Does Fodor’s lemma fail for countable ordinals?
For the usual statement of Fodor’s lemma to make sense, one needs well-behaved notions of club and stationary sets, which fail for countable ordinals, so let me be more precise.
Question: Let $alpha$ be a countable limit ordinal. Does there exist a weakly increasing, regressive function $f: alpha to alpha$ such that the image of $f$ is cofinal in $alpha$?
Here “regressive” means “$f(beta) < beta$ for all $0 < beta < alpha$“. I’m also interested in the version of the question where $f$ is defined only on limit ordinals $<alpha$.