Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.

But the conjecture is that $\omega_4$ can be provably replaced by $\omega_1$. Namely, $2^{\aleph_\omega}<\aleph_{\omega_1}$ holds, assuming that $\aleph_\omega$ is a strong limit cardinal.

As far as I understand it, we know that from large cardinal assumptions it is consistent that $2^{\aleph_\omega}$ is arbitrarily large below $\aleph_{\omega_1}$ (and it is a strong limit, of course). But there is no current way to go beyond $\aleph_{\omega_1}$. Even Gitik's work on the subject does not translate to the $\aleph_n$'s.

Question.Suppose that the PCF Conjecture fails. Namely, $\aleph_\omega$ is a strong limit cardinal, but $2^{\aleph_\omega}>\aleph_{\omega_1}$. What kind of large cardinals can we expect to find in inner models?

(Of course large cardinals are necessary, since $2^{\aleph_\omega}>\aleph_{\omega+1}$ with $\aleph_\omega$ as a strong limit was shown by Gitik to be equiconsistent with the existence of a measurable $\kappa$ of Mitchell order $\kappa^{++}$.)

notthat. $\endgroup$ – Asaf Karagila Feb 11 at 18:02