Cofinality changes required for a large set of unapproachable ordinals below $\aleph _{\omega +1}$

Abstract:

In $V$, assume that $\aleph _{\omega }$ is a strong limit cardinal and $2^{\aleph _{\omega }}=\aleph _{\omega +1}$. Let $A$ be the set of approachable ordinals less than $\aleph _{\omega +1}$. An open question of M. Foreman is whether $A$ can be non-stationary in some $\aleph _{\omega }$ and $\aleph _{\omega +1}$ preserving extension of $V$. It is shown here that if $W$ is such an outer model, then ${\{ k<\omega :\text {cf}^{W}(\aleph ^{V}_{k})=\aleph ^{W}_{n} \}}$ is infinite, for each positive integer $n$.