Alternative Definition of Ordinal

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


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.


Sufficient Condition

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

Let $a \in S$.


\(\ds a\) \(=\) \(\ds S \cap a\) $S$ is transitive, $a \subseteq S$; then apply Intersection with Subset is Subset
\(\ds \) \(=\) \(\ds \set {x \in S: x \in a}\)
\(\ds \) \(=\) \(\ds S_a\) Definition of Initial Segment

That is, $S$ is an ordinal.


Also see