Sartre said that the great philosophical question is, why is there something instead of nothing? (This does not mean, why is there something in philosophy. Why is there something in the real world? Why does anything physically exist?) If your philosophy doesn't explain the real world, it isn't much of a philosophy. But I don't think set theory can explain that.
Two other great philosophical questions are: Where do we humans find meaning? And, what is the basis for morals and ethics? How do we determine what actions are right and wrong? Set theory isn't going to answer those questions at all.
Two other great philosophical questions are: Where do we humans find meaning? And, what is the basis for morals and ethics? How do we determine what actions are right and wrong?
And Camus said that the one truly interesting philosophical problem is suicide.
It was a bit tongue-in-cheek, but I'd argue that mathematical foundations (maybe not set theory interpreted strictly) does take aim at answers to "why is there something?" and "where do humans find meaning?". Possibly by unifying the questions!
Only if you're biased towards a realist philosophical view. Though admittedly it makes many feel warm and fuzzy, there's really no evidence for it over positions like intuitionism.
You have no idea. The more I study this stuff the more I feel that all philosophical questions might be somehow encoded in there.