Forking and dimensions in pseudofinite structures Daro Garca - - PowerPoint PPT Presentation

Forking and dimensions in pseudofinite structures

Darío García University of Leeds. Logic Seminar - School of Mathematics. Leeds, UK. February 22, 2017

Model theory, forking and dimension

Model Theory: tame vs. wild structures

Example (The quintessential example of a tame structure)

Consider the theory of the complex field which can be effectively axiomatized by a finite number of axiom schemes:

• 1. Field axioms (finite in number).
• 2. ∀x1, . . . , xn∃y(y n + x1y n−1 + · · · + xn = 0), for n = 1, 2, 3, . . .
• 3. 1 + · · · + 1 = 0
• n-times

, for n = 1, 2, 3, . . ..

Example (The quintessential example of a wild structure)

Gödel proved that Th(Z, +, ·) cannot be effectively described in any reasonable way, so in contrast to the field of complex numbers, the ring of integers is wild. (But Z as ordered additive group is tame again!)

◮ We use here “tame” and “wild” very informally, to suggest the

distinction between good and bad model-theoretic behavior.

◮ Shelah defined several dividing lines (stability, simplicity, NIP,

NTP2, etc.) to distinguish “tame” from “wild” structures in terms of the (non-)existence of combinatorial configurations of their definable sets.

◮ Gödel’s work is often characterized as saying that those

structures M for which Th(M) can be effectively axiomatizable are uninteresting.

◮ However, even in extremely “wild” subjects (such as number

theory), the solution to difficult problems often uses illuminating explorations into tame territory!

◮ We use here “tame” and “wild” very informally, to suggest the

distinction between good and bad model-theoretic behavior.

◮ Shelah defined several dividing lines (stability, simplicity, NIP,

NTP2, etc.) to distinguish “tame” from “wild” structures in terms of the (non-)existence of combinatorial configurations of their definable sets.

◮ Gödel’s work is often characterized as saying that those

structures M for which Th(M) can be effectively axiomatizable are uninteresting.

◮ However, even in extremely “wild” subjects (such as number

theory), the solution to difficult problems often uses illuminating explorations into tame territory! model theory = geography of tame mathematics.

• E. Hrushovski c

Model theory=geography of tame mathematics

Model theory=geography of tame mathematics

(C, +, ·, 0, 1) (Q, <) (R, +, ·, <) Random Graph Pseudofinite fields Pseudoreal closed fields (Z, +, ·, <) (Z, +) (Z, +, <) (Qp, +, ·) ZFC

Simple NIP Stable

Strongly minimal

Rosy NTP2

O-minimal A more detailed map at www. forkinganddividing. com (due to Gabriel Conant)

Dividing and forking.

Definition

Let φ(x, b) be a formula and A ⊆ M be a set of parameters.

• 1. We say that φ(x, b) divides over A if there is an infinite

sequence bi : i < ω of elements such that:

◮ tp(bi/A) = tp(b/A). ◮ The set of formulas {φ(x, bi) : i < ω} is k-inconsistent for

some k < ω.

• 2. We say that a formula θ(x) (possibly with parameters) forks
• ver A if θ(x) implies a finite disjunction of formulas that

divide over A.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A whenever b ∈ acl(A).

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A whenever b ∈ acl(A). It is enough to take a sequence bi : i < ω of different conjugates

• f b over A. The set of formulas {x = bi : i < ω} will be

2-inconsistent.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q. If π1, π2, . . . , is a sequence of infinitely many distinct transcendental numbers, then we have:

◮ tp(πi/Q) = tp(π/Q). ◮ The set of formulas {x2 = πi : i < ω} is 3-inconsistent.

In fact, one can show that in ACF forking can be characterized by the algebraic formulas.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q. In fact, one can show that in ACF forking can be characterized by the algebraic formulas.

• 3. In the theory DLO of dense linear orders, the formula

φ(x; ab) ≡ a < x < b divides over A = ∅.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q. In fact, one can show that in ACF forking can be characterized by the algebraic formulas.

