There are a variety of stronger axiomatic systems (typically ZFC + some other axioms). Some of them prove the continuum hypothesis (which is what I stated), oth

There are a variety of stronger axiomatic systems (typically ZFC + some other axioms). Some of them prove the continuum hypothesis (which is what I stated), others prove the negation. There's no way to decide which one is "right", though, just like there's no way to tell if the Axiom of Choice is "right". Your second paragraph is confused. Given an axiom system like ZFC, there are (a) statements that can be proved true or false using it, (b) statements that can't be proved that are true in a particular model, (c) statements that can't be proved that are false in that model. It's set (b) that the incompleteness theorem tells us must exist. The theorem doesn't "find" statements, it proves that (b) exists by constructing a particular one, which is necessarily meta because it applies to every formal system. However, we do not have access to a model which tells us the answers for things like the CH. So all you can do is decide on the axiom scheme you like, and then some things are provable. You can always add more axioms, like CH, but you can add their negation instead, if you want. So there's no sense in which there's really a right answer for CH but we haven't found it yet.