Failure of Approachability at the Successor of the first Singular for any Cofinality

Abstract

We solve two long-standing open problems regarding the combinatorics of ${\aleph }_{\omega +1}$. We answer a question of Shelah by showing that it is consistent for any $n\ge 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points of cofinality ${\aleph }_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality ${\aleph }_1$, answering an open question of Cummings, Foreman, and Magidor.

22 pages