Definition:Minimal Uncountable Well-Ordered Set

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

Then $\Omega$ is the minimal uncountable well-ordered set if every initial segment in $\Omega$ is countable.

Also denoted as
This set is sometimes denoted $S_\Omega$, matching the notation of initial segments.

Also known as
The set $\Omega$ is also known as the set of countable ordinals.

Also see

 * Existence of Minimal Uncountable Well-Ordered Set
 * Set of Countable Ordinals Unique up to Isomorphism, justifying the use of "the" in the definition.