Definition:Ordinal

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Definition

Definition 1

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 $S$.


Definition 2

Let $A$ be a set.

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

transitive
epsilon-connected, that is:
$\forall x, y \in A: x \ne y \implies x \in y \lor y \in x$
well-founded


Definition 3

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 $\struct {X, \preceq}$, the expression:

$\map {\operatorname {Ord} } X$

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.


Sources