# 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 and only 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.