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In a series of lectures in 1993, mathematician [[Andrew Wiles]] announced a proof using techniques in algebraic geometry, relying on the nonconstructive [[Axiom of Choice]].<ref name="Occam">[http://www.occampress.com/fermat.pdf Is There a "Simple" Proof of Fermat's Last Theorem, Page 5(occampress.com)]</ref> A flaw was found before publication, and Wiles spent a year on fixing the flaw. Then, in September 1994, he and Richard Taylor announced a new version of the proof. However, criticism does continue on the internet.<ref name="Occam" /> Further criticism came from [[Marilyn vos Savant]], known for her very high [[IQ]] and commentary on [[mathematics]], in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref> She questioned the use of [[Non-Euclidean geometry]] and the Axiom of Choice, among other points. She retracted her argument in a 1995 addendum to the book.

The Wiles-Taylor proof also makes use of some [[Grothendieck]] tools in cohomological number theory that use an axiom beyond the standard [[Zermelo-Fraenkel|ZFC]] axioms. It is an open question whether these tools can be formalized into a ZFC proof.<ref>Colin Mclarty [http://www.cwru.edu/artsci/phil/Proving_FLT.pdf - Is There a “Simple” Proof ofFermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu)[http://www.cwru.edu/artsci/phil/Proving_FLT.pdf]</ref>