Weak Reflection Principle Definition 1. X : non-empty set. S [ X - - PowerPoint PPT Presentation

Reflection principles on [λ]ω

Toshimichi Usuba Nagoya University

• Oct. 26, 2010

RIMS Set Theory Workshop 2010, Kyoto

Weak Reflection Principle

✓ ✏

Definition 1. X: non-empty set. S ⊆ [X]ω is stationary (in [X]ω) ⇐ ⇒ For every function f : <ωX → X there is x ∈ S such that f“<ωx ⊆ x.

✒ ✑ ✓ ✏

Definition 2. λ: cardinal ≥ ω2. WRP(λ) ⇐ ⇒ For every stationary S ⊆ [λ]ω, there is X ⊆ λ such that |X| = ω1 ⊆ X and S reflects at X, i.e., S ∩ [X]ω is stationary in [X]ω. WRP ⇐ ⇒ WRP(λ) for every λ ≥ ω2.

✒ ✑

1

Weak Reflection Principle

There are many useful consequences from WRP(λ):

✓ ✏

Fact 3 (Foreman-Magidor-Shelah, Shelah, Todorcevic). ➀ WRP(ω2) ⇒ 2ω ≤ ω2. ➁ WRP ⇒ NSω1 is presaturated. ➂ WRP ⇒ SCH.

✒ ✑

2

Reflection Principle

✓ ✏

Definition 4. λ: regular cardinal ≥ ω2. RP(λ) ⇐ ⇒ For every stationary S ⊆ [λ]ω, there is X ⊆ λ such that |X| = ω1 ⊆ X, cf(sup(X)) = ω1, and S reflects at X.

✒ ✑

Martin’s Maximum ⇒ RP(λ) ⇒ WRP(λ).

3

Fodor-type Reflection Principle

S: set of ordinals ⃗ c = ⟨cα : α ∈ S⟩ is a ladder on S if for every α ∈ S, cα ⊆ α is unbounded in α and ot(cα) = cf(α).

✓ ✏

Definition 5 (Fuchino-Juh´ asz-Soukup-Szentmikl´

• ssy-U.).

λ ≥ ω2: regular FRP(λ) ⇐ ⇒ For every stationary E ⊆ {α < λ : cf(α) = ω} and every ladder ⃗ c = {cα : α ∈ E}, there is I ⊆ E such that: ➀ |I| = ω1 = cf(sup(I)). ➁ For every g : I → λ with g(α) ∈ cα, there is ξ < λ with {α ∈ I : g(α) = ξ} stationary in sup(I). FRP ⇐ ⇒ FRP(λ) for every λ ≥ ω2.

✒ ✑

4

Fodor-type Reflection Principle

✓ ✏

Fact 6 (F.-J.-So.-Sz.-U., Fuchino-Soukup-Sakai-U.). FRP ⇐ ⇒ For every locally countably compact topo- logical space X, if every subspace of X with size ω1 is metrizable, then X itself is also metrizable.

✒ ✑

5

Semi-Stationary Reflection principle

✓ ✏

Definition 7 (Shelah). X: a set with ω1 ⊆ X. S ⊆ [X]ω is semi-stationary (in [X]ω) if the set {x ∈ [λ]ω : ∃a ∈ S (a ⊆ x ∧ a ∩ ω1 = x ∩ ω1)} is stationary.

✒ ✑ ✓ ✏

Definition 8 (Shelah). λ: cardinal ≥ ω2. SSR(λ) ⇐ ⇒ For every semi-stationary (or stationary) S ⊆ [λ]ω, there is X ⊆ λ such that |X| = ω1 ⊆ X and S ∩ [X]ω is semi- stationary in [X]ω. SSR ⇐ ⇒ SSR(λ) for every λ ≥ ω2.

✒ ✑

6

Semi-Stationary Reflection principle

✓ ✏

Fact 9 (Shelah). SSR ⇐ ⇒ every ω1-stationary preserving forcing notion is semi-proper.

✒ ✑ ✓ ✏

Fact 10 (F.-M.-Sh., Sakai, Todorcevic). ➀ SSR ⇒ NSω1 is precipitous. ➁ SSR ⇒ strong Chang’s conjecture. ➂ SSR(ω2) ⇐ ⇒ WRP(ω2).

