I'm not really that familiar with AC, I've just started talking about it in my classes. But from what I understand, one of its formulations is that it is possible to create a set made from choosing elements from a series (potentially infinite) of other sets. Are we doing this in the Diagonal argument? We have a countably infinite number of sets (each with a real number) and we create a different one by choosing a digit from each of the subsets. Is this correct? Or am I misunderstanding AC, making some other error?
EDIT: It seems that someone asked which axioms of ZF did the argument use and it got answered. My question then becomes: Why doesn't Cantor's Diagonal argument depend on the Axiom of Choice?