In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.


Here's the article


* Richard Elwes, *[To infinity and beyond: The struggle to save arithmetic](http://www.newscientist.com/article/mg20727731.300-to-infinity-and-beyond-the-struggle-to-save-arithmetic.html)*, New Scientist, August 2010.


Here's the relevant bit from the article:



> 
> "Large cardinals have been studied by logicians for a century, but their intangibility means they have seldom featured in mainstream mathematics. A notable exception is the most celebrated result of recent years, the proof of Fermat's last theorem by the British mathematician Andrew Wiles in 1994 [...]
> To complete his proof, Wiles assumed the existence of a type of large cardinal known as an inaccessible cardinal, technically overstepping the bounds of conventional arithmetic"
> 
> 
> 


Is this true ?
If so, could someone please outline how they are used ?



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](http://en.wikipedia.org/wiki/Grothendieck_universe). It has since been clarified that the existence of such a thing is equivalent to the existence of an [inaccessible cardinal](http://en.wikipedia.org/wiki/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:


* Colin McLarty, *What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory*, Bull. Symb. Log. **16** No. 3 (2010) pp. 359–377, doi:[10.2178/bsl/1286284558](https://doi.org/10.2178/bsl/1286284558), [archived author pdf](https://web.archive.org/web/20100528052913/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?


