The combinatorial essence of supercompactness Christoph Wei (Joint - - PowerPoint PPT Presentation

The combinatorial essence of supercompactness

Christoph Weiß

(Joint work with Matteo Viale)

October 26, 2010

Christoph Weiß The combinatorial essence of supercompactness

Goal: Con(PFA) = ⇒ Con(there is a supercompact cardinal)

Christoph Weiß The combinatorial essence of supercompactness

Goal: Con(PFA) = ⇒ Con(there is a supercompact cardinal) Problem: Inner model theory

Christoph Weiß The combinatorial essence of supercompactness

Goal: Con(PFA) = ⇒ Con(there is a supercompact cardinal) Problem: Inner model theory Revised Goal: Show that if we force a model of PFA, then we need a supercompact cardinal for it.

Christoph Weiß The combinatorial essence of supercompactness

Common understanding: Weak compactness Strong compactness Measurability Supercompactness

Christoph Weiß The combinatorial essence of supercompactness

Common understanding: Weak compactness Strong compactness Measurability Supercompactness But this depends on the definition of λ-strongly compact and λ-supercompact, for it is equally plausible that Measurability Strong compactness

Christoph Weiß The combinatorial essence of supercompactness

Common understanding: Weak compactness Strong compactness Measurability Supercompactness But this depends on the definition of λ-strongly compact and λ-supercompact, for it is equally plausible that Measurability Strong compactness So one should look for the “minimal” principle that generalizes to supercompactness under a suitable choice of λ-supercompact: Ineffability Supercompactness

Christoph Weiß The combinatorial essence of supercompactness

Definition dα | α < κ is called a κ-list iff dα ⊂ α for all α < κ.

Christoph Weiß The combinatorial essence of supercompactness

Definition dα | α < κ is called a κ-list iff dα ⊂ α for all α < κ. Definition Let D be a κ-list. d ⊂ κ is a branch for D iff for all α < κ there is β < κ, β ≥ α, such that dβ ∩ α = d ∩ α.

Christoph Weiß The combinatorial essence of supercompactness

Definition dα | α < κ is called a κ-list iff dα ⊂ α for all α < κ. Definition Let D be a κ-list. d ⊂ κ is a branch for D iff for all α < κ there is β < κ, β ≥ α, such that dβ ∩ α = d ∩ α. Fact A cardinal κ is weakly compact iff every κ-list has a branch.

Christoph Weiß The combinatorial essence of supercompactness

Now we write up ineffability the same way. Definition Let D be a κ-list. d ⊂ κ is an ineffable branch for D iff there is a stationary set S ⊂ κ such that dα = d ∩ α for all α ∈ S.

Christoph Weiß The combinatorial essence of supercompactness

Now we write up ineffability the same way. Definition Let D be a κ-list. d ⊂ κ is an ineffable branch for D iff there is a stationary set S ⊂ κ such that dα = d ∩ α for all α ∈ S. Fact A cardinal κ is ineffable iff every κ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

The concepts of lists and branches generalize to Pκλ. Definition da | a ∈ Pκλ is called a Pκλ-list iff da ⊂ a for all a ∈ Pκλ. Definition Let D be a Pκλ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκλ there is b ∈ Pκλ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκλ such that da = d ∩ a for all a ∈ S.

Christoph Weiß The combinatorial essence of supercompactness

The concepts of lists and branches generalize to Pκλ. Definition da | a ∈ Pκλ is called a Pκλ-list iff da ⊂ a for all a ∈ Pκλ. Definition Let D be a Pκλ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκλ there is b ∈ Pκλ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκλ such that da = d ∩ a for all a ∈ S. Theorem (Jech) A cardinal κ is strongly compact iff for all λ ≥ κ every Pκλ-list has a branch. Theorem (Magidor) A cardinal κ is supercompact iff for all λ ≥ κ every Pκλ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

Definition Let D = dα | α < κ be a κ-list. D is called thin iff for all δ < κ we have that |{dα ∩ δ | α < κ}| < κ. Note that if κ is inaccessible, then every κ-list is thin. Observed a long time ago: If we restrict ourselves to thin lists, then the principle of κ-lists having branches also makes sense for accessible cardinals κ. (For this is just the tree property.)

