Gabriel Conant UIC - nd.edugconant/Talks/GSC_Model_Theory_handout.pdfA partial type p(x ) forks over...

Model Theory and Forking Independence

Gabriel ConantUIC

UIC Graduate Student ColloquiumApril 22, 2013

Gabriel Conant (UIC) Model Theory and Forking Independence April 22, 2013 1 / 24

Types

We fix a first order language L and a complete L-theory T .Fix a modelM |= T and a subset B ⊆M.

DefinitionA partial type over B, p(x , b), is a consistent set of formulas ϕ(x , b),where the parameters of the formulas are taken from the set B. If|x | = n, this is also called an n-type.

Given a ∈M, we have the complete type over B

tp(a/B) := {ϕ(x , b) :M |= ϕ(a, b), b ∈ B}.

If tp(a/B) = tp(b/B) then we write a ≡B b.


Saturation

DefinitionGiven n > 0 we define the Stone space

SMn (B) := {tp(a/B) : a ∈ N �M}.

This is a compact Hausdorff space with the basis of clopen sets

[ϕ(x , b)] := {p ∈ SMn (B) : ϕ(x , b) ∈ p}.

DefinitionFix an infinite cardinal κ. M is κ-saturated if for any subset B ⊆M ofsize less than κ, every type in SMn (B) has a realization inM.


Saturation

ExampleConsider the theory of algebraically closed fields of characteristic 0 inthe language L = {+, ·,0,1}.• Qalg is not ℵ0-saturated, e.g. the type p(x) of a transcendental

element is not realized, where p(x) := {f (x) 6= 0 : f (x) ∈ Z[x ]}.• C is ℵ1-saturated.

Fact(a) IfM is κ-saturated then |M| ≥ κ. In particular, the type{x 6= a : a ∈M} cannot be realized inM.

(b) For any complete theory T we can find κ-saturated models forarbitrarily large κ. In general, these models could be much largerthan κ. We can ensure κ-saturated models of size κ by making settheoretic assumptions (e.g. κ is inaccessible) or assumptions on T(e.g. stability ).


The Monster Model

For the rest of this talk, we will work in a very large κ-saturated modelM, for some very large cardinal κ.

What this means is that all modelsM of our theory, of size smallerthan κ, will be elementary submodels of M.

In particular, any consistent partial type over a parameter set of sizeless than κ will be realized in M.

Unless otherwise stated, all modelsM, N and sets A, B, C will be ofsize less than κ.

Another consequence of saturation is that for any set B and tuples a,b, we have a ≡B b if and only if there is an automorphism of M, whichfixes B pointwise, and sends a to b.


Forking and Dividing

DefinitionA formula ϕ(x , b) divides over C if there is a sequence of tuples(bi)i<ω such that:• bi ≡C b for all i < ω,• there is an integer k ≥ 1 such that every k -element subset of{ϕ(x , bi) : i < ω} is inconsistent.

A partial type p(x) divides over C if it contains a formula that dividesover C.

A partial type p(x) forks over C if there are finitely many formulasϕ1(x , b), . . . , ϕn(x , b) such that:• each ϕi(x , b) divides over C,• any realization of p(x) also realizes some ϕi(x , b).


Circular Order on Q

Consider L = {cyc}, where cyc is a ternary relation.

Interpret cyc in Q by

Q |= cyc(x , y , z) ⇔ x ≤ y ≤ z or z ≤ x ≤ y or y ≤ z ≤ x .

Consider the formula cyc(0, x ,1).For n < ω, (2n,2n + 1) ≡∅ (0,1). Moreover any 2-element subset of

{cyc(2n, x ,2n + 1) : n < ω}

is inconsistent.

So cyc(0, x ,1) divides over ∅.


Forking and Independence

We define a ternary relation | f on subsets of M. In particular,

A | fC

B ⇔ for all a ∈ A, tp(a/BC) does not fork over C.

This is supposed to capture some kind of notion of “freeness” or“independence”. If A | f

CB, we often say “A is free from B over C.”

If A 6 | fC

B, we say “A forks with B over C”.

Another common slogan:

A 6 | fC

B ⇔ BC knows more about A than C knows alone.


