What does it take to prove fermat's last theorem?

Abstract
Does the proof of Fermat’s Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by “proof ” and “use,” and are not entirely known. This paper surveys the current state of these questions and briefly sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles [1995] plus improvements that do not yet change its character. Far from self-contained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are “used in fact in the published proof.” It is currently unknown what assumptions are “used in principle” in the sense of being proof-theoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck uni- verses.1 On ZFC foundations a universe is an uncountable transitive set U..
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