As I understand it, the Banach-Tarski construction depends on the axiom of choice. And Banach-Tarski violates my intuitions so thoroughly that I think I might prefer to dispense with the axiom of choice than accept B-T.
If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up?
If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up?