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:


 * $\card \N < \card \Omega \le \mathfrak c$

where $\card \N$ 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:


 * $\card \N < \card \Omega$

by the definition of uncountable.

Furthermore:


 * $\card \Omega \le \card {\powerset \N}$ follows the from construction of $\Omega$ in the main proofs; $\Omega$ is a subset of $\powerset \N$.

That $\card {\powerset \N} = \mathfrak c$ is showed in Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers.

Combining the above statements yields:


 * $\card \N < \card \Omega \le \mathfrak c$