Let $\alpha$ and $\beta$ be ordinals. Is it true that $\mathrm{cf}(\beta)\le \mathrm{cf}(\alpha)$, if there is a function from $\alpha$ to $\beta$ with cofinal image?
2026-04-03 16:47:02.1775234822
Cofinality of an ordinal
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Note that for any ordinals $\alpha,\beta$, if $\alpha>\beta$ then there is a function from $\alpha$ to $\beta$ with cofinal image (why?). Now, can you think of a pair of ordinals $\alpha>\beta$ with $cf(\alpha)<cf(\beta)$?