The Random Graph

Let L = {R}, where R is a binary relation. A random graph is anonempty graph G with the property that for any finite, disjoint subsetsA,B ⊂ G, there is a vertex in G that is connected to every point in Aand no point in B.

This can be axiomatized in the language L.

TheoremIn the theory of the random graph,

A | fC

B ⇔ A ∩ B ⊆ C


Additive Group of the Integers

Consider the theory of Z in the language L = {+,0,1}.

Given a set A, let cl(A) be the divisible hull of the subgroup generatedby A ∪ Z.

In other words, x ∈ cl(A) if and only if there is some integer n > 0 suchthat nx is in the subgroup generated by A ∪ Z.

TheoremIn Th(Z,+,0,1),

A | fC

B ⇔ cl(AC) ∩ cl(BC) = cl(C)


Vector spaces over Q

Consider the theory of Q in the language L = {+,0,1}. A model ofTh(Q,+,0,1) can be thought of as a torsion free divisible abeliangroup or, equivalently, as a vector space over Q.

Given sets A and C, let dim(A/C) be the cardinality of a basis for Aover C. Let 〈C〉 be the vector space span of C.

Note that if C ⊆ D then dim(A/D) ≤ dim(A/C).

TheoremIn Th(Q,+,0,1),

A | fC

B ⇔ dim(A/BC) = dim(A/C)

⇔ 〈AC〉 ∩ 〈BC〉 = 〈C〉


Algebraically Closed Fields

Consider the theory ACF0 of algebraically closed fields ofcharacteristic 0.

Given a set B, let KB be the algebraically closed field generated by B.Given a ∈M, let trdg(a/B) be the transcendence degree of a over KB.

Note that if C ⊆ D then trdg(a/D) ≤ trdg(a/C).

TheoremIn algebraically closed fields,

a | fC

B ⇔ for every ideal I ⊆ KBC [x ],if a ∈ V (I) then V (I) contains a point in K n

C

⇔ trdg(a/BC) = trdg(a/C)

Remark: A | fC

B ⇒ KAC ∩ KBC = KC is still true, but the implicationis strict.


The Random Kn-free Graph

Let L = {R}, where R is a binary relation. Fix n ≥ 3 and let Kn be thecomplete graph on n vertices.

A Kn-free random graph is a nonempty graph G with the property thatfor any finite, disjoint subsets A,B ⊂ G, if A is Kn−1-free then there is avertex in G that is connected to every point in A and no point in B.

Given disjoint sets B and C, we say B is n-bound to C if there is agraph X ⊆ BC of size n, intersecting both B and C, such that the onlyedges missing in X are between two points in B.

Theorem (C.)In the theory of the Kn-free random graph,

A | fC

B ⇔ A ∩ B ⊆ C and for all b ⊆ B\C,b is either n-bound to C or not n-bound to AC.


The Urysohn Sphere

The Urysohn sphere is a countably universal and homogeneous metricspace in the sense that:• every finite metric space (with distances bounded by 1) can be

isometrically embedded into it.• every isometry between finite subspaces can be extended to an

isometry of the whole space.

This theory can be quantified in a generalization of classical first orderlogic called continuous logic. In this setting, we let U be a κ-saturatedmodel of the theory of the Urysohn sphere. Then U has the twoproperties above, where we can replace “finite” with “size less than κ”.


The Urysohn Sphere

Fix C ⊂ U and b1,b2 ∈ U. Define

dmin(b1,b2/C) = max{

13d(b1,b2), sup

c∈C|d(b1, c)− d(b2, c)|

}and

dmax(b1,b2/C) = infc∈C

(d(b1, c) u d(b2, c)).

Interpretation: Considering C ∪ {b1} and C ∪ {b2} as individualmetric spaces, we can amalgamate them into a new metric spaceC ∪ {b1,b2} by choosing d(b1,b2). We only need

supc∈C|d(b1, c)− d(b2, c)| ≤ d(b1,b2) ≤ inf

c∈C(d(b1, c) u d(b2, c)).

The 13d(b1,b2) term is due to more technical issues.


