# Definition:Ordinal/Definition 3

## Definition

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 $\beta$:

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

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From Initial Segment of Ordinal is Ordinal we have that $\alpha_\beta$ 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.

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

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- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Exercise $2.3$