Definition:Ordinal

Definition
An ordinal is a well-ordered set $\left({S, \preceq}\right)$ 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.

Notation
The class of all ordinals can be found 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.

For a given well-ordered set $\left({X, \preceq}\right)$, the expression:
 * $\operatorname{Ord} \left({X}\right)$

can be used to denote the unique ordinal which is order isomorphic to $\left({X, \preceq}\right)$.

Also see

 * Ordering on Ordinal is Subset Relation where it is shown that $\forall a, b \in S$, the following statements are equivalent:
 * $b \prec a$
 * $b \subsetneqq a$
 * $b \in a$

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