# Definition:Ordinal/Definition 2

## Definition

Let $\alpha$ be a set.

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

 $(1)$ $:$ $\alpha$ is a transitive set $(2)$ $:$ the epsilon relation is connected on $\alpha$: $\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x$ $(3)$ $:$ $\alpha$ is well-founded.

## Notation

The class of all ordinals can be found denoted $\On$.

In order to indicate that a set $S$ is an ordinal, this notation is often seen:

$\Ord S$

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.

A tradition has grown up in set theory to use lowercase Greek letters $\alpha, \ \beta, \ \gamma, \ldots$ as symbols to represent variables over ordinals.

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

## Also known as

An ordinal is also known as an ordinal number.

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

$\map {\mathrm {Ord} } S$

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

## Also see

• Results about ordinals can be found here.