An ordinal $\alpha>0$ is a limit ordinal if it is not a successor ordinal, i.e. there is no $\beta$ 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.