Definition:Ordinal/Definition 1

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Definition

Let $S$ be a set.

Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.


Then $S$ is an ordinal if and only if:

$S$ is a transitive set
$\Epsilon \! \restriction_S$ strictly well-orders $S$.


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.


Sources