Definition:Ordinal/Definition 3

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An ordinal is a strictly well-ordered set $\struct {S, \prec}$ such that:

$\forall a \in S: S_a = a$

where $S_a$ is the initial segment of $S$ determined by $a$.

From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that:

$S_a = \set {x \in S: x \subsetneqq a}$

From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.


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.

Also known as

An ordinal is also known as an ordinal number.

For a given well-ordered set $\struct {X, \preceq}$, the expression:

$\Ord X$

can be used to denote the unique ordinal which is order isomorphic to $\struct {X, \preceq}$.

Also see

$b \prec a$
$b \subsetneqq a$
$b \in a$

It is customary to denote the ordering relation between ordinals as $\le$ rather than $\subseteq$ or $\preceq$.

  • Results about ordinals can be found here.