Definition:Minimal Uncountable Well-Ordered Set

Definition
Let $\Omega$ be an uncountable well-ordered set.

Then $\Omega$ is the set of countable ordinals if every initial segment in $\Omega$ is countable.

Also see

 * Definition:Ordinal
 * Existence of Set of Countable Ordinals
 * Set of Countable Ordinals Unique up to Isomorphism