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Introduction to Model Theory Introduction to Model Theory Martin Hils Équipe de Logique Mathématique, Institut de Mathématiques de Jussieu Université Paris Diderot – Paris 7 Second International Conference and Workshop on Valuation Theory Segovia / El Escorial (Spain), 18th – 29th July 2011 Introduction to Model Theory Outline Basic Concepts Languages, Structures and Theories Definable Sets and Quantifier Elimination Types and Saturation Some Model Theory of Valued Fields Algebraically Closed Valued Fields The Ax-Kochen-Eršov Principle Imaginaries Imaginary Galois theory and Elimination of Imaginaries Imaginaries in valued fields Definable Types Basic Properties and examples Stable theories Prodefinability Introduction to Model Theory Basic Concepts Languages, Structures and Theories First order languages A first order language L is given by I constant symbols fci gi2I ; I relation symbols fRj gj2J (Rj of some fixed arity nj ); I function symbols ffk gk2K (fk of some fixed arity nk ); I a distinguished binary relation "=" for equality; I an infinite set of variables fvi j i 2 Ng (we also use x; y etc.); I the connectives :, ^, _, !, $, and I the quantifiers 8; 9. Introduction to Model Theory Basic Concepts Languages, Structures and Theories First order languages (continued) L-formulas are built inductively (in the obvious manner). Let ' be an L-formula. I A variable x is free in ' if it is not bound by a quantifier. I ' is called a sentence if it contains no free variables. I We write ' = '(x1;:::; xn) to indicate that the free variables of ' are among fx1;:::; xng. In what follows, we will only consider countable languages. Introduction to Model Theory Basic Concepts Languages, Structures and Theories First order structures Definition M M M An L-structure M is a tuple M = (M; ci ; Rj ; fk ), where I M is a non-empty set, the domain of M; M M nj M nk I ci 2 M , Rj ⊆ M , and fk : M ! M are interpretations of the symbols in L. To interpret an L-formula ' in M, note that the quantified variables run over M. n Let '(x1;:::; xn) and a 2 M be given. We set M j= '(a) if and only if ' holds for a in M. Introduction to Model Theory Basic Concepts Languages, Structures and Theories Examples of languages and structures I Lrings = f0; 1; +; −; ·} (language of rings). Any (unitary) ring is naturally an Lrings -structure, e.g. C = (C; 0; 1; +; −; ·) and R = (R; 0; 1; +; −; ·). ' ≡ 8x9y y · y = x is an Lrings -formula (even a sentence), with C j= ' and R j= :'. I Loag = f0; +; <g (language of ordered abelian groups) Let Z = (Z; 0; +; <) and Q = (Q; 0; +; <). Let (x; y) ≡ 9z(x < z ^ z < y). Then Q j= (1; 2) , Z 6j= (1; 2) and Z j= (0; 2). We will often write M instead of M, if the structure we mean is clear from the context. Introduction to Model Theory Basic Concepts Languages, Structures and Theories First order theories An L-theory T is a set of L-sentences. I An L-structure M is a model of T if M j= ' for every ' 2 T . We denote this by M j= T . I T is called consistent if it has a model. Examples 1. The usual field axioms, in Lrings , give rise a theory Tfields , with M j= Tfields if and only if M = (M; 0; 1; +; −; ·) is a field. n n−1 2. Let 'n ≡ 8z0 · · · 8zn−19x x + zn−1x + ::: + z0 = 0. ACF=Tfields [ f'n j n ≥ 2g. (Models are alg. closed fields.) 3. There is an Loag-theory DOAG whose models are preciseley the non-trivial divisible ordered abelian groups. 4. If M is an L-structure, Th(M) = f' L-sentence j M j= 'g. Introduction to Model Theory Basic Concepts Languages, Structures and Theories The expressive power of first order logic Theorem (Compactness Theorem) Let T be a theory. Suppose that any finite subtheory T0 of T has a model. Then T has a model. Corollary 1. If T has arbitrarily large finite models, it has an infinite model. Thus, there is e.g. no theory whose models are the finite fields. 2. If T has an infinite model, it has models of arbitrarily large cardinality. In particular, an infinite L-structure is not determined (up to L-isomorphism) by its theory. V To prove (1), consider n ≡ 9x1;:::; xn i<j xi 6= xj , and apply 0 compactness to T = T [ f n j n 2 Ng. Introduction to Model Theory Basic Concepts Languages, Structures and Theories Complete theories Let T be a theory. A sentence is a consequence of T , denoted T j= , if every model of T is also a model of . M and N are called elementarily equivalent if Th(M) = Th(N ). We write M ≡ N . A consistent theory T is complete if all its models are elementarily equivalent. Alternatively, for every ', either T j= ' or T j= :'. Examples 1. Th(M) is complete, for any structure M. 2. ACFp is a complete Lrings -theory, for p = 0 or a prime. 