Minimal Uncountable Well-Ordered Set Unique up to Isomorphism

From ProofWiki
Jump to navigation Jump to search

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 set, $\Omega$ and $\Omega'$ are uncountable.

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

An uncountable set cannot be order isomorphic to a countable set.



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

The result follows from the definition of uniqueness.

$\blacksquare$


Sources