How do you show that the set of prime numbers is countably infinite?
Intuitively, I know that it should be countably infinite because a set of natural numbers is countably infinite. However, I don't know how to show it formally.
2026-04-24 02:26:10.1776997570
How to prove that the set of prime numbers is countably infinite?
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Given that the Prime Numbers are a subset of the Natural Numbers and (by definition) the latter are countably infinite, the Primes cannot be uncountably infinite; their cardinality must be less than or equal to that of $\mathbb{N}$. By Euclid's proof there are infinitely many primes, therefore there can only be countably infinitely many primes.