NTP 1 Artem Chernikov (IMJ-PRG) Paris, June 9, 2015
Joint work with Nicholas Ramsey (UC Berkeley).
Shelah’s classification
Tree properties Let T be a complete theory and ϕ ( x ; y ) ∈ L a formula in the language of T . ◮ ϕ ( x ; y ) has the tree property (TP) if there is k < ω and a tree of tuples ( a η ) η ∈ ω <ω in M such that: ◮ for all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent, ◮ for all η ∈ ω <ω , { ϕ ( x ; a η⌢ � i � ) : i < ω } is k -inconsistent. ◮ ϕ ( x ; y ) has the tree property of the first kind (TP 1 ) if there is a tree of tuples ( a η ) η ∈ ω <ω in M such that: ◮ for all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent, ◮ for all η ⊥ ν in ω <ω , { ϕ ( x ; a η ) , ϕ ( x ; a ν ) } is inconsistent. ◮ ϕ ( x ; y ) has the tree property of the second kind (TP 2 ) if there is a k < ω and an array ( a α, i ) α<ω, i <ω in M such that: ◮ for all functions f : ω → ω , { ϕ ( x ; a α, f ( α ) ) : α < ω } is consistent, ◮ for all α , { ϕ ( x ; a α, i ) : i < ω } is k -inconsistent. ◮ T has one of the above properties if some formula does modulo T .
Shelah’s theorem, 1 ◮ So TP 1 and TP 2 are two extreme forms in which TP can occur. In TP 1 , everything that is not forced to be consistent by the definition of TP, is inconsistent. In TP 2 , everything that is not forced to be inconsistent by the definition of TP, is consistent. Fact [Shelah] If T has TP , then it either has TP 1 or TP 2 . ◮ To each theory T , one associates cardinal invariants κ cdt , κ sct , κ inp measuring how much of TP, TP 1 and TP 2 (respectively) it contains. Namely, we allow different formulas at each level in the definition above, and take the first cardinal such that there is no tree with that many levels. ◮ E.g. κ cdt = ∞ iff T has TP, and T is supersimple iff κ cdt = ℵ 0 . Similarly, κ inp = ∞ iff T has TP 2 , and T is strong iff κ inp = ℵ 0 . ◮ Shelah asked for a quantitative refinement of the above theorem: does κ cdt = κ sct + κ inp hold?
Shelah’s theorem, 2 Theorem If T is countable, then κ cdt = κ sct + κ inp . ◮ In fact if T is countable, then κ cdt , κ sct , κ inp ∈ {ℵ 0 , ℵ 1 , ∞} . We treat each of ℵ 0 and ℵ 1 separately, the ∞ case follows from Shelah’s theorem. Theorem [Ramsey] There are theories (in an uncountable language) with κ cdt > κ inp + κ sct . ◮ Constructs a theory reducing the question to a deep result of Shelah and Juhász on the non-existence of homogeneous partitions for certain colorings of families of finite subsets of 2 λ � ++ + ω 4 for some � certain cardinals (one can take κ = infinite cardinal λ , then there is T with | T | = κ and such that κ cdt = κ + but κ sct ≤ κ and κ inp ≤ κ ).
So what is known about NTP 1 ? ◮ [Kim, Kim] In the definition of TP 1 , one can replace 2-inconsistency by k -inconsistency, for any k ≥ 2. Also, there is a characterization of NTP 1 via counting certain families of partial types. ◮ [Malliaris, Shelah] If T has TP 1 , then it is maximal in the Keisler order (via equivalence to SOP 2 , see later). ◮ Not much more. For example, any kind of a basic theory of forking is missing. ◮ Another question from Shelah’s book, in the special case: is TP 1 always witnessed by a formula in a single variable? ◮ As usual for this kind of questions, to simplify combinatorics we would like to work with “indiscernible” witnesses of our properties.
Indiscernible trees, 1 ◮ Fix a theory T in a language L and M | = T a monster model. ◮ Consider the language L 0 = { ⊳ , ∧ , < lex } . We view the tree κ <λ as an L 0 -structure in a natural way, interpreting ⊳ as the tree partial order, ∧ as the binary meet function and < lex as the lexicographic order. ◮ Suppose that ( a η ) η ∈ κ <λ is collection of tuples and C a set of parameters in some model. ◮ We say that ( a η ) η ∈ κ <λ is a strongly indiscernible tree over C if qftp L 0 ( η 0 , . . . , η n − 1 ) = qftp L 0 ( ν 0 , . . . , ν n − 1 ) implies tp L ( a η 0 , . . . , a η n − 1 / C ) = tp L ( a ν 0 , . . . , a ν n − 1 / C ) , for all n ∈ ω .
Indiscernible trees, 2 Using some results from structural Ramsey theory of trees, one can show that indiscernible trees “exist”. More precisely, let I 0 be the L 0 -structure ( ω <ω , � , < lex , ∧ ) with all symbols given their intended interpretations. Fact [Takeuchi, Tsuboi], [Kim, Kim, Scow] Given any tree ( a i : i ∈ I 0 ) of tuples from M , there is a strongly indiscernible tree ( b i : i ∈ I 0 ) in M locally based on the ( a i ) : given any finite set of formulas ∆ from L and a finite tuple ( t 0 , . . . , t n − 1 ) from I 0 , there is a tuple ( s 0 , . . . , s n − 1 ) from I 0 such that qftp L 0 ( t 0 , . . . , t n − 1 ) = qftp L 0 ( s 0 , . . . , s n − 1 ) and tp ∆ ( b t 0 , . . . , b t n − 1 ) = tp ∆ ( a s 0 , . . . , a s n − 1 ) .
