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Sniffnoy comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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No they don't (*). Your saying this explicitly somewhat confirms my brain's natural, automatic assumption that your error here (and in similar comments in the past -- "infinite set atheism" and all that business) is as much sociological as philosophical: all along, I instinctively thought, "he doesn't seem to realize that that's a low-status position".
ZFC is considered the standard axiom system of modern mathematics. I have no doubt that if an international body (say, the IMU) were to take a vote and choose a set of "official rules of mathematics", the way (say) FIDE decides on the official rules of chess, they would pick ZFC (or something equivalent).
Now it's true, there are some mathematicians who are contrarians and think that AC is somehow "wrong". They are philosophically confused, of course; but, more to the point here in this comment, they are a marginal group. (In fact, even worrying about foundational issues too much -- whatever your "position" -- is kind of a low-status marker itself: the sociological reality of the mathematical profession is that members are expected to get on with the business of proving impressive-looking new theorems in mainstream, high-status fields, and not to spend time fussing about foundations except at dinner parties.)
See also this comment of mine.
(*) I don't know the numbers, or how you define "lots", and there are a large number of mathematicians in the world, so technically I don't know if it's literally false that "lots" of mathematicians would say that they "reject AC" . But the clear implication of the statement -- that constructivism is a mainstream stance -- most definitely is false.
And anecdotally it seems that the AC skepticism that does exist seems to largely come from constructivism, so if we rule out that (since it doesn't seem that Eliezer wants to go all constructivist on us :) ), it's even less so.
I'm not sure what you mean by "constructivism" here; I usually hear that term referring to doubting the law of excluded middle (when applied to statements quantified over infinite sets), but I know several mathematicians who doubt the axiom of choice without doubting excluded middle.
I should also clarify the difference between doubting AC and denying AC. If you deny AC, then you believe that it is false, and hence any theorem whose only known proofs use AC is no theorem at all; it might be true, but it has not been proved. (And if AC follows from it, then it must in fact be false.) If you only doubt AC, however, then you simply believe that a theorem with a proof that uses AC is a weaker result than the same theorem with a proof that doesn't, and so the former theorem is still worth publishing but the latter is naturally preferred.
This seems such an obvious position to me that I doubt everything in mathematics (although there is a core which I generally assume since mathematics without it seems uninteresting (although I'm open to being proved wrong about this)).