STATIONARY REFLECTION AND THE FAILURE OF THE SCH

Omer Ben-Neria; Yair Hayut; Spencer Unger

doi:10.1017/jsl.2023.80

STATIONARY REFLECTION AND THE FAILURE OF THE SCH

Omer Ben-Neria, Yair Hayut, Spencer Unger

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal v such that the singular cardinal hypothesis fails at v and every collection of fewer than cf(v) stationary subsets of v⁺ reflects simultaneously. For cf(v) > ω, 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 cf(v) = ω to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

Original language	American English
Journal	Journal of Symbolic Logic
DOIs	https://doi.org/10.1017/jsl.2023.80
State	Accepted/In press - 2023

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© 2023 Cambridge University Press. All rights reserved.

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abstract = "In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal v such that the singular cardinal hypothesis fails at v and every collection of fewer than cf(v) stationary subsets of v+ reflects simultaneously. For cf(v) > ω, 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 cf(v) = 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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AB - In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal v such that the singular cardinal hypothesis fails at v and every collection of fewer than cf(v) stationary subsets of v+ reflects simultaneously. For cf(v) > ω, 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 cf(v) = 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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STATIONARY REFLECTION AND THE FAILURE OF THE SCH

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