• 3. In the theory DLO of dense linear orders, the formula

φ(x; ab) ≡ a < x < b divides over A = ∅. ( ) ( ) ( ) a1 b1 a2 b2 a3 b3

b b b

The set of formulas {ai < x < bi : i < ω} is 2-inconsistent.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q. In fact, one can show that in ACF forking can be characterized by the algebraic formulas.

• 3. In the theory DLO of dense linear orders, the formula

φ(x; ab) ≡ a < x < b divides over A = ∅.

• 4. In the theory TE of an equivalence relation with infinitely many

infinite classes, the formula φ(x, b) ≡ xEb divides over A = ∅.

Example

• 1. In any theory T with infinite models, the formula

φ(x, b) ≡ x = b divides over A = ∅.

• 2. For the theory ACF of algebraically closed fields, the formula

φ(x, π) ≡ x2 = π divides over A = Q. In fact, one can show that in ACF forking can be characterized by the algebraic formulas.

• 3. In the theory DLO of dense linear orders, the formula

φ(x; ab) ≡ a < x < b divides over A = ∅.

• 4. In the theory TE of an equivalence relation with infinitely many

infinite classes, the formula φ(x, b) ≡ xEb divides over A = ∅. If bi : i < ω is a sequence of element in different equivalence classes, then the set of formulas {xEbi : i < ω} is 2-inconsistent (by the transitivity of E).

Non-Forking independence

Definition

Given a tuple a, we say that a is independent from B over A if there is no formula φ(x, b) ∈ tp(a/B) that forks over A. We denote this by a | ⌣

A

B

◮ The concept of forking and forking-independence played a

crucial role in the Theory of Classification developed by Shelah, especially for stable theories.

◮ Even today, a recurrent theme in model theory is characterize

forking in certain known structures in terms of combinatorial

• r algebraic invariants.

◮ The notion of non-forking independence generalizes several

classic notions of independence (algebraic independence, linear independence, among others).

Forking and dimension.

Example

Consider the formula φ(x, y; b) ≡ y = x + b defined in C2.

b b b

. . . b1 b2 b3 φ(x, b1) φ(x, b2) φ(x, b3) 2-inconsistent

This formula divides over ∅. Intuitively, the set by φ(x, y; b) is small as it defines a set of dimension 1 inside a space of dimension 2.

This formula divides over ∅. Intuitively, the set by φ(x, y; b) is small as it defines a set of dimension 1 inside a space of dimension 2.

Question

In those examples when there is a established notion of dimension, can forking-independence be detected by changes (decreasing) of this dimension? As the examples of the complex numbers suggest, an ideal answer to this question would be the following:

This formula divides over ∅. Intuitively, the set by φ(x, y; b) is small as it defines a set of dimension 1 inside a space of dimension 2.

Question

In those examples when there is a established notion of dimension, can forking-independence be detected by changes (decreasing) of this dimension? As the examples of the complex numbers suggest, an ideal answer to this question would be the following: c | ⌣

A

B ⇔ dim(c/B) < dim(c/A) ⇔There is a set X = φ(M; b) such that c ∈ X and dim(X) < dim(Y ) for every A-definable set Y containing c.

Example

• 1. In ACF, the notion of dimension is transcendence degree:

dim(a/A) := transc. deg(Q(A, a)/Q(A)). We have √π | ⌣Q Q(π), but also dim(√π/Q) = transc. deg

• Q(√π)/Q
• = 1

< dim(√π/Q(π)) = 0.

