# Alternative Definition of Ordinal

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Equivalence of Definitions of Ordinal.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## 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

## Sources

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