The semiproperness of Namba forcings in Prikry-type extensions

The semiproperness of Namba forcings in
Prikry-type extensions
Kenta Tsukuura
Hosei university
Department of Industrial & Systems Engineering
Faculty of Science and Engineering

2023/10/26
RIMS Set Theory Workshop 2023

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The semiproperness of Namba forcings in Prikry-type extensions

1 Introduction

2 Namba forcing over Pκ λ

3 Proof of Main Theorem

4 Namba forcings and ideals in Prikry extensions

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

This research is motivated from the context of saturated ideals.
For an ideal I over Z ⊆ P(X ),P(Z )/I is the quotient of the set algebra ⟨P(Z ), ⊆, ∩, ∪, ∅, Z ⟩ by anequivalence A ≃I B iff A △ B ∈ I.
For a property Ψ of Boolean algebra, I is Ψ iff P(Z )/I is Ψ.
I is saturated if I is |X |+ -saturated (that is having the |X |+ -c.c.).
We always assume that an ideal is fine and normal unless otherwise
stated. Our fineness includes suitable completeness.
Theorem (T. [15])
If j : V → M is an almost huge embedding with critical point κ. For aregular cardinals µ < κ ≤ λ < j (κ), there is a generic extension in which
there is a normal, fine, and κ-complete ideal I over Pκ λ such thatµ is regular.
µ+ = κ and λ+ = j (κ).
I is saturated.
If µ > ℵ0 then I is proper ( in the sense of forcing).
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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Theorem (Shelah)
If µ is singular, λ is regular and I is a saturated ideal over Pµ+ λ then I is
not proper ( in the sense of forcing).
Proof.
We only see in the case of cf(µ) = ω. It is known that P(Pµ+ λ)/I forces
cf(λ) = ω if I preserves λ+ . Then P(Pµ+ λ)/I cannot be proper.
□
Question
For singular µ with countable cofinality and regular λ, is it possible thatPµ+ λ carries a saturated and semiproper ideal?
Every saturated ideal I over Pκ λ is ω1 -stationary preserving if κ ≥ ℵ2 .
(Sakai [3]) For supercompact κ, Coll(µ+ , < κ) forces that µ+ carries
a semiproper ideal even if µ singular.
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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

How do we construct a model in which µ+ carries a saturated ideal for
singular µ?
Lemma (Foreman [11])Suppose that P is Prikry forcing over measurable µ. If I is a saturated
ideal over Pµ+ λ then P forces that I is saturated.
I denotes the ideal generated by I.
Note that
Foreman proved that the same thing for more loud class of posetsthat covers another Prikry-type forcings, for example,
Woodin’s modification that forces µ = ℵω and
Magidor forcing that forces ω < cf(µ) < µ.

All of the known constructions of models with saturated ideals oversuccessors of singular cardinals use Foreman’s lemma andPrikry-type forcings.

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Question
Does Prikry forcing force that I is semiproper?
We give an answer to this by showing:
Theorem (T.)
Suppose that P is Prikry forcing over a measurable cardinal µ. If I is asaturated ideal over Pµ+ λ then P forces that I is not semiproper.
Our argument also shows that all of the known saturated ideals are notsemiproper.
⇝ The key is Namba forcing.

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

We have the following observations now:If cf(µ) = ω then P(Pµ+ λ)/I forces cf(λ) = ω.

P(Pµ+ λ)/I is ω1 -stationary preserving
Saturated ideals are resembling Namba forcings. The following lists basic
properties of Namba forcing Nm(λ).
1

Nm(λ) forces cf(λ) = ω.

2

Nm(λ) is ω1 -stationary preserving.

Theorem (Shelah [13])The following are equivalent:1

There is a semiproper poset that forces cf(λ) = ω.

2

Namba forcing Nm(λ) is semiproper.

⇝ If Nm(µ+ ) is forced to be non semiproper by Prikry forcing then so I is.
Goal: Showing that Nm(µ+ ) is not semiproper in Prikry-type extensions
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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Definition (Foreman–Magidor–Shelah [12])(†) is the following statement:(†) Every ω1 -stationary preserving poset is semiproper.
Many reflection principles (e.g. MM, SR) imply (†).
Theorem (Shelah [13] for (1) ⇔ (2), Doebler–Schindler [6] for (3) ⇒ (2))
The following are equivalent:
1
2(†)
SSR holds. That is, for every λ, the following statement SSR([λ]ω )holds:For every semistationary subset S of [λ]ω , there is an R ∈ [λ]<ω2such that ω1 ⊆ R and S ∩ [R ]ω is semistationary.