Christoph Weiß The combinatorial essence of supercompactness

Definition Let D = dα | α < κ be a κ-list. D is called thin iff for all δ < κ we have that |{dα ∩ δ | α < κ}| < κ. Note that if κ is inaccessible, then every κ-list is thin. Observed a long time ago: If we restrict ourselves to thin lists, then the principle of κ-lists having branches also makes sense for accessible cardinals κ. (For this is just the tree property.) But of course so does the principle of κ-lists having ineffable branches! We use the following abbreviations. Definition TP(κ) holds iff every thin κ-list has a branch. ITP(κ) holds iff every thin κ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

Now we are able to rephrase the facts above. Fact A cardinal κ is weakly compact iff it is inaccessible and TP(κ) holds. Fact A cardinal κ is ineffable iff it is inaccessible and ITP(κ) holds. The advantage is the principles TP and ITP make sense for accessible cardinals! For example, it is consistent up to a weakly compact (an ineffable) that TP(ω2) (ITP(ω2)) holds.

Christoph Weiß The combinatorial essence of supercompactness

Thin can also be defined for Pκλ. Definition Let D = da | a ∈ Pκλ be a Pκλ-list. D is called thin iff there is a club C ⊂ Pκλ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκλ}| < κ. Definition TP(κ, λ) holds iff every thin Pκλ-list has a branch. ITP(κ, λ) holds iff every thin Pκλ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

Thin can also be defined for Pκλ. Definition Let D = da | a ∈ Pκλ be a Pκλ-list. D is called thin iff there is a club C ⊂ Pκλ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκλ}| < κ. Definition TP(κ, λ) holds iff every thin Pκλ-list has a branch. ITP(κ, λ) holds iff every thin Pκλ-list has an ineffable branch. Theorem (Jech) A cardinal κ is strongly compact iff it is inaccessible and TP(κ, λ) holds for all λ ≥ κ. Theorem (Magidor) A cardinal κ is supercompact iff it is inaccessible and ITP(κ, λ) holds for all λ ≥ κ.

Christoph Weiß The combinatorial essence of supercompactness

But there is something better than thin. Definition Let D = dα | α < κ be a κ-list. D is called slender iff there is a club C ⊂ κ such that for every γ ∈ C and every δ < γ there is β < γ such that dγ ∩ δ = dβ ∩ δ. It is easy to see that if a κ-list D is thin, then D is slender. Definition SP(κ) holds iff every slender κ-list has a branch. ISP(κ, λ) holds iff every slender κ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

Slender also makes sense for Pκλ-lists. Definition Let D = da | a ∈ Pκλ be a Pκλ-list. D is called slender iff for every sufficiently large θ there is a club C ⊂ PκHθ such that for all M ∈ C and all b ∈ M ∩ Pκλ we have dM∩λ ∩ b ∈ M. Again if a Pκλ-list is thin, then it is slender. Definition SP(κ, λ) holds iff every slender Pκλ-list has a branch. ISP(κ, λ) holds iff every slender Pκλ-list has an ineffable branch.

