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Andrew Wiles' proof of Fermat's Last Theorem, with an assist from RichardTaylor, focused renewed attention on the foundational question of whether theuse of Grothendieck's Universes in number theory entails that the resultsproved therewith make essential use of the large cardinal axiom that there isan uncountable strongly inaccessible cardinal, or more generally, that everycardinal is less than a strongly inaccessible cardinal. If one traces backthrough the references in Wiles' proof, one finds that the proof does dependupon explicit use of Grothendieck's Universes. Thus, prima facie, it appearsthat the proof of Fermat's Last Theorem depends upon a foundation that isstrictly stronger than ZFC. Colin McLarty removes this appearance bydemonstrating that all of Grothendieck's large tools, i.e., entities whoseconstruction depended upon Grothendieck's Universes, can instead be founded ona fragment of ZFC with the logical strength of Finite-Order Arithmetic. Thegoal of this article is to present overviews both of the history of Fermat'sLast Theorem and of McLarty's foundation for Grothendieck's large tools.

Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not  Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large  Structures of Grothendieck Founded on Finite-Order Arithmetic"