# Definition:Ordinal

## Definition

### Informal Definition

A natural number considered as an indication of the position in a sequence of objects is referred to as an **ordinal number**.

Let $\alpha$ be a set.

### Definition 1

$\alpha$ 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 $\alpha$.

### Definition 2

$\alpha$ 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. |

### Definition 3

An **ordinal** is a strictly well-ordered set $\struct {\alpha, \prec}$ such that:

- $\forall \beta \in \alpha: \alpha_\beta = \beta$

where $\alpha_\beta$ is the initial segment of $\alpha$ determined by $\beta$:

- $\alpha_\beta = \set {x \in \alpha: x \subsetneqq \beta}$

### Definition 4

$\alpha$ is an **ordinal** if and only if:

- $\alpha$ is an element of every superinductive class.

## 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

- Equivalence of Definitions of Ordinal
- Counting Theorem
- Ordering on Ordinal is Subset Relation
- Ordinal is Set of all Smaller Ordinals

- Results about
**ordinals**can be found**here**.

## Historical Note

The concept of a **ordinal** was first introduced by Georg Cantor.

## Sources

- 1939: E.G. Phillips:
*A Course of Analysis*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.1$ Introduction - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**ordinal number**:**2.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**number**:**4.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**ordinal number**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**number**:**4.**