Christoph Weiß The combinatorial essence of supercompactness

The principles ITP(κ, λ) and ISP(κ, λ) give rise to natural ideals. Definition IIT[κ, λ] := {A ⊂ Pκλ | there is a thin Pκλ-list D without an ineffable branch living on A} IIS[κ, λ] := {A ⊂ Pκλ | there is a slender Pκλ-list D without an ineffable branch living on A}

Christoph Weiß The combinatorial essence of supercompactness

The principles ITP(κ, λ) and ISP(κ, λ) give rise to natural ideals. Definition IIT[κ, λ] := {A ⊂ Pκλ | there is a thin Pκλ-list D without an ineffable branch living on A} IIS[κ, λ] := {A ⊂ Pκλ | there is a slender Pκλ-list D without an ineffable branch living on A} Thus the principles ITP(κ, λ) and ISP(κ, λ) say that the ideals IIT[κ, λ] and IIS[κ, λ] are proper ideals respectively. The ideals IIT[κ, λ] and IIS[κ, λ] are normal. It is easy to see that I[κ] ⊂ IIS[κ, κ], where I[κ] denotes the approachability ideal on κ. Therefore ISP(κ) implies the failure of the approachability property

• n the predecessor of κ. Furthermore ITP(κ, λ) implies the failure
• f weak versions of square.

Christoph Weiß The combinatorial essence of supercompactness

Theorem PFA implies ISP(ω2, λ) holds for all λ ≥ ω2. This in some sense says that PFA shows ω2 is supercompact, apart from its missing inaccessibility. If we could go to an inner model that thinks ω2 is inaccessible but inherits ITP(ω2, λ) for all λ ≥ ω2 from V , then we would have an inner model for a supercompact.

Christoph Weiß The combinatorial essence of supercompactness

We now want to pull back the principle ISP from a larger model W to a smaller model V . Definition Let V ⊆ W be a pair of transitive models of ZFC. (V , W ) satisfies the µ-covering property if the class PV

µ V is

cofinal in PW

µ V , that is, for every x ∈ W with x ⊂ V and

|x| < µ there is z ∈ PV

µ V such that x ⊂ z.

(V , W ) satisfies the µ-approximation property if for all x ∈ W , x ⊂ V , it holds that if x ∩ z ∈ V for all z ∈ PV

µ V , then x ∈ V .

A forcing P is said to satisfy the µ-covering property or the µ-approximation property if for every V -generic G ⊂ P the pair (V , V [G]) satisfies the µ-covering property or the µ-approximation property respectively.

Christoph Weiß The combinatorial essence of supercompactness

Why are we interested in pairs of models that satisfy the κ-covering and the κ-approximation properties? Definition A forcing P is called a standard iteration of length κ if

1 P is the direct limit of an iteration Pα | α < κ that takes

direct limits stationarily often,

2 |Pα| < κ for all α < κ.

Note that the usual forcing constructions for creating models of PFA or MM are standard iterations of length κ, where κ is a large cardinal that is collapsed to ω2. Lemma Let P be a standard iteration of length κ. Then V and V [G] satisfy the κ-covering and the κ-approximation properties for V -generic G ⊂ P.

Christoph Weiß The combinatorial essence of supercompactness

Proposition Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Then I V

IT[κ, λ] ⊂ I W IT [κ, λ].

Christoph Weiß The combinatorial essence of supercompactness

Proposition Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Then I V

IT[κ, λ] ⊂ I W IT [κ, λ].

Theorem Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering property and the τ-approximation property for some τ < κ, and suppose κ is inaccessible in V . Then PW

κ λ − PV κ λ ∈ IIT[κ, λ],

so also F V

IT[κ, λ] ⊂ F W IT [κ, λ].

Thus, if W | = ITP(κ, λ), then V | = ITP(κ, λ).

Christoph Weiß The combinatorial essence of supercompactness

Hope Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Then PW

κ λ − PV κ λ ∈ IIT[κ, λ].

Christoph Weiß The combinatorial essence of supercompactness

Hope Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Then PW

κ λ − PV κ λ ∈ IIT[κ, λ].

For the proof of “PFA = ⇒ ∀λ ≥ ω2 ISP(ω2, λ)” the following set is used. MP,θ := {M ∈ Pω2Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2Hθ for proper P. Our hope would imply that MP,θ ↾ λ is a subset of V , possibly modulo a nonstationary part.

