# Definition:Ordinal/Definition 3

## Definition

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:

- $\Ord X$

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.

## Sources

- 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