• 2. In the theory TE, there is a combinatorial notion of dimension

that can be defined recursively by the following rule:

Definition

The dimension of a point is equal to zero, and dim(X) ≥ n + 1 if and only if there are infinitely many disjoints sets Yi contained in X with dim(Yi) ≥ n.

b b b

dim = 2

• 2. In the theory TE, there is a combinatorial notion of dimension

that can be defined recursively by the following rule:

Definition

The dimension of a point is equal to zero, and dim(X) ≥ n + 1 if and only if there are infinitely many disjoints sets Yi contained in X with dim(Yi) ≥ n.

b b b

dim = 2 xEb

b b

dim = 1

• 2. In the theory TE, there is a combinatorial notion of dimension

that can be defined recursively by the following rule:

Definition

The dimension of a point is equal to zero, and dim(X) ≥ n + 1 if and only if there are infinitely many disjoints sets Yi contained in X with dim(Yi) ≥ n.

b b b

dim = 2 xEb

b b

dim = 1 . . . also the formula xEb divides over A = ∅.

• 3. In the theory DLO there is a natural notion of dimension given

by the topological dimension. ( ) ( ) ( ) a b

b b b

The formula a < x < b divides over ∅. However, there is no decrease of dimension as dim((a, b)) = 1 = dim(R). ✗

Pseudofinite structures

Given a collection of L-structures {Mi : i ∈ I} and an ultrafilter U

• n I, we can consider the ultraproduct

M =

• U

Mi :=

• i∈I

Mi/ ∼U .

Theorem (Łoś, 1955)

For every L-formula φ(x) and every tuple c = ([ci

1]i∈U, . . . , [ci n]i∈U)

• f elements in M, we have that

M | = φ(c) if and only if {i ∈ I : Mi | = φ(ci

1, . . . , ci n)} ∈ U.

Roughly speaking, a formula is true in the ultraproduct M if and

• nly if it is true for “almost every” Mi.

Definition

A structure M is said to be pseudofinite if it is elementary equivalent to an ultraproduct of finite structures.

◮ This kind of finite/infinite connection can sometimes be used

to prove qualitative properties of large finite structures, and in the other direction, quantitative properties in the finite structures often induce desirable qualitative properties in their ultraproducts.

◮ The idea is that the counting measure on a class of finite

structures can be lifted using Łoś’ theorem to give notions of dimension and measure on their ultraproduct.

Notation

◮ L will denote a first-order language. ◮ C = {Mi : i ∈ I} will denote a class of finite L-structures, and

U will denote an ultrafilter on I.

◮ We will assume that for every n ∈ N, {i ∈ I : |Mi| ≥ n} ∈ U.

That is, lim

i→U |Mi| = ∞.

Notation

◮ L will denote a first-order language. ◮ C = {Mi : i ∈ I} will denote a class of finite L-structures, and

U will denote an ultrafilter on I.

◮ We will assume that for every n ∈ N, {i ∈ I : |Mi| ≥ n} ∈ U.

That is, lim

i→U |Mi| = ∞. ◮ We enrich L to a 2-sorted language L+ with sorts D (carrying

the language L) and OF (carrying the language of ordered fields). Also, for every L-formula φ(x, y), a function symbol fφ : D|y| → OF.

◮ Every finite L-structure Mi ∈ C gives rise to an L+-structure

Ki = Mi; (R, +, ·, 0, 1, <) with the functions fφ interpreted as: f Ki

φ

: M|y|

i

− → R b − →

• φ(M|x|

i

, b)

Notation

◮ Consider now the ultraproduct of the structures Ki with

respect to U, K :=

• i∈U

Ki =

• i∈U

Mi, R∗

• .

◮ We put M := i∈U Mi, and T = Th(M). ◮ If X = φ(Mr; b) is an L-definable set in M, |X| := fφ(b) will

denote its (non-standard) cardinality.

◮ Counting measures: For a non-empty definable subset D of M,

there is a finitely-additive real valued probability measure µD defined as: µD(X) := st |X ∩ D| |D|

• = st
• lim

i→U

|X(Mi) ∩ D(Mi)| |D(Mi)|

Pseudofinite dimension

Let C := Conv(Z) will denote the convex hull of the integers.

