An ordinal $\alpha > 0$ is a limit ordinal if it is not a successor ordinal, i.e. there is no ordinal $\beta$ such that $\alpha = \beta + 1$.
Let $\alpha > 0$ be an ordinal. The following statements are equivalent:
$\alpha$ is a limit ordinal.
- \[\alpha = \sup\{\beta:\beta<\alpha\}.\]
- \[\alpha = \bigcup\alpha.\]
$\alpha$ has no maximum.
$\alpha$ is inductive.