Alternative Definition of Ordinal in Well-Founded Theory
Theorem
A set $S$ is an ordinal if and only if $S$ is transitive and $\forall x, y \in S: \paren {x \in y \lor x = y \lor y \in x}$.
Proof
Forward Implication
Let $S$ be an ordinal.
By Alternative Definition of Ordinal, $S$ is transitive and strictly well-ordered by the epsilon relation.
By Strict Well-Ordering is Strict Total Ordering, $S$ is strictly totally ordered by $\in$.
Thus:
- $\forall x, y \in S: \paren {x \in y \lor x = y \lor y \in x}$
$\Box$
Reverse Implication
Let $S$ be a transitive set such that for any $x, y \in S$, $x \in y \lor y \in x \lor x = y$.
We first show that $\in$ is a strict ordering of $S$.
Asymmetric: Let $x, y \in S$.
By Epsilon is Foundational, $\{ x,y \}$ has an $\Epsilon$-minimal element.
Thus $x \notin t$ or $y \notin x$.
Transitive: Let $x, y, z \in S$ with $x \in y$ and $y \in z$.
By assumption, $x = z$, $x \in z$, or $z \in x$.
Suppose for the sake of contradiction that $x = z$.
Then $x \in y$ and $y \in x$, contradicting the fact that $\Epsilon$ is asymmetric.
Suppose that $z \in x$.
Then $x \in y$, $y \in z$, and $z \in x$.
Thus the set $\set {x, y, z}$ has no $\Epsilon$-minimal element, contradicting Epsilon is Foundational.
Thus $x \in z$.
Thus $\in$ is a strict ordering of $S$.
Let $T$ be a non-empty subset of $S$.
By Epsilon is Foundational, $S$ has an $\Epsilon$-minimal element, $m$.
Since a minimal element of a strictly totally ordered set is the smallest element, $\Epsilon$ strictly well-orders $S$.
Thus by Alternative Definition of Ordinal, $S$ is an ordinal.
$\blacksquare$
Also see
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.3$, $\S 7.4$