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.

💡 Research Summary

This paper, “Total Failure of Approachability at Successors of Singulars of Countable Cofinality” by Hannes Jakob, establishes a consistency result in set theory concerning the approachability property at successors of singular cardinals. Assuming the existence of a proper class of supercompact cardinals, the author constructs a model of ZFC + GCH where, for every singular cardinal δ with countable cofinality and every regular uncountable cardinal μ < δ, the set E_{δ⁺}^{μ} (points of cofinality μ in δ⁺) is not a member of the approachability ideal I