# Alternative Definition of Ordinal

## 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

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