Christoph Weiß The combinatorial essence of supercompactness

Hope Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Then PW

κ λ − PV κ λ ∈ IIT[κ, λ].

For the proof of “PFA = ⇒ ∀λ ≥ ω2 ISP(ω2, λ)” the following set is used. MP,θ := {M ∈ Pω2Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2Hθ for proper P. Our hope would imply that MP,θ ↾ λ is a subset of V , possibly modulo a nonstationary part. But our hope was shattered. Theorem (Sakai, 2010) Let κ be a supercompact cardinal, and let µ > κ be a Woodin

• cardinal. Then in W , where W is the standard extension of V such

that W | = MM + κ = ω2, it holds that {M ∈ MP,θ | M ∩ κ+ / ∈ V } is stationary for any stationary preserving P ∈ W .

Christoph Weiß The combinatorial essence of supercompactness

But the following works. Theorem Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . If W | = TP(κ, λ), then V | = TP(κ, λ). Corollary Let P be a standard iteration of length κ and suppose κ is

• inaccessible. Suppose P “κ = ω2 ∧ PFA.” Then κ is strongly

compact.

Christoph Weiß The combinatorial essence of supercompactness

Theorem Let V ⊂ W be a pair of models of ZFC that satisfies the κ-covering and the κ-approximation properties, and suppose κ is inaccessible in V . Let λ be regular in V , and suppose that for all γ < λ and every S ⊂ cof(ω) ∩ γ it holds that V | = “S is stationary in γ” iff W | = “S is stationary in γ.” Suppose W | = λ<κ = λ. If W | = ISP(κ, λ), then V | = ISP(κ, λ). Corollary Let P be a standard iteration of length κ and suppose κ is

• inaccessible. Suppose P “κ = ω2 ∧ PFA.” If P is proper, then κ is

supercompact.

Christoph Weiß The combinatorial essence of supercompactness

Open problems:

Christoph Weiß The combinatorial essence of supercompactness

Open problems: Question Can we get supercompactness instead of strong compactness by pulling back ITP also for semiproper (or arbitrary if semiproper does not help) forcings?

Christoph Weiß The combinatorial essence of supercompactness

Question Suppose κ is regular uncountable and λ ≥ κ is such that cf λ ≥ κ. Is there an A ∈ FIT[κ, λ] such that sup a | a ∈ A is injective?

Christoph Weiß The combinatorial essence of supercompactness

Question Suppose κ is regular uncountable and λ ≥ κ is such that cf λ ≥ κ. Is there an A ∈ FIT[κ, λ] such that sup a | a ∈ A is injective? This would answer the following question affirmatively. Question Suppose κ is regular uncountable such that 2<κ = κ and λ ≥ κ is such that cf λ ≥ κ. Does ISP(κ, λ) imply λ<κ = λ? This would in particular mean that “∀λ ≥ ω2 ISP(ω2, λ)” = ⇒ SCH. Note that SCH can be proved from “∀λ ≥ ω2 ISP(ω2, λ)” using some additional cardinal invariant assumption which also follows from PFA.

Christoph Weiß The combinatorial essence of supercompactness

Question Suppose ISP(ω2, λ) holds for all λ ≥ ω2. Does this imply 2ω = ω2? (Or, does TP(ω2), the tree property on ω2, imply 2ω = ω2? Everybody thinks it should not, but nobody seems to know.)

Christoph Weiß The combinatorial essence of supercompactness

Question Suppose ISP(ω2, λ) holds for all λ ≥ ω2. Does this imply 2ω = ω2? (Or, does TP(ω2), the tree property on ω2, imply 2ω = ω2? Everybody thinks it should not, but nobody seems to know.) Question How can we get an inner model for a supercompact? Start with L and adjoin all necessary witnesses for ITP until we are done? Is core model theory really the right way?

Christoph Weiß The combinatorial essence of supercompactness

Thank you for your attention!

Christoph Weiß The combinatorial essence of supercompactness