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 $a$.

That is, the strict well-ordering on $S$ is given by the $\in$-relation.

From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that:


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

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

Hence we can define an ordinal $S$ as:


 * $S = \left\{{x: x \subsetneqq S}\right\}$

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:
 * $b < a$
 * $b \subsetneqq a$
 * $b \in a$

Notation
The class of all ordinals is denoted $\operatorname{On}$.

In order to indicate that a set $S$ is an ordinal, this notation is often seen:
 * $\operatorname{Ord} S$

whose meaning is:
 * $S$ is an ordinal.

Thus $\operatorname{Ord}$ can be used as a propositional function whose domain is the class of all sets.

Also known as
An ordinal is also known as an ordinal number.