Possible cardinality of power sets

Question

What cardinals can provably not occur as the cardinality of a power set? I know that $\mathbb N$ and natural numbers that are not powers of two are such cardinals. What else is out there?

Answer 1

The Konig theorem says that $$\kappa < cf(2^{\kappa})$$ Thus if $cf(\lambda)=\omega$, such as $\aleph_{\omega}$ then $\lambda$ is not a power set.

Answer 2

Let $S$ be an infinite set of cardinals and let $x=\cup_{y\in S}2^y.$ Then $x$ is a strong limit cardinal: That is, $\forall z<x\;(2^z<x).$ So $x$ cannot be a power cardinal.