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 $\beta$:
- $\alpha_\beta = \set {x \in \alpha: x \prec \beta}$
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
- Equivalence of Definitions of Ordinal
- Initial Segment of Ordinal is Ordinal: $\alpha_\beta$ is itself an ordinal
- 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 $2.3$