Alternative Definition of Ordinal

 It has been suggested that this page or section be merged into Equivalence of Definitions of Ordinal. (Discuss)

Theorem

A set $S$ is an ordinal if and only if $S$ is transitive and is strictly well-ordered by the $\in$-relation.

Proof

Necessary Condition

Suppose that $S$ is an ordinal.

Then $S$ is transitive.

By definition, the strict well-ordering on $S$ is given by the $\in$-relation.

Hence, the necessary condition is satisfied.

$\Box$

Sufficient Condition

Suppose that $S$ is a transitive set that is strictly well-ordered by the $\in$-relation.

Let $a \in S$. Then:

 $\displaystyle a$ $=$ $\displaystyle S \cap a$ $S$ is transitive, $a \subseteq S$; then apply Intersection with Subset is Subset $\displaystyle$ $=$ $\displaystyle \left\{ {x \in S: x \in a} \right\}$ $\displaystyle$ $=$ $\displaystyle S_a$ Definition of Initial Segment

That is, $S$ is an ordinal.

$\blacksquare$