Definition

For a definable subset X of M, we define the pseudofinite dimension of X to be δ(X) = log |X| + C ∈ R∗/C

Pseudofinite dimension

Let C := Conv(Z) will denote the convex hull of the integers.

Definition

For a definable subset X of M, we define the pseudofinite dimension of X to be δ(X) = log |X| + C ∈ R∗/C For non-empty definable subsets X, Y in M, we have δ(X) = δ(Y ) ⇔ log |X| − log |Y | ∈ C ⇔ 1 n ≤ |Y | |X| ≤ n (for some n ∈ N)

◮ For a type-definable set X, we define

δ(X) := inf{δ(D) : D ⊇ X, D definable}

◮ Given B ⊆ M and a tuple a from M, δ(a/B) := δ(tp(a/B)).

Properties of the pseudofinite dimension δ

Proposition

The pseudofinite dimension δ satisfies the following properties:

• 1. δ(∅) = −∞, and δ(X) = 0 for any finite set X.
• 2. If X1, X2 are -definable, then δ(X ∪ Y ) = max{δ(X), δ(Y )}.
• 3. If X1, X2 are -definable, then δ(X × Y ) = δ(X) + δ(Y )
• 4. If X = Xn with X1 ⊃ X2 ⊃ · · · all -definable, then

δ(X) = infn δ(Xn).

• 5. If X is -definable, f is a definable map, γ ∈ V0 and

δ(f −1(a) ∩ X) ≤ γ for all a, then δ(X) ≤ δ(f (X)) + γ.

Dividing and drop of dimension

Dividing and drop of dimension

Proposition (G., Macpherson, Steinhorn)

Let D = ψ(x, a) be a definable subset of M and φ(x, b) a formula implying ψ(x, a). If φ(x, b) divides over a, then there exists b

′ |

= tp(b/a) such that δ(φ(x, b

′)) < δ(D).

Dividing and drop of dimension

Proposition (G., Macpherson, Steinhorn)

Let D = ψ(x, a) be a definable subset of M and φ(x, b) a formula implying ψ(x, a). If φ(x, b) divides over a, then there exists b

′ |

= tp(b/a) such that δ(φ(x, b

′)) < δ(D).

Proof: Towards a contradiction, assume that for every b

′ |

= tp(b/a) we have δ(φ(x, b′)) = δ(D). Then, for every b

′ |

= tp(b/a), there is nb′ such that log |D| − log |φ(x, b

′)| ≤ nb′.

Dividing and drop of dimension

Proposition (G., Macpherson, Steinhorn)

Let D = ψ(x, a) be a definable subset of M and φ(x, b) a formula implying ψ(x, a). If φ(x, b) divides over a, then there exists b

′ |

= tp(b/a) such that δ(φ(x, b

′)) < δ(D).

Proof: Towards a contradiction, assume that for every b

′ |

= tp(b/a) we have δ(φ(x, b′)) = δ(D). Then, for every b

′ |

= tp(b/a), there is nb′ such that log |D| − log |φ(x, b

′)| ≤ nb′.

(Compactness)⇒ there is a uniform bound N such that log |D| − log |(φ(x, b′))| ≤ N ⇔ |φ(x, b′)| |D| ≥ 1 eN = ǫ > 0

In particular, µD(φ(x, b′)) ≥ ǫ > 0 for every b′ | = tp(b/a).

In particular, µD(φ(x, b′)) ≥ ǫ > 0 for every b′ | = tp(b/a).

Lemma

Let X be a probability space and fix 0 < ǫ ≤ 1

• 2. Let Ai : i < ω be

a sequence of measurable subsets of X such that µ(Ai) ≥ ǫ for every i. Then, for every k < ω there are i1 < i2 < . . . < i2k such that µ  

2k

• j=1

Aij   ≥ ǫ3k.

In particular, µD(φ(x, b′)) ≥ ǫ > 0 for every b′ | = tp(b/a).

