The combinatorial essence of supercompactness - Semantic Scholar
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 Measurability
Strong compactness
Supercompactness
Christoph Weiß
The combinatorial essence of supercompactness
Common understanding: Weak compactness Measurability
Strong compactness
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 Measurability
Strong compactness
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 hdα | α < κi is called a κ-list iff dα ⊂ α for all α < κ.
Christoph Weiß
The combinatorial essence of supercompactness
Definition hdα | α < κi 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 hdα | α < κi 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 Pκ λ. Definition hda | a ∈ Pκ λi is called a Pκ λ-list iff da ⊂ a for all a ∈ Pκ λ. Definition Let D be a Pκ λ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκ λ there is b ∈ Pκ λ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκ λ such that da = d ∩ a for all a ∈ S.
Christoph Weiß
The combinatorial essence of supercompactness
The concepts of lists and branches generalize to Pκ λ. Definition hda | a ∈ Pκ λi is called a Pκ λ-list iff da ⊂ a for all a ∈ Pκ λ. Definition Let D be a Pκ λ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκ λ there is b ∈ Pκ λ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκ λ such that da = d ∩ a for all a ∈ S. Theorem (Jech) A cardinal κ is strongly compact iff for all λ ≥ κ every Pκ λ-list has a branch. Theorem (Magidor) A cardinal κ is supercompact iff for all λ ≥ κ every Pκ λ-list has an ineffable branch. Christoph Weiß
The combinatorial essence of supercompactness
Definition Let D = hdα | α < κi 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 = hdα | α < κi 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 Pκ λ. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called thin iff there is a club C ⊂ Pκ λ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκ λ}| < κ. Definition TP(κ, λ) holds iff every thin Pκ λ-list has a branch. ITP(κ, λ) holds iff every thin Pκ λ-list has an ineffable branch.
Christoph Weiß
The combinatorial essence of supercompactness
Thin can also be defined for Pκ λ. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called thin iff there is a club C ⊂ Pκ λ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκ λ}| < κ. Definition TP(κ, λ) holds iff every thin Pκ λ-list has a branch. ITP(κ, λ) holds iff every thin Pκ 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 = hdα | α < κi 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 Pκ λ-lists. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called slender iff for every sufficiently large θ there is a club C ⊂ Pκ Hθ such that for all M ∈ C and all b ∈ M ∩ Pκ λ we have dM∩λ ∩ b ∈ M. Again if a Pκ λ-list is thin, then it is slender. Definition SP(κ, λ) holds iff every slender Pκ λ-list has a branch. ISP(κ, λ) holds iff every slender Pκ 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 IIT [κ, λ] := {A ⊂ Pκ λ | there is a thin Pκ λ-list D without an ineffable branch living on A} IIS [κ, λ] := {A ⊂ Pκ λ | there is a slender Pκ 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 IIT [κ, λ] := {A ⊂ Pκ λ | there is a thin Pκ λ-list D without an ineffable branch living on A} IIS [κ, λ] := {A ⊂ Pκ λ | there is a slender Pκ λ-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 on the predecessor of κ. Furthermore ITP(κ, λ) implies the failure of 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 PµV V is cofinal in PµW V , that is, for every x ∈ W with x ⊂ V and |x| < µ there is z ∈ PµV 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 ∈ PµV 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 hPα | α < κi 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 V W IIT [κ, λ] ⊂ IIT [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 V W IIT [κ, λ] ⊂ IIT [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 PκW λ − PκV λ ∈ IIT [κ, λ], so also V W FIT [κ, λ] ⊂ FIT [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 PκW λ − PκV 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 PκW λ − PκV λ ∈ IIT [κ, λ]. For the proof of “PFA =⇒ ∀λ ≥ ω2 ISP(ω2 , λ)” the following set is used. MP,θ := {M ∈ Pω2 Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2 Hθ 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 PκW λ − PκV λ ∈ IIT [κ, λ]. For the proof of “PFA =⇒ ∀λ ≥ ω2 ISP(ω2 , λ)” the following set is used. MP,θ := {M ∈ Pω2 Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2 Hθ 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
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 Measurability
Strong compactness
Supercompactness
Christoph Weiß
The combinatorial essence of supercompactness
Common understanding: Weak compactness Measurability
Strong compactness
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 Measurability
Strong compactness
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 hdα | α < κi is called a κ-list iff dα ⊂ α for all α < κ.
