on definability of types in dependent theoriesguingona/thesisdefense.pdfon deﬁnability of types in...

On Definability of Types in Dependent Theories

Vincent Guingona

University of Maryland, College Park

March 14, 2011

Doctoral Thesis Defense, University of Maryland, College Park

Outline

1 IntroductionTypes and Type Spaces

2 UDTFSIntroduction to UDTFSThree Main Theorems on UDTFS

3 VC-Minimal TheoriesIntroduction to VC-Minimal TheoriesThe Kueker Conjecture

Vincent Guingona (UMCP) Definability of Types March 14, 2011

Introduction Types and Type Spaces

ϕ-Types

For this talk, we work in a complete, first-order theory T with infinitemodels and let C be a large saturated model of T .

Fix a partitioned formula ϕ(x ; y) and a set B ⊆ Clg(y).

Definition.

A ϕ-type p over B is a maximal collection of consistent formulas of theform ±ϕ(x ; b) for various b ∈ B.

Example.

Consider T = Th(R; +, ·, <, 0, 1) and let

ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

A set B ⊆ R3 gives a list of parameters corresponding to closed disks inR2. A ϕ-type over B is a consistent region of R2 carved out by the disks.

ϕ-Types

Definition.

Example.

ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

ϕ-Types

Definition.

Example.

ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

ϕ-Types

Definition.

Example.

ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

Example of ϕ-Types

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

There are 6 ϕ-types over this B. That is, |Sϕ(B)| = 6. More generally...

Definition.

The ϕ-Stone Space over B, denoted Sϕ(B), is the set of all ϕ-types overB.

Example of ϕ-Types

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

Definition.

Example of ϕ-Types

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

Definition.

Stability and Dependence

Definition.

We say a partitioned formula ϕ(x ; y) is stable if there do not exist 〈ai : i <ω〉 and 〈bj : j < ω〉 such that, for all i , j < ω

|= ϕ(ai ; bj) if and only if i < j .

Definition.

We say a partitioned formula ϕ(x ; y) is dependent (or sometimes NIP)if there do not exist 〈as : s ∈ P(ω)〉 and 〈bj : j < ω〉 such that, for alls ∈ P(ω), j < ω

|= ϕ(as ; bj) if and only if j ∈ s.

Stability and Dependence

Definition.

We say a partitioned formula ϕ(x ; y) is stable if there do not exist 〈ai : i <ω〉 and 〈bj : j < ω〉 such that, for all i , j < ω

|= ϕ(ai ; bj) if and only if i < j .

Definition.

We say a partitioned formula ϕ(x ; y) is dependent (or sometimes NIP)if there do not exist 〈as : s ∈ P(ω)〉 and 〈bj : j < ω〉 such that, for alls ∈ P(ω), j < ω

|= ϕ(as ; bj) if and only if j ∈ s.

Stable vs. Dependent

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

Then ϕ is not stable. For example, take bj = (0, 0, j + 1) for j < ω andai = (0, i + 3/2) for i < ω.

However, ϕ is dependent. In fact, all formulas in Th(R; +, ·, <, 0, 1) aredependent.

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

Let ϕ(x0, x1; y0, y1, y2) =[(x0 − y0)

2 + (x1 − y1)2 ≤ y2

]as before.

Examples of Stable and Dependent Theories

Examples.

The following theories are stable:

1 The theory of a vector space over a field.

2 Th(C; +, ·, 0, 1).

3 The theory of differentially closed fields.

Examples.

The following theories are unstable and dependent:

1 Th(R; +, ·, <, 0, 1).

2 Th(Q;<).

3 Th(Z;<,+).

Examples of Stable and Dependent Theories

Examples.

The following theories are stable:

1 The theory of a vector space over a field.

2 Th(C; +, ·, 0, 1).

3 The theory of differentially closed fields.

Examples.

The following theories are unstable and dependent:

1 Th(R; +, ·, <, 0, 1).

2 Th(Q;<).

3 Th(Z;<,+).

Definability of Types

Definition.

Fix a formula ϕ(x ; y), a ϕ-type p, and a formula ψ(y) defined over dom(p).We say that ψ defines p if, for all b ∈ dom(p), we have that

ϕ(x ; b) ∈ p(x) if and only if |= ψ(b).

Definition.

We say that ϕ has uniform definability of types if there exists a finitenumber of ∅-definable formulas ψk(y ; z1, ..., zn) for k = 1, ...,K such that,for all non-empty sets B ⊆ Clg(y) and ϕ-types p ∈ Sϕ(B), there existsc1, ..., cn ∈ B and k0 ∈ {1, ...,K} such that

ψk0(y ; c1, ..., cn) defines p.

