On generalized notion of higher stationarity Hiroshi Sakai Kobe - PowerPoint PPT Presentation

On generalized notion of higher stationarity Hiroshi Sakai Kobe University Reflections on Set Theoretic Reflection 2018 Bagaria’s 60th Birthday Conference November 16–19. 2018 Joint work with Saka´ e Fuchino and Hazel Brickhill H. Sakai (Kobe) Higher Stationarity Bagaria 60 1 / 18

Section 1 Higher stationary sets of ordinals H. Sakai (Kobe) Higher Stationarity Bagaria 60 2 / 18

n -stationary subsets of ordinals Definition ( n -stationary subsets of ordinals) By induction on n < ω , define the notion of n -stationary subsets of κ ∈ On as follows: S ⊆ κ is 0-stationary in κ if S is unbouned in κ . S is n -stationary in κ if for all m < n and all m -stationary T ⊆ κ there is µ ∈ S s.t. T ∩ µ is m -stationary in µ . κ is n -stationary if κ is n -stationary in κ . S is 1-stationary in κ iff S is stationary in κ . S is 2-stationary in κ iff every stationary subset of κ reflects to some µ ∈ S . In particular, κ is 2-stationary iff the stationary reflection in κ holds. This notion of n -stationary sets is relevant to the proof theory: ▶ topological semantics of provability logic. (Beklemishev et al.) ▶ ordinal analysis of the theory ZFC + Π 1 n -Indescribable Card. Axiom. (Arai) Bagaria also considers ξ -stationary sets for infinite ξ ’s. H. Sakai (Kobe) Higher Stationarity Bagaria 60 3 / 18

Family of non- n -stationary sets Definition For κ ∈ On and n < ω , let NS n κ := { S ⊆ κ | S is not n -stationary in κ } . NS 1 κ is the non-stationary ideal over κ . For n ≥ 2, NS n κ may not be an ideal: Suppose every stationary subset of κ reflects, but there are stationary T 0 , T 1 ⊆ κ which do not reflect simultaneously. Let S i := { µ < κ | T i ∩ µ is not stationary in µ } . Then S 0 , S 1 ∈ NS 2 ∈ NS 2 κ , but S 0 ∪ S 1 = κ / κ . H. Sakai (Kobe) Higher Stationarity Bagaria 60 4 / 18

Π 1 n -indescribable sets Definition (Π 1 n -indescribability) Suppose κ ∈ On and n < ω . S ⊆ κ is Π 1 n -indescribable in κ if for all P ⊆ V κ and all Π 1 n -sentence ϕ with ( V κ , ∈ , P ) | = ϕ , there is µ ∈ S with ( V µ , ∈ , P ∩ V µ ) | = ϕ . κ is Π 1 n -indescribable if κ is Π 1 n -indescribable in κ . NI n κ := { S ⊆ κ | S is not Π 1 n -indescribable in κ } . Fact ((1),(2),(4):L´ evy, (3):Scott) κ is Π 1 0 -indescribable iff κ is inaccessible. 1 For an inaccessible cardinal κ , S ⊆ κ is Π 1 0 -indescribable in κ iff S is 2 stationary in κ . κ is Π 1 1 -indescribable iff κ is weakly compact. 3 NI n κ is a normal ideal over κ . 4 Bagaria defined the generalized notion of Π 1 ξ -indescribability for infinite ξ . H. Sakai (Kobe) Higher Stationarity Bagaria 60 5 / 18

Π 1 1 -indescribability and 2-stationarity The following is easy: Fact If S is Π 1 1 -indescribable in κ , then S is 2-stationary in κ . In L , the converse also holds: Theorem (Jensen) Assume V = L . Let κ be a regular uncountable cardinal. If S is 2-stationary in κ , then S is Π 1 1 -indescribable in κ . Kunen proved that the 2-stationarity does not imply the Π 1 1 -indescribability in general. In fact, the consistency strengths are different: Theorem (Shelah-Mekler) The consistency strength of the existence of a 2-stationary cardinal is strictly weaker than that of a Π 1 1 -indescribable cardinal. H. Sakai (Kobe) Higher Stationarity Bagaria 60 6 / 18

