Math people:

It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. Cantor's diagonal proof that the real numbers are uncountable involves showing that there is no surjection from $\mathbb{N}$ to $(0,1)$. So do I need the Axiom of Choice to accept Cantor's Diagonal Proof?

I browsed the Similar Questions and I could not find an answer. I apologize if this is a duplicate.

StEFAN (Stack Exchange FAN)