As I understand it, most math is done implicitly within $ZFC$, but sometimes stronger systems are used--for instance, the initial proof of Fermat's Last Theorem used Grothendieck universes, which imply the existence of large cardinals.

I am curious how powerful axioms you could use before a proof of some $\Pi_1$ statement (so that only consistency, not soundness, need be considered) using them wouldn't be generally accepted by the math community. If there were a proof of, say, the Riemann Hypothesis in $ZFC+X$, how weak would $X$ need to be for the authors to get the Millenium Prize money?


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