Does $\operatorname{cf}(\alpha)=\operatorname{cf}(\beta)$ if there is a monotone cofinal function $\alpha\to\beta$?

Question

It is a theorem in Kunen's Set Theory that if $\alpha$ is a limit ordinal and $\beta$ an ordinal such that there is a strictly increasing cofinal function $\alpha\to\beta$, then $\operatorname{cf}(\alpha)=\operatorname{cf}(\beta)$. I am wondering what can be said if we weaken the hypothesis to there existing a monotone increasing cofinal function $\alpha\to\beta$. Can we still conclude that $\operatorname{cf}(\alpha)=\operatorname{cf}(\beta)$?

Answer 1

I think you need to assume $\beta$ is limit.

Let $f: \operatorname{cf}(\alpha) \to \beta$ be increasing and cofinal. If $\operatorname{cf}(\beta) \lt \operatorname{cf}(\alpha)$, define $g:\operatorname{cf}(\beta) \to \operatorname{cf}(\alpha)$ as follows:

$g(\xi)$ is the smallest $\eta$ such that $f(\eta) \gt h(\xi)$, where $h: \operatorname{cf}(\beta) \to \beta$ is cofinal. Then $f \circ g: \operatorname{cf} (\beta) \to \alpha$ is cofinal in $\alpha$, which is a contradiction.