personal theory: I think there's going to turn out to be a parallel development of math that is basically strictly finitist and never contends with the concept of an infinite set, much less the axiom of choice or any of its ilk. Which would require the foundation being something other than set theory. You basically do away with referring to the real numbers or the set of all natural numbers or anything like that, and skip all the parts of math that require them. I suspect that for any real-world purpose you basically don't lose anything. (This is a stance that I keep finding reinforced as a learn more math, but I don't really feel like I can defend it... it's a hunch I guess.)