The Urysohn Sphere

Note that if C ⊆ D then

dmin(b1,b2/C) ≤ dmin(b1,b2/D) ≤ dmax(b1,b2/D) ≤ dmax(b1,b2/C).

Theorem (C., Terry)In the Urysohn sphere,

A | fC

B ⇔for every b1,b2 ∈ B,dmin(b1,b2/C) = dmin(b1,b2/AC) anddmax(b1,b2/C) = dmax(b1,b2/AC).


Forking and DividingExcept for the random Kn-free graph and the Urysohn sphere, all of theprevious examples have the property that forking and dividing are thesame for formulas.

Theorem (C.)In the random Kn-free graph, forking and dividing are the same forcomplete types. However, there is a formula that forks and does notdivide.

Theorem (C.,Terry)In the Urysohn sphere, forking and dividing are the same for completetypes.

QuestionAre forking and dividing the same for formulas in the Urysohn sphere?


Good Behavior of Forking

The property of forking equaling dividing for formulas can beconsidered an example of good or desirable behavior for a theory. It isalso stronger than forking equaling dividing for complete types.

Even better behavior is when forking is symmetric, i.e.,A | f

CB ⇔ B | f

CA. This is stronger than forking equaling dividing

for formulas, and such theories are called simple.

Except for the random Kn-free graph and Urysohn sphere, all of theprevious examples (random graph, Th(Z,+,0,1), Th(Q,+,0,1), ACF)are simple.


Bad Behavior of Forking

Recall the slogan:

A 6 | fC

B ⇔ BC knows more about A than C knows alone.

With this in mind, we see that A 6 | fC

C should be considered badbehavior.

This cannot happen in a simple theory, or in a theory where forkingand dividing are the same (even just for complete types).

A 6 | fC

C can also be described as a type “forking over its own set ofparameters”. It is not hard to see that a type can never divide over itsown set of parameters.


Circular Order on Q

Recall

Q |= cyc(x , y , z) ⇔ x ≤ y ≤ z or z ≤ x ≤ y or y ≤ z ≤ x .

We showed that cyc(0, x ,1) divides over ∅. A similar argument showsthat cyc(1, x ,0) divides over ∅.

Note that cyc(0, x ,1) ∨ cyc(1, x ,0) holds for any element of M.

Therefore the partial type {x = x} forks over ∅ by definition.

a 6 | f∅∅ for any a ∈M.


Classifying Lines Among Theories

Definition(a) A theory T has the independence property if there is a formula

ϕ(x , y) and tuples (bi)i<ω such that if Ai := {a : M |= ϕ(a, bi)} thenfor all I ⊆ ω ⋂

i∈I

Ai ∩⋂i 6∈I

¬Ai 6= ∅.

(b) A theory T has the strict order property if there is a formulaϕ(x , y) and tuples (bi)i<ω such that

A0 ( A1 ( A2 . . . .

If T does not have the independence property then T is NIP.If T does not have the strict order property then T is NsOP.


Map of the Universe

NIP

NsOP

stable ←− →simple

(Q,+,0,1)

(C,+, ·,0,1)

(Z,+,0,1)

randomgraph

randomKn-free graph

Urysohnsphere

(Q, cyc)

(R,+, ·,0,1)(Z,+, ·,0,1)

ZFC


A Coincidental Question

Theorem (Chernikov, Kaplan)Suppose T is an NIP theory in which no type forks over its ownparameter set. Then forking and dividing are the same for formulas.

Question (Chernikov, Kaplan)Can the same result be shown for NsOP theories?

Answer (C.)No. The random Kn-free graph is an NsOP theory in which no typeforks over its parameter set, but there is a formula that forks and doesnot divide.


References

Chernikov A. & Kaplan I., Forking and dividing in NTP2 theories,Journal of Symbolic Logic 77 (2012) 1-20.

B. Kim & A. Pillay, Simple theories, Annals of Pure and AppliedLogic 88 (1997) 149-164.

D. Marker, Model Theory: An Introduction, Springer, 2002.

S. Shelah, Classification Theory and the Number ofNon-Isomorphic Models, North-Holland, 1978.

K. Tent & M. Ziegler, A Course in Model Theory, Cambridge, 2012.

Map of the Universe, http://www.forkinganddividing.com


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