SLIDE 1
1.1 Stationary reflection
- For a cardinal λ ≥ ω2, the stationary reflection in [λ]ω is
the following statement:
SR([λ]ω)
≡ for all stationary S ⊆ [λ]ω there is W ⊆ λ s.t. (i) |W| = ω1 ⊆ W, (ii) S ∩ [W]ω is stationary.
- SR
Semi-stationary reflection, stationary reflection and combinatorics - - PowerPoint PPT Presentation
Semi-stationary reflection, stationary reflection and combinatorics - - PowerPoint PPT Presentation
Semi-stationary reflection, stationary reflection and combinatorics Hiroshi Sakai (joint work with Boban Veli ckovi c) October 28, 2010 1.1 Stationary reflection For a cardinal 2 , the stationary reflection in [ ] is
SLIDE 2
SLIDE 3
1.2 (†) and Semi-stationary reflection
(†) ≡ Every ω1-stationary preserving forcing notion is semi-proper. Thm (Foreman-Magidor-Shelah)
SR ⇒ (†).
Thm (†) implies the following: (i) precipitousness of NSω1 (Foreman-Magidor-Shelah) (ii) (Strong) Chang’s Conjecture (Foreman-Magidor-Shelah) (iii) 2ω ≤ ω2 (Todorˇ cevi´ c)
SLIDE 4
- Let W be a set ⊇ ω1.
S ⊆ [W]ω is semi-stationary if the set {y ∈ [W]ω | (∃x ∈ S) x ⊆ y ∧ x ∩ ω1 = y ∩ ω1} 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 ⊇ ω1. 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 ⊇ ω1.
SLIDE 5
- For a cardinal λ ≥ ω1, the semi-stationary reflection in [λ]ω
is the following statement:
SSR([λ]ω)
≡ for all semi-stationary S ⊆ [λ]ω there is W ⊆ λ s.t. (i) |W| = ω1 ⊆ W, (ii) S ∩ [W]ω is semi-stationary.
- SSR
≡
SR([λ]ω) holds for all λ ≥ ω2.
Thm (Shelah)
SSR ⇔ (†).
Thm (Todorˇ cevi´ c)
SSR([ω2]ω) ⇔ SR([ω2]ω).
SLIDE 6
1.3 More consequences of SSR
Thm 1 If SSR([λ]ω) holds, then the following hold for every regular cardinal κ with ω2 ≤ κ ≤ λ: (i) reflection of stationary subsets of {α ∈ κ | cf(α) = ω} (Sakai) (ii) the failure of (κ) (Sakai-Veliˇ ckovi´ c) (iii) κω = κ (Sakai-Veliˇ ckovi´ c)
(κ) ≡ there is ⟨cα | α ∈ Lim(κ)⟩ with the following properties:
- cα is a club subset of α
- cβ = cα ∩ β if β ∈ Lim(α)
- there are no club C ⊆ κ such that cα = C ∩ α
for all α ∈ Lim(C).
SLIDE 7
1.4 Reflection principles and compact cardinals
1.4.1 L´ evy collapse of compact cardinals Thm (Foreman-Magidor-Shelah) κ: supercompact ⇒ Col(ω1,<κ) SR. Thm (Shelah) κ: strongly compact ⇒ Col(ω1,<κ) SSR. Thm (Sakai) κ: strongly compact ̸⇒ Col(ω1,<κ) SR.
SLIDE 8
1.4.2 TP and ITP
- A list on Pκ(λ) is a seq. ⃗
d = ⟨dx | x ∈ Pκ(λ)⟩ s.t. dx : x → 2.
- A list ⃗
d = ⟨dx | x ∈ Pκ(λ)⟩ is said to be thin if |{dy ↾x | y ⊇ x}| < κ for all x ∈ Pκ(λ).
- D : λ → 2 is an ineffable branch of a list ⃗
d = ⟨dx | x ∈ Pκ(λ)⟩ if there are stationary many x ∈ Pκ(λ) with D ↾x = dx.
- D : λ → 2 is a cofinal branch of a list ⃗
d = ⟨dx | x ∈ Pκ(λ)⟩ if for any x ∈ Pκ(λ) there is y ⊇ x with D ↾x = dy ↾x.
