Existence of Minimal Uncountable Well-Ordered Set/Corollary 1

Corollary to Existence of Minimal Uncountable Well-Ordered Set
Let $\Omega$ denote the minimal uncountable well-ordered set.

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
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:


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