# Power Set of Natural Numbers Not Countable

## Theorem

The power set $\mathcal P \left({\N}\right)$ of the natural numbers $\N$ is not countable.

## Proof

From Injection from Set to Power Set, we have that there exists an injection $f: \N \to \mathcal P \left({\N}\right)$.

From the Cantor-Bernstein-Schroeder Theorem, there can be no injection $g: \mathcal P \left({\N}\right) \to \N$.

So, by definition, $\mathcal P \left({\N}\right)$ is not countable.

$\blacksquare$