Independence in model theory Asa˚ Hirvonen Department of Mathematics and Statistics University of Helsinki March 5, 2014 Amsterdam A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 1 / 24 Model theoretic motivation Classification of structures find a system of invariants that will determine the structure up to isomorphism Examples: transcendence degree of algebraically closed fields of fixed characteristic Ulm invariants of countable Abelian groups A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 2 / 24 Shelah's Main Gap Theorem (Shelah) A countable first order theory T falls into one of two categories: it is intractable, i.e. it has the maximal number of models in all sufficiently large cardinalities or every model of T is decomposed as a tree of countable models A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 3 / 24 Independence calculus Shelah's proof builds on notions of independence and generation. Shelah used nonforking independence. Proving structure theorems for more general classes involve developing suitable independence notions. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 4 / 24 Model theoretic definitions: type Definition When studying the class of models of a first order theory T a type over A is a set of formulas with parameters from A that is consistent with T . We write tp(¯a=A) for the type over A realised by ¯a. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 5 / 24 Model theoretic definitions: monster model M In elementary model theory a monster model is just a large saturated model. When studying nonelementary classes (with amalgamation), a monster model is a µ-universal, strongly µ-homogeneous model for a large enough µ. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 6 / 24 Model-theoretic definitions: Galois-type When working inside a strongly homogeneous monster model the type of ¯a over A is the orbit of ¯a under automorphisms of M fixing A pointwise. tp(a=A) = ff (a): f 2 Aut(M=A)g: In the elementary case this coincides with the syntactic type. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 7 / 24 Model theoretic definitions: Stability Definition A theory T is stable in λ if there are only λ types (n-types for some n < !) over parameter sets of size λ. A theory T is stable if it is stable in some λ. A theory T is superstable if it is stable from some κ onwards. A theory T is !-stable if it is stable in @0 (and then stable in all infinite cardinalities). A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 8 / 24 Forking Definition The formula '(¯x; ¯a) divides over A if there are n < ! and tuples ¯ai , i < !, such that 1 tp(¯a=A) = tp(¯ai =A), 2 f'(¯x; ¯ai ): i < !g is n-inconsistent (i.e. every n-size subset is inconsistent wrt the theory) Definition p = tp(¯a=A) forks over A if there are formulas '0(¯x0; ¯a0);:::;'n−1(¯xn−1; ¯an−1) withx ¯k ⊆ x¯ for k < n, such that 1 W p ` k<n 'k (¯xk ; ¯ak ), 2 'k (¯xk ; ¯ak ) divides over A for all k < n. In stable theories we have nonforking independence: We write A #B C if for all finite tuples ¯a from A, tp(¯a=C) does not fork over B. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 9 / 24 Properties of nonforking If T is a stable first order theory, then nonforking has the following properties 1 Invariance If ¯a #A B and F is an isomorphism, then f (¯a) #f (A) f (B). 2 Monotonicity If A ⊆ B ⊆ C ⊆ D and ¯a #A D then ¯a #B C. 3 Transitivity If A ⊆ B ⊆ C, ¯a #A B and ¯a #B C then ¯a #A C. 4 Symmetry If ¯a #A b¯ then b¯ #A ¯a. 5 Existence/Extension For any ¯a and A ⊂ B there is b¯ satisfying tp(¯a=A) such that b¯ #A B. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 10 / 24 Properties of nonforking 6 Finite character If ¯a 6#A B and A ⊂ B then there is a formula '(¯x; b¯) 2 tp(a=B) such that no type containing '(¯x; b¯) is independent over A. 7 Local character There is a cardinal κ(T ) such that for any ¯a and B there is A ⊆ B such that jAj < κ(T ) and ¯a #A B. 8 Reflexivity If A ⊂ B, b¯ 2 BnA and tp(b=A) is not algebraic, then b¯ 6#A B. 9 Stationarity over models If A is a model, tp(¯a=A) = tp(b¯=A), ¯a #A B and b¯ #A B, then tp(¯a=B) = tp(b¯=B). A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 11 / 24 Towards greater generality Generalising the framework elementary classes elementary submodels of a strongly homogeneous model abstract elementary classes \elementary" classes wrt continuous logic metric abstract elementary classes Results require various stability assumptions (!