TITLE: Is there an axiom of ZFC expressing that GCH fails as badly as possible?
QUESTION [13 upvotes]: The GCH axiom basically says that for all infinite cardinal numbers $\kappa$, the number of cardinals lying strictly between $\kappa$ and $2^\kappa$ is as small as possible. Namely, there are none.
Is there an axiom which claims the opposite, in other words that the number of cardinal numbers lying strictly between $\kappa$ and $2^\kappa$ is as large (in some sense) as possible?
Edit. For example - and I don't know if this is a silly suggestions, I know very little set theory - is the following axiom for infinite cardinals $\kappa$ consistent with ZFC? And if so, is it interesting? $$|\{\mbox{cardinals } \nu \mid \kappa<\nu<2^\kappa\}|=2^\kappa$$

REPLY [4 votes]: Let me [partially] address your edit, as the original question was well addressed by Arthur in his answer.
We start with a model of $\sf ZFC+GCH$. Consider the function defined for regular $\kappa$ as $F(\kappa)=\min\{\lambda\mid\lambda=\aleph_\lambda\land\operatorname{cf}(\lambda)>\kappa\}$, then this function satisfies the requirements of Easton's theorem. Therefore there exists a model of set theory such that for every regular $\kappa$ it holds:$$2^\kappa=F(\kappa)=\aleph_{F(\kappa)}=\aleph_{2^\kappa}.$$
It's not hard to see that the gap between $\kappa$ and $2^\kappa$ contains $2^\kappa$ cardinals, and in fact the gap itself is unbounded.
On the axiom of choice side of events, Truss showed that if there exists $\alpha$ such that for every $X$, $\alpha$ cannot be embedded into the cardinals between $X$ and $2^X$, then the axiom of choice holds. That is, a bounded gap between a set and its power set is a strengthening of the axiom of choice, much like $\sf GCH$ is.