On Shelah's Approachability Ideal

Hannes Jakob; Alejandro Poveda

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arXiv · 2508.04374

On Shelah's Approachability Ideal

Hannes Jakob·Alejandro Poveda

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Abstract

We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[κ^+]$. Given a singular cardinal $\aleph_γ$, a regular cardinal $μ\in (\mathrm{cf}(γ),\aleph_γ)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which $\aleph_{γ+1} \cap \mathrm{cof}(μ) \notin I[\aleph_{γ+1}]$. This provides a definitive answer to a question of Shelah from the 80's. In addition, assuming large cardinals, we construct a model of $\mathsf{ZFC}$ in which the approachability property fails, simultaneously, at every singular cardinal. This is a major milestone in the solution of a question of Foreman and Magidor from the 80's.

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BibTeX RIS

Hannes Jakob, Alejandro Poveda. 2025-08-06. On Shelah's Approachability Ideal. https://arxiv.org/abs/2508.04374

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