✒ ✑

7

Implications

It is easy to check that: RP(λ) ⇒ WRP(λ). RP(λ) ⇒ FRP(λ). WRP(λ) ⇒ SSR(λ). RP(λ)

• WRP(λ)

SSR(λ)

FRP(λ)

8

Implications

RP(λ) ⇒ WRP(λ). RP(λ) ⇒ FRP(λ). WRP(λ) ⇒ SSR(λ). Question: What about other directions? Especially does WRP(λ) ⇒ RP(λ)? RP(λ)

• WRP(λ)

?

• ?
• SSR(λ)

?

• ?
• FRP(λ)

?

• ?
• 9

Facts

✓ ✏

Fact 11 (Sakai). SSR ̸⇒ WRP(ω3). Fact 12 (F.-J.-So.-Sz.-U.). FRP ̸⇒ WRP(ω2).

✒ ✑

RP(λ)

• WRP(λ)

?

• ?
• SSR(λ)

⧸

?

• FRP(λ)
• ⧸
• ⧸

10

Facts about WRP and RP

✓ ✏

Fact 13 (K¨

• nig-Larson-Yoshinobu).

Under GCH, WRP(ωn) ⇒ RP(ωn) for every 2 ≤ n < ω.

✒ ✑ ✓ ✏

Fact 14 (Krueger). Con(ZFC + ∃ κ+-supercompact cardinal κ) ⇒ Con(ZFC + WRP(ω2) + ¬RP(ω2))

✒ ✑

RP(λ)

• WRP(λ)

⧸

• ?
• SSR(λ)

⧸

?

• FRP(λ)
• ⧸
• ⧸

11

Theorem 1

✓ ✏

Theorem 1. Suppose that there exists a weakly compact

• cardinal. Then there exists a forcing extension in which

the following hold: ➀ WRP(ω2). ➁ ¬FRP(ω2).

✒ ✑

RP(λ)

• WRP(λ)

⧸

• ⧸

SSR(λ)

⧸

• ⧸

FRP(λ)

• ⧸
• ⧸
• ⧸

12

Theorem 1

✓ ✏

Fact 15 (Shelah, Velickovic). Con(ZFC + ∃ weakly compact cardinal) ⇐ ⇒ Con(ZFC + WRP(ω2))

✒ ✑ ✓ ✏

Corollary 16. ➀ Con(ZFC + ∃ weakly compact cardinal) ⇐ ⇒ Con(ZFC + WRP(ω2) + ¬RP(ω2)). ➁ WRP(ω2) ̸⇒ FRP(ω2)

✒ ✑

13

Theorem 2

✓ ✏

Theorem 17. Suppose that there exists a supercompact

• cardinal. Then there exists a forcing extension in which

the following hold: ➀ SSR. ➁ ¬FRP(ω2).

✒ ✑

Hence SSR does not imply FRP(ω2).

14

Stationary set induced by ladder

For i < 2, let S2

i = {α < ω2 : cf(α) = ωi} \ (ω1 + 1).

Fix a surjection πα : ω1 → α for ω1 < α < ω2 and let C∗ be the set of all x ∈ [ω2]ω such that: sup(x) > ω1 x ∩ ω1 ∈ ω1 and sup(x) / ∈ x. ∀α ∈ x, πα“(x ∩ ω1) = x ∩ α. C∗ forms a club in [ω2]ω.

✓ ✏

Definition 18. For a ladder ⃗ c on S2

0, let S⃗ c be the set of

all x ∈ C∗ with csup(x) ⊆ x.

✒ ✑

15

Basic properties of S⃗

c

✓ ✏

Lemma 19. S⃗

c and [ω2]ω \ S⃗ c are stationary.

✒ ✑ ✓ ✏

Lemma 20. [ω2]ω \ S⃗

c does not reflect at any α ∈ S2 0.

✒ ✑

Idea of Proof of theorems: Collapse a weakly compact κ to ω2. By forcing, we make S⃗

c non-reflecting at any α ∈ S2 1.

