Alternative Definition of Ordinal

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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:

\(\ds a\)

\(=\)

\(\ds S \cap a\)

$S$ 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.

$\blacksquare$

Also see

• Alternative Definition of Ordinal in Well-Founded Theory

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.3$, $\S 7.4$