Bind all lowercase greek letters to cardinal numbers.
Question. Is it true that for all infinite cardinal numbers $\nu,$ we have that $\{\kappa \mid \kappa^\nu > \kappa\}$ is unbounded?
For example, is $\{\kappa \mid \kappa^{\aleph_0} > \kappa\}$ unbounded?
A bit of conceptualizing. The function $\kappa \mapsto \kappa^{\aleph_0}$ of "raising to the power of ${\aleph_0}$" is a closure operator. Furthermore, it clearly has arbitrarily large fixed points; in particular, observe that for all cardinal numbers $\gamma$, we have that $\gamma^{\aleph_0}$ is a fixed point of the aforementioned function. What I'd like to know is, does it have arbitrarily large non-fixed points? More generall, does $\kappa \mapsto \kappa^{\nu}$ have arbitrarily large non-fixed points?
If $\kappa$ is a strong limit cardinal and $\text{cf}(\kappa) \leq \lambda < \kappa$, then $\kappa^\lambda = \kappa^{\text{cf}(\kappa)} \geq \kappa^+$. There are unbounded many strong limit cardinals of a particular cofinality.