On the consistency of local and global versions of Chang’s Conjecture

Eskew, Monroe; Hayut, Yair

doi:10.1090/tran/7260

Cited by 20 publications

(32 citation statements)

References 16 publications

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“…Proof By [2] there exists some

ρ < λ

such that forcing with

Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false)

gives us

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

. We claim that more is true.…”

Section: A Pseudo Prikry Forcingmentioning

confidence: 93%

“…There is, in fact, a

ρ < λ

such that forcing with

P * {(Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false))}^{V^{P}}

gives us

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

for any

P \subseteq V_{κ}

. This requires only minor alterations to the proof of Theorem 17 from [2]. Counterexamples will now be pairs

⟨ {double-struckP}_{ρ}, {\overset{f}{true}}_{ρ} ⟩

where

{double-struckP}_{ρ} \subset V_{κ}

is a partial order, and

{\ddot{f}}_{ρ}

is a

P_{ρ}

‐name for

{\dot{f}}_{ρ}

, a

Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false)

‐name for a counterexample to

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

.…”

Section: A Pseudo Prikry Forcingmentioning

confidence: 99%

“…Assume

false(κ^{+ + +}, κ^{+ +} false) ↠ false(κ^{+}, κ false)

. (A more optimal consistency proof for this property has recently appeared in [2]). The question given

X \subset κ^{+ + +}

witnessing this and a cardinal

α \leq κ^{+ + +}

is: What is

cof (X \cap α)

?…”

Section: Introductionmentioning

confidence: 99%

“…We will use these results together with arguments from [2] to prove the following: Theorem Let

κ < λ < δ

be three cardinals such that κ is λ‐supercompact, λ is +2‐subcompact, and δ is Woodin or strongly compact. Then there exists a set generic extension of the universe

V [G]

containing partially ordered sets

P_{α}

for any regular cardinal

α < κ

with the following properties:

P_{α}

does not add bounded subsets to κ, it changes

cof (κ)

to ω and

cof (κ^{+})

to α, but does not collapse

κ^{+ + +}

.…”

Section: Introductionmentioning

confidence: 99%

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Some basic thoughts on the cofinalities of Chang structures with an application to forcing

Adolf

2021

Mathematical Logic Qtrly

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Consider false(κ+++,κ++false)↠false(κ+,κfalse) where κ is an uncountable regular cardinal. By a result of Shelah's we have cof(X∩κ++)=κ for almost all X⊂κ+++ witnessing this. Here we consider the question if there could be a similar result for X∩κ+. We will discuss some basic facts implying that this cannot hold in general. We will use these facts to construct an interesting example of a pseudo Prikry forcing, answering a question of Sinapova.

show abstract

“…Proof By [2] there exists some

ρ < λ

such that forcing with

Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false)

gives us

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

. We claim that more is true.…”

Section: A Pseudo Prikry Forcingmentioning

confidence: 93%

“…There is, in fact, a

ρ < λ

such that forcing with

P * {(Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false))}^{V^{P}}

gives us

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

for any

P \subseteq V_{κ}

. This requires only minor alterations to the proof of Theorem 17 from [2]. Counterexamples will now be pairs

⟨ {double-struckP}_{ρ}, {\overset{f}{true}}_{ρ} ⟩

where

{double-struckP}_{ρ} \subset V_{κ}

is a partial order, and

{\ddot{f}}_{ρ}

is a

P_{ρ}

‐name for

{\dot{f}}_{ρ}

, a

Col false(κ, ρ^{+} false) \times Col false(ρ^{+ +}, λ false)

‐name for a counterexample to

false(κ^{+ + +}, κ^{+ +} false) ↠_{κ} false(κ^{+}, κ false)

.…”

Section: A Pseudo Prikry Forcingmentioning

confidence: 99%

“…Assume

false(κ^{+ + +}, κ^{+ +} false) ↠ false(κ^{+}, κ false)

. (A more optimal consistency proof for this property has recently appeared in [2]). The question given

X \subset κ^{+ + +}

witnessing this and a cardinal

α \leq κ^{+ + +}

is: What is

cof (X \cap α)

?…”

Section: Introductionmentioning

confidence: 99%

“…We will use these results together with arguments from [2] to prove the following: Theorem Let

κ < λ < δ

be three cardinals such that κ is λ‐supercompact, λ is +2‐subcompact, and δ is Woodin or strongly compact. Then there exists a set generic extension of the universe

V [G]

containing partially ordered sets

P_{α}

for any regular cardinal

α < κ

with the following properties:

P_{α}

does not add bounded subsets to κ, it changes

cof (κ)

to ω and

cof (κ^{+})

to α, but does not collapse

κ^{+ + +}

.…”

Section: Introductionmentioning

confidence: 99%

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Some basic thoughts on the cofinalities of Chang structures with an application to forcing

Adolf

2021

Mathematical Logic Qtrly

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show abstract

“…As requested by the referee, we provide a proof for completeness. We follow [6]. We prove the lemma by induction on κ w .…”

Section: Definition Of Forcingmentioning

confidence: 99%

(Weak) diamond can fail at the least inaccessible cardinal

Golshani

2022

Fund. Math.

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Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.1. Introduction. We study the combinatorial principles diamond, introduced by Jensen [9], and weak diamond, introduced by Devlin-Shelah [5], and prove the consistency of their failure at the least inaccessible cardinal.Suppose κ is an uncountable regular cardinal. Recall that diamond at κ, denoted ♦ κ , asserts the existence of a sequence S α | α < κ such that for each α < κ, S α ⊆ α, and if X ⊆ κ, then {α < κ | X ∩ α = S α } is stationary in κ.Also, weak diamond at κ, denoted Φ κ , is the assertion "For every c : 2 <κ → 2, there exists g : κ → 2 such that for all f : κ → 2, the setIt is easily seen that ♦ κ + implies 2 κ = κ + , and in fact by a celebrated theorem of Shelah [12], for all uncountable cardinals κ, ♦ κ + is equivalent to 2 κ = κ + . It follows that it is easy to force the failure of diamond at successor cardinals.By [5], 2 κ < 2 κ + implies Φ κ + . It was later observed by Abraham and Baumgartner that Φ κ + implies 2 κ < 2 κ + : see for example [7], [11] where the proof is attributed to Abraham, or [13] where the proof is attributed to Baumgartner. It follows that Φ κ + and 2 κ < 2 κ + are equivalent, and ♦ κ + implies Φ κ + . It also shows that it is easy to force Φ κ + to fail.Unlike the case of successor cardinals, it is difficult to force the failure of diamond or weak diamond at an inaccessible cardinal.

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Easton collapses and a strongly saturated filter

Shioya

2020

Arch. Math. Logic

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On the consistency of local and global versions of Chang’s Conjecture

Cited by 20 publications

References 16 publications

Some basic thoughts on the cofinalities of Chang structures with an application to forcing

Some basic thoughts on the cofinalities of Chang structures with an application to forcing

(Weak) diamond can fail at the least inaccessible cardinal

Easton collapses and a strongly saturated filter

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