Semi-stationary reflection, stationary reflection and combinatorics - PowerPoint PPT Presentation

Semi-stationary reflection, stationary reflection and combinatorics Hiroshi Sakai (joint work with Boban Veliˇ ckovi´ c) October 28, 2010

1.1 Stationary reflection • For a cardinal λ ≥ ω 2 , the stationary reflection in [ λ ] ω is the following statement: for all stationary S ⊆ [ λ ] ω there is W ⊆ λ s.t. SR ([ λ ] ω ) ≡ (i) | W | = ω 1 ⊆ W , (ii) S ∩ [ W ] ω is stationary. SR ([ λ ] ω ) holds for all λ ≥ ω 2 . • SR ≡

1.2 ( † ) and Semi-stationary reflection ( † ) Every ω 1 -stationary preserving forcing notion is ≡ semi-proper. Thm (Foreman-Magidor-Shelah) SR ⇒ ( † ). Thm ( † ) implies the following: (i) precipitousness of NS ω 1 (Foreman-Magidor-Shelah) (ii) (Strong) Chang’s Conjecture (Foreman-Magidor-Shelah) 2 ω ≤ ω 2 (iii) (Todorˇ cevi´ c)

• Let W be a set ⊇ ω 1 . S ⊆ [ W ] ω is semi-stationary if the set { y ∈ [ W ] ω | ( ∃ x ∈ S ) x ⊆ y ∧ x ∩ ω 1 = y ∩ ω 1 } is stationary. Thm (Shelah) TFAE for a forcing notion P : (1) P is semi-proper. (2) P preserves semi-stationary subsets of [ W ] ω for all W ⊇ ω 1 . Here recall the following: Thm (Shelah) TFAE for a forcing notion P : (1) P is proper. (2) P preserves stationary subsets of [ W ] ω for all W ⊇ ω 1 .

• For a cardinal λ ≥ ω 1 , the semi-stationary reflection in [ λ ] ω is the following statement: for all semi-stationary S ⊆ [ λ ] ω there is SSR ([ λ ] ω ) ≡ W ⊆ λ s.t. (i) | W | = ω 1 ⊆ W , (ii) S ∩ [ W ] ω is semi-stationary. SR ([ λ ] ω ) holds for all λ ≥ ω 2 . • SSR ≡ Thm (Shelah) SSR ⇔ ( † ). SSR ([ ω 2 ] ω ) ⇔ SR ([ ω 2 ] ω ). Thm (Todorˇ cevi´ c)

1.3 More consequences of SSR If SSR ([ λ ] ω ) holds, then the following hold for every Thm 1 regular cardinal κ with ω 2 ≤ κ ≤ λ : (i) reflection of stationary subsets of { α ∈ κ | cf( α ) = ω } (Sakai) (ii) the failure of � ( κ ) (Sakai-Veliˇ ckovi´ c) (iii) κ ω = κ (Sakai-Veliˇ ckovi´ c) � ( κ ) ≡ there is ⟨ c α | α ∈ Lim( κ ) ⟩ with the following properties: - c α is a club subset of α - c β = c α ∩ β if β ∈ Lim( α ) - there are no club C ⊆ κ such that c α = C ∩ α for all α ∈ Lim( C ).

1.4 Reflection principles and compact cardinals 1.4.1 L´ evy collapse of compact cardinals Thm (Foreman-Magidor-Shelah) κ : supercompact ⇒ � Col( ω 1 ,<κ ) SR . Thm (Shelah) κ : strongly compact ⇒ � Col( ω 1 ,<κ ) SSR . Thm (Sakai) κ : strongly compact ̸⇒ � Col( ω 1 ,<κ ) SR .

1.4.2 TP and ITP • A list on P κ ( λ ) is a seq. ⃗ d = ⟨ d x | x ∈ P κ ( λ ) ⟩ s.t. d x : x → 2. • A list ⃗ d = ⟨ d x | x ∈ P κ ( λ ) ⟩ is said to be thin if |{ d y ↾ x | y ⊇ x }| < κ for all x ∈ P κ ( λ ). • D : λ → 2 is an ineffable branch of a list ⃗ d = ⟨ d x | x ∈ P κ ( λ ) ⟩ if there are stationary many x ∈ P κ ( λ ) with D ↾ x = d x . • D : λ → 2 is a cofinal branch of a list ⃗ d = ⟨ d x | x ∈ P κ ( λ ) ⟩ if for any x ∈ P κ ( λ ) there is y ⊇ x with D ↾ x = d y ↾ x . ITP ( κ, λ ) Every thin list on P κ ( λ ) has an ineffable branch. ≡ TP ( κ, λ ) ≡ Every thin list on P κ ( λ ) has a cofinal branch.