3

Nm(λ) is semiproper for all λ ≥ ℵ2 .

It is known that Prikry forcing over µ forces ¬(†). So Nm(λ) is notsemiproper for some λ. What about Nm(µ+ )?
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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Theorem (Shelah [13] for (1) ⇔ (2), Doebler–Schindler [6] for (3) ⇒ (2))
The following are equivalent:
1
2(†)
SSR holds. That is, for every λ, the following statement SSR([λ]ω )holds:For every semistationary subset S of [λ]ω , there is an R ∈ [λ]<ω2such that ω1 ⊆ R and S ∩ [R ]ω is semistationary.

3

1
2

3

Nm(λ) is semiproper for all λ ≥ ℵ2 .
1 → 3 is trivial.
2 → 1 and 3 uses Shelah’s claim. Shelah showed that SSR([2|P | ]ω )implies P is semiproper for ω1 -stationary preserving poset P.
3 → 2 uses that Strong Chang’s conjecture implies SSR.

Proposition
Prikry forcing over µ+ forces ¬SSR([µ+ ]ω ).
Non-semiproperness of Nm(µ+ ) in Prikry extensions follows from theproof of 3 → 2 in [6] and this proposition.

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

SSR([λ]ω ) and the semistationarity of Nm(λ) are consequences of
λ-strong compatness of some cardinal κ ≤ λ.
Proposition
Suppose that there is a λ-strongly compact cardinal κ. Then Coll(ℵ1 , < κ)forces1

SSR([λ]ω ).

2

Nm(λ) is semiproper.

By κ is λ-strongly compact, we mean that Pκ λ carries a fine ultrafilter.

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Question
Are the following equivalent?
1

SSR([λ]ω ).

2

Nm(λ) is semiproper.

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Question
Are the following equivalent?
1

SSR([λ]ω ).

2

Nm(λ) is semiproper.

⇝ NO! This follows from the following results:(Baumgartner [5] for (⇐), Velic̆ković [2] for (⇒))
Con(SR([ω2 ]ω )) ⇔ Con(∃weakly compact).
(Sakai [4]) SR([ω2 ]ω ) ↔ SSR([ω2 ]ω ).
(Todorčević [14]) The semiproperness of Nm(ω2 ) implies the
Chang’s conjecture.

Con(Nm(ℵ2 ) is semiproper) is strictly larger than Con(SSR([ω2 ]ω )).

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The semiproperness of Namba forcings in Prikry-type extensions
Introduction

Theorem (Todorčević [14])If Nm(ℵ2 ) is semiproper then SSR([ℵ2 ]ω ) holds.
We will introduce the following variations to analyze the torrelance.

λNm(λ)
SSR([λ]ω )Pκ λNm(κ, λ)SSR([λ]ω , < κ)

SSR([λ]ω , < κ) is introduced by Sakai [4].
Theorem (T.)
For regular cardinals κ ≤ λ,If Nm(κ, λ) preserves the semistationarity of any S ⊆ [λ]ωthen SSR([λ]ω , < κ) holds.
If αω < κ for all ℵ2 ≤ α < κ then the inverse direction holds too.
Non-semiproperness of Nm(µ+ ) in Prikry-type extensions will follow fromthis theorem.
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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcing over Pκ λ

1 Introduction

2 Namba forcing over Pκ λ

3 Proof of Main Theorem

4 Namba forcings and ideals in Prikry extensions

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcing over Pκ λ

[Pκ λ]<ω⊆ is the set of all {a0 , ..., an } ⊆ Pκ λ that is ⊆-increasing.
For a fine ideal I over Pκ λ,<ω
1 (I-)Namba tree is p ⊆ [Pκ λ]⊆ such thatp is closed under initial segment.
p has a trunk tr(p ), that is maximal s ∈ p such that∀t ∈ p (s ⊆ t ∨ t ⊆ s ).
For every s ∈ p, if s ⊆ tr(p ) then Sucp (s ) = {a | s ⌢ ⟨a ⟩ ∈ p } ∈ I+ .
2

3

Nm(κ, λ, I) is the set of all I-Namba trees. Nm(κ, λ, I) is ordered byinclusion.
We write Nm(κ, λ) for Nm(κ, λ, I) when I is the bounded ideal overPκ λ.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcing over Pκ λLemma

Nm(κ, λ, I) forces cf(δ) = ω for all regular κ ≤ δ ≤ λ.
Proof.
Let ġ be a Nm(κ, λ, I)-name for the set
following properties:1

p ⊩ ġ ∩ tr(p ) = tr(p ).

