# Definition:Ordinal

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## Definition

### Definition 1

Let $S$ be a set.

Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.

Then $S$ is an **ordinal** if and only if:

- $S$ is a transitive set
- $\Epsilon \! \restriction_S$ strictly well-orders $S$.

### Definition 2

Let $A$ be a set.

Then $A$ is an **ordinal** if and only if $A$ is:

### Definition 3

An **ordinal** is a strictly well-ordered set $\struct {S, \prec}$ 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 Ordering on Ordinal is Subset Relation, we have that:

- $S_a = \set {x \in S: x \subsetneqq a}$

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

## 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.

According to 1993: Keith Devlin: *The Joy of Sets: Fundamentals of Contemporary Set Theory* (2nd ed.), it is common practice in set theory to use lowercase Greek letters $\alpha, \ \beta, \ \gamma, \ldots$ for **ordinals**.

## Also known as

An **ordinal** is also known as an **ordinal number**.

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

- $\map {\operatorname {Ord} } X$

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

## Also see

- Ordering on Ordinal is Subset Relation where it is shown that $\forall a, b \in S$, the following statements are equivalent:

- $b \prec a$
- $b \subsetneqq a$
- $b \in a$

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

- 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.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**ordinal number**:**2.**