Model Theory and Forking Independence

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

2 Types We fix a first order language L and a complete L-theory T. Fix a model M = T and a subset B M. Definition A 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 ā M, we have the complete type over B tp(ā/b) := {ϕ( x, b) : M = ϕ(ā, b), b B}. If tp(ā/b) = tp( b/b) then we write ā B b. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

3 Saturation Definition Given n > 0 we define the Stone space S M n (B) := {tp(ā/b) : ā N M}. This is a compact Hausdorff space with the basis of clopen sets [ϕ( x, b)] := {p S M n (B) : ϕ( x, b) p}. Definition Fix an infinite cardinal κ. M is κ-saturated if for any subset B M of size less than κ, every type in S M n (B) has a realization in M. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

4 Saturation Example Consider the theory of algebraically closed fields of characteristic 0 in the language L = {+,, 0, 1}. Q alg is not ℵ 0 -saturated, e.g. the type p(x) of a transcendental element is not realized, where p(x) := {f (x) 0 : f (x) Z[x]}. C is ℵ 1 -saturated. Fact (a) If M is κ-saturated then M κ. In particular, the type {x a : a M} cannot be realized in M. (b) For any complete theory T we can find κ-saturated models for arbitrarily large κ. In general, these models could be much larger than κ. We can ensure κ-saturated models of size κ by making set theoretic assumptions (e.g. κ is inaccessible) or assumptions on T (e.g. stability). Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

5 The Monster Model For the rest of this talk, we will work in a very large κ-saturated model M, for some very large cardinal κ. What this means is that all models M of our theory, of size smaller than κ, will be elementary submodels of M. In particular, any consistent partial type over a parameter set of size less than κ will be realized in M. Unless otherwise stated, all models M, N and sets A, B, C will be of size less than κ. Another consequence of saturation is that for any set B and tuples ā, b, we have ā B b if and only if there is an automorphism of M, which fixes B pointwise, and sends ā to b. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

6 Forking and Dividing Definition A formula ϕ( x, b) divides over C if there is a sequence of tuples ( b i ) i<ω such that: b i C b for all i < ω, there is an integer k 1 such that every k-element subset of {ϕ( x, b i ) : i < ω} is inconsistent. A partial type p( x) divides over C if it contains a formula that divides over C. A partial type p( x) forks over C if there are finitely many formulas ϕ 1 ( x, b),..., ϕ n ( x, b) such that: each ϕ i ( x, b) divides over C, any realization of p( x) also realizes some ϕ i ( x, b). Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

7 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 is inconsistent. So cyc(0, x, 1) divides over. {cyc(2n, x, 2n + 1) : n < ω} Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

8 Forking and Independence We define a ternary relation f on subsets of M. In particular, A f B for all ā A, tp(ā/bc) does not fork over C. C This is supposed to capture some kind of notion of freeness or independence. If A f B, we often say A is free from B over C. C If A f B, we say A forks with B over C. C Another common slogan: A f B BC knows more about A than C knows alone. C Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

9 The Random Graph Let L = {R}, where R is a binary relation. A random graph is a nonempty graph G with the property that for any finite, disjoint subsets A, B G, there is a vertex in G that is connected to every point in A and no point in B. This can be axiomatized in the language L. Theorem In the theory of the random graph, A f C B A B C Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

10 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 generated by A Z. In other words, x cl(a) if and only if there is some integer n > 0 such that nx is in the subgroup generated by A Z. Theorem In Th(Z, +, 0, 1), A f B cl(ac) cl(bc) = cl(c) C Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

11 Vector spaces over Q Consider the theory of Q in the language L = {+, 0, 1}. A model of Th(Q, +, 0, 1) can be thought of as a torsion free divisible abelian group or, equivalently, as a vector space over Q. Given sets A and C, let dim(a/c) be the cardinality of a basis for A over C. Let C be the vector space span of C. Note that if C D then dim(a/d) dim(a/c). Theorem In Th(Q, +, 0, 1), A B C dim(a/bc) = dim(a/c) AC BC = C Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

12 Algebraically Closed Fields Consider the theory ACF 0 of algebraically closed fields of characteristic 0. Given a set B, let K B be the algebraically closed field generated by B. Given ā M, let trdg(ā/b) be the transcendence degree of ā over K B. Note that if C D then trdg(ā/d) trdg(ā/c). Theorem In algebraically closed fields, ā f B for every ideal I K BC[ x], C if ā V (I) then V (I) contains a point in KC n trdg(ā/bc) = trdg(ā/c) Remark: A f C B K AC K BC = K C is still true, but the implication is strict. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

