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:

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

That is, $S$ is an ordinal.

$\blacksquare$


Also see


Sources