Ordinal is Transitive/Proof 1
Jump to navigation
Jump to search
Theorem
Every ordinal is a transitive set.
Proof
Let $\alpha$ be an ordinal by Definition 1:
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
\((1)\) | $:$ | $\alpha$ is a transitive set | |||||||
\((2)\) | $:$ | $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$ |
where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.
Thus $\alpha$ is a priori transitive.
$\blacksquare$