# Definition:Ordinal/Definition 1

## 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 | |||||||

\((2)\) | $:$ | $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$ |

where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.

## 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**.

## Sources

- 1980: Kenneth Kunen:
*Set Theory: An Introduction to Independence Proofs*: Definition $\text{I}.7.2$ - 1977: Herbert B. Enderton:
*Elements of Set Theory*: $\S 7.6$: Theorem $7 \text{L}$