And honestly, the story isn't over. We brought axioms into set theory after that, but Godel showed that that was never going to be a cure-all, and people like Woodin later on have added to the pile. At this point, you can have two totally reasonable axioms which don't just prove different things, but actually can prove opposite answers about the same thing.

I think it's fair to say even platonism is starting to look a bit threatened at this point, and there's people (the Sydney school) who want to go back to looking at math as descriptive rather than ideal. Finitism is also worth a look, I think, and avoids things like Russel's paradox easily, although interestingly MIP*=RE implies that there may be directly measurable infinities in quantum mechanics.