Definition:Ordinal

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.

According to, it is common practice in set theory to use lowercase Greek letters $\alpha, \, \beta, \, \gamma, \ldots$ for ordinals.

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

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

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

Also see

 * Equivalence of Definitions of Ordinal


 * Woset is Isomorphic to Unique Ordinal


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


 * Ordinal is Set of all Smaller Ordinals

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