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RichardKennaway comments on St. Petersburg Mugging Implies You Have Bounded Utility - Less Wrong
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Limited comprehension in ZF set theory is the example I had in mind in coining the term "principled hack". Russell said to Frege, "what about the set of sets not members of themselves?", whereupon Frege was embarrassed, and eventually a way was found of limiting self-reference enough to avoid the contradiction. There's a principle there -- unrestricted self-reference can't be done -- but all the methods of limiting self-reference that have yet been devised look like hacks. They work, though. ZF appears to be consistent, and all of mathematics can be expressed in it. As a universal language, it completely solves the problem of formalising mathematics.
(I am aware that there are mathematicians who would disagree with that triumphalist claim, but as far as I know none of them are mainstream.)
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.
The idea of programming ZF into an AGI horrifies my aesthetics, but that is no reason not to use it (well it is an indication that it might not be a good idea but in this specific case ZF does have the evidence on its side). If expected utility, or anything else necessary for an AGI, could benefit from a principled hack as well-tested as limited comprehension, I would accept it.