The continuum hypothesis asserts that $\aleph_{1}=\beth_{1}$. Both it and its negation can be consistent with ZFC, if ZFC is consistent itself.
The generalised continuum hypothesis asserts that $\aleph_{\alpha}=\beth_{\alpha}$ for all ordinals $\alpha$. Similarly, it and its negation can be consistent with ZFC.
Say we have an "nth continuum hypothesis", which asserts $\aleph_{\alpha}=\beth_{\alpha}$ for $\alpha\leq n$ and $\aleph_{\alpha}<\beth_{\alpha}$ for $\alpha>n$. Could any of these (and presumably their negations) be consistent with ZFC? Furthermore, would it work for $\alpha$ being a limit ordinal, or would we have to restrict $\alpha$ to successor ordinals to make these to meaningfully work?
