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

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Higher Stationarity Bagaria 60 1 / 18

Section 1 Higher stationary sets of ordinals

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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 + Π1

n-Indescribable Card. Axiom. (Arai)

Bagaria also considers ξ-stationary sets for infinite ξ’s.

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Family of non-n-stationary sets

Definition

For κ ∈ On and n < ω, let NSn

κ := {S ⊆ κ | S is not n-stationary in κ} .

NS1

κ is the non-stationary ideal over κ.

For n ≥ 2, NSn

κ may not be an ideal:

Suppose every stationary subset of κ reflects, but there are stationary T0, T1 ⊆ κ which do not reflect simultaneously. Let Si := {µ < κ | Ti ∩ µ is not stationary in µ} . Then S0, S1 ∈ NS2

κ, but S0 ∪ S1 = κ /

∈ NS2

κ.

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Π1

n-indescribable sets

Definition (Π1

n-indescribability)

Suppose κ ∈ On and n < ω. S ⊆ κ is Π1

n-indescribable in κ if for all P ⊆ Vκ and all Π1 n-sentence ϕ with

(Vκ, ∈, P) | = ϕ, there is µ ∈ S with (Vµ, ∈, P ∩ Vµ) | = ϕ. κ is Π1

n-indescribable if κ is Π1 n-indescribable in κ.

NIn

κ := {S ⊆ κ | S is not Π1 n-indescribable in κ}.

Fact ((1),(2),(4):L´ evy, (3):Scott)

1

κ is Π1

0-indescribable iff κ is inaccessible.

2

For an inaccessible cardinal κ, S ⊆ κ is Π1

0-indescribable in κ iff S is

stationary in κ.

3

κ is Π1

1-indescribable iff κ is weakly compact.

4

NIn

κ is a normal ideal over κ.

Bagaria defined the generalized notion of Π1

ξ-indescribability for infinite ξ.

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Π1

1-indescribability and 2-stationarity The following is easy:

Fact

If S is Π1

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 Π1

1-indescribable in κ.

Kunen proved that the 2-stationarity does not imply the Π1

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 Π1

1-indescribable cardinal.

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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 ≤ 2ω which is 2-stationary.

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Higher Stationarity Bagaria 60 7 / 18

Π1

n-indescribability and n + 1-stationarity

Fact

If S is Π1

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 Π1

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 Π1

n-indescribable cardinal.

Bagaria generalizes these results to those of Π1

ξ-indescribability and

ξ + 1-stationarity for infinite ξ.

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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 NSm

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. Note that, in L, NSm

µ = NIm−1 µ

is a normal ideal. So the assumption of Theorem holds in L. Thus we have the following corollary by a forcing over L:

Corollary

It is consistent that there is a cardinal κ ≤ 2ω which is n-stationary for all n < ω.

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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: NSn

µ = (NSn µ)V .

[Proof of “⊆” (i.e. S / ∈ (NSn

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. It suffices to prove that the following C is in the dual filter F of (NSn

µ)V :

C := {ν < µ | ⊩P ˙ T ∩ ν is m-stationary }. For each p ∈ P, the following Tp is m-stationary in µ: Tp := {α < µ | ∃q ≤ p, q ⊩P α ∈ ˙ T}. By the normality of (NSn

µ)V ,

D := {ν < µ | ∀p < ν, Tp ∩ ν is m-stat. & P ∩ ν ⊆c P} ∈ F. Moreover, D ⊆ C. □

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Section 2 Higher stationary sets in Pκ(λ)

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n-stationary subsets of Pκ(A)

Definition (n-stationary subsets of Pκ(A))

For a regular cardinal κ, a set A ⊇ κ and n < ω: S ⊆ Pκ(A) is 0-stationary in Pκ(A) if S is ⊆-cofinal in Pκ(A). S ⊆ Pκ(A) is n-stationary in Pκ(A) if for all m < n and all m-stationary T ⊆ Pκ(A), there is B ∈ S s.t.

• µ := B ∩ κ is a regular cardinal,
• T ∩ Pµ(B) is m-stationary in Pµ(B).

Pκ(A) is n-stationary if Pκ(A) is n-stationary in Pκ(A). NSn

κ,A := {S ⊆ Pκ(A) | S is not n-stationary in Pκ(A)}.