Using a iteration of club shootings, destroy the sta- tionarity of non-reflecting subset of S⃗

c.

(Hence every stationary subset of S⃗

c is reflecting).

Using the weak compactness of κ, show that every sta- tionary subset of [ω2]ω \ S⃗

c reflects at α ∈ S2 1.

16

Basic properties of S⃗

c

✓ ✏

Lemma 21. For x, y ∈ S⃗

c, if x ∩ ω1 = y ∩ ω1 and sup(x) =

sup(y) then x = y.

✒ ✑ ✓ ✏

Lemma 22. θ: large regular cardinal, M ≺ Hθ: a countable model with C∗,⃗ c ∈ M. X = {x ∈ S⃗

c : x ⊆ M ∩ ω2}.

➀ X is countable. ➁ For x ∈ X, if x∩ω1 = M ∩ω1 then x = M ∩α for some α ∈ (M ∩ ω2) ∪ {ω2}. ➂ If S ∈ M is non-reflecting subset of S⃗

c, x ∈ S ∩X, and

x ∩ ω1 = M ∩ ω1, then x = M ∩ ω2.

✒ ✑

17

Club Shooting

✓ ✏

Definition 23. For X ⊆ [ω2]ω, C(X) is the poset consists

• f all functions p such that:

➀ p : d(p) × d(p) → ω1 for some d(p) ∈ [ω2]ω. ➁ For every x ∈ X, if x ⊆ d(p) then x is not closed under p. For p, q ∈ C(X), p ≤ q ⇐ ⇒ p ⊇ q.

✒ ✑

18

Club Shooting

✓ ✏

Lemma 24. ➀ C(X) satisfies the (2ω)+-c.c. ➁ For every a ∈ [ω2]ω, {p ∈ C(X) : a ⊆ d(p)} is dense

• pen.

➂ If G is (V, C(X))-generic and F = ∪ G, then F : ωV

2 ×

ωV

2 → ω1 and there is no x ∈ X closed under F.

✒ ✑

19

Complete forcing

✓ ✏

Definition 25 (Shelah). Let P be a poset and θ a large regular cardinal. For a countable M ≺ Hθ with P ∈ M, a sequence ⟨pi : i < ω⟩ is generic sequence of M ⇐ ⇒ ⟨pi : i < ω⟩ is a descending sequence of P and for every dense open set D ∈ M in P, there is i < ω with pi ∈ D ∩ M. T ⊆ [λ]ω: stationary for some λ. A poset P is T-complete ⇐ ⇒ every countable M ≺ Hθ with P ∈ M ∩ λ ∈ T and every generic sequence of M has a lower bound.

✒ ✑

20

Complete forcing

✓ ✏

Lemma 26 (Shelah). ➀ T-complete poset is σ-Baire. ➁ For every stationary T ′ ⊆ T, T-complete poset pre- serves the stationarity of T ′. ➂ If P is a countable support iteration of T-complete posets, then P is T-complete.

✒ ✑

21

Non-reflecting ladder

✓ ✏

Definition 27. A ladder ⃗ c on S2

0 is non-reflecting

⇐ ⇒ for every α ∈ S2

1, there is a club d ⊆ α ∩ S2 0 in α such

that {cβ : β ∈ d} has an injective choice function.

✒ ✑ ✓ ✏

Lemma 28. If ⃗ c is a non-reflecting ladder, then FRP(ω2)

• fails. In fact S⃗

c and ⃗

c witnesses that ¬FRP(ω2).

✒ ✑

22

✓ ✏

Lemma 29. Suppose ⃗ c is a non-reflecting ladder and S ⊆ S⃗

c a non-reflecting. Let θ be a large regular cardinal and

M ≺ Hθ a countable model with C∗,⃗ c ∈ M. Suppose M ∩ω2 / ∈ S. Then for every x ∈ S, x ⊆ M ∩ω2 ⇐ ⇒ x ∈ M.

✒ ✑ ✓ ✏

Lemma 30. If ⃗ c is non-reflecting ladder and S ⊆ S⃗

c is

non-reflecting subset, then C(S) is [ω2]ω \ S⃗

c-complete.

