I -Indexed Indiscernible Sets and Trees Lynn Scow Vassar College - - PowerPoint PPT Presentation

background the modeling property a dictionary theorem

I-Indexed Indiscernible Sets and Trees

Lynn Scow

Vassar College

Harvard/MIT Logic Seminar December 3, 2013

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background the modeling property a dictionary theorem

Outline

1 background 2 the modeling property 3 a dictionary theorem

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background the modeling property a dictionary theorem

• rder indiscernible sets

Fix a linear order O and an L-structure M (we assume M is sufficiently saturated.) Let bi be same-length finite tuples from M: Definition B = {bi | i ∈ O} is an order-indiscernible set if for all n ≥ 1, for all i1, . . . , in, j1, . . . , jn from O, (i1, . . . , in) → (j1, . . . , jn) is an order-isomorphism ⇒ tpL(bi1, . . . , bin; M) = tpL(bj1, . . . , bjn; M)

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typical application

Suppose we have parameters A = {ai | i < ω} and i < j ⇒ ϕ(ai, aj) (let’s assume ϕ(x, x) is unsatisfiable) In a typical application, we use Ramsey’s theorem to find an

• rder-indiscernible set B = {bi | i < ω} such that

i < j ⇒ ϕ(bi, bj) Because B is indiscernible, for some t ∈ {0, 1} (ϕ0 = ϕ, ϕ1 = ¬ϕ) i > j ⇒ ϕ(bi, bj)t In a well-known characterization: Th(M) is stable ⇔ t = 0 for all such B

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generalizing order-indiscernible sets

Consider O as a structure in its own right, O = (O, <) in the language L′ = {<}, and re-write the definition: Definition B = {bi : i ∈ O} is an order-indiscernible set if for all n ≥ 1, for all i1, . . . , in, j1, . . . , jn from O, (i1, . . . , in) ∼ (j1, . . . , jn) ⇒ tpL(bi1, . . . , bin; M) = tpL(bj1, . . . , bjn; M) Here (i1, . . . , in) ∼ (j1, . . . , jn) means qftpL′(i1, . . . , in; O) = qftpL′(j1, . . . , jn; O)

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I-indexed indiscernible sets

Now we fix an arbitrary language L′, and an L′-structure I in the place of O. Definition ([She90]) B = {bi : i ∈ I} is an I-indexed indiscernible set if for all n ≥ 1, for all i1, . . . , in, j1, . . . , jn from I, (i1, . . . , in) ∼ (j1, . . . , jn) ⇒ tpL(bi1, . . . , bin; M) = tpL(bj1, . . . , bjn; M) Here (i1, . . . , in) ∼ (j1, . . . , jn) means qftpL′(i1, . . . , in; I) = qftpL′(j1, . . . , jn; I) Say that B is ∆-I-indexed indiscernible for ∆ ⊆ L if we replace L above by ∆.

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• verview

Suppose ϕ(x, x) is unsatisfiable. Then the “type” of a {ϕ}-I-indexed indiscernible set B is determined entirely by the data t = (t0, t1, . . .) If B is an order-indiscernible set: i < j ⇒ ϕ(bi, bj)t0 i > j ⇒ ϕ(bi, bj)t1 If B is an ordered-graph indexed indiscernible set i < j ∧ iRj ⇒ ϕ(bi, bj)t0 i > j ∧ iRj ⇒ ϕ(bi, bj)t1 i < j ∧ ¬iRj ⇒ ϕ(bi, bj)t2 i > j ∧ ¬iRj ⇒ ϕ(bi, bj)t3

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• rdered graphs

Consider the example I = (I, <, E) for an order relation < and an edge relation E. Suppose we only consider I that are weakly saturated, i.e., that embed all possible ordered graphs. The above kind of I-indexed indiscernible can be applied to characterize NIP theories. We call it an ordered graph-indiscernible set. Suppose we have an ordered graph-indexed set B such that i < j ∧ iRj ⇒ ϕ(bi, bj) i < j ∧ ¬iRj ⇒ ϕ(bi, bj)t T is NIP ⇔ t = 0 for all such B In a characterization from [Sco12]: T is NIP iff any ordered graph indiscernible set in a model of T is an order-indiscernible set.

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different partition properties

Fix a coloring on n-tuples from I, where coloring is uniform on pairs: ⇒ ∃ large homogeneous B ⊆ I s.t. ∀(i, j) from B: iRj ¬iRj i < j red blue i > j green purple − → iRj ¬iRj i < j r (b) r (b) i > j p (g) p (g) (Ramsey’s theorem) − → iRj ¬iRj i < j red blue i > j green purple (Neˇ setˇ ril-R¨

• dl theorem)

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trees

Is = (ω<ω, , ∧, <lex, (Pn)n<ω) where is the partial tree-order, ∧ is the meet function in this

• rder, <lex is the lexicographical order, and the Pn are predicates

picking out the n-th level of the tree I1 = (ω<ω, , ∧, <lex, <lev) where η <lev ν ⇔ ℓ(η) < ℓ(ν) I0 = (ω<ω, , ∧, <lex) It = (ω<ω, , <lex)

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a typical dichotomy result

The structure Is is ideal to study TP Definition A theory T has the (2-)tree property (TP) if there is a model M T, a formula ϕ(x; y) and parameters aη from M with ℓ(aη) = ℓ(y) such that:

1 {ϕ(x; aσ↾n) : σ ∈ ωω} is consistent

(nodes on a path “are consistent”), and

2 for all η ∈ ω<ω, pairs from {ϕ(x; aηi) : i < ω} are inconsistent

(siblings “are inconsistent”)

By a well-known result, if a theory has TP, then it has TP as witnessed by B = {bη | η ∈ ω>ω} where B is Is-indexed indiscernible. By a series of reductions, one proves the well-known theorem that TP comes in one of two extremal versions...TP1 and TP2.

