Definition:Ordinal/Definition 3

Definition
Let $\left({S, \preceq}\right)$ be a well-ordered set.

Then $\left({S, \preceq}\right)$ is an ordinal iff:
 * $\forall a \in S: S_a = a$

where $S_a$ is the $\preceq$-initial segment of $S$ determined by $a$.