|Abstract||Does the proof of Fermat’s Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by “proof ” and “use,” and are not entirely known. This paper surveys the current state of these questions and brieﬂy sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles  plus improvements that do not yet change its character. Far from self-contained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are “used in fact in the published proof.” It is currently unknown what assumptions are “used in principle” in the sense of being proof-theoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck uni- verses.1 On ZFC foundations a universe is an uncountable transitive set U..|
|Keywords||No keywords specified (fix it)|
|Categories||No categories specified (fix it)|
|Through your library||Only published papers are available at libraries|
Similar books and articles
Colin Mclarty (2010). What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory. Bulletin of Symbolic Logic 16 (3):359-377.
J. Todd Wilson (2001). An Intuitionistic Version of Zermelo's Proof That Every Choice Set Can Be Well-Ordered. Journal of Symbolic Logic 66 (3):1121-1126.
Matt Kaufmann (1983). Blunt and Topless End Extensions of Models of Set Theory. Journal of Symbolic Logic 48 (4):1053-1073.
Bryan W. Roberts (2011). How Galileo Dropped the Ball and Fermat Picked It Up. Synthese 180 (3):337-356.
Ali Enayat (2001). Power-Like Models of Set Theory. Journal of Symbolic Logic 66 (4):1766-1782.
Mateja Jamnik, Alan Bundy & Ian Green (1999). On Automating Diagrammatic Proofs of Arithmetic Arguments. Journal of Logic, Language and Information 8 (3):297-321.
Chris Freiling (1986). Axioms of Symmetry: Throwing Darts at the Real Number Line. Journal of Symbolic Logic 51 (1):190-200.
Paolo Gentilini (1999). Proof-Theoretic Modal PA-Completeness III: The Syntactic Proof. Studia Logica 63 (3):301-310.
Jan Krajíček (1997). Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic. Journal of Symbolic Logic 62 (2):457-486.
Anita Wasilewska (1984). DFC-Algorithms for Suszko Logic and One-to-One Gentzen Type Formalizations. Studia Logica 43 (4):395 - 404.
George Tourlakis (2010). On the Proof-Theory of Two Formalisations of Modal First-Order Logic. Studia Logica 96 (3):349-373.
Sachio Hirokawa, Yuichi Komori & Misao Nagayama (2000). A Lambda Proof of the P-W Theorem. Journal of Symbolic Logic 65 (4):1841-1849.
Added to index2010-12-22
Total downloads16 ( #75,695 of 556,895 )
Recent downloads (6 months)1 ( #64,931 of 556,895 )
How can I increase my downloads?