Christoph Weiß
The combinatorial essence of supercompactness
Definition hdα | α < κi 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 hdα | α < κi 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 Pκ λ. Definition hda | a ∈ Pκ λi is called a Pκ λ-list iff da ⊂ a for all a ∈ Pκ λ. Definition Let D be a Pκ λ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκ λ there is b ∈ Pκ λ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκ λ such that da = d ∩ a for all a ∈ S.
Christoph Weiß
The combinatorial essence of supercompactness
The concepts of lists and branches generalize to Pκ λ. Definition hda | a ∈ Pκ λi is called a Pκ λ-list iff da ⊂ a for all a ∈ Pκ λ. Definition Let D be a Pκ λ-list. d ⊂ λ is called a branch for D iff for all a ∈ Pκ λ there is b ∈ Pκ λ, b ⊃ a, such that db ∩ a = d ∩ a. d ⊂ λ is called an ineffable branch for D iff there is a stationary set S ⊂ Pκ λ such that da = d ∩ a for all a ∈ S. Theorem (Jech) A cardinal κ is strongly compact iff for all λ ≥ κ every Pκ λ-list has a branch. Theorem (Magidor) A cardinal κ is supercompact iff for all λ ≥ κ every Pκ λ-list has an ineffable branch. Christoph Weiß
The combinatorial essence of supercompactness
Definition Let D = hdα | α < κi 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 = hdα | α < κi 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 Pκ λ. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called thin iff there is a club C ⊂ Pκ λ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκ λ}| < κ. Definition TP(κ, λ) holds iff every thin Pκ λ-list has a branch. ITP(κ, λ) holds iff every thin Pκ λ-list has an ineffable branch.
Christoph Weiß
The combinatorial essence of supercompactness
Thin can also be defined for Pκ λ. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called thin iff there is a club C ⊂ Pκ λ such that for all c ∈ C we have {da ∩ c | c ⊂ a ∈ Pκ λ}| < κ. Definition TP(κ, λ) holds iff every thin Pκ λ-list has a branch. ITP(κ, λ) holds iff every thin Pκ 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 = hdα | α < κi 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 Pκ λ-lists. Definition Let D = hda | a ∈ Pκ λi be a Pκ λ-list. D is called slender iff for every sufficiently large θ there is a club C ⊂ Pκ Hθ such that for all M ∈ C and all b ∈ M ∩ Pκ λ we have dM∩λ ∩ b ∈ M. Again if a Pκ λ-list is thin, then it is slender. Definition SP(κ, λ) holds iff every slender Pκ λ-list has a branch. ISP(κ, λ) holds iff every slender Pκ 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 IIT [κ, λ] := {A ⊂ Pκ λ | there is a thin Pκ λ-list D without an ineffable branch living on A} IIS [κ, λ] := {A ⊂ Pκ λ | there is a slender Pκ 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 IIT [κ, λ] := {A ⊂ Pκ λ | there is a thin Pκ λ-list D without an ineffable branch living on A} IIS [κ, λ] := {A ⊂ Pκ λ | there is a slender Pκ λ-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 on the predecessor of κ. Furthermore ITP(κ, λ) implies the failure of 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 PµV V is cofinal in PµW V , that is, for every x ∈ W with x ⊂ V and |x| < µ there is z ∈ PµV 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 ∈ PµV 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 hPα | α < κi 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 V W IIT [κ, λ] ⊂ IIT [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 V W IIT [κ, λ] ⊂ IIT [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 PκW λ − PκV λ ∈ IIT [κ, λ], so also V W FIT [κ, λ] ⊂ FIT [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 PκW λ − PκV 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 PκW λ − PκV λ ∈ IIT [κ, λ]. For the proof of “PFA =⇒ ∀λ ≥ ω2 ISP(ω2 , λ)” the following set is used. MP,θ := {M ∈ Pω2 Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2 Hθ 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 PκW λ − PκV λ ∈ IIT [κ, λ]. For the proof of “PFA =⇒ ∀λ ≥ ω2 ISP(ω2 , λ)” the following set is used. MP,θ := {M ∈ Pω2 Hθ | ∃G ⊂ P M-generic}. By an argument of Woodin, PFA implies the set MP,θ is stationary in Pω2 Hθ 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
