Did Hardy prove that there are countably, or uncountably many zeros on the line Re$(s)=1/2$ of $\zeta(s)$?

Question

It's known that Hardy proved that there are infinitely many zeros of $\zeta(s)$ on the line Re$(s)=\frac{1}{2}$, but did he prove it's countably infinite? Or uncountable?

Answer 1

There's a nifty property of meromorphic functions: they have isolated zeros. Construct a ball of radius $\delta$ around each zero such that no two zeros of $\zeta$ belong to the same ball. The set comprising all these neighborhoods must have countably many connected components, since each open neighborhood contains $x+iy$ with $x$ and $y$ rational; thus each component has a rational representative, and there are countably many rational complex numbers.

Answer 2

By the identity theorem and the fact that uncountable subsets of $\mathbb{R}^n$ must have at least one limit point, any holomorphic function having uncountably many zeroes must vanish identically on its domain. $\zeta$ is holomorphic on a region excluding the origin, so it must not have uncountably many zeroes.