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 � d α | α < κ � is called a κ -list iff d α ⊂ α for all α < κ . Christoph Weiß The combinatorial essence of supercompactness

Definition � d α | α < κ � is called a κ -list iff d α ⊂ 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 d β ∩ α = d ∩ α . Christoph Weiß The combinatorial essence of supercompactness

Definition � d α | α < κ � is called a κ -list iff d α ⊂ 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 d β ∩ 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 d α = 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 d α = 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 P κ λ . Definition � d a | a ∈ P κ λ � is called a P κ λ -list iff d a ⊂ a for all a ∈ P κ λ . Definition Let D be a P κ λ -list. d ⊂ λ is called a branch for D iff for all a ∈ P κ λ there is b ∈ P κ λ , b ⊃ a , such that d b ∩ a = d ∩ a . d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ P κ λ such that d a = d ∩ a for all a ∈ S . Christoph Weiß The combinatorial essence of supercompactness

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

Definition Let D = � d α | α < κ � be a κ -list. D is called thin iff for all δ < κ we have that |{ d α ∩ 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 = � d α | α < κ � be a κ -list. D is called thin iff for all δ < κ we have that |{ d α ∩ 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 ( ω 2 ) (ITP ( ω 2 ) ) holds. Christoph Weiß The combinatorial essence of supercompactness

Thin can also be defined for P κ λ . Definition Let D = � d a | a ∈ P κ λ � be a P κ λ -list. D is called thin iff there is a club C ⊂ P κ λ such that for all c ∈ C we have { d a ∩ c | c ⊂ a ∈ P κ λ }| < κ . Definition TP ( κ, λ ) holds iff every thin P κ λ -list has a branch. ITP ( κ, λ ) holds iff every thin P κ λ -list has an ineffable branch. Christoph Weiß The combinatorial essence of supercompactness

Thin can also be defined for P κ λ . Definition Let D = � d a | a ∈ P κ λ � be a P κ λ -list. D is called thin iff there is a club C ⊂ P κ λ such that for all c ∈ C we have { d a ∩ c | c ⊂ a ∈ P κ λ }| < κ . Definition TP ( κ, λ ) holds iff every thin P κ λ -list has a branch. ITP ( κ, λ ) holds iff every thin P κ λ -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 = � d α | 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 d γ ∩ δ = d β ∩ 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 P κ λ -lists. Definition Let D = � d a | a ∈ P κ λ � be a P κ λ -list. D is called slender iff for every sufficiently large θ there is a club C ⊂ P κ H θ such that for all M ∈ C and all b ∈ M ∩ P κ λ we have d M ∩ λ ∩ b ∈ M . Again if a P κ λ -list is thin, then it is slender. Definition SP ( κ, λ ) holds iff every slender P κ λ -list has a branch. ISP ( κ, λ ) holds iff every slender P κ λ -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 I IT [ κ, λ ] := { A ⊂ P κ λ | there is a thin P κ λ -list D without an ineffable branch living on A } I IS [ κ, λ ] := { A ⊂ P κ λ | there is a slender P κ λ -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 I IT [ κ, λ ] := { A ⊂ P κ λ | there is a thin P κ λ -list D without an ineffable branch living on A } I IS [ κ, λ ] := { A ⊂ P κ λ | there is a slender P κ λ -list D without an ineffable branch living on A } Thus the principles ITP ( κ, λ ) and ISP ( κ, λ ) say that the ideals I IT [ κ, λ ] and I IS [ κ, λ ] are proper ideals respectively. The ideals I IT [ κ, λ ] and I IS [ κ, λ ] are normal. It is easy to see that I [ κ ] ⊂ I IS [ κ, κ ] , where I [ κ ] denotes the approachability ideal on κ . Therefore ISP ( κ ) implies the failure of the approachability property on the predecessor of κ . Furthermore ITP ( κ, λ ) implies the failure of weak versions of square. Christoph Weiß The combinatorial essence of supercompactness