Independence in model theory
Åsa Hirvonen
Department of Mathematics and Statistics
University of Helsinki
March 5, 2014
Amsterdam
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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
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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
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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.
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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(ā/A) for the type over A realised by ā.
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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.
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Model-theoretic definitions: Galois-type
When working inside a strongly homogeneous monster model the type of ā
over A is the orbit of ā 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.
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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).
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Forking
Definition
The formula ϕ(x̄, ā) divides over A if there are n < ω and tuples āi , i < ω,
such that
1
tp(ā/A) = tp(āi /A),
2
{ϕ(x̄, āi ) : i < ω} is n-inconsistent (i.e. every n-size subset is
inconsistent wrt the theory)
Definition
p = tp(ā/A) forks over A if there are formulas
ϕ0 (x̄0 , ā0 ), . . . , ϕn−1 (x̄n−1 ; ān−1 ) with x̄k ⊆ x̄ for k < n, such that
W
1 p `
k<n ϕk (x̄k , āk ),
2
ϕk (x̄k , āk ) divides over A for all k < n.
In stable theories we have nonforking independence:
We write A ↓B C if for all finite tuples ā from A, tp(ā/C ) does not fork
over B.
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Properties of nonforking
If T is a stable first order theory, then nonforking has the following
properties
1
Invariance If ā ↓A B and F is an isomorphism, then f (ā) ↓f (A) f (B).
2
Monotonicity If A ⊆ B ⊆ C ⊆ D and ā ↓A D then ā ↓B C .
3
Transitivity If A ⊆ B ⊆ C , ā ↓A B and ā ↓B C then ā ↓A C .
4
Symmetry If ā ↓A b̄ then b̄ ↓A ā.
5
Existence/Extension For any ā and A ⊂ B there is b̄ satisfying
tp(ā/A) such that b̄ ↓A B.
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Properties of nonforking
6
Finite character If ā 6↓A B and A ⊂ B then there is a formula
ϕ(x̄, b̄) ∈ tp(a/B) such that no type containing ϕ(x̄, b̄) is
independent over A.
7
Local character There is a cardinal κ(T ) such that for any ā and B
there is A ⊆ B such that |A| < κ(T ) and ā ↓A B.
8
Reflexivity If A ⊂ B, b̄ ∈ B\A and tp(b/A) is not algebraic, then
b̄ 6↓A B.
9
Stationarity over models If A is a model, tp(ā/A) = tp(b̄/A),
ā ↓A B and b̄ ↓A B, then tp(ā/B) = tp(b̄/B).
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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
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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
iS< κ(K) such that tp(a/ j≤i (bj ∪ cj )) splits strongly over
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 .
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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
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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.
If hAi : i < δi is an 4-increasing chain, then
4
1
2
3
S
i<δ Ai ∈ K,
S
for each j < δ, Aj 4 i<δ A
Si ,
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 Löwenheim-Skolem number LS(K) such that if A ∈ K and
B ⊂ A, then there is A0 ∈ K such that B ⊆ A0 4 A and
|A0 | = |B| + LS(K).
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Independence in finitary abstract elementary classes
Definition
In a simple, superstable, finitary AEC one can define
ā ↓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 t w (b/AB) = t w (a/AB) and t w (b/D) does not
Lascar-split over E .
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Independence in finitary abstract elementary classes
Theorem (Hyttinen-Kesälä)
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
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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.
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Dividing
Definition
A type p(x, B) divides over A if there
S exists an A-indiscernible sequence
(Bi : i < ω) in tp(B/A) such that i<ω p(x, Bi ) is inconsistent with T .
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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
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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 -saturated models
Stationarity over Fλ(M)
M -saturated models (i.e. if we have two independence
is unique over Fλ(M)
notions, ↓ and ↓0 satisfying the properties, ā ↓A B and A is
M -saturated, then ā ↓0 B).
Fλ(M)
A
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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 ā in A and b̄
in B tp(ā/C b̄) does not divide over A.
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Uniqueness
Theorem (Hyttinen-Kesälä)
In a superstable finitary AEC any independence notion satisfying
Invariance
Reflexivity
Extension
Stationarity
Monotonicity
Transitivity
is unique.
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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.
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