2

For all t ∈ p, p · ||t ⊆ ġ || , 0.

S
{tr(p ) | p ∈ Ġ }. ġ has theThen ⊩ supi sup(ġi ∩ δ) = δ. Here, ġi is the i-th element of ġ.
Note that ⊩ ġi ∩ δ ∈ (Pκ δ)V , and thus, sup(ġi ∩ δ) is an ordinal below δ.

□

Corollary
If Nm(κ, λ) is semiproper then Nm(δ) is semiproper for every regularκ ≤ δ ≤ λ.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcing over Pκ λTheorem (Shelah)Nm(λ) is ω1 -stationary preserving.
An importance is σ-completeness of ideals in Shelah’s proof. The
bounded ideal over Pκ λ has κ-completeness. We can extend this byLemma
If S ⊆ [W ]ω is semi-stationary (resp. stationary) and [W ]ω contains a club
subset of size < κ. Then Nm(κ, λ, I) forces that S remains
semi-stationary (resp. stationary).
CorollaryNm(κ, λ, I) is ω1 -stationary preserving.
If µω < κ and µ is regular then, for every δ with cf(δ) = µ, Nm(κ, λ, I)
forces cf(δ) > ω.
It is known that Nm(κ, λ, I) preserves all cardinals below κ if I is maximal.
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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcing over Pκ λ

We conclude this section with the following lemma.
Lemma
The following are equivalent.
1

Nm(λ, λ) is semiproper.

2

Nm(λ) is semiproper.

We don’t know whether these Namba forcings are forcing equivalent ornot.

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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

Towards showing the main theorem, we recall semistationarity of sets.
Definition (Shelah [13])For a set W with ω1 ⊆ W , S ⊆ [W ]ω is semistationary in [W ]ω if and only
if, for every F : [W ]<ω → W , there is an x ∈ S and y ∈ [W ]ω with thefollowing properties:x ⊆ y and x ∩ ω1 = y ∩ ω1 .
F “[y ]<ω ⊆ y,
In other word, {y ∈ [W ]ω | ∃x ∈ S (x ⊆ y ∧ x ∩ ω1 = y ∩ ω1 )} is stationary.
Lemma (Shelah [13])For sets ω1 ⊆ W1 ⊆ W2 ,1

2

If S ⊆ [W1 ]ω is semistationary in [W1 ]ω then S is semistationary in[W2 ]ω .
For S ⊆ T ⊆ [W1 ]ω , if S is semistationary then so T is.

By 1, we can omit a phrase “in [W ]ω ” when we discuss about thesemistationarity of some set S ⊆ [W ]ω .
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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

In particular,
Lemma
For a set S ⊆ [W ]ω , if S ∩ [a ]ω is semistationary for some a ⊆ W then S
is semistationary.
Note that
Lemma (Shelah [13])The following are equivalent.
1

P is semiproper.

2

P preserves the semistationarity of any sets.

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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

Let us show the main theorem.
Theorem
For regular cardinals κ < λ,If Nm(κ, λ) preserves the semistationarity of any S ⊆ [λ]ωthen SSR([λ]ω , < κ) holds.

SSR([λ]ω , < κ) is the following statement:
For every semistationary subset S of [λ]ω , there is an R ∈ [λ]<κ such that
ω1 ⊆ R and S ∩ [R ]ω is semistationary.

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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

In the proof, we use a Nm(κ, λ)-name ġ for the generic countable cofinal
subset of Pκ λ. We recall the properties of ġ by
1
2

p ⊩ ġ ∩ tr(p ) = tr(T ).
For all t ∈ p, p · ||t ⊆ ġ || , 0.

