Total Failure of Approachability at Successors of Singulars of Countable Cofinality

Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $δ$ of countable cofinality and every regular uncountable $μ<δ$ there are stationarily many non-approachable points of cofinality $μ$ in $δ^+$. This answers a question of Mitchell and provides a decisive answer to a question of Foreman and Shelah.

Publication Details

Journal
arXiv (Cornell University)
Published
2025-08-04
Primary Topic
Advanced Topology and Set Theory
Type
article
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article

Total Failure of Approachability at Successors of Singulars of Countable Cofinality

arXiv (Cornell University)
Advanced Topology and Set Theory
article

Total Failure of Approachability at Successors of Singulars of Countable Cofinality

article en

Abstract

Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $δ$ of countable cofinality and every regular uncountable $μ<δ$ there are stationarily many non-approachable points of cofinality $μ$ in $δ^+$. This answers a question of Mitchell and provides a decisive answer to a question of Foreman and Shelah.

arXiv (Cornell University)
Decent work and economic growth
Openalex Percentile: Top 43%
Advanced Topology and Set Theory
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Total Failure of Approachability at Successors of Singulars of Countable Cofinality · arXiv (Cornell University) (2025) | TGRS Research Map | TGRS