Limit Ordinal

Definition 1. An ordinal $\alpha > 0$ is a limit ordinal if it is not a successor ordinal, i.e. there is no ordinal $\beta$ such that $\alpha = \beta + 1$.

Proposition 1. Let $\alpha > 0$ be an ordinal. The following statements are equivalent:

1. $\alpha$ is a limit ordinal.

2. \[\alpha = \sup\{\beta:\beta<\alpha\}.\]
3. \[\alpha = \bigcup\alpha.\]

4. $\alpha$ has no maximum.

5. $\alpha$ is inductive.