✒ ✑

23

✓ ✏

Lemma 31. For every stationary E ⊆ ω1, the set {x ∈ C∗ : x ∩ ω1, csup(x) ⊈ x} is stationary.

✒ ✑ ✓ ✏

Lemma 32. If ⃗ c is non-reflecting ladder and S ⊆ S⃗

c is non-

reflecting subset, then C(S) is ω1-stationary preserving.

✒ ✑

24

✓ ✏

Lemma 33. Suppose CH. Let ⃗ c is non-reflecting ladder and α an ordinal. Suppose that Pα is a countable support iteration of the ˙ Qβ’s such that for every β < α, ⊩Pβ“ ˙ Qβ = C( ˙ Sβ) for some non-reflecting ˙ Sβ ⊆ S⃗

c”. Then Pα satisfies

the ω2-c.c.

✒ ✑

25

Adding non-reflecting ladder

✓ ✏

Lemma 34. Suppose CH. There is a poset R such that ➀ R is σ-closed, ➁ |R| = ω2, ➂ R satisfies ω2-c.c. ➃ R forces that “there exists a non-reflecting ladder on S2

0”.

✒ ✑

26

Definition of R

R is the set of all pair ⟨p, q⟩ such that: ➀ p : dom(p) → [ω2]ω for some dom(p) ∈ [S2

0]ω.

➁ For α ∈ dom(p), p(α) ⊆ α is unbounded in α and

• t(p(α)) = ω.

➂ q : dom(q) → V for some dom(q) ∈ [S2

1]ω.

➃ For α ∈ dom(q), ➊ q(α) : dom(q(α)) → α for some closed dom(q(α)) ∈ [dom(p) ∩ α]ω. ➋ q(α) is injective and q(α)(β) ∈ p(β) for every β ∈ dom(q(α)). For ⟨p0, q0⟩, ⟨p1, q1⟩ ∈ R, ⟨p0, q0⟩ ≤ ⟨p1, q1⟩ ⇐ ⇒ p0 ⊇ p1. dom(q0) ⊇ dom(q1).

27

For every α ∈ dom(q1), q0(α) ⊇ q1(α) and dom(q0(α)) is an end extension of dom(q1(α)).

✓ ✏

Corollary 35. Suppose CH and 2ω = ω1. Then there is a forcing notion P such that: ➀ P satisfies ω2-c.c. and σ-Baire. ➁ P forces that There exists a non-reflecting ladder ⃗ c on S2

0 (hence

¬FRP(ω2)) Every stationary subset of S⃗

c is reflecting at some

α ∈ S2

0.

✒ ✑

Note: Sakai already established a similar result using □ω1- sequence.

28

Outline of the Proof of Corollary

Force with R. Let ⃗ c be the non-reflecting ladder induced by (V, R)- generic. In the generic extension, choose a ω3-stage countable suppose iteration Pω3 of the ˙ Qα’s such that For each β < ω3, ⊩Pβ“ ˙ Qβ = C( ˙ Sβ) for some non-reflecting ˙ S ⊆ S⃗

c”.

For every β < ω3 and Pβ-name ˙ S of non-reflecting subset of S⃗

c, there is γ < ω3 with ⊩Pγ“ ˙

S = ˙ Sγ”. Pω3 satisfies the ω3-c.c., is σ-Baire, and ω1-preserving. Using those properties we know that: Pω3 forces that “every stationary subset of S⃗

c is reflect-

ing”.

29

Proof of Theorem 1 (Outline)

κ is weakly compact and 2κ = κ+. Take a (V, Col(ω1, < κ))-generic G and work in V [G]. In V [G], take a (V [G], R)-generic R and work in V [G][R]. Let ⃗ c be the non-reflecting ladder induced by R. As before, define a κ+-stage countable support itera- tion Pκ+ of the club shootings which forces “every sta- tionary subset of S⃗

c is reflecting”.

Let H be (V [G][R], Pκ+)-generic. In V [G][R][H], it holds that ¬FRP(ω2). Every stationary subset of S⃗

c is reflecting.

30

Proof of Theorem 1 (Outline)

Thus the rest is to show that every stationary subset of [ω2]ω\S⃗

c is reflecting in V [G][R][H].