ITP(κ, λ)
≡ Every thin list on Pκ(λ) has an ineffable branch.
TP(κ, λ)
≡ Every thin list on Pκ(λ) has a cofinal branch.
SLIDE 9
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)
PFA
⇒ ITP(ω2, λ) for all λ ≥ ω2. Thm 2 (Sakai-Veliˇ ckovi´ c) (1) SR + MAℵ1(Cohen) ⇒ ITP(ω2, λ) for all λ ≥ ω2. (2) SSR + MAℵ1(Cohen) ⇒ TP(ω2, λ) for all λ ≥ ω2. (3) SSR + MAℵ1(Cohen) ̸⇒ ITP(ω2, ω3).
SLIDE 10
Proof skech of (1) and (2) of Thm 2 The following lemma is a key: Lem Assume MAℵ1(Cohen). Let λ be a cardinal ≥ ω2, ⃗ d = ⟨dx | x ∈ Pω2(λ)⟩ be a thin list and θ be a regular cardinal >> λ. For each M ∈ [Hθ]ω let xM :=
∪(Pω2(λ) ∩ M) ∈ Pω2(λ).
Moreover let S be the set of all countable M ≺ ⟨Hθ, ∈, ⃗ d⟩ such that either of the following holds for any y ∈ Pω2(λ) with y ⊇ xM: (I) There is D ∈ λ2 ∩ M with D ↾xM = dy ∩ xM. (II) There is x ∈ Pω2(λ) ∩ M with dy ↾x / ∈ M. Then S is stationary in [Hθ]ω.
SLIDE 11
[Proof of (1)] Assume SR + MAℵ1(Cohen), and suppose that ⃗ d = ⟨dx | x ∈ Pω2(λ)⟩ is a thin list. Let θ be a regular cardinal >> λ. It suffices to find W ∈ Pω2(Hθ) such that W ≺ ⟨Hθ, ∈, ⃗ d⟩, such that W ∩ ω2 ∈ ω2 and such that there is D ∈ λ2 ∩ W with D ↾(W ∩ λ) = dW∩λ. Let S be as in Lemma. Then we can take W ∈ Pω2(Hθ) which reflects S being stationary. Then W ≺ ⟨Hθ, ∈, ⃗ d⟩, and W ∩ω2 ∈ ω2. Note that dW∩λ ↾x ∈ W for each x ∈ Pω2(λ)∩W because ⃗ d is thin. Hence there are club many M ∈ [W]ω such that dW∩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 DM ∈ λ2 ∩ M with DM ↾xM = dW∩λ ↾xM. Then by the pressing down lemma we can take D such that D = DM for stationary many M ∈ [W]ω. This D is as desired.
SLIDE 12
[Proof of (2)] Assume SSR + MAℵ1(Cohen), and suppose that ⃗ d = ⟨dx | x ∈ Pω2(λ)⟩ is a thin list. Let θ be a regular cardinal >> λ. It suffices to find a countable M ≺ ⟨Hθ, ∈, ⃗ d⟩ and y ∈ Pω2(λ) with y ⊇ xM such that there is D ∈ λ2 ∩ M with D ↾x = dy ↾x for all x ∈ Pω2(λ) ∩ M. Let S be as in Lemma, and take W ∈ Pω2(Hθ) which reflects S being semi-stationary. We can take such W with W ≺ ⟨Hθ, ∈, ⃗ d⟩. Let y := W ∩ λ. Then there are club many N ∈ [W]ω which does not satisfy (II) for y. So we can take M ∈ S ∩ [W]ω and N ∈ [W]ω such that M ⊆ N, M ∩ ω1 = N ∩ ω1, N ≺ ⟨Hθ, ∈, ⃗ 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 ∩ω1 = N ∩ω1. Hence M satisfies (I) for y. Let D ∈ λ2∩M be such that D ↾xM = dy ↾xM. Then D witnesses that M and y are as desired.
SLIDE 13
Viale and Weiss introduced a stronger principle ISP(κ, λ) which implies that κ / ∈ I[κ]. It is not hard to show that SR + MAℵ1(Cohen) is consistent with ω2 ∈ I[ω2]. Hence we have the following: Remark
SR + MAℵ1(Cohen)
̸⇒ ISP(ω2, ω2).
SLIDE 14