set theory - Does the Continuum Hypothesis hold for sets of a certain rank?

Adding to the previous answers: Note that $V_{aleph_omega}$ is not a model of ZFC.
Hence, what holds or not in $V_{aleph_omega}$ does not formally tell you what follows from ZFC and what doesn't. This issue aside, as mentioned above, $V_{aleph_omega}$ knows whether or not CH holds in the real world.

What Gödel did to show the consistency of CH was to construct a "narrow" class $L$ inside a model of ZFC (resp. ZF if you also want to also prove the consistency of AC with ZF) such that
$L$ satisfies ZFC+CH. The point here is that even if CH fails in the large model of set theory, $L$ contains so few subsets of $omega$ that CH actually holds (roughly).

So, L is narrow, while $V_{aleph_omega}$ is short and wide (it contains all set up to a certain rank). A short wide structure (as long it is long
enough) is correct about CH, a narrow structure might not be.