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ramsey classes: I

Fix a class K of finite L′-structures. Definition For A, B ∈ K, a copy of A in B is an embedding f : A → B modulo the equivalence relation of being the same embedding up to an automorphism of A From now on, assume L′ contains a relation < linearly ordering all members of K. Then we may think of a copy of A in B as being the range of an embedding from A into B. We denote the copies of A in B as B

A

• .

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ramsey classes: II

Given a finite set X of cardinality k, We refer to a map c : C

A

• → X as a k-coloring of the copies of A in C.

We say that B′ ⊆ C is homogeneous for c if there is an element x0 ∈ X such that for all A′ ∈ B′

A

• , c(A′) = x0.

Definition A class K of finite L′-structures is a Ramsey class if for all A, B ∈ K there is a C ∈ K such that for any 2-coloring of C

A

• , there is a

B′ ⊆ C, isomorphic to B that is homogeneous for this coloring.

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EM-types

For A = {ai | i ∈ I} we can formally define a type in variables {xi | i ∈ I} called the Ehrenfeucht-Mostowski type of A, EM(A) If ϕ(ai1, . . . , ain) for all (i1, . . . , in) ∼ (j1, . . . , jn), then ϕ(xj1, . . . , xjn) ∈ EM(A) If B EM(A), and q is a complete quantifier free type in the language of I, then if ∀ı (q(ı) ⇒ ϕ(ai)) then ∀ı

• q(ı) ⇒ ϕ(bi)
• In fact B will have a rule such as the above for all quantifier-free

types q ; whereas A could have rules for none.

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the modeling property

Definition I-indexed indiscernibles have the modeling property if for all I-indexed parameters A = (ai : i ∈ I) in any structure M, there exists I-indexed indiscernible parameters B EM(A) For which I do I-indexed indiscernibles have the modeling property?

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translation

Theorem (dictionary theorem) Suppose that I is a qfi, locally finite structure in a language L′ with a relation < linearly ordering I. Then I-indexed indiscernible sets have the modeling property just in case age(I) is a Ramsey class. Recall I0 = (ω<ω, , ∧, <lex) Theorem (Takeuchi-Tsuboi) I0-indexed indiscernibles have the modeling property. Corollary age(I0) is a Ramsey class. Removing ∧ destroys the Ramsey property.

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K = age(It) not a Ramsey class

Proof. By [Neˇ s05], if K is a Ramsey class, then K has the amalgamation

• property. However, an example analyzed in Takeuchi-Tsuboi provides

a counterexample to amalgamation. A Lt-embeds into B1, B2 by ai → bi, ci. A: a0 a2 a1 a3 B1: b0 b2 b1 b3 b4 B2: c0 c2 c1 c3 c4 Suppose there exists some amalgam C for (A, B1, B2). Observe that b4, c4 in C must be -comparable in C, as both points are

• predecessors of the same point, b2(= c2). If b4 c4, then

b4 c4 c3 = b3, contradicting the data in B1. If c4 b4, then c4 b4 b1 = c1, contradicting the data in B2.

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finitary infinitary

Theorem ([She90]) For every n, m < ω there is some k = k(n, m) < ω such that for any infinite cardinal χ, the following is true of λ := k(χ)+: for every f : (n≥λ)m → χ there is a level-preserving, orientation-preserving subtree I ⊆ n≥λ such that (i) ∈ I and whenever η ∈ I ∩ n>λ, ||{α < λ : η α ∈ I}|| ≥ χ+. (ii)f If ¯ η, ¯ ν ∈ I are such that ¯ η ∼Is ¯ ν then f(η0, . . . , ηm−1) = f(ν0, . . . , νm−1). Theorem ([Fou99]) age(Is) is a Ramsey class Both yield that Is-indexed indiscernibles have the modeling property, the second by way of the dictionary theorem. The first result yields a height-n indiscernible subtree with m-types from the original tree.

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Thanks

Thanks for your attention!

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background the modeling property a dictionary theorem

• W. L. Fouch´

e. Symmetries and Ramsey properties of trees. Discrete Mathematics, 197/198:325–330, 1999. 16th British Combinatorial Conference (London, 1997).

• J. Neˇ

setˇ ril. Homogeneous structures and Ramsey classes. Combinatorics, Probability and Computing, 14:171–189, 2005.

• L. Scow.

Characterization of NIP theories by ordered graph-indiscernibles. Annals of Pure and Applied Logic, 163:1624–1641, 2012.

• S. Shelah.

Classification Theory and the number of non-isomorphic models (revised edition). North-Holland, Amsterdam-New York, 1990.

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