TITLE: Are the nontrivial zeroes of the Riemann zeta function countable?
QUESTION [13 upvotes]: It is known that the set of non trivial zeros is an infinite set. But is it known if it is a countable, or uncountable infinite set?

REPLY [41 votes]: If the set $Z$ of zeroes of $\zeta(s)$ were uncountable, then it would have an accumulation point. Now, by certain version of identity theorem, this implies that $\zeta(s)$ is identically zero on its domain, which is absurd.