An interesting note about axiomatics

Andrew Wiles’ proof of Fermat’s Last Theorem relied on some very heavy machinery in algebraic geometry. What I had not realized (this is not at all my field) was that this machinery (built by Grothendieck in the middle of the 20th century) involved an additional axiom beyond those of ordinary set theory. Thus Wiles’ proof FLT relied on an axiom well beyond those needed to state the theorem. This is disconcerting.

Colin McLarty of Case Western Reserve University has just shown that Grothendieck’s work (or maybe just that part needed for FLT, I’m not sure) could in fact be built on a more familiar, standard set of axioms, meaning that the proof of FLT is a little more concerting (which should be the opposite of disconcerting, n’est-ce pas?).

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One thought on “An interesting note about axiomatics

  1. Brian

    For anyone who is interested in this axiom that Grothendiek used, it’s the use of Grothendiek universes ( http://en.wikipedia.org/wiki/Grothendieck_universe ) which is equivalent to strongly inaccessible cardinals ( http://en.wikipedia.org/wiki/Inaccessible_cardinal ).

    If you’d like to read a couple of papers by McLarty on this subject, check out the following:
    “WHAT DOES IT TAKE TO PROVE FERMAT’S LAST THEOREM?” ( http://www.cwru.edu/artsci/phil/Proving_FLT.pdf )
    “A FINITE ORDER ARITHMETIC FOUNDATION FOR COHOMOLOGY” ( http://arxiv.org/pdf/1102.1773.pdf )

    I believe that the second paper is where the result that McLarty presented is proven, since its title is very similar to the title of the talk that he gave at the Joint Mathematics Meetings.

    Reply

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