Definition:Ordinal
Definition
Informal Definition
A natural number considered as an indication of the position in a sequence of objects is referred to as an ordinal number.
Let $\alpha$ be a set.
Definition 1
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
| \((1)\) | $:$ | $\alpha$ is a transitive set | |||||||
| \((2)\) | $:$ | $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$ |
where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.
Definition 2
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
| \((1)\) | $:$ | $\alpha$ is a transitive set | |||||||
| \((2)\) | $:$ | the epsilon relation is connected on $\alpha$: | \(\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x \) | ||||||
| \((3)\) | $:$ | $\alpha$ is well-founded. |
Definition 3
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 $\beta$:
- $\alpha_\beta = \set {x \in \alpha: x \prec \beta}$
Definition 4
$\alpha$ is an ordinal if and only if:
- $\alpha$ is an element of every superinductive class.
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
- Counting Theorem
- Ordering on Ordinal is Subset Relation
- Ordinal is Set of all Smaller Ordinals
- Definition:Ordinal Variable: a variable that can be ordered
- 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): number: 4.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ordinal number: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): number: 4.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordinal number: 2.