Preservation of 2-stationarity and continuum Theorem (Shelah) Suppose κ ∈ On . Then every c.c.c. forcing preserves 2-stationary subsets of κ . Corollary It is consistent that there is κ ≤ 2 ω which is 2-stationary. H. Sakai (Kobe) Higher Stationarity Bagaria 60 7 / 18

Π 1 n -indescribability and n + 1-stationarity Fact If S is Π 1 n -indescribable in κ , then S is n + 1-stationary in κ . Theorem (Bagaria-Magidor-S.) Assume V = L . Let κ be a regular uncountable cardinal. If S is n + 1-stationary in κ , then S is Π 1 n -indescribable in κ . Theorem (Bagaria-Magidor-Mancilla) For n ∈ ω \ { 0 } , the consistency strength of the existence of an n + 1-stationary cardinal is strictly weaker than that of a Π 1 n -indescribable cardinal. Bagaria generalizes these results to those of Π 1 ξ -indescribability and ξ + 1-stationarity for infinite ξ . H. Sakai (Kobe) Higher Stationarity Bagaria 60 8 / 18

Preservation of n -stationarity and continuum We have the following preservation theorem for n -stationary sets: Theorem Assume GCH. Suppose n ∈ ω , κ is a regular uncountable cardinal and ρ < κ . Assume NS m µ is a normal ideal over µ for all regular µ ≤ κ and all m with 1 ≤ m ≤ n . Then every ρ -c.c. forcing preserves n -stationary subsets of κ . µ = NI m − 1 Note that, in L , NS m is a normal ideal. So the assumption of Theorem µ holds in L . Thus we have the following corollary by a forcing over L : Corollary It is consistent that there is a cardinal κ ≤ 2 ω which is n -stationary for all n < ω . H. Sakai (Kobe) Higher Stationarity Bagaria 60 9 / 18

Outline of Proof of Theorem By induction on n , we prove that for all regular µ with ρ < µ ≤ κ and all ρ -c.c. poset P , we have the following in V P : µ ) V . NS n µ = ( NS n ∈ ( NS n [Proof of “ ⊆ ” (i.e. S / µ ) V ⇒ S is n -stationary)] We may assume | P | ≤ µ . Suppose P = µ . We work in V . Suppose m < n and ˙ T is a P -name for an m -stationary subset of µ . µ ) V : It suffices to prove that the following C is in the dual filter F of ( NS n C := { ν < µ | ⊩ P ˙ T ∩ ν is m -stationary } . For each p ∈ P , the following T p is m -stationary in µ : T p := { α < µ | ∃ q ≤ p , q ⊩ P α ∈ ˙ T } . µ ) V , By the normality of ( NS n D := { ν < µ | ∀ p < ν, T p ∩ ν is m -stat. & P ∩ ν ⊆ c P } ∈ F . Moreover, D ⊆ C . □ H. Sakai (Kobe) Higher Stationarity Bagaria 60 10 / 18

Section 2 Higher stationary sets in P κ ( λ ) H. Sakai (Kobe) Higher Stationarity Bagaria 60 11 / 18

