One of the famous applications of pcf theory is that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. I'm curious whether any weaker result with the same flavor can be gotten without pcf theory. It's a bit tricky to phrase this precisely, but here's one attempt: can the following result be proved in $\mathsf{ZFC}$ without pcf theory?
$(*)\quad$ Suppose $\aleph_\omega$ is a strong limit cardinal. Then there is some cardinal $\kappa$ such that for every cardinal-preserving set-generic extension $V[G]$ satisfying "$\aleph_\omega$ is a strong limit cardinal" we have $2^{\aleph_\omega}<\kappa$.
Certainly $(*)$ is implied by Shelah's result, since if $V$ and $V[G]$ have the same cardinals they agree on $\aleph_{\omega_4}$. But in principle it seems like it could be much weaker.
I'd also be interested in pcf-free proofs of further weakenings of $(*)$; to me, the most natural weakening seems to be gotten by restricting attention to some particularly nice class of forcings. (Basically, that "tame" forcings can't move $2^{\aleph_\omega}$ too much while keeping it a strong limit.)