Path collapse lemma, 1 ◮ In particular, if φ ( x ; y ) has TP 1 , then there is a strongly indiscernible tree witnessing this. ◮ (Path Collapse lemma) Suppose κ is an infinite cardinal, ( a η ) η ∈ 2 <κ is a tree strongly indiscernible over a set of parameters C and, moreover, ( a 0 α : 0 < α < κ ) is an indiscernible sequence over cC . Let p ( y ; z ) = tp ( c ; ( a 0 ⌢ 0 γ : γ < κ ) / C ) . Then if p ( y ; ( a 0 ⌢ 0 γ ) γ<κ ) ∪ p ( y ; ( a 1 ⌢ 0 γ ) γ<κ ) is not consistent, then T has TP 1 , witnessed by a formula with free variables y .
Path collapse lemma, 2 The proof requires in particular a (rather tedious) demonstration that various operations on strongly indiscernible trees preserve strong indiscernibility, e.g.
Application 1: TP 1 is witnessed by a formula in a single variable Theorem Suppose T witnesses TP 1 via ϕ ( x , y ; z ) . Then there is a formula ϕ 0 ( x ; v ) with free variables x and parameter variables v, or a formula ϕ 1 ( y ; w ) with free variables y and parameter variables w so that one of ϕ 0 and ϕ 1 witness TP 1 . ◮ Proof idea. Start with a strongly indiscernible tree witnessing that ϕ has TP 1 . Assume that no formula in the free variable y has TP 1 , and let bc 0 realize a branch of the tree. Then iteratively applying the path collapse lemma to the type of c 0 over that branch in increasing fattenings of the tree we can conclude by compactness that there is some c such that ϕ ( x ; c , z ) has TP 1 , which is enough.
Application 2: Weak k − TP 1 is equivalent to TP 1 ◮ Say that a subset { η i : i < k } ⊆ ω <ω is a collection of distant siblings if given i � = i ′ , j � = j ′ , all of which are < k , η i ∧ η i ′ = η j ∧ η j ′ . Definition [Kim, Kim] ϕ ( x ; y ) has weak k − TP 1 if there is a a collection of tuples ( a η ) η ∈ ω <ω such that: ◮ for all η ∈ ω ω , { ϕ ( x ; a η | α ) : α < ω } is consistent. ◮ if { η i : i < k } ⊆ ω <ω is a collection of distinct distant siblings, then { ϕ ( x ; a η i ) : i < k } is inconsistent. ◮ TP 1 ⇐ ⇒ weak 2-TP 1 = ⇒ weak 3-TP 1 = ⇒ . . . ◮ [Kim, Kim] Do the converse implications hold? Theorem T has weak k- TP 1 iff it has TP 1 , for all k ≥ 2 .
SOP n hierarchy, 1 Definition [Shelah], [Dzamonja, Shelah] ◮ Fix n ≥ 3. We say that a formula φ ( x ; y ) has SOP n if: ◮ there are pairwise different ( a i ) i ∈ ω such that | = φ ( a i , a j ) for all i < j < ω , ◮ | = ¬∃ x 0 . . . x n − 1 � j = i + 1 ( mod n ) φ ( x i , x j ) . ◮ ϕ ( x ; y ) has SOP 2 if there is a collection of tuples ( a η ) η ∈ 2 <ω such that: ◮ for all η ∈ 2 ω , { ϕ ( x ; a η | α ) : α < ω } is consistent, ◮ If η, ν ∈ 2 <ω and η ⊥ ν , then { ϕ ( x ; a η ) , ϕ ( x ; a ν ) } is inconsistent. ◮ ϕ ( x ; y ) has SOP 1 if there are ( a η ) η ∈ 2 <ω such that: ◮ for all η ∈ 2 ω , { ϕ ( x ; a η | n ) : n < ω } is consistent, ◮ if η ⌢ 0 � ν ∈ 2 <ω , then { ϕ ( x ; a η⌢ 1 ) , ϕ ( x ; a ν ) } is inconsistent. ◮ Motivated by the Keisler order and related questions.
SOP n hierarchy, 2 ◮ What is known: ◮ NTP ⊆ NSOP 1 ⊆ NSOP 2 = NTP 1 ⊆ NSOP 3 ⊆ . . . ⊆ NSOP. ◮ NSOP n + 1 \ NSOP n � = ∅ for all n ≥ 3, and NSOP \ ( � n NSOP n ) � = ∅ . ◮ NSOP 2 ∩ NTP 2 = NTP (Shelah’s theorem). ◮ [Shelah, Usvyatsov] give an example showing that NTP � NSOP 1 , however their proof appears to be wrong. Yet their example is correct, as follows from our theorem. ◮ Open problems: ◮ NSOP 2 � NSOP 3 ? NSOP 1 � NSOP 2 ? ◮ Does NSOP n ∩ NTP 2 collapse for n ≥ 3? At least, NTP � NSOP ∩ NTP 2 ?
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