13 The Random K n -free Graph Let L = {R}, where R is a binary relation. Fix n 3 and let K n be the complete graph on n vertices. A K n -free random graph is a nonempty graph G with the property that for any finite, disjoint subsets A, B G, if A is K n 1 -free then there is a vertex 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 a graph X BC of size n, intersecting both B and C, such that the only edges missing in X are between two points in B. Theorem (C.) In the theory of the K n -free random graph, A f C B A B C and for all b B\C, b is either n-bound to C or not n-bound to AC. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

14 The Urysohn Sphere The Urysohn sphere is a countably universal and homogeneous metric space 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 order logic called continuous logic. In this setting, we let U be a κ-saturated model of the theory of the Urysohn sphere. Then U has the two properties above, where we can replace finite with size less than κ. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

15 The Urysohn Sphere Fix C U and b 1, b 2 U. Define d min (b 1, b 2 /C) = max { 1 3 d(b 1, b 2 ), sup c C d max (b 1, b 2 /C) = inf c C (d(b 1, c) d(b 2, c)). } d(b 1, c) d(b 2, c) and Interpretation: Considering C {b 1 } and C {b 2 } as individual metric spaces, we can amalgamate them into a new metric space C {b 1, b 2 } by choosing d(b 1, b 2 ). We only need sup d(b 1, c) d(b 2, c) d(b 1, b 2 ) c C inf (d(b 1, c) d(b 2, c)). c C The 1 3 d(b 1, b 2 ) term is due to more technical issues. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

16 The Urysohn Sphere Note that if C D then d min (b 1, b 2 /C) d min (b 1, b 2 /D) d max (b 1, b 2 /D) d max (b 1, b 2 /C). Theorem (C., Terry) In the Urysohn sphere, for every b 1, b 2 B, A f B d C min (b 1, b 2 /C) = d min (b 1, b 2 /AC) and d max (b 1, b 2 /C) = d max (b 1, b 2 /AC). Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

17 Forking and Dividing Except for the random K n -free graph and the Urysohn sphere, all of the previous examples have the property that forking and dividing are the same for formulas. Theorem (C.) In the random K n -free graph, forking and dividing are the same for complete types. However, there is a formula that forks and does not divide. Theorem (C.,Terry) In the Urysohn sphere, forking and dividing are the same for complete types. Question Are forking and dividing the same for formulas in the Urysohn sphere? Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

18 Good Behavior of Forking The property of forking equaling dividing for formulas can be considered an example of good or desirable behavior for a theory. It is also stronger than forking equaling dividing for complete types. Even better behavior is when forking is symmetric, i.e., A f B B C f A. This is stronger than forking equaling dividing C for formulas, and such theories are called simple. Except for the random K n -free graph and Urysohn sphere, all of the previous examples (random graph, Th(Z, +, 0, 1), Th(Q, +, 0, 1), ACF) are simple. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

19 Bad Behavior of Forking Recall the slogan: A f B BC knows more about A than C knows alone. C With this in mind, we see that A behavior. f C C should be considered bad This cannot happen in a simple theory, or in a theory where forking and dividing are the same (even just for complete types). A f C can also be described as a type forking over its own set of C parameters. It is not hard to see that a type can never divide over its own set of parameters. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

20 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 shows that 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 f for any a M. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

21 Classifying Lines Among Theories Definition (a) A theory T has the independence property if there is a formula ϕ( x, ȳ) and tuples ( b i ) i<ω such that if A i := {ā : M = ϕ(ā, b i )} then for all I ω A i. i I A i i I (b) A theory T has the strict order property if there is a formula ϕ( x, ȳ) and tuples ( b i ) i<ω such that A 0 A 1 A 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. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

22 Map of the Universe (Q, cyc) NIP ZFC (R, +,, 0, 1) (Z, +,, 0, 1) (Q, +, 0, 1) (C, +,, 0, 1) (Z, +, 0, 1) random graph random K n -free graph NsOP Urysohn sphere stable simple Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

23 A Coincidental Question Theorem (Chernikov, Kaplan) Suppose T is an NIP theory in which no type forks over its own parameter 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 K n -free graph is an NsOP theory in which no type forks over its parameter set, but there is a formula that forks and does not divide. Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

24 References Chernikov A. & Kaplan I., Forking and dividing in NTP 2 theories, Journal of Symbolic Logic 77 (2012) B. Kim & A. Pillay, Simple theories, Annals of Pure and Applied Logic 88 (1997) D. Marker, Model Theory: An Introduction, Springer, S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, North-Holland, K. Tent & M. Ziegler, A Course in Model Theory, Cambridge, Map of the Universe, Gabriel Conant (UIC) Model Theory and Forking Independence April 22, / 24

Forking and Dividing in Random Graphs

Forking and Dividing in Random Graphs Gabriel Conant UIC Graduate Student Conference in Logic University of Notre Dame April 28-29, 2012 Gabriel Conant (UIC) Forking and Dividing in Random Graphs April