Stationary Reflection

Hayut, Yair; Unger, Spencer

doi:10.1017/jsl.2020.64

Cited by 4 publications

(31 citation statements)

References 20 publications

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“…A theorem of Magidor [12] shows that it is consistent relative to the existence of infinitely many supercompact cardinals that every finite collection of stationary subsets of ℵ ω+1 reflects. Recently, the second and third author [11] were able show the same result from an assumption below the existence of a cardinal κ which is κ + -supercompact. Both of these models satisfy GCH.…”

Section: Introductionmentioning

confidence: 59%

“…This study was prompted by two other recent studies. First, the work of second and third authors in [11] on stationary reflection in Prikry forcing extensions from subcompactness assumptions. The arguments of [11] show how to examine the stationary reflection in the extension by Prikry type forcing by studying suitable iterated ultrapowers of V .…”

Section: Introductionmentioning

confidence: 99%

“…First, the work of second and third authors in [11] on stationary reflection in Prikry forcing extensions from subcompactness assumptions. The arguments of [11] show how to examine the stationary reflection in the extension by Prikry type forcing by studying suitable iterated ultrapowers of V . This approach and method has been highly effective in our situation and we follow it here.…”

Section: Introductionmentioning

confidence: 99%

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Stationary Reflection and the failure of SCH

Ben-Neria¹,

Hayut²,

Unger³

2019

Preprint

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In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal ν such that the singular cardinal hypothesis fails at ν and every collection of fewer than cf(ν) stationary subsets of ν + reflects simultaneously. For cf(ν) > ω, this situation was not previously known to be consistent. Using different methods, we reduce the upperbound on the consistency strength of this situation for cf(ν) = ω to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

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Section: Introductionmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stationary Reflection and the failure of SCH

Ben-Neria¹,

Hayut²,

Unger³

2019

Preprint

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“…It was introduced by Neeman and Steel in [NS16], where it goes by the name "Π 2 1subcompact". We use the alternative name introduced by Hayut and Unger in [HU18].…”

Section: Definition 32 ([Lh17]mentioning

confidence: 99%

“…It is easily proven (see [HU18,Lemma 36]) that, if λ is λ + -Π 1 1 -subcompact, then λ is measurable.…”

Section: Definition 32 ([Lh17]mentioning

confidence: 99%

Knaster and friends II: The C-sequence number

Lambie-Hanson¹,

Rinot²

2019

Preprint

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Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(. . .), introduced in Part I of this series. Contents 1. Introduction 1.1. Organization of this paper 1.2. Notation and conventions 2. The C-sequence number 3. Changing the value of the C-sequence number 3.1. Inaccessibles 3.2. Successors of singulars 4. The C-sequence spectrum 5. The C-sequence spectrum and closed colorings 5.1. From C-sequences to closed colorings 5.2. From closed colorings to C-sequences 5.3. From colorings to closed colorings 5.4. The structure of the C-sequence spectrum 6. Concluding remarks

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Stationary Reflection and the Failure of the SCH

BEN-NERIA,

HAYUT,

UNGER

2023

J. symb. log.

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In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $ , this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\operatorname {\mathrm {cf}}(\nu ) = \omega $ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

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Stationary Reflection

Abstract: We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

Cited by 4 publications

References 20 publications

Stationary Reflection and the failure of SCH

Stationary Reflection and the failure of SCH

Knaster and friends II: The C-sequence number

Stationary Reflection and the Failure of the SCH

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