n -stationary subsets of P κ ( A ) Definition ( n -stationary subsets of P κ ( A )) For a regular cardinal κ , a set A ⊇ κ and n < ω : S ⊆ P κ ( A ) is 0-stationary in P κ ( A ) if S is ⊆ -cofinal in P κ ( A ). S ⊆ P κ ( A ) is n -stationary in P κ ( A ) if for all m < n and all m -stationary T ⊆ P κ ( A ), there is B ∈ S s.t. - µ := B ∩ κ is a regular cardinal, - T ∩ P µ ( B ) is m -stationary in P µ ( B ). P κ ( A ) is n -stationary if P κ ( A ) is n -stationary in P κ ( A ). NS n κ, A := { S ⊆ P κ ( A ) | S is not n -stationary in P κ ( A ) } . If P κ ( A ) is 1-stationary, then κ is weakly Mahlo. Suppose κ is Mahlo. Then NS 1 κ, A is the smallest strongly normal ideal over P κ ( A ). If |P κ ( A ) | = | A | and f : P κ ( A ) → A is a bijection, then NS 1 κ, A = NS κ, A | S , where S = { x ∈ P κ ( λ ) | µ := x ∩ κ ∈ Reg & f [ P µ ( x )] ⊆ y } . H. Sakai (Kobe) Higher Stationarity Bagaria 60 12 / 18

Π 1 n -indescribability in P κ ( A ) For a regular κ and a set A ⊇ κ , we define V α ( κ, A ) by induction on α . V 0 ( κ, A ) := A , V α +1 ( κ, A ) := P κ ( V α ( κ, A )) ∪ V α ( κ, A ), V α ( κ, A ) := ∪ β<α V β ( κ, A ) for a limit α . Definition (Baumgartner) Suppose κ is a regular cardinal, A ⊇ κ and n < ω . S ⊆ P κ ( A ) is Π 1 n -indescribable in P κ ( A ) if for all P ⊆ V κ ( κ, A ) and all Π 1 n -sentence ϕ with ( V κ ( κ, A ) , ∈ , P ) | = ϕ , there is B ∈ S such that - µ := B ∩ κ is a regular cardinal, - ( V µ ( µ, B ) , ∈ , P ∩ V µ ( µ, B )) | = ϕ . P κ ( A ) is Π 1 n -indescribable if P κ ( A ) is Π 1 n -indescribable in P κ ( A ). NI n κ, A := { S ⊆ P κ ( A ) | S is not Π 1 n -indescribable in P κ ( A ) } . H. Sakai (Kobe) Higher Stationarity Bagaria 60 13 / 18

Theorem (Abe, Car) P κ (2 λ <κ ) is Π 1 1 -indescribable. ⇒ κ is λ -supercompact. 1 ⇒ P κ ( λ ) is Π 1 n -indescribable for all n ∈ ω . NI n κ, A is a strongly normal ideal over P κ ( A ). 2 H. Sakai (Kobe) Higher Stationarity Bagaria 60 14 / 18

Π 1 n -indescribability and n + 1-stationarity Proposition If S is Π 1 n -indescribable in P κ ( λ ), then S is n + 1-stationary in P κ ( λ ). Suppose κ is Mahlo. Then S is Π 1 0 -indescribable in P κ ( λ ) iff S is 1-stationary in P κ ( λ ). Recall that, in L , S is Π 1 n -indescribable in κ iff S is n + 1-stationary in κ . We do not know whether its analogy is consistent in a non-trivial way: Question Is the following consistent? For all regular κ , all λ ≥ κ , all S ⊆ P κ ( λ ) and all n < ω , S is Π 1 n -indescribable in P κ ( λ ) iff S is n + 1-stationary in P κ ( λ ). There is a supercompact cardinal. H. Sakai (Kobe) Higher Stationarity Bagaria 60 15 / 18

Strongly compact cardinal Recall that if κ is λ -supercompact, then P κ ( λ ) is n -stationary for all n < ω . Question Is P κ ( λ ) n -stationary for all n < ω if κ is λ -strongly compact ? We can prove that the strong compactness of κ does not imply the stationary reflection in P κ ( λ ): For all stationary S ⊆ P κ ( λ ) , there is B ∈ P κ ( λ ) s.t. - µ := B ∩ κ is a regular cardinal, - S ∩ P µ ( B ) is stationary in P µ ( B ) . Proposition It is consistent that there is a strongly compact cardinal κ s.t. the stationary reflection in P κ ( κ + ) fails. But, we do not know the answer of the above question. H. Sakai (Kobe) Higher Stationarity Bagaria 60 16 / 18