We also use a bijection φ : <ω λ → λ.
Proof(1/2).
We show the contraposition. For a non-reflecting semistationary subset
S ⊆ [λ]ω , we claim that S is forced to be not semistationary.
For every a ∈ Pκ λ, we can choose Fa : [a ]<ω → a that avoids S. Thismeans, for every x ∈ S ∩ [a ]ω ,HullFa (x ) ∩ ω1 ⊋ x ∩ ω1 .
Define a Nm(κ, λ)-name Ḟ by
φ(Fġ0 (ξ0 , ..., ξi ), ..., Fġj (ξ0 , ..., ξi )) n = 2i 3jḞ (ξ0 , ..., ξn ) = 

0
o.w.
Here, ġi is the i-th element of ġ.

□
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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

Proof(2/2).
For every p ∈ Nm(κ, λ) and x ∈ S, we may assume that there is ana ∈ tr(p ) such that a ⊇ x. Since Fa is avoiding S ∩ [a ]ω , Puttingy = HullFa (x ) theny ∩ ω1 ⊋ x ∩ ω1 .
So there are ξ0 , ..., ξn ∈ x such that Fa (ξ0 , ...., ξn ) ∈ ω1 \ x. Then p forcesthatḞ (ξ0 , ..., ξk ) = φ(..., Fa (ξ0 , ..., ξn ), ...) ∈ HullḞ (x )for some k and ξn+1 , ..., ξk . We may assume that every hull closed under
φ and φ−1 . Then p also forces that
Fa (ξ0 , ..., ξn ) ∈ ω1 ∩ HullḞ (x ) \ x.
Therefore Ḟ avoids S, and thus, S is forced to be a non-semistationary
subset, as desired.
□
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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

Let us show the inverse direction, that is,
Theorem
For regular cardinals κ ≤ λ, we assume that αω < λ for all ℵ2 ≤ α < λ andSSR([λ]ω , < κ) holds. Then Nm(κ, λ) preserves the semistationarity of
any S ⊆ [λ]ω .
Proof.
If S reflects to a ∈ Pκ λ then Nm(κ, λ) preserves the semistationarity ofS ∩ [a ]ω . This follows from:
If |a | ≥ ℵ2 then the assumption shows |a |ω < κ.
If |a | = ℵ1 then Nm(κ, λ) is ω1 -stationary preserving.
Recall that if S1 ⊆ S2 and S1 is semistationary then so S2 is. Therefore it
is forced that S ∩ [a ]ω ⊆ S are both semistationary.

□

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The semiproperness of Namba forcings in Prikry-type extensions
Proof of Main Theorem

We proved the following theorem.
Theorem
For regular cardinals κ ≤ λ, if suppose that αω < λ for all ℵ2 ≤ α < κ thenthe following are equivalent:1

Nm(κ, λ) preserves the semistationarity of any S ⊆ [λ]ω .

2

SSR([λ]ω , < κ) holds.

If κ = ℵ2 then SSR([λ]ω , < κ) is the same as SSR([λ]ω ) and the cardinalarithmetical assumtion of the theorem above is meaningless. Therefore
Corollary
For a regular cardinal λ, the following are equivalent:1

Nm(ℵ2 , λ) preserves the semistationarity of any S ⊆ [λ]ω .

2

SSR([λ]ω ) holds.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

Let us see the semiproperness of Nm(κ, λ) in Prikry extensions. The key
theorem is
Theorem (Sakai [3])SSR([λ]ω , < κ) implies that, for every stationary subset S ⊆ Eωλ , there isan α < λ with ω < cf(α) < κ such that S ∩ α is stationary in α.
To negate SSR([λ]ω , < κ), it sufficies to give a stationary subset S ⊆ Eωλthat does not reflect to any α < λ with cf(α) < κ.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

We make a list of properties of Prikry forcing P over µ:12

P has the µ+ -c.c. Therefore S ⊆ λ remains stationary for all
stationary S and λ ≥ µ+ .

˙P ⊩ ω < cf(δ)= cf ˇ(δ) < µ if ω < cf(δ) < µThen, for every α < λ and S = Eµλ , if cf(α) < µ+ thenS ∩ α is non-stationary in α.
This is preserved by Prikry forcing. By P ⊩ cf(µ) = ω,P ⊩ S ⊆ Ėωλ .
In particular, P ⊩ ¬SSR([λ]ω , < µ+ ), and thus,
Theorem
P forces that Nm(µ+ , λ) is non-semiproper for all λ ≥ µ+ . In particular, it
is also forced that Nm(µ+ ) is non-semiproper.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

We also have an analogie of main theorem for precipitous ideals.
Theorem+Suppose that I is a normal, fine and precipitous ideal over Z ⊆ [X ]≤µ and⊩ crit(j̇ ) = µ+ . Then the following are equivalent:1

SSR([µ+ ]ω , < µ+ ).

