Minimal Uncountable Well-Ordered Set Unique up to Isomorphism
(Redirected from Set of Countable Ordinals 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:
- $\Omega$ 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.
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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 $\S \text P.17$