background the modeling property a dictionary theorem I -Indexed Indiscernible Sets and Trees Lynn Scow Vassar College Harvard/MIT Logic Seminar December 3, 2013 1 / 20
background the modeling property a dictionary theorem Outline 1 background 2 the modeling property 3 a dictionary theorem 2 / 20
background the modeling property a dictionary theorem order indiscernible sets Fix a linear order O and an L -structure M (we assume M is sufficiently saturated.) Let b i be same-length finite tuples from M : Definition B = { b i | i ∈ O } is an order-indiscernible set if for all n ≥ 1, for all i 1 , . . . , i n , j 1 , . . . , j n from O , ( i 1 , . . . , i n ) �→ ( j 1 , . . . , j n ) is an order-isomorphism ⇒ tp L ( b i 1 , . . . , b i n ; M ) = tp L ( b j 1 , . . . , b j n ; M ) 3 / 20
background the modeling property a dictionary theorem typical application Suppose we have parameters A = { a i | i < ω } and i < j ⇒ � ϕ ( a i , a j ) (let’s assume ϕ ( x, x ) is unsatisfiable) In a typical application, we use Ramsey’s theorem to find an order-indiscernible set B = { b i | i < ω } such that i < j ⇒ � ϕ ( b i , b j ) Because B is indiscernible, for some t ∈ { 0 , 1 } ( ϕ 0 = ϕ, ϕ 1 = ¬ ϕ ) i > j ⇒ � ϕ ( b i , b j ) t In a well-known characterization: Th( M ) is stable ⇔ t = 0 for all such B 4 / 20
background the modeling property a dictionary theorem 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 = { b i : i ∈ O } is an order-indiscernible set if for all n ≥ 1, for all i 1 , . . . , i n , j 1 , . . . , j n from O , ( i 1 , . . . , i n ) ∼ ( j 1 , . . . , j n ) ⇒ tp L ( b i 1 , . . . , b i n ; M ) = tp L ( b j 1 , . . . , b j n ; M ) Here ( i 1 , . . . , i n ) ∼ ( j 1 , . . . , j n ) means qftp L ′ ( i 1 , . . . , i n ; O ) = qftp L ′ ( j 1 , . . . , j n ; O ) 5 / 20
background the modeling property a dictionary theorem I -indexed indiscernible sets Now we fix an arbitrary language L ′ , and an L ′ -structure I in the place of O . Definition ([She90]) B = { b i : i ∈ I } is an I - indexed indiscernible set if for all n ≥ 1, for all i 1 , . . . , i n , j 1 , . . . , j n from I , ( i 1 , . . . , i n ) ∼ ( j 1 , . . . , j n ) ⇒ tp L ( b i 1 , . . . , b i n ; M ) = tp L ( b j 1 , . . . , b j n ; M ) Here ( i 1 , . . . , i n ) ∼ ( j 1 , . . . , j n ) means qftp L ′ ( i 1 , . . . , i n ; I ) = qftp L ′ ( j 1 , . . . , j n ; I ) Say that B is ∆- I -indexed indiscernible for ∆ ⊆ L if we replace L above by ∆. 6 / 20
background the modeling property a dictionary theorem overview Suppose ϕ ( x, x ) is unsatisfiable. Then the “type” of a { ϕ } - I -indexed indiscernible set B is determined entirely by the data t = ( t 0 , t 1 , . . . ) If B is an order-indiscernible set: � ϕ ( b i , b j ) t 0 i < j ⇒ � ϕ ( b i , b j ) t 1 i > j ⇒ If B is an ordered-graph indexed indiscernible set � ϕ ( b i , b j ) t 0 i < j ∧ iRj ⇒ � ϕ ( b i , b j ) t 1 i > j ∧ iRj ⇒ � ϕ ( b i , b j ) t 2 i < j ∧ ¬ iRj ⇒ � ϕ ( b i , b j ) t 3 i > j ∧ ¬ iRj ⇒ 7 / 20
background the modeling property a dictionary theorem ordered 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 ( b i , b j ) i < j ∧ ¬ iRj ⇒ ϕ ( b i , b j ) 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. 8 / 20
background the modeling property a dictionary theorem 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 iRj ¬ iRj i < j red blue − → i < j r (b) r (b) green purple p (g) p (g) i > j i > j (Ramsey’s theorem) ¬ iRj iRj − → i < j red blue i > j green purple (Neˇ setˇ ril-R¨ odl theorem) 9 / 20
