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paper-machine comments on St. Petersburg Mugging Implies You Have Bounded Utility - Less Wrong
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Being a mathematician who at least considers himself mainstream, I would think that ZFC and the existence of a large cardinal is probably the minimum one would need to express a reasonable fragment of mathematics.
If you can't talk about the set of all subsets of the set of all subsets of the real numbers, I think analysis would become a bit... bondage and discipline.
Surely the power set axiom gets you that?
That it exists, yes. But what good is that without choice?
Ok, ZFC is a more convenient background theory than ZF (although I'm not sure where it becomes awkward to do without choice). That's still short of needing large cardinal axioms.