What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory

This paper explores the set theoretic assumptions used in the current published proof of Fermat’s Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. 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 universes.1 On ZFC foundations a universe is an uncountable transitive set U such that 〈U,∈〉 satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of U to U the range is also an element of U. This is much stronger than merely saying 〈U,∈〉 satisfies the ZFC axioms. We do not merely say the powerset axiom “every set has a powerset” is true with all quantifiers relativized to U. Rather, we require “for every set x ∈ U , the powerset of x is also in U” Received August 16, 2009. I thank Jeremy Avigad, Angus Macintyre, Barry Mazur, and an anonymous referee for advice, which is certainly not to say any of them agrees with everything here. See Grothendieck [1971] and the fuller account Artin et al., [1972, vol. I, pp. 185–217]. We abbreviate these books as SGA 1 and SGA 4 respectively. c © 0000, Association for Symbolic Logic 1079-8986/00/0000-0000/$00.00