Towards the dependence spectra

Towards the dependence spectra Artem Chernikov ¨ (Humboldt Universitat zu Berlin) ´ ´ ´ ”Seminaire general de logique” Paris, 29 June 2009

Introduction: Type spaces T is a complete first-order theory, M |= T . Let B(A) be a boolean algebra of A-definable subsets of M. Then S(A) is its Stone dual (that is, space of ultrafilters on B(A)) - space of types over A. Examples:

Introduction: Stability and NIP φ(x, y ) is unstable if there is a sequence (ai bi )i<ω such that φ(ai , bj ) ⇐⇒ i ≤ j. φ(x, y) has IP (independence property) if there is a sequence (ai )i<ω such that for any s ⊂ ω there is some bs with φ(ai , bs ) ⇐⇒ i ∈ S. T is said to be stable (dependent) if no formula is unstable (has IP). Every stable theory is dependent. Examples:

Introduction: Stability function ST (κ) = sup{|S(M)| : M |= T , |M| = κ} Characterization of stability: TFAE 1. T is stable 2. ST (κ) ≤ κ for some κ 3. ST (κ) ≤ κ for every κ = κ|T | T is said to be superstable if it is κ-stable for all sufﬁcientely big κ’s. Question: what is κ-dependence and superdependence?

Introduction: Possible stability functions Shelah, Keisler: For a countable ﬁrst-order T the only possible stability functions are κ —— total transcendence —— κ + 2ℵ0 —— superstability —— κℵ 0 —— stability —— dedκ —— non multi-order —— (dedκ)ℵ0 —— dependence —— 2κ

Possible stability functions: set-theoretical issue For any κ: κ < dedκ ≤ (dedκ)ℵ0 ≤ 2κ So assuming GCH everything collapses. Mithchel: for any κ with cof κ > ℵ0 it is consistent that dedκ < 2κ . It seems not to be known whether dedκ < (dedκ)ℵ0 is consistent for any κ. =⇒ To capture dependence need to count types more carefully.

Counting types: How to count? Morally the only obstacle for having few types in NIP are cuts. 1. Restrict types that we count, that is measure the size of certain subspaces S (M) ⊆ S(M). 2. Restrict domains over which we count types. 3. Count types up to certain equivalence relations weaker than equality.

1. Localizable types We say that a type p ∈ S(M) is localizable if it does not split over some A ⊂ M such that M is saturated over A. A type p ∈ S(M) is simple if it does not fork over some A ⊂ M with |A| ≤ |T |. Examples: Let S loc (A), S smpl ⊆ S(A) be the spaces of localizable and simple types respectively. In stable theories S loc (A) = S smpl (A) = S(A). And let S loc (κ) = sup{|S loc (M)||M| = κ}

1. Localizable types: capturing dependence λ Theorem: Let κ be nice (such that 22 > κ for some λ < κ and κ<<κ+|T | = κ). TFAE 1. T is dependent 2. S loc (κ) ≤ κ Question: So seems < 2κ localizable types in NIP for orthogonal reasons?

1. Localizable types: Brushing up Recall: TFAE 1. T is stable 2. ST (κ) ≤ κ for some κ 3. ST (κ) ≤ κ for every κ = κ|T | Desire: real analogue for dependent theories. Trouble: always few localizable types at some cardinals (e.g. at inaccessible), for any theory.

1. Interlude: Bounded non-forking T is said to have bounded non-forking if every type p ∈ S(M) has boudedly-many global non-forking extensions. Adler: T has bounded non-forking ⇐⇒ non-forking = |M| invariance ⇐⇒ every p ∈ S(M) has at most 22 global non-forking extensions. Poizat: T is dependent ⇐⇒ every p ∈ S(M) has at most 2|M| global non-forking extensions.

1. Interlude: Bounded non-forking κ Does this gap between 2κ and 22 actually exist? Unbounded non-forking conjecture: Non-forking is either exponentially bounded (and T is dependent) or it is unbounded (and T has IP). Strong unbounded non-forking conjecture: T has IP =⇒ there is p ∈ S(M) such that for any κ and some N ⊃ M, |N| = κ there are > κ non-forking extensions of p over N. Question: are they equivalent?

1. Interlude: NTP2 We say that φ(x, y) has TP2 if: T is NTP2 if no formula has TP2 . NTP2 theories generalize simple, dependent, without dense forking chains - so cover quite a lot of theories.

1. Interlude: Strong unbounded non-forking conjecture in NTP2 In joint work with Itay Kaplan: Unbounded non-forking conjecture holds in NTP2 . Theorem: Strong unbounded non-forking conjecture holds in NTP2 .

1. Right claim modulo NTP2 (or strong unbounded forking conjecture) Assume T is NTP2 . TFAE: 1. T is dependent 2. S smpl (κ) ≤ κ for some κ 3. S smpl (κ) ≤ κ for every κ = κ|T | Good because stable + κ-dependent = κ-stable. Not good enough since already DLO has exactly this spectra, so there are not going to be any unstable ”superdependent” theories.

2. Counting types over complete sets We say that A is complete if every type over A is deﬁnable. Most of dependent theories (even o-minimal) don’t have complete models. Baldwin-Benedikt: T dependent. Then every indiscernible sequence indexed by Dedekind-complete linear order is complete.

2. Counting types over complete sets: NIP Observation: T is dependent if and only if for every complete indiscernible sequence I there are at most |I|-many types over it. In particular, every dependent theory has a model with few types. Good because there are ”superdependent” theories, bad because superdependence|stable = superstability any longer.

3. Counting types up to isomorphism: S aut (M) Let p, q ∈ S(M). Then deﬁne p ≡aut q if there is an automorphism of M sending p to q. Let S aut (M) = S(M)/ ≡aut .

3. Counting types up to isomorphism: Shelah’s dependent recounting types 950 Let λ = λ<λ be large enough, M a saturated model of T of cardinality λ. Then T is dependent ⇐⇒ S aut (M) ≤ λ. (Moreover it is ≤ α when λ = ℵα and λ not too small).

Counting types up to isomorphism: easier counterpart We look at S aut (I) with I indiscernible sequence indexed by a saturated dense linear order of size κ. T is dependent ⇐⇒ |S aut (I)| ≤ |I| T is stable ⇐⇒ |S aut (I)| is eventually constant. Essentially one direction is Baldwin-Benedikt, another is existence of many non-isomorphic linear orders.

... So each way leads to a characterization of dependence by counting, but none seems sensitive enough to capture what one wishes to be ”superdependence”. Shelah suggested a solution, strong dependence, which turned out to be essentially ”ﬁnite weight” rather than ”superdependence”. Question still open: is there a good notion of superdependence?

Qestions Find ranks in dependent theories majorising dp-rank and restricting to U-rank in stable theories. Develop theory of external imaginaries. Find a characterization of NTP2 theories by counting small contradictory localizable types (parallel to the simple case).

Towards the dependence spectra

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