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