# Cardinality of prime numbers

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?

• – lulu Feb 27 '20 at 11:34
• The mapping you're looking for is simply an enumeration of the prime numbers, for instance the one sending 1 to 2, 2 to 3, 3 to 5, and so on. – Ben Steffan Feb 27 '20 at 11:36
• Welcome to Mathematics Stack Exchange. Did you mean aleph naught? – J. W. Tanner Feb 27 '20 at 12:09
• I will point out that the function described by BenSteffan certainly exists. That it is difficult to compute is irrelevant. It doesn't bother us that we have a difficult time saying within a short amount of time what the four-billionth prime number is. It is obvious that there is a four-billionth prime number (and indeed an $n$'th prime number for any finite $n$) and that is all that we needed to convince ourselves of in regards to the validity of the existence of the described function. – JMoravitz Feb 27 '20 at 14:15

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 $$|A|=|\Bbb N|=\aleph_0$$.