Definition:Ordinal

Definition
An ordinal is a well-ordered set $S$ such that:
 * $\forall a \in S: S_a = a$

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

From the definition of an initial segment, and the fact that the ordering on an ordinal is the subset relation, we have that:


 * $S_a = \left\{{x \in S: x \subset a}\right\}$

From Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.

Hence we can define an ordinal $S$ as:
 * $S = \left\{{x: x \subset S}\right\}$

So we can define an ordinal as the set of all smaller ordinals.

It is customary to denote the ordering relation between ordinals as $\le$ rather than $\subseteq$.

Thus, $\forall a, b \in S$, the following statements are equivalent:
 * $b < a$
 * $b \subset a$
 * $b \in a$