background the modeling property a dictionary theorem trees I s = ( ω <ω , � , ∧ , < lex , ( P n ) n<ω ) where � is the partial tree-order, ∧ is the meet function in this order, < lex is the lexicographical order, and the P n are predicates picking out the n -th level of the tree I 1 = ( ω <ω , � , ∧ , < lex , < lev ) where η < lev ν ⇔ ℓ ( η ) < ℓ ( ν ) I 0 = ( ω <ω , � , ∧ , < lex ) I t = ( ω <ω , � , < lex ) 10 / 20
background the modeling property a dictionary theorem a typical dichotomy result The structure I s 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 a η from M with ℓ ( a η ) = ℓ ( y ) such that: 1 { ϕ ( x ; a σ ↾ n ) : σ ∈ ω ω } is consistent (nodes on a path “are consistent”), and 2 for all η ∈ ω <ω , pairs from { ϕ ( x ; a η � � i � ) : 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 = { b η | η ∈ ω> ω } where B is I s -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. 11 / 20
background the modeling property a dictionary theorem 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 . � B � We denote the copies of A in B as . A 12 / 20
background the modeling property a dictionary theorem ramsey classes: II Given a finite set X of cardinality k , We refer to a map � C � c : → X as a k -coloring of the copies of A in C . A We say that B ′ ⊆ C is homogeneous for c if there is an element x 0 ∈ X such that for all A ′ ∈ � B ′ � , c ( A ′ ) = x 0 . A Definition A class K of finite L ′ -structures is a Ramsey class if for all A, B ∈ K � C � there is a C ∈ K such that for any 2-coloring of , there is a A B ′ ⊆ C , isomorphic to B that is homogeneous for this coloring. 13 / 20
background the modeling property a dictionary theorem EM-types For A = { a i | i ∈ I } we can formally define a type in variables { x i | i ∈ I } called the Ehrenfeucht-Mostowski type of A , EM( A ) If � ϕ ( a i 1 , . . . , a i n ) for all ( i 1 , . . . , i n ) ∼ ( j 1 , . . . , j n ), then ϕ ( x j 1 , . . . , x j n ) ∈ 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 ( a i )) then � � ∀ ı q ( ı ) ⇒ � ϕ ( b i ) In fact B will have a rule such as the above for all quantifier-free types q ; whereas A could have rules for none. 14 / 20
background the modeling property a dictionary theorem the modeling property Definition I -indexed indiscernibles have the modeling property if for all I -indexed parameters A = ( a i : 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? 15 / 20
background the modeling property a dictionary theorem 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 I 0 = ( ω <ω , � , ∧ , < lex ) Theorem (Takeuchi-Tsuboi) I 0 -indexed indiscernibles have the modeling property. Corollary age ( I 0 ) is a Ramsey class. Removing ∧ destroys the Ramsey property. 16 / 20
background the modeling property a dictionary theorem K = age( I t ) 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 L t -embeds into B 1 , B 2 by a i �→ b i , c i . a 1 a 2 a 3 c 1 c 2 c 3 b 1 b 2 b 3 B 1 : B 2 : A : c 4 b 4 a 0 c 0 b 0 Suppose there exists some amalgam C for ( A, B 1 , B 2 ). Observe that b 4 , c 4 in C must be � -comparable in C , as both points are � -predecessors of the same point, b 2 (= c 2 ). If b 4 � c 4 , then b 4 � c 4 � c 3 = b 3 , contradicting the data in B 1 . If c 4 � b 4 , then c 4 � b 4 � b 1 = c 1 , contradicting the data in B 2 . 17 / 20
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