If Pκ(A) is 1-stationary, then κ is weakly Mahlo. Suppose κ is Mahlo. Then NS1

κ,A is the smallest strongly normal ideal over

Pκ(A). If |Pκ(A)| = |A| and f : Pκ(A) → A is a bijection, then NS1

κ,A = NSκ,A | S, where

S = {x ∈ Pκ(λ) | µ := x ∩ κ ∈ Reg & f [Pµ(x)] ⊆ y} .

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Π1

n-indescribability in Pκ(A) For a regular κ and a set A ⊇ κ, we define Vα(κ, A) by induction on α. V0(κ, A) := A, Vα+1(κ, A) := Pκ(Vα(κ, A)) ∪ Vα(κ, A), Vα(κ, A) := ∪

β<α Vβ(κ, A) for a limit α.

Definition (Baumgartner)

Suppose κ is a regular cardinal, A ⊇ κ and n < ω. S ⊆ Pκ(A) is Π1

n-indescribable in Pκ(A) if for all P ⊆ Vκ(κ, A) and all

Π1

n-sentence ϕ with (Vκ(κ, A), ∈, P) |

= ϕ, there is B ∈ S such that

• µ := B ∩ κ is a regular cardinal,
• (Vµ(µ, B), ∈, P ∩ Vµ(µ, B)) |

= ϕ.

Pκ(A) is Π1

n-indescribable if Pκ(A) is Π1 n-indescribable in Pκ(A).

NIn

κ,A := {S ⊆ Pκ(A) | S is not Π1 n-indescribable in Pκ(A)}.

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Theorem (Abe, Car)

1

Pκ(2λ<κ) is Π1

1-indescribable. ⇒ κ is λ-supercompact.

⇒ Pκ(λ) is Π1

n-indescribable for all n ∈ ω.

2

NIn

κ,A is a strongly normal ideal over Pκ(A).

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Π1

n-indescribability and n + 1-stationarity

Proposition

If S is Π1

n-indescribable in Pκ(λ), then S is n + 1-stationary in Pκ(λ).

Suppose κ is Mahlo. Then S is Π1

0-indescribable in Pκ(λ) iff S is 1-stationary

in Pκ(λ). Recall that, in L, S is Π1

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 ⊆ Pκ(λ) and all n < ω, S is Π1

n-indescribable in Pκ(λ) iff S is n + 1-stationary in Pκ(λ).

There is a supercompact cardinal.

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Strongly compact cardinal

Recall that if κ is λ-supercompact, then Pκ(λ) is n-stationary for all n < ω.

Question

Is Pκ(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 Pκ(λ): For all stationary S ⊆ Pκ(λ), there is B ∈ Pκ(λ) s.t.

• µ := B ∩ κ is a regular cardinal,
• S ∩ Pµ(B) is stationary in Pµ(B).

Proposition

It is consistent that there is a strongly compact cardinal κ s.t. the stationary reflection in Pκ(κ+) fails. But, we do not know the answer of the above question.

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Higher Stationarity Bagaria 60 16 / 18

Preservation of n-stationary sets under forcing

We have the following preservation theorem:

Theorem

Assume GCH. Suppose n < ω, κ is a Mahlo cardinal and ρ < κ ≤ λ. Assume NSm

µ,ν is a strongly normal ideal over Pµ(ν) for all m with 1 ≤ m ≤ n and

all µ, ν with µ ≤ κ and ν ≤ λ. Then every ρ-c.c. forcing preserves n-stationary subsets of Pκ(λ). But, we do not know the assumption of Theorem is consistent... One of difficulties to deal with Pκ(λ) in forcing extensions is that Pκ(λ) changes after a forcing. But, in our context, this is not problematic:

Lemma

Assume GCH. Suppose κ is a Mahlo cardinal and λ ≥ κ. Let W be a ρ-c.c. forcing extension of V for some ρ < κ. Then Pκ(λ)W \ Pκ(λ)V is not 1-stationary in W . This is almost immediate from the fact that every ρ-c.c. forcing extension has the ρ+-approximation property.

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n-stationarity and continuum

It is not hard to prove the following:

Proposition

It is consistent that there is a cardinal κ ≤ 2ω such that Pκ(λ) is 2-stationary for all λ ≥ κ. But, we do not know whether this can be generalized to n-stationarity:

Question

For n ≥ 3, is it conssistent that there is a cardinal κ ≤ 2ω such that Pκ(λ) is n-stationary for all λ ≥ κ ?

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Higher Stationarity Bagaria 60 18 / 18