I am looking for an answer to the question:
"Show there exists an infinite cardinal $\kappa$ with $2^{cf(\kappa)}$ < $\kappa$ "
Where $cf(\kappa)$ is defined as the least $\alpha$ such that there is a map from $\alpha$ cofinally into $\beta$ and if f: $\alpha$ $\rightarrow$ $\beta$, f maps $\alpha$ cofinally iff ran(f) is unbounded in $\beta$.
I have looked in both Kunen's book and Jech's, but although there's lots of helpful things about the topic I can't quite get to the answer.
I know that $\kappa$ cannot be regular for the inequality to hold. The question is from a past exam, and the examiner's report indicates there's quite an easy proof but I can't seem to find it.
On a related note, is there a simplified expression for $cf(\kappa)^{cf(\kappa)}$?