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) = {f (a): f ∈ Aut(M/A)}.

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 < ω) 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 ω-stable if it is stable in ℵ0 (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 ϕ(¯x, ¯a) divides over A if there are n < ω and tuples ¯ai , i < ω, such that

1 tp(¯a/A) = tp(¯ai /A),

2 {ϕ(¯x, ¯ai ): i < ω} 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 ϕ0(¯x0, ¯a0), . . . , ϕn−1(¯xn−1; ¯an−1) withx ¯k ⊆ x¯ for k < n, such that

1 W p ` k

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 ϕ(¯x, b¯) ∈ tp(a/B) such that no type containing ϕ(¯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 |A| < κ(T ) and ¯a ↓A B. 8 Reflexivity If A ⊂ B, b¯ ∈ B\A 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 (ω-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 ∈ B and an infinite sequence I , indiscernible over A, with b, c ∈ 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

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 ∈ K, i<δ S 2 for each j < δ, Aj 4 i<δ Ai , S 3 if each Ai 4 M ∈ K then i<δ Ai 4 M. 5 If A, B, C ∈ 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 ∈ K and 0 0 B ⊂ A, then there is A ∈ K such that B ⊆ A 4 A and 0 |A | = |B| + 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 ∈ 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 < ω) in tp(B/A) such that i<ω 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. Theorem Assume M is stable and strongly homogeneous. Then any independence notion satisfying Invariance Monotonicity Existence/Extension M Stationarity over Fλ(M)-saturated models M is unique over Fλ(M)-saturated models (i.e. if we have two independence 0 notions, ↓ and ↓ satisfying the properties, ¯a ↓A B and A is M 0 Fλ(M)-saturated, then ¯a ↓A B).

A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 21 / 24 Uniqueness

Theorem (Hyttinen-Lessmann) Suppose M is a stable homogeneous monster model and there exists an independence relation satisfying: Invariance Monotonicity Local character Finite character Symmetry Transitivity Extension Bounded extensions

Then M is superstable and A ↓C B if and only if for all finite ¯a in A and b¯ in B tp(¯a/Cb¯) does not divide over A.

A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 22 / 24 Uniqueness

Theorem (Hyttinen-Kes¨al¨a) In a superstable finitary AEC any independence notion satisfying Invariance Reflexivity Extension Stationarity Monotonicity Transitivity is unique.

A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 23 / 24 Connection to probability

Theorem (Ben Yaacov)

1 The class of probability algebras is axiomatisable in continuous logic. 2 Random variables are interpretable in the event space and vice versa. 3 Model theoretic independence coincides with probabilistic independence.

A.˚ Hirvonen (University of Helsinki) Independence in model theory March 5, 2014 24 / 24