Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-05T20:41:00.311Z Has data issue: false hasContentIssue false
  • English
  • Français

What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory

Published online by Cambridge University Press:  15 January 2014

Colin McLarty
Colin McLarty*
Affiliation:
Department of Philosophy, Case Western Reserve University, Cleveland, OH 44106, USA, E-mail: colin.mclarty@case.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 3 , September 2010 , pp. 359 - 377
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1972] Artin, M., Grothendieck, A., and Verdier, J.-L., Théorie des topos et cohomologie etale des schémas, Séminaire de Géométrie Algébrique du Bois-Marie, 4, Springer-Verlag, 1972, three volumes, generally cited as SGA 4.Google Scholar
[2003] Avigad, J., Number theory and elementary arithmetic, Philosophia Mathematica, vol. 11 (2003), pp. 257–284.CrossRefGoogle Scholar
[2009] Basbois, N., La naissance de la cohomologie des groupes, Ph.D. thesis, Université de Nice Sophia-Antipolis, 2009.Google Scholar
[2008] Carter, J., Categories for the working mathematician: Making the impossible possible, Synthese, vol. 162 (2008), pp. 1–13.CrossRefGoogle Scholar
[1997] Cornell, G., Silverman, J., and Stevens, G. (editors), Modular forms and Fermat's Last Theorem, Springer-Verlag, 1997.CrossRefGoogle Scholar
[1974] Deligne, P., La conjecture de Weil I, Publications Mathématiques. Institut de Hautes Études Scientifiques, (1974), no. 43, pp. 273–307.Google Scholar
[1977] Deligne, P. (editor), Cohomologie étale, 1977, Séminaire de Géométrie Algébrique du Bois-Marie; SGA 4 1/2, Springer-Verlag. Generally cited as SGA 4 1/2, this is not strictly a report on Grothendieck's Seminar.CrossRefGoogle Scholar
[1998] Deligne, P., Quelques idées maîtresses de l'œuvre de A. Grothendieck, Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996), Société Mathématique de France, 1998, pp. 11–19.Google Scholar
[2009] Deligne, P., Colloque Grothendieck, Pierre Deligne, video by IHES Science, on-line at www.dailymotion.com/us, 2009.Google Scholar
[1973] Deligne, P. and Rapoport, M., Les schémas de modules de courbes elliptiques, Modular functions of one variable, II, Lecture Notes in Mathematics, vol. 349, Springer-Verlag, New York, 1973, pp. 143–316.CrossRefGoogle Scholar
[2008] Ellenberg, J., Arithmetic geometry, Princeton companion to mathematics (Gowers, T., Barrow-Green, J., and Leader, I., editors), Princeton University Press, 2008, pp. 372–383.Google Scholar
[1983] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae, vol. 73 (1983), pp. 349–366.CrossRefGoogle Scholar
[2005] Fantechi, B., Vistoli, A., Gottsche, L., Kleiman, S. L., Illusie, L., and Nitsure, N., Fundamental algebraic geometry: Grothendieck's FGA explained, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, 2005.Google Scholar
[1964] Freyd, P., Abelian categories: An introduction to the theory of functors, Harper and Row, 1964, reprinted with author commentary in: Reprints in Theory and Applications of Categories , (2003), no. 3, pp. –25–164, available on-line at www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html.Google Scholar
[1957] Grothendieck, A., Sur quelques points d'algèbre homologique, Tôhoku Mathematical Journal, vol. 9 (1957), pp. 119–221.Google Scholar
[1971] Grothendieck, A., Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie, 1, Springer-Verlag, 1971, generally cited as SGA1.Google Scholar
[1985] Grothendieck, A., Récoltes et semailles, Université des Sciences et Techniques du Languedoc, Montpellier, 1985, published in several successive volumes.Google Scholar
[1961] Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique III: Étude cohomologique des faisceaux cohérents, Publications Mathématiques. Institut des Hautes Études Scientifiques, Paris, (1961), no. 11.Google Scholar
[1971] Grothendieck, A., Éléments de géométrie algébrique I, Springer-Verlag, 1971.Google Scholar
[1966] Hartshorne, R., Residues and duality, lecture notes of a seminar on the work of A. Grothendieck given at Harvard 1963–64, Lecture Notes in Mathematics, no. 20, Springer-Verlag, New York, 1966.Google Scholar
[1977] Hartshorne, R., Algebraic geometry, Springer-Verlag, 1977.CrossRefGoogle Scholar
[1973] Herrlich, H. and Strecker, G., Category theory, Allyn and Bacon, Boston, 1973.Google Scholar
[2009] Lipman, J. and Hashimoto, M., Foundations of Grothendieck duality for diagrams of schemes, Springer-Verlag, 2009.CrossRefGoogle Scholar
[2009] Lurie, J., Higher topos theory, Annals of Mathematics Studies, no. 170, Princeton University Press, Princeton, 2009.CrossRefGoogle Scholar
[1988] Lane, S. Mac, Group extensions for 45 years, Mathematical Intelligencer, vol. 10 (1988), no. 2, pp. 29–35.CrossRefGoogle Scholar
[2003] Macintyre, A., Model theory: Geometrical and set-theoretic aspects and prospects, this Bulletin, vol. 9 (2003), no. 2, pp. 197–212.Google Scholar
[forthcoming] Macintyre, A., The impact of Gödel's incompleteness theorems on mathematics, Horizons of truth: Proceedings of Gödel centenary, Vienna, 2006 , forthcoming.Google Scholar
[1977] Mazur, B., Modular curves and the Eisenstein ideal, Publications Mathématiques. Institut des Hautes Études Scientifiques, vol. 47 (1977), pp. 133–186.Google