TITLE: Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?
QUESTION [6 upvotes]: I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot be true for all limits, right?

REPLY [4 votes]: In fact, one can prove in ZFC that some infinite cardinals are not of form $2^\kappa$, regardless of whether the GCH holds or not. For example, $\aleph_\omega$ cannot be $2^\kappa$ for any $\kappa$, since it has cofinality $\omega$, and so $(\aleph_\omega)^\omega>\aleph_\omega$ by König's theorem, whereas $(2^\kappa)^\omega=2^{\kappa\cdot\omega}=2^\kappa$ for any infinite $\kappa$.