✓ ✏

Lemma 36. For ξ < κ+, let Hξ be the (V [G][R], Pξ)- generic filter induced by H. Then in V [G][R][Hξ], every stationary subset of [ω2]ω \ S⃗

c is reflecting.

✒ ✑

(Why?) If S ⊆ [ω2]ω \ S⃗

c is stationary, there is ξ < κ+

with S ∈ V [G][R][Hξ]. By the lemma, S ∩ [α]ω is stationary for some α < κ. By the ω1-preservation of Pξκ+, S ∩ [α]ω remains a stationary.

31

Proof of Theorem 1 (Outline)

Fix ξ < κ+ and let Q = Col(ω1, < κ) ∗ R ∗ Pξ. Take a Q-name ˙ S of a stationary subset of [ω2]ω \ S⃗

c.

In V , take M ≺ Hθ such that |M| = κ, <κM ⊆ M, and κ, ξ, Q, ˙ S, . . . ∈ M. Because κ is weakly compact, there is a transitive N and an elementary embedding j : M → N with the critical point of j is κ. Since Q satisfies the κ-c.c., i = j|Q is a complete em- bedding from Q to j(Q).

✓ ✏

Claim 37. In V [G][R][Hξ], j(Q)/(G ∗ R ∗ Hξ) is [ω2]ω \ S⃗

c-

complete.

✒ ✑

32

Choose a (V [G][R][Hξ], j(Q)/(G ∗ R ∗ Hξ))-generic j(G ∗ R ∗ Hξ). j : M → N can be extended to j : M[G][R][Hξ] → N[j(G ∗ R ∗ Hξ)]. N[j(G ∗ R ∗ Hξ)] ⊨ j(S) ∩ κ = S is stationary in [κ]ω N[j(G∗R∗Hξ)] ⊨ ∃α < j(κ) (j(S) ∩ α is stationary in [α]ω) By the elementarity, M[G][R][Hξ] ⊨ ∃α < κ (S ∩ α is stationary in [α]ω) Thus S reflects at some α < κ in V [G][R][Hξ].

33

Proof of Theorem 2

κ is supercompact and 2κ = κ+. As before, we force with Q = Col(ω1, < κ) ∗ R ∗ Pκ+. Take a (V, Q)-generic G ∗ R ∗ H. Fix λ > ω1. In V , take a λ-supercompact embedding j : V → M. Consider j(Q)/(G ∗ H ∗ R).

✓ ✏

Lemma 38. Let T = {x ∈ [λ]ω : x ∩ ω2 / ∈ S⃗

c}.

Then j(Q)/(G ∗ R ∗ H) is T-complete

✒ ✑ ✓ ✏

Lemma 39. In V [G ∗ R ∗ H], every stationary S ⊆ {x ∈ [λ]ω : x ∩ ω2 / ∈ S⃗

c} is reflecting.

✒ ✑

34

✓ ✏

Lemma 40. In V [G ∗ R ∗ H], let S be a stationary subset

• f {x ∈ [λ]ω : x ∩ ω2 ∈ S⃗

c}. Then

{x ∈ [λ]ω : x ∩ ω2 / ∈ S⃗

c, ∃a ∈ S (a ⊆ x ∧ a ∩ ω1 = x ∩ ω1)}

is stationary.

✒ ✑

Fix f : <ωλ → λ. We claim that there is a club D in [λ]ω such that for all a ∈ D with a∩ω2 ∈ S⃗

c, clf(a∪{sup(a∩ω2)})∩ω1 = a∩ω1.

This follows from the reflection of S⃗

c.

Repeating this argument, we can find a ∈ S and x ∈ [λ]ω which is f-closed, a ⊆ x, x ∩ ω2 / ∈ S⃗

c, and a ∩ ω1 = x ∩ ω1.

Combining those lemmas, we can conclude that “every stationary subset of [λ]ω is semi-reflecting”.

35

Question

➀ Does WRP ⇒ RP(ω2)? What about FRP(ω2)? ➁ Does WRP(ω3) ⇒ RP(ω3)?

36

Thank you for your attention!

37