# set theory – Countable Fodor’s Lemma?

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 $$.