On Shelah's Approachability Ideal

TL;DR

This paper resolves a long-standing open problem by Shelah regarding the approachability ideal at singular cardinals, demonstrating models where certain sets are not in the ideal and where the approachability property fails universally.

Contribution

It constructs models of ZFC with specific properties of the approachability ideal at singular cardinals, answering key questions from Shelah, Foreman, and Magidor.

Findings

Constructed a model where a specific cofinal subset is not in the approachability ideal.

Established the failure of the approachability property at all singular cardinals under large cardinal assumptions.

Provided a definitive answer to Shelah's 1980s question about the approachability ideal.

Abstract

We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[\kappa^+]$. Given a singular cardinal $\aleph_\gamma$, a regular cardinal $\mu\in (\mathrm{cf}(\gamma),\aleph_\gamma)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which $\aleph_{\gamma+1} \cap \mathrm{cof}(\mu) \notin I[\aleph_{\gamma+1}]$. 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.