Inaccessible cardinals and Andrew Wiles's proof

The basic contention here is that Wiles' work uses cohomology of sheaves on certain Grothendieck topologies, the general theory of which was first developed in Grothendieck's SGAIV and which requires the existence of an uncountable Grothendieck universe. It has since been clarified that the existence of such a thing is equivalent to the existence of an inaccessible cardinal, and that this existence -- or even the consistency of the existence of an inaccessible cardinal! -- cannot be proved from ZFC alone, assuming that ZFC is consistent.

Many working arithmetic and algebraic geometers however take it as an article of faith that in any use of Grothendieck cohomology theories to solve a "reasonable problem", the appeal to the universe axiom can be bypassed. Doubtless this faith is predicated on abetted by the fact that most arithmetic and algebraic geometers (present company included) are not really conversant or willing to wade into the intricacies of set theory. I have not really thought about such things myself so have no independent opinion, but I have heard from one or two mathematicians that I respect that removing this set-theoretic edifice is not as straightforward as one might think. (Added: here I meant removing it from general constructions, not just from applications to some particular number-theoretic result. And I wasn't thinking solely about the small etale site -- see e.g. the comments on crystalline stuff below.)

Here is an article which gives more details on the matter.

http://www.cwru.edu/artsci/phil/Proving_FLT.pdf

Note that I do not necessarily endorse the claims in this article, although I think there is something to the idea that written work by number theorists and algebraic geometers usually does not discuss what set-theoretic assumptions are necessary for the results to hold, so that when a generic mathematician tries to trace this back through standard references, there may seem to be at least an apparent dependency on Grothendieck universes.

P.S.: If a mathematician of the caliber of Y.I. Manin made a point of asking in public whether the proof of the Weil conjectures depends in some essential way on inaccessible cardinals, is this not a sign that "Of course not; don't be stupid" may not be the most helpful reply?

I'm writing a new community wiki answer because it seems to me the consensus in the comments is that the accepted answer doesn't really tell the right story, and since this is something that pops up all the time it'd be good to have a single place to point people without making them read through all the comments. Please improve my answer.

In the most naive sense Wiles proof does depend on existence of Grothendieck universes (and thus on existence of inaccessible cardinals). By this I mean if you took every reference in Wiles proof and read the first published proof of that fact you'd certainly end up somewhere in SGA where, due to Grothendieck's love of generalization, you'd find universes popping up.

However, this certainly doesn't mean the proof really uses universes. It's widely believed (though for some people this belief may not come from much direct evidence) that in any practical situation you don't actually need universes. However, there are some concrete situations (BCnrd mentions some involving sheafification on the crystalline site) where it's not necessarily known how to eliminate the use of universes.

As a result, in order to figure out if Wiles's proof uses universes, or whether it's relatively easy to avoid them, you'd need to either read the proofs yourself or find someone who was both deeply familiar with the details of the proof, and someone who cares a lot about details. One person who comes quickly to mind is BCnrd. BCnrd was one of the mathematicians who proved the Modularity Theorem, which showed that all elliptic curves over $\mathbb{Q}$ are modular. This is a strengthening of Taylor and Wiles' result, which applied only to semi-stable elliptic curves, and the proof involved understanding and building on Taylor and Wiles' work. BCnrd is also famous for his attention to detail and for consulting underlying foundational sources; he is the author of a book dedicated to simplifying and correcting the presentation of Grothendieck duality in Hartshorne's book Residues and Duality.

As explained in the comments to Pete's answer, BCnrd says there's really no issue at all in Wiles's proof. All of the specific things that Wiles uses stay far away from any of the difficult issues where you might be worried about needing to invoke universes.

Can I draw attention to the continuation of that quotation? "But there is a general consensus among mathematicians that this was just a convenient short cut rather than a logical necessity. With a little work, Wiles's proof should be translatable into Peano arithmetic or some slight extension of it."

For the record, McLarty's paper in the Bulletin of Symbolic Logic was indeed the source for my claim that Wiles used an inaccessible cardinal (via Universes). If anyone fancies debating the exact definition of 'using' an axiom without 'needing' it, and whether or not that definition applies in the current case, we should probably do it somewhere else (Pedantry Overflow?). But I explicitly opposed the claim that the proof 'needed' large cardinal assumptions, so I don't propose to be too apologetic.

Anyway, it's an interesting question, and I'd be pleased if my article goes some way to bringing an answer into the public domain (even if I do get my head shot off in the process ;) ).

(Incidentally I would rather have left this as a comment than an answer, but don't have enough (any) reputation points. If anyone with superpowers wants to move it, please go ahead.)