Lemma

Let X be a probability space and fix 0 < ǫ ≤ 1

• 2. Let Ai : i < ω be

a sequence of measurable subsets of X such that µ(Ai) ≥ ǫ for every i. Then, for every k < ω there are i1 < i2 < . . . < i2k such that µ  

2k

• j=1

Aij   ≥ ǫ3k. Assume now that {bi : i < ω} is an a-indiscernible sequence witnessing dividing for φ(x, b). By taking Ai = φ(M, bi) and using the lemma above, we will obtain a non-empty k-intersection.

Example: it is necessary to move to a conjugate

Consider the class C = {Mn : n < ω} with Mn = ([1, n · 2n], En), where En is an equivalence relation with classes as follows:

b

bn = 0 |E1| = n · 2n−1 |E2| = n · 2n−2 n · 2n−3 ...

Example: dividing cannot be replaced by forking

Consider the structure Mn = (Z/(3n)Z, R) where R is a ternary relation interpreted in Mn as follows: Mn | = R(w; a, c) if and only if there are integers a′, w′, c′ congruent with a, w, c ( mod 3n) respectively, such that a′ < w′ < c′ and |c′ − a′| ≥ n. Take M =

U Mn.

a1 = b3 b1 = a2 b2 = a3

b b b

( ) ( ) ( )

Example: dividing cannot be replaced by forking

Consider the structure Mn = (Z/(3n)Z, R) where R is a ternary relation interpreted in Mn as follows: Mn | = R(w; a, c) if and only if there are integers a′, w′, c′ congruent with a, w, c ( mod 3n) respectively, such that a′ < w′ < c′ and |c′ − a′| ≥ n. Take M =

U Mn.

a1 = b3 b1 = a2 b2 = a3

b b b

( ) ( ) ( )

Each formula R(x; ai, bi) divides over ∅

Example: dividing cannot be replaced by forking

Consider the structure Mn = (Z/(3n)Z, R) where R is a ternary relation interpreted in Mn as follows: Mn | = R(w; a, c) if and only if there are integers a′, w′, c′ congruent with a, w, c ( mod 3n) respectively, such that a′ < w′ < c′ and |c′ − a′| ≥ n. Take M =

U Mn.

a1 = b3 b1 = a2 b2 = a3

b b b

( ) ( ) ( )

Each formula R(x; ai, bi) divides over ∅ Thus, the formula x = x forks over ∅

Example: dividing cannot be replaced by forking

Consider the structure Mn = (Z/(3n)Z, R) where R is a ternary relation interpreted in Mn as follows: Mn | = R(w; a, c) if and only if there are integers a′, w′, c′ congruent with a, w, c ( mod 3n) respectively, such that a′ < w′ < c′ and |c′ − a′| ≥ n. Take M =

U Mn.

a1 = b3 b1 = a2 b2 = a3

b b b

( ) ( ) ( )

Each formula R(x; ai, bi) divides over ∅ Thus, the formula x = x forks over ∅ But, the only conjugate of x = x is x = x, and δ(x = x) = δ(M) is maximal.

Conditions on δ.

◮ Attainability (Aφ): There is no sequence (pi : i < ω) of finite

partial positive φ-types such that pi ⊆ pi+1 (as sets of formulas) and δ(pi) > δ(pi+1) for each i < ω.

◮ (A∗ φ): same as attainability, but for finite partial φ-types (not

necessarily positive)

◮ Strong attainability (SA) For each partial type p(x) over a

parameter set B there is a finite subtype p0 of p such that δ(p(x)) = δ(p0(x)).

◮ Dimension Comparison in L (DCL): For all formulas φ(x, y)

and ψ(x, z) there is an L-formula χφ,ψ(y, z) such that for all a ∈ M|y|, b ∈ M|z|, M | = χφ,ψ(a, b) ⇔ δ(φ(x; a)) ≤ δ(ψ(x; b)).

◮ Finitely many values (FMVφ): There is a finite set

{δ1, . . . , δk} such that for each b ∈ M|y| there is i ∈ {1, . . . , k} with δ(φ(M|x|, b)) = δi.

