Possible cardinality of power sets 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?

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. 
