complete theory T equals Th A) for every model A , and hence at heory T is complete if a nd onl y if a ny two mod els of T a re elementarilyequivalent .

Definition 1 6 9 A theory T is model complete if every embedding of models of T is elemen tary.

I t is easy to exhibit t heo ries which are not model complete . For instance,t he previous examples 1.3.14 ensure that the t heory of (N ,


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1.7. D EFINABLE SETS1 7 Definable sets

35

Formulas include equat ions, and Algebra aims at finding solut ions of equations . More generally, given a language L, a formula


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36 CHAPTER 1. STRUCTURESAn element a E A is X -definable if it s singleton is. Of course , every a isA-definable (by v = a). But , when a rJ. X , things are not so t rivial.Remark 1 7 2 Fi x a struct ure A of L and a positive integer n.

1. Let X be a subset of A . The X-definable subset of An form a subalgebra of the Boolean algebra of all t he subset of An.In other words, both An and 0are X-definable (by t he formulas VI = VIand (V I = vd respectively) , and , if Do and DI are two X-definablesubsets of An, t hen even th eir union DoUDI , their intersection DonD Iand the complement An - Do are X-definable (if


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1.7. DEFINABLE SETS 37An explicit example of an infinite structure wit h a non-definable subsetis the following. Let L = 0, so t he struct ures of L a re ju st t he nonempty sets A . Take an infinite set A. We have see n that every finiteor cofinite subset of A is definable. We claim t hat no other subset ofA is definable. In fact, let D be a subset of A such t hat both D andits complement A - D are infinite. Suppose towards a cont radictiont ha t D is definable, and so D =


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38 CHAPTER 1. STRUCTURES The union of two (and consequently of finitely many) algebraic

varieties in an algebraic variety. This is a simple exercise of Algebra, essenti ally using the fact that , in a field K , t he product oftwo nonzero elements is different from O.

T he intersection of two, or finitely many, or even infinitely manyalgebraic varieties is st ill an algebraic variety. This is a trivialexercise in the finite case, and a deep theorem in Algebra -knownas Hilbert Basis Theorem- otherwise.

Notice that these properties (together with the easy observation t hatK and 0 are algebraic varieties -for, they are th e zero sets of t hezero polynomial, and of a ny non zero constant polynomial in K[x Jrespectively-) show t hat th e algebraic varieties of K ar e the closedsets in a suitable topology of K (the Zariski topology) . However

t he complement of an algebraic vari et y of K is not necessarilyan algebraic variety of K :

So there are definabl e sets of K which are not a lgebraic varieties.Indeed Algebraic Geometry introduces the notion of constructible sett o define a finit e Boolean combination of algebraic variet ies of K :Remark 1.7.2 ,1 before ensures that every constructible set is definable.In certain fields K the converse is also true, and hence definable j ustmeans constructible. For instance, this is what happens when K is analgebraically closed field (and so, in particular , when K is the complexfield). This is not a t rivial result , but a deep t heorem of Tarski andChevalley, and will be discussed in the next Chapter .

2. (Definable set s a n d Real Algebraic G eome t r y ) Let L = {O , 1, +, - ::;} be our language for ord ered fields. Fix an ordered field K, anda positive integer n . Algebraic Geometry studies the sets of the elements of K satisfying disequations like q x) ~ 0 where q x) E I


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1.7. DEFINABLE SETS 39Notice a lso t hat the order relation is definable in the real field Reven within the language of fields {O, 1, +, - }: in fact, it suffices torecall that the non negative reals are exactly the squares, and hence todefine

by the formula:3W(VI - V2 = w2).

Consequently every semialgebraic set D in R is definabl e in the realfield even wit hin the language of fields, just by replacing any formula

q iJ ~ 0(with q x) E R[X]) by the equivalent formula

w q iJ) = w2 ) .However , notice that the latt er formula requires a quantifier.

annalisa marcja, carlo toffalori a guide to classical and modern model theory trends in logic 2003

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