Examples

• 1. Asymptotic classes:

Every infinite ultraproduct of an asymptotic class satisfies (SA), (DCL) and (FMV). The following are examples of asymptotic classes:

◮ Finite fields. ◮ Finite cyclic groups. ◮ Finite simple groups of fixed Lie type. ◮ Finite fields with a Frobenius automorphism. ◮ Paley graphs

Examples

• 1. Asymptotic classes:

Every infinite ultraproduct of an asymptotic class satisfies (SA), (DCL) and (FMV). The following are examples of asymptotic classes:

◮ Finite fields. ◮ Finite cyclic groups. ◮ Finite simple groups of fixed Lie type. ◮ Finite fields with a Frobenius automorphism. ◮ Paley graphs

• 2. Pseudofinite vector spaces: If C is the class of all structures

(V , F) where V is a finite-dimensional vector space over the finite field F, then M satisfies (SA) and (DCL)

Examples

• 1. Asymptotic classes:

Every infinite ultraproduct of an asymptotic class satisfies (SA), (DCL) and (FMV). The following are examples of asymptotic classes:

◮ Finite fields. ◮ Finite cyclic groups. ◮ Finite simple groups of fixed Lie type. ◮ Finite fields with a Frobenius automorphism. ◮ Paley graphs

• 2. Pseudofinite vector spaces: If C is the class of all structures

(V , F) where V is a finite-dimensional vector space over the finite field F, then M satisfies (SA) and (DCL)

• 3. Homocyclic groups: Any infinite ultraproduct G of groups

(Z/pnZ)n : n < ω) satisfies (A) and (DCL).

Pseudofinite dimension and simplicity

Proposition

• 1. Assume (DCL). If FMVφ fails for some φ, then T has the

strict order property.

• 2. If Th(M) has the SOP then (FMV) fails.

Theorem (G.,Macpherson, Steinhorn)

• 1. Assume that (A) holds. Then Th(M) is simple and low.
• 2. Assume (SA). Then Th(M) is supersimple.

δ-independence.

Definition

Let a be a tuple and A, B be countable subsets of M. We define a notion of δ-independence as follows: a

δ

| ⌣

A

B ⇔ δ(a/AB) = δ(a/A).

Theorem (G., Macpherson, Steinhorn)

Let A, B be countable subsets of M, and a a tuple from M. If (SA) and (DCL) hold, then a

δ

| ⌣

A

B if and only if a | ⌣

A

B.

Pseudofinite dimension and stability

Proposition (G., Macpherson, Steinhorn)

Assume (A∗

φ). Then the following are equivalent:

• 1. φ(x, y) is unstable.
• 2. For some d there is a d-definable set D ⊆ M|x| and a

sequence (ai : i < ω), L+-indiscernible over d, such that δ(D) = δ

• D ∧
• i<ω

φ(x, ai)

• , and

µD(φ(x, ai) ∧ φ(x, aj)) < µD(φ(x, ai) for all i < j.

• 3. φ(x, y) has the independence property.

Further problems

• 1. Model-theoretic properties of the pairs K = (

U Mi, R∗).

• 2. Develop geometric model theory of pseudofinite structures.

2.1 What is the relationship between the different concepts in model theory (stability, NIP, simplicity, geometries coming from independence relations, etc) once the assumption of pseudofiniteness is added? 2.2 How these classical model-theoretic properties on the ultraproducts of a class of finite structures reflect on quantitative properties for the definable sets along the class?

Further problems

• 1. Model-theoretic properties of the pairs K = (

U Mi, R∗).

• 2. Develop geometric model theory of pseudofinite structures.

2.1 What is the relationship between the different concepts in model theory (stability, NIP, simplicity, geometries coming from independence relations, etc) once the assumption of pseudofiniteness is added? 2.2 How these classical model-theoretic properties on the ultraproducts of a class of finite structures reflect on quantitative properties for the definable sets along the class?

