I want to know the cardinality of the set of prime numbers. Is it aleph not? The cardinality of natural numbers and all countably infinite sets? But, how can we make a mapping of the set of prime numbers with the set of natural numbers? Is there such a mapping to impose the same cardinality on the set of primes as that of natural numbers?
Any infinite subset $A$ of the natural numbers $\Bbb N$ must have cardinality $\aleph_0$: being infinite implies there is an injection $\Bbb N\to A$, and since $A$ is a subset of $\Bbb N$, the identity map forms an injection $A\to \Bbb N$.
By the Cantor-Schröder-Bernstein theorem it follows that $|A|=|\Bbb N|=\aleph_0$.