Definition:Ordinal/Definition 3

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Definition

An ordinal is a strictly well-ordered set $\left({S, \prec}\right)$ 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:

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


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


Notation

The class of all ordinals can be found denoted $\operatorname{On}$.

In order to indicate that a set $S$ is an ordinal, this notation is often seen:

$\operatorname{Ord} S$

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 $\left({X, \preceq}\right)$, the expression:

$\operatorname{Ord} \left({X}\right)$

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


Also see

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


  • Results about ordinals can be found here.


Sources