Underlying philosophy (belief): A geography of tame fragments and tame classes of finite struc- tures may yield some insight into finite model theory and more applications to finite (extremal) combinatorics and/or computer science.

References

• R. Elwes, H.D. Macpherson, ‘A survey of asymptotic classes

and measurable structures’, in Model Theory with Applications to Algebra and Analysis, Vol. 2. 2008

• D. García, H.D. Macpherson, C. Steinhorn. Pseudofinite

structures and simplicity. Journal of Mathematical Logic. Vol. 15, No. 1 (2015)

• E. Hrushovski, ‘Stable group theory and approximate

subgroups’, J. Amer. Math. Soc. 25 (2012), 189–243.

• E. Hrushovski. On Pseudo-Finite Dimensions. Notre Dame

Journal of Formal Logic. Volume 54 (2013), no. 3-4, 463–495.

• E. Hrushovski, F. Wagner, ‘Counting and dimensions’, Model

Theory with Applications to Algebra and Analysis, Vol. 2 2008.

GRACIAS!!

Some consequences of (A), (SA), (DCL) and (FMV)

Proposition

• 1. Assume (Aφ) holds. Then there is m = mφ ∈ N such that

there are no tuples a1, . . . , am so that δ(φ(x, a1)) > δ (φ(x, a1) ∧ φ(x, a2)) > · · · > δ  

1≤i≤m

φ(x, ai)   .

• 2. Assume (SA) holds. Then there is no sequence of definable

sets (Sn : n < ω) such that Sn+1 ⊆ Sn and δ(Sn+1) < δ(Sn) for all n < ω.

Some consequences of (A), (SA), (DCL) and (FMV)

Proposition

• 1. Assume (Aφ) holds. Then there is m = mφ ∈ N such that

there are no tuples a1, . . . , am so that δ(φ(x, a1)) > δ (φ(x, a1) ∧ φ(x, a2)) > · · · > δ  

1≤i≤m

φ(x, ai)   .

• 2. Assume (SA) holds. Then there is no sequence of definable

sets (Sn : n < ω) such that Sn+1 ⊆ Sn and δ(Sn+1) < δ(Sn) for all n < ω.

Proof.

Essentially, Compactness + types in L+ + ω1-saturation of M.

δ-independence

Properties of δ-independence

Proposition

The following are properties of δ-independence (with A, B, D all countable):

• 1. Existence: Given sets A ⊆ B and p ∈ S(A) there is a |

= p with a | ⌣

δ A B.

• 2. Monotonicity and transitivity: If A ⊆ D ⊆ B, then

a

δ

| ⌣

A

B ⇔

• a

δ

| ⌣

A

D and a

δ

| ⌣

D

B

• .
• 3. Finite character: If a

δ

| ⌣

A

B, then there is a finite subset b ⊆ B such that a | ⌣

δ A b.

Proposition (G., Macpherson, Steinhorn)

Under further assumptions, we have:

• 4. (SA) ⇒ Local character: For every a and B ⊆ M, there is a

finite subset A ⊆ B such that a | ⌣

δ A B.

• 5. (DCL) ⇒ Invariance: If α ∈ Aut(M), then

a

δ

| ⌣

A

B ⇔ α(a)

δ

| ⌣

α(A)

α(B)

• 6. (DCL+FMV)⇒ Symmetry:

a

δ

| ⌣

A

b ⇔ b

δ

| ⌣

A

a

Theorem (G., Macpherson, Steinhorn)

Let A, B be countable subsets of M, and a a tuple from M.

• 1. Assume (A) and (DCL). Then a |

⌣

A

B ⇒ a

δ

| ⌣

A

B.

• 2. Assume (SA) and (DCL). Then a

δ

| ⌣

A

B ⇒ a | ⌣

A

B. In particular, under (SA) and (DCL) we have a | ⌣

A

B ⇔ a

δ

| ⌣

A

B.