Definability of Types

Definition.

Fix a formula ϕ(x ; y), a ϕ-type p, and a formula ψ(y) defined over dom(p).We say that ψ defines p if, for all b ∈ dom(p), we have that

ϕ(x ; b) ∈ p(x) if and only if |= ψ(b).

Definition.

We say that ϕ has uniform definability of types if there exists a finitenumber of ∅-definable formulas ψk(y ; z1, ..., zn) for k = 1, ...,K such that,for all non-empty sets B ⊆ Clg(y) and ϕ-types p ∈ Sϕ(B), there existsc1, ..., cn ∈ B and k0 ∈ {1, ...,K} such that

Definability of Types for Stable Formulas

Theorem (Shelah).

The following are equivalent for a partitioned formula ϕ(x ; y):

1 ϕ is stable.

2 For all ϕ-types p, p is definable over dom(p).

3 ϕ has uniform definability of types.

Goal: Generalize this to dependent formulas.

Definability of Types for Stable Formulas

Theorem (Shelah).

The following are equivalent for a partitioned formula ϕ(x ; y):

1 ϕ is stable.

2 For all ϕ-types p, p is definable over dom(p).

3 ϕ has uniform definability of types.

Goal: Generalize this to dependent formulas.

Stable Theories Have...

Definability of types has many consequences for stable theories, includingthe following:

1 (local type space bounds) For any ϕ(x ; y) and any infinite B ⊆ Clg(y),|Sϕ(B)| ≤ |B|.

2 (λ-stability) There exists λ such that, for all B with |B| ≤ λ,|S1(B)| ≤ λ.

3 (stable embeddedness) For any B, every externally definable subsetC ⊆ B is definable with parameters in B.

4 (non-forking extensions) For any type p and any B ⊇ dom(p), thereexists a non-forking extension q ∈ S(B) of p.

5 etc.

Stable Theories Have...

Definability of types has many consequences for stable theories, includingthe following:

1 (local type space bounds) For any ϕ(x ; y) and any infinite B ⊆ Clg(y),|Sϕ(B)| ≤ |B|.

2 (λ-stability) There exists λ such that, for all B with |B| ≤ λ,|S1(B)| ≤ λ.

3 (stable embeddedness) For any B, every externally definable subsetC ⊆ B is definable with parameters in B.

4 (non-forking extensions) For any type p and any B ⊇ dom(p), thereexists a non-forking extension q ∈ S(B) of p.

5 etc.

Counting Type Spaces

Corollary.

If ϕ(x ; y) is stable, then there exists K , n < ω such that, for anynon-empty set B ⊆ Clg(y), |Sϕ(B)| ≤ K · |B|n.

Theorem (Sauer’s Lemma).

If ϕ(x ; y) is dependent, then there exists K , n < ω such that, for anynon-empty FINITE set B ⊆ Clg(y), |Sϕ(B)| ≤ K · |B|n.

Definition.

We say that a dependent formula ϕ has VC-density ` if ` is the infimumof all n ∈ R+ such that the condition in the above theorem holds.

Corollary.

Definition.

Corollary.

Definition.

UDTFS Introduction

Definability of Types over Finite Sets

For any formula ϕ(x ; y) and any ϕ-type p with a finite domain, p isdefinable.Let B1 = {b ∈ dom(p) : ϕ(x ; b) ∈ p(x)}. Then, since dom(p) is finite, B1

is finite, so the following is a formula:

ψ(y) =

∨b∈B1

.Notice that ψ defines p.

UDTFS Introduction

Uniform Definability of Types over Finite Sets

One generalization of definability of types to some dependent theories isUDTFS.

Definition.

We say that ϕ has uniform definability of types over finite sets (UDTFS)if there exists a finite number of ∅-definable formulas ψk(y ; z1, ..., zn) fork = 1, ...,K such that, for all non-empty FINITE sets B ⊆ Clg(y) andϕ-types p ∈ Sϕ(B), there exists c1, ..., cn ∈ B and k0 ∈ {1, ...,K} suchthat

UDTFS Introduction

Uniform Definability of Types over Finite Sets

One generalization of definability of types to some dependent theories isUDTFS.

Definition.

