Definition:Ordinal/Definition 2

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
\((2)\)	$:$		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.

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

• Results about ordinals can be found here.

Sources

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Work In Progress In particular: S&F attribute this to Raphael Robinson but do not cite a source; it's not clear whether he admitted On as an ordinal. I don't understand, S&F is a source. We don't demand that every source we cite cites a source. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.