A note on a question of Woodin

A question of Woodin from the 1980s asks, assuming there is no inner model of ZFC with a strong cardinal, if it is possible for there to be a model $M$ of ZFC such that $M \vDash “2^{\aleph _\omega } \gt \aleph _{\omega + 2}$ and $2^{\aleph _n} = \aleph _{n + 1}$ for every $n \lt \omega $”, together with the existence of an inner model $N^* \subseteq M$ of ZFC such that for the $\gamma , \delta $ satisfying $\gamma = (\aleph _\omega )^M$ and $\delta = (\aleph _{\omega + 3})^M$, $N^* \vDash “\gamma $ is measurable and $2^\gamma \ge \delta $”. We show that this is the case for a choiceless version of Woodin’s question, where we assume AC fails in $M$ but holds in $N^*$. We also prove analogous results for $\aleph _{\omega _1}$ and $\aleph _{\omega _2}$. The methods used allow for equiconsistencies in certain cases.\looseness -1