Regular cardinal

Remarks

Example: in ZFC, every infinite successor cardinal $\kappa^+$ is regular. Check: if $\kappa^+$ were the union of fewer than $\kappa^+$ sets each of size less than $\kappa^+$, it would have size at most $\kappa \cdot \kappa = \kappa$, using the axiom of choice. Non-example: $\aleph_\omega$ is singular, since $\set{\aleph_n : n < \omega}$ is a cofinal subset of order type $\omega$. Equivalently, in ZFC, $\kappa$ is regular if and only if it is not the union of fewer than $\kappa$ sets each of cardinality less than $\kappa$. $\aleph_0$ is regular, and a limit cardinal $\aleph_\lambda$ with $\lambda$ a limit ordinal has cofinality $\operatorname{cf}(\lambda)$. A regular uncountable limit cardinal is called weakly inaccessible; its existence cannot be proved in ZFC, if ZFC is consistent.