Minimal Uncountable Well-Ordered Set Unique up to Isomorphism

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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:

or:

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.

$\blacksquare$


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