Total Failure of Approachability at Successors of Singulars of Countable Cofinality

TL;DR

The paper constructs a model in set theory where many singular cardinals of countable cofinality have stationarily many non-approachable points, answering longstanding open questions.

Contribution

It demonstrates the total failure of approachability at successors of singulars of countable cofinality under certain large cardinal assumptions.

Findings

Constructed a model with stationarily many non-approachable points in $oldsymbol{ ext{delta}}^+$

Answered a question of Mitchell regarding approachability

Provided a decisive answer to a question of Foreman and Shelah

Abstract

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