# Minimal Uncountable Well-Ordered Set Unique up to Isomorphism

## Theorem

Let $\Omega, \Omega'$ be minimal uncountable well-ordered sets.

Then $\Omega$ is order isomorphic to $\Omega'$.

That is, the minimal uncountable well-ordered set is unique up to order isomorphism.

## Proof

From Wosets are Isomorphic to Each Other or Initial Segments, precisely one of the following holds:

- $\Omega$ is order isomorphic to $\Omega'$

or:

- $\Omega$ is order isomorphic to an initial segment in $\Omega'$

or:

- $\Omega'$ is order isomorphic to an initial segment in $\Omega$.

By the definition of minimal uncountable well-ordered sets, $\Omega$ and $\Omega'$ are uncountable.

The initial segments of $\Omega$ and $\Omega'$ are countable.

An uncountable set can't be isomorphic to a countable set.

Thus $\Omega$ must be order isomorphic to $\Omega'$.

The result follows from the definition of uniqueness.

$\blacksquare$

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*$\S P.17$