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” where no quantifier in the definition of the powerset of x is relativized to U. (引用終り) 以上
