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 segment of $$S$$ determined by $$a$$.

From the definition of a 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:
 * $$b < a$$;
 * $$b \subset a$$;
 * $$b \in a$$.