# Definition:Minimal Uncountable Well-Ordered Set

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## 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**.

## 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.

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*$\S \text P.18$ - 2000: James R. Munkres:
*Topology*(2nd ed.) $\S 1.2$