Thm (Magidor) κ is supercompact if and only if - κ is inaccessible, - ITP ( κ, λ ) holds for all λ ≥ κ . Thm (Jech) κ is strongly compact if and only if - κ is inaccessible, - TP ( κ, λ ) holds for all λ ≥ κ . Thm (Weiss) ⇒ ITP ( ω 2 , λ ) for all λ ≥ ω 2 . PFA Thm 2 (Sakai-Veliˇ ckovi´ c) (1) SR + MA ℵ 1 (Cohen) ⇒ ITP ( ω 2 , λ ) for all λ ≥ ω 2 . (2) SSR + MA ℵ 1 (Cohen) ⇒ TP ( ω 2 , λ ) for all λ ≥ ω 2 . (3) SSR + MA ℵ 1 (Cohen) ̸⇒ ITP ( ω 2 , ω 3 ).

Proof skech of (1) and (2) of Thm 2 The following lemma is a key: Assume MA ℵ 1 (Cohen). Let λ be a cardinal ≥ ω 2 , ⃗ Lem d = ⟨ d x | x ∈ P ω 2 ( λ ) ⟩ be a thin list and θ be a regular cardinal >> λ . For each M ∈ [ H θ ] ω let x M := ∪ ( P ω 2 ( λ ) ∩ M ) ∈ P ω 2 ( λ ). Moreover let S be the set of all countable M ≺ ⟨H θ , ∈ , ⃗ d ⟩ such that either of the following holds for any y ∈ P ω 2 ( λ ) with y ⊇ x M : (I) There is D ∈ λ 2 ∩ M with D ↾ x M = d y ∩ x M . (II) There is x ∈ P ω 2 ( λ ) ∩ M with d y ↾ x / ∈ M . Then S is stationary in [ H θ ] ω .

[Proof of (1)] Assume SR + MA ℵ 1 (Cohen), and suppose that ⃗ d = ⟨ d x | x ∈ P ω 2 ( λ ) ⟩ is a thin list. Let θ be a regular cardinal >> λ . It suffices to find W ∈ P ω 2 ( H θ ) such that W ≺ ⟨H θ , ∈ , ⃗ d ⟩ , such that W ∩ ω 2 ∈ ω 2 and such that there is D ∈ λ 2 ∩ W with D ↾ ( W ∩ λ ) = d W ∩ λ . Let S be as in Lemma. Then we can take W ∈ P ω 2 ( H θ ) which reflects S being stationary. Then W ≺ ⟨H θ , ∈ , ⃗ d ⟩ , and W ∩ ω 2 ∈ ω 2 . Note that d W ∩ λ ↾ x ∈ W for each x ∈ P ω 2 ( λ ) ∩ W because ⃗ d is thin. Hence there are club many M ∈ [ W ] ω such that d W ∩ λ ↾ x ∈ M for each x ∈ M , i.e. M does not satisfy (II) for y = W ∩ λ . Then there are stationary many M ∈ [ W ] ω which satisfies (I) for y = W ∩ λ , i.e. there is D M ∈ λ 2 ∩ M with D M ↾ x M = d W ∩ λ ↾ x M . Then by the pressing down lemma we can take D such that D = D M for stationary many M ∈ [ W ] ω . This D is as desired. �

[Proof of (2)] Assume SSR + MA ℵ 1 (Cohen), and suppose that ⃗ d = ⟨ d x | x ∈ P ω 2 ( λ ) ⟩ is a thin list. Let θ be a regular cardinal >> λ . It suffices to find a countable M ≺ ⟨H θ , ∈ , ⃗ d ⟩ and y ∈ P ω 2 ( λ ) with y ⊇ x M such that there is D ∈ λ 2 ∩ M with D ↾ x = d y ↾ x for all x ∈ P ω 2 ( λ ) ∩ M . Let S be as in Lemma, and take W ∈ P ω 2 ( H θ ) which reflects S being semi-stationary. We can take such W with W ≺ ⟨H θ , ∈ , ⃗ d ⟩ . Let y := W ∩ λ . Then there are club many N ∈ [ W ] ω which does not satisfy (II) So we can take M ∈ S ∩ [ W ] ω and N ∈ [ W ] ω such that for y . M ⊆ N , M ∩ ω 1 = N ∩ ω 1 , N ≺ ⟨H θ , ∈ , ⃗ d ⟩ , and N does not satisfy (II) for y . Note that M does not satisfy (II) for y , too, because ⃗ d is thin and M ∩ ω 1 = N ∩ ω 1 . Hence M satisfies (I) for y . Let D ∈ λ 2 ∩ M be such that D ↾ x M = d y ↾ x M . Then D witnesses that M and y are as desired. �

Viale and Weiss introduced a stronger principle ISP ( κ, λ ) which implies that κ / ∈ I [ κ ]. It is not hard to show that SR + MA ℵ 1 (Cohen) is consistent with ω 2 ∈ I [ ω 2 ]. Hence we have the following: Remark SR + MA ℵ 1 (Cohen) ̸⇒ ISP ( ω 2 , ω 2 ).

Remark ITP ( ω 2 , ω 2 ) ⇒ TP ( ω 2 , ω 2 ) ⇔ ̸ ∃ ω 2 -Aronszajn tree ⇒ ¬ CH . Question The assumption “ MA ℵ 1 (Cohen)” can be weakened to “ ¬ CH ” ? What influence on cardinal invariants do TP and ITP have ?