3. DOAG is a complete Loag-theory. Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination Definable sets Let M be an L-structure. A set D ⊆ Mn is said to be definable if there is a formula '(x; y) and parameters b from M such that n o D = '(M; b) := a 2 Mn j M j= '(a; b) : If b may be taken from B ⊆ M, we say D is B-definable. Convenient to add parameters, passing to LB = L [ fcb j b 2 Bg. Then M expands naturally to an LB -structure MB . Examples 1. In R, the set R≥0 is Lrings -definable, as the set of squares. 2. Let K j= ACF, and let V = V (K) ⊆ K n be an affine variety. Then V is definable in Lrings by a quantifier free formula. More generally, this is the case for every constructible subset of K n. Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination Elementary substructures I M ⊆ N is a substructure if M N N M N n M c = c ; f Mn = f and R \ M = R : I We say M is an elementary substructure of N , M 4 N if for every L-formula '(x) and every tuple a 2 Mn one has M j= '(a) iff N j= '(a): In other words, the embedding respects all definable sets. Note: M 4 N ) M ≡ N . Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination Quantifier elimination Definition A theory T has quantifier elimination (QE) if for every formula '(x) there is a quantifier free (q.f.) formula (x) such that T j= 8x ('(x) $ (x)) : Proposition Let T be a (consistent) theory with QE. I In M j= T , every definable set is q.f. definable. Equivalently, projections of q.f. definable sets are q.f. definable. I Let M and N be models of T . Then M ⊆ N ) M 4 N . (T is model complete). I If any two models of T contain a common substructure, then T is complete. Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination Examples of theories with QE Theorem (Chevalley-Tarski Theorem) ACF has quantifier elimination. Corollary In algebraically closed fields, a set is definable iff it is constructible. Corollary ACFp is complete and strongly minimal: in every model M j= ACFp, every definable subset of M is finite or cofinite. Remark Model-completeness of ACF ^= Hilbert’s Nullstellensatz. Example The theory of the real field R = (R; 0; 1; +; −; ·) does not have QE. (The set of squares is not q.f. definable.) Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination Tarski’s theorem Let Lo:rings = Lrings [ f<g, and let RCF (the theory of real closed fields) be the Lo:rings -theory whose models are I ordered fields F such that I every positive element in F is a square in F and I every polynomial of odd degree over F has a zero in F . Theorem (Tarski 1951) RCF is complete (so equal to Th(R)) and has QE. Corollary The definable sets in RCF are precisely the semi-algebraic sets (sets defined by boolean combinations of polynomial inequalities). Introduction to Model Theory Basic Concepts Definable Sets and Quantifier Elimination 0-minimal theories Definition Let L = f<; : : :g. An L-theory T is o-minimal if in any M j= T , any definable subset of M is a finite union of intervals and points. Corollary RCF is an o-minimal theory. Proof. Clearly, p(X ) ≥ 0 defines a set of the right form, for p a polynomial. We are done by Tarski’s QE result. Proposition 1. DOAG is complete and has QE (in Loag). 2. Definable sets in DOAG are piecewise linear (given by bool. comb. of linear inequalities). In particular, DOAG is o-minimal. Introduction to Model Theory Basic Concepts Types and Saturation The notion of a complete type Definition Let M be a structure and B ⊆ M. A set p(x) of LB -formulas '(x1;:::; xn) is a (complete) n-type over B if I p(x) is finitely satisfiable, i.e. for any '1;:::;'k 2 p there is n a 2 M such that M j= 'i (a) for all i; I p(x) is maximal with this property. Example n Let N < M. For a 2 N , tp(a=B) := f'(x) 2 LB j N j= '(a)g is a complete n-type over B, the type of a over B. Lemma Every complete type p is of the form p(x) = tp(a=B).
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