Did I make any incorrect assumptions in my proof?

Question

Prove that the set of prime numbers is countable. My proof: Let P denote the set of prime numbers. We know that $P\subseteq \mathbb{Z}$, which means that $|P|\leq |\mathbb{Z}|$. But since $\mathbb{N}\subseteq \mathbb{Z}$, $|\mathbb{N}|\leq |\mathbb{Z}|$. The cardinality of $\mathbb{N}$ is $|\mathbb{N}|=\aleph_0$, and we know that $\aleph_0+\aleph_0=\aleph_0$, $\aleph_0*\aleph_0=\aleph_0$, and $\aleph_0+k=\aleph_0$ for some $k\in \mathbb{N}$, so $|\mathbb{Z}|=\aleph_0$. Therefore, since $\mathbb{N}$ is countably infinite, $\mathbb{Z}$ is countably infinite. Since $|\mathbb{Z}|=\aleph_0$, the cardinality of the set of prime numbers is $|P|=\aleph_0$. These results imply that $|P|=|\mathbb{N}|$. Therefore, there is a bijection $f:P\rightarrow \mathbb{N}$. Hence, the set of prime numbers is countable and infinite.

Answer 1

You way overcomplicated the problem. There was no need to consider $|\mathbb Z|$ at all. And the stuff you were doing with cardinal arithmetic was quite unnecessary.

Note that if there exists an injection $f:X\to Y$, then $|X|\leq |Y|$. But, there's an obvious injection from $P\to \mathbb N$, the identity injection.

So, the set of primes is countable. (If your definition of countable also implies infinite, simply refer to Euclid's proof of the existence of infinitely many primes. I don't know how you concluded that $P$ is not finite via cardinal arithmetic).