Given that the cardinality of the naturals is the smallest infinite cardinal ( $|\mathbb{N}| = \aleph_0$ ), and the primes $\mathbb{P}$ are an infinite subset of the naturals $\mathbb{N}$, we know that that:
$$|\mathbb{P}| = |\mathbb{N}| = \aleph_0$$
However, Cantor's theorem states that for any set $A$ where $|A|=n$, its powerset has a greater cardinality; i.e., $|\mathcal{P}(A)| > n$. There is a simple injection from $\mathcal{P}(\mathbb{P})$ to $\mathbb{N}$: just multiply each set of primes together to get a unique natural (sorry, I'm not sure how to notate this). Therefore, it would seem that:
$$|\mathbb{P}| < |\mathcal{P}(\mathbb{P})|\le|\mathbb{N}| = \aleph_0$$
Where am I going wrong?