Here's a question I'm having some trouble answering:
Say we have an ordinal $\omega^\alpha$ and suppose it has cofinality $\omega^\beta$, i.e., $\omega^\beta$ is the smallest order type of a cofinal subset. (Yes, it must be an initial ordinal; it's not clear to me whether that's relevant for this particular question.) Let $\gamma$ be such that $\beta\le\gamma\le\alpha$. The question is: Must $\omega^\alpha$ have a cofinal subset of order type exactly $\omega^\gamma$?
Notes: This is obviously false if we were to replace $\omega^\alpha$ with an ordinal that might not be a power of $\omega$.