Definition:Ordinal/Definition 3

Definition
An ordinal is a strictly well-ordered set $\struct {S, \prec}$ 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 Ordering on Ordinal is Subset Relation, we have that:


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

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

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

For a given well-ordered set $\struct {X, \preceq}$, the expression:
 * $\Ord X$

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

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 $\preceq$.