Cofinality

Remarks

Equivalently, $\operatorname{cf}(\alpha)$ is the least $\beta$ for which there is a function $f \colon \beta \to \alpha$ whose range is cofinal in $\alpha$. Example: $\operatorname{cf}(\aleph_\omega) = \omega$. Check: the set $\set{\aleph_n : n < \omega}$ has order type $\omega$ and is cofinal in $\aleph_\omega$, and no finite subset of a limit ordinal is cofinal. Non-example of a cofinal subset: a finite subset $A$ of $\omega$ is not cofinal, since $\max A + 1$ has no element of $A$ above it. For a successor ordinal $\alpha + 1$, the set $\set{\alpha}$ is cofinal, so $\operatorname{cf}(\alpha + 1) = 1$, and $\operatorname{cf}(0) = 0$. Always $\operatorname{cf}(\operatorname{cf}(\alpha)) = \operatorname{cf}(\alpha)$, and the cofinality of a limit ordinal is a regular cardinal. In ZFC, König's theorem gives $\operatorname{cf}(2^\kappa) > \kappa$ for each infinite cardinal $\kappa$.