We say that ϕ has uniform definability of types over finite sets (UDTFS)if there exists a finite number of ∅-definable formulas ψk(y ; z1, ..., zn) fork = 1, ...,K such that, for all non-empty FINITE sets B ⊆ Clg(y) andϕ-types p ∈ Sϕ(B), there exists c1, ..., cn ∈ B and k0 ∈ {1, ...,K} suchthat

UDTFS Introduction

UDTFS Implies VC-Density Bound

If ϕ has UDTFS, witnessed by ψk(y ; z1, ..., zn) for k = 1, ...,K , then ϕhas VC-density ≤ n.

Why?For any fixed finite B ⊆ Clg(y), ϕ-types over B are determined by:

1 elements c1, ..., cn ∈ B, and

2 some k0 ∈ {1, ...,K}.So there are, at most, K · |Bn| = K · |B|n such ϕ-types.

UDTFS Introduction

UDTFS Implies VC-Density Bound

If ϕ has UDTFS, witnessed by ψk(y ; z1, ..., zn) for k = 1, ...,K , then ϕhas VC-density ≤ n.

Why?For any fixed finite B ⊆ Clg(y), ϕ-types over B are determined by:

1 elements c1, ..., cn ∈ B, and

2 some k0 ∈ {1, ...,K}.So there are, at most, K · |Bn| = K · |B|n such ϕ-types.

UDTFS Introduction

Facts about UDTFS

Facts.

1 If ϕ(x ; y) is stable, then ϕ has UDTFS.

2 If ϕ(x ; y) has UDTFS, then ϕ is dependent.

Theorem (Johnson, Laskowski).

If T is o-minimal, then T has UDTFS.

stable⇓

o-minimal ⇒ UDTFS ⇒ dependent

UDTFS Introduction

Facts about UDTFS

Facts.

stable⇓

UDTFS Introduction

Facts about UDTFS

Facts.

stable⇓

UDTFS Three Main Theorems

dp-Minimal Theories

Definition.

A theory T is dp-minimal if there do not exist ϕ(x ; y), ψ(x ; z), 〈bi : i < ω〉,and 〈c j : j < ω〉 such that, for all i0, j0 < ω, the type

{¬ϕ(x ; bi0),¬ψ(x ; c j0)} ∪ {ϕ(x ; bi ) : i 6= i0} ∪ {ψ(x ; c j) : j 6= j0}.

is consistent.

Examples of dp-Minimal Theories

Examples.

The following theories are dp-minimal:

1 Any o-minimal theory or weakly o-minimal theory,

2 Th(Z;<,+),

3 Th(Qp; +, ·, |, 0, 1) (where x |y iff. vp(x) ≤ vp(y)),

4 Algebraically closed valued fields.

5 In general, any VC-minimal theory is dp-minimal.

6 Any theory with VC-density ≤ 1 is dp-minimal.

dp-Minimal Theories have UDTFS

Theorem (G.).

If T is dp-minimal, then T has UDTFS.

Theorem (G.).

If ϕ(x ; y) and N < ω are such that, for all B ⊆ Clg(y) with |B| = N,|Sϕ(B)| ≤ N(N + 1)/2, then ϕ has UDTFS (in particular if ϕ hasVC-density < 2, then ϕ has UDTFS).

dp-Minimal Theories have UDTFS

Theorem (G.).

If T is dp-minimal, then T has UDTFS.

Theorem (G.).

If ϕ(x ; y) and N < ω are such that, for all B ⊆ Clg(y) with |B| = N,|Sϕ(B)| ≤ N(N + 1)/2, then ϕ has UDTFS (in particular if ϕ hasVC-density < 2, then ϕ has UDTFS).

Valued Fields and UDTFS

Theorem (G.).

If (K , k, Γ) is a Henselian valued field that has elimination of fieldquantifiers in the Denef-Pas language, Th(k) has UDTFS, and Th(Γ) hasUDTFS, then the full theory in the Denef-Pas language has UDTFS.

Examples.

The theories of the following structures in the Denef-Pas language haveUDTFS:

2 R((t)),

3 C((t)),

4 C((tQ)).

Valued Fields and UDTFS

Theorem (G.).

If (K , k, Γ) is a Henselian valued field that has elimination of fieldquantifiers in the Denef-Pas language, Th(k) has UDTFS, and Th(Γ) hasUDTFS, then the full theory in the Denef-Pas language has UDTFS.

Examples.

The theories of the following structures in the Denef-Pas language haveUDTFS:

2 R((t)),

3 C((t)),

4 C((tQ)).

The UDTFS Conjecture

o-minimal ⇒ VC-density ≤ 1 stable⇓ ⇓ ⇓

VC-minimal ⇒ dp-minimal ⇒ UDTFS ⇒ dependent

Open Question (Laskowski).

If ϕ is dependent, then does ϕ have UDTFS?

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