Existence of Minimal Uncountable Well-Ordered Set

Theorem
There exists a minimal uncountable well-ordered set.

That is, there exists an uncountable well-ordered set $\Omega$ with the property that every initial segment in $\Omega$ is countable.

Corollary
The cardinality of $\Omega$ satisfies:


 * $\operatorname{card} \left({\N}\right) < \operatorname{card} \left({\Omega}\right) \le \mathfrak c$

where $\operatorname{card}\left({\N}\right)$ is the cardinality of the natural numbers and $\mathfrak c$ is the cardinality of the continuum.

Proof using choice
By the axiom of powers, there exists the power set $\mathcal P \left({\N}\right)$.

By Power Set of Natural Numbers Not Countable, this set is uncountable

By the well-ordering theorem, $\mathcal P \left({\N}\right)$ can be endowed with a well-ordering. Call such an ordering $\preccurlyeq$.

Let $\mathcal P \left({\N}\right)_a$ denote the initial segments of $\mathcal P \left({\N}\right)$ determined by $a \in P \left({\N}\right)$

Suppose $\left({P \left({\N}\right),\preccurlyeq}\right)$ has the property:


 * ''$\mathcal P \left({\N}\right)_a$ is countable for every $a \in \mathcal P \left({\N}\right)$

Then set $\Omega = \mathcal P\left({\N}\right)$.

Otherwise, suppose $\left({P \left({\N}\right),\preccurlyeq}\right)$ does not have the above property.'

Consider the subset of $\mathcal P\left({\N}\right)$


 * $P \subseteq \left\{ { a \in \mathcal P\left({\N}\right) : \mathcal P \left({\N}\right)_a \text{ is uncountable} } \right\}$

Then $P$ has a smallest element, by the definition of a well-ordered set. Call such an element $a_0$.

That is, $a_0 \in \mathcal P \left({\N}\right)$ is the smallest $a$ such that $\mathcal P \left({\N}\right)_{a_0}$ is uncountable.

Then the segment $\mathcal P \left({\N}\right)_{a_0}$ is itself uncountable, by virtue of $a_0$ being in $P$.

Thus every initial segment in $\mathcal P \left({\N}\right)_{a_0}$ is countable, because it is not uncountable.

Then set $\Omega = \mathcal P \left({\N}\right)_{a_0}$.

Proof without using choice
By the axiom of powers, there exists the power set $\mathcal P \left({\N}\right)$.

By Power Set of Natural Numbers Not Countable, this set is uncountable.

We construct a well-ordering $\left({P \left({\N}\right), \preccurlyeq}\right)$ that has the desired defining properties of $\Omega$.

Proof of Corollary
By the definition of $\Omega$ as a minimal uncountable well-ordered set:


 * $\operatorname{card} \left({\N}\right) < \operatorname{card} \left({\Omega}\right)$

by the definition of uncountable.

Furthermore:


 * $\operatorname{card} \left({\Omega}\right) \le \operatorname{card}\left({\mathcal P \left({\N}\right)}\right)$ follows the from construction of $\Omega$ in the main proof; $\Omega$ is a subset of $\mathcal P \left({\N}\right)$.

That $\operatorname{card} \left({\mathcal P \left({\N}\right) }\right) = \mathfrak c$ is showed in Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers.

Combining the above statements yields:


 * $\operatorname{card} \left({\N}\right) < \operatorname{card} \left({\Omega}\right) \le \mathfrak c$

Also see

 * Minimal Uncountable Well-Ordered Set Unique up to Isomorphism