Definition:Ordinal

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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 $a$.

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

Hence we can define an ordinal $S$ as:

$S = \left\{{x: x \subset S}\right\}$

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:

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.