Alternative Definition of Ordinal
 It has been suggested that this page or section be merged into Equivalence of Definitions of Ordinal. (Discuss) |
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: $\S 7.3$, $\S 7.4$
