Let $A\_k$ be the set of all subsets of $P(\mathbb{N})$ with size $k$.

$A\_k\in\mathbb{N}^k$ is countable for $k\in\mathbb{N}$.

Since $$P(\mathbb{N})=\bigcup\_{k=1}^\infty A\_k$$
which is a countable union of countable sets, $P(\mathbb{N})$ must also be countable.

However, $P(\mathbb{N})$ is known to be uncountable. Which step of the proof is wrong?