-stable, superstable, stable, weakly stable, simple) Study different notions of types types Lascar types Lascar strong types types in continuous logic A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 12 / 24 Independence in homogeneous model theory When working in a stable homogeneous class, one can define independence based on strong splitting. Definition 1 A type tp(a=B) is said to split strongly over A ⊂ B if there are b; c 2 B and an infinite sequence I , indiscernible over A, with b; c 2 I such that tp(b=A [ a) 6= tp(c=A [ a). 2 κ(K) denotes the least cardinal such that there are no a, bi and ci for S i < κ(K) such that tp(a= j≤i (bj [ cj )) splits strongly over S j<i (bj [ cj ) for each i < κ(K). 3 a #A B if there is C ⊆ A of cardinality < κ(M) such that for all D ⊇ A [ B there is b with tp(b=A [ B) = tp(a=A [ B) such that tp(b=D) does not split strongly over C. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 13 / 24 Independence in homogeneous model theory Theorem (Hyttinen-Shelah) In a simple stable homogeneous class # satisfies Monotonicity Extension Finite character Symmetry Transitivity Stationarity for Lascar strong types A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 14 / 24 Independence in finitary abstract elementary classes Definition A class of structures of a fixed vocabulary (K; 4) is an abstract elementary class if 1 Both K and the binary relation 4 are closed under isomorphism. 2 If A 4 B then A is a substructure of B. 3 4 is a partial order on K. 4 If hAi : i < δi is an 4-increasing chain, then S 1 Ai 2 K, i<δ S 2 for each j < δ, Aj 4 i<δ Ai , S 3 if each Ai 4 M 2 K then i<δ Ai 4 M. 5 If A; B; C 2 K, A 4 C, B 4 C and A ⊆ B then A 4 B. 6 There is a L¨owenheim-Skolem number LS(K) such that if A 2 K and 0 0 B ⊂ A, then there is A 2 K such that B ⊆ A 4 A and 0 jA j = jBj + LS(K). A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 15 / 24 Independence in finitary abstract elementary classes Definition In a simple, superstable, finitary AEC one can define ¯a #A B if there is a finite E ⊆ A such that for all monster models M0 extending M and D ⊂ M0 such that A [ B ⊂ D there is a monster model M00 extending M and b 2 M00 such that tw (b=AB) = tw (a=AB) and tw (b=D) does not Lascar-split over E. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 16 / 24 Independence in finitary abstract elementary classes Theorem (Hyttinen-Kes¨al¨a) If (K; 4) is a simple, superstable, finitary AEC with the Tarksi-Vaught property, then the relation # has the following properties Invariance Monotonicity Transitivity Symmetry Extension Finite character Local character Reflexivity Stationarity for Lascar types A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 17 / 24 Continuous logic Continuous logic takes truth values in the interval [0; 1] has continuous functions [0; 1]n ! [0; 1] as connectives has sup and inf as quantifiers Continuous logic is used to study bounded metric structures. A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 18 / 24 Dividing Definition A type p(x; B) divides over A if there exists an A-indiscernible sequence S (Bi : i < !) in tp(B=A) such that i<! p(x; Bi ) is inconsistent with T . A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 19 / 24 Independence in continuous logic In continuous logic independence is defined via dividing A #B C if and only if tp(A=BC) does not divide over B. Theorem (Ben Yaacov, Berenstein, Henson, Usvyatsov) If T is a continuous theory, stable in the metric sense (i.e. considering the density character of the type set), then # satisfies Invariance Symmetry Transitivity Finite character Extension Local character Stationarity over models A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 20 / 24 Uniqueness Well-behaved independence notions are unique.