Power Set of Natural Numbers is Uncountable

Theorem
The power set $\powerset \N$ of the natural numbers $\N$ is not countable.

Proof
There is no bijection from a set to its power set.

From Injection from Set to Power Set, we have that there exists an injection $f: \N \to \powerset \N$.

From the Cantor-Bernstein-Schröder Theorem, there can be no injection $g: \powerset \N \to \N$.

So, by definition, $\powerset \N$ is not countable.