On Shelah's Approachability Ideal

Abstract

We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[\hat{I}{º}^{+}]$. Given a singular cardinal ${\aleph }_{\hat{I}}^3$, a regular cardinal $\hat{I}¼\in (\mathrm{cf}({\hat{I}}^3),{\aleph }_{\hat{I}}^3)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which ${\aleph }_{{\hat{I}}^3+1}\cap \mathrm{cof}(\hat{I}¼)\notin I[{\aleph }_{{\hat{I}}^3+1}]$. 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.