Definition:Ordinal/Definition 1
Definition
Let $\alpha$ be a set.
$\alpha$ 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 $\alpha$.
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
- 1980: Kenneth Kunen: Set Theory: An Introduction to Independence Proofs: Definition $\text{I}.7.2$
- 1977: Herbert B. Enderton: Elements of Set Theory: $\S 7.6$: Theorem $7 \text{L}$