# Definition:Ordinal

(Redirected from Definition:Ordinal Number)

## 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

• Results about ordinals can be found here.

## Historical Note

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