2P(Z )/I preserves all semistationary subset of [µ+ ]ω .

We have
TheoremSuppose that V ⊆ W are inner models and µ < λ are cardinals such that:
1

In V, µ and λ are regular.

2

In W , λ remains regular but µ is singular of cofinality ω.

3(Eµλ )V remains a stationary subset in W .

Then, in W , Nm(µ+ , λ) is not semiproper and there is no semiproper
precipitous ideals over Pµ+ λ.
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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

TheoremSuppose that µ is a singular cardinal with cofinality ω and λ > µ is a+regular cardinal. If [λ+ ]µ carries a normal, fine, and µ+ -complete
λ+ -saturated ideal I then I is not semiproper.
Proof.
If I is semiproper then SSR([µ+ ]ω , < µ+ ) holds.
Let j : V → M ⊆ V [G ] be a generic ultrapower mapping by I. Thenthe following holds:
1234

crit(j ) = µ+ and j (µ+ ) = λ+ .
+S = Eλλ remains stationary in V [G ].
+λM ∩ V [G ] ⊆ M. Therefore S ∈ M.
++
cf M (λ) = cf V [G ] (λ) = ω. Therefore S ⊆ Eωλ = j (Eωµ ).

By the elementarity, there is an T ∈ V such thatµ+T ⊆ Eω is a stationary subset.
For every α < µ+ , T ∩ α is non-stationary.

Therefore SSR([µ+ ]ω , < µ+ ) fails.
This is a contradiction.

□

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

We also have new mutual inconsistency.
Corollary
The following are mutually inconsistent:1[λ+ ]µ carries a normal, fine, and µ+ -complete λ+ -saturated ideal.

2µ+ carries a semiproper ideal.

+

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

We proved that the existence of ℵω+3 -saturated ideals over [ℵω+3 ]ℵω+1
implies the existence of a non-reflecting stationary subset in the previous
proof. The similar argument shows that □∗ℵω holds too.
More precisely,
Theorem (Eskew–Hayut [8])(λ+ , λ) ↠ (ℵω+1 , ℵω ) implies □∗ℵω holds.

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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

We list conclusions for ideals at this moment.
All1 of Prikry-type forcings that can preserve saturation of ideals
destroy the semiproperness.
It is consistent that [ℵω+3 ]ℵω+1 can carries an ℵω+3 -saturated ideal.
All of such ideals are not semiproper.
The existence of saturated ideals over ℵω+1 , Pℵω+1 λ or [λ+ ]ℵω+1 claims
that ℵω+1 is a generic large cardinal. We ask
Question1

Can ℵω+1 carries a semiproper saturated ideal?

2

Does the existence of saturated ideals over ℵω+1 imply some kind ofanti-compactness around ℵω+1 ?

1

As far as I know
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The semiproperness of Namba forcings in Prikry-type extensions
Namba forcings and ideals in Prikry extensions

Lastly, we give some remarks from the view of strong compactness.
Theorem
For every regular cardinals κ ≤ λ with λ<κ = λ, the following holds.

⇒⇒

SSR([2λ ]ω , < κ)
the semiproperness of Nm(κ, λ)SSR([λ]ω , < κ)Each inverse directions cannot be shown in ZFC.
If 2λ = λ+ , we have∃ fine ult. over Pκ λ+∃ fine ult. over Pκ λ⇒⇒SSR([λ+ ]ω , < κ)
the semiproperness of Nm(κ, λ)SSR([λ]ω , < κ)

Under the GCH, there is a hierarchy of ((†)-aspects of) strong
compactness of κ such that

· · · → Nm(κ, λ++ ) is s.p. → SSR([λ++ ]ω , < κ) → Nm(κ, λ+ ) is s.p.
→ SSR([λ+ ]ω , < κ) → · · ·32 / 34

The semiproperness of Namba forcings in Prikry-type extensions
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Namba forcings and ideals in Prikry extensions[14] S.Todorčević, Conjectures of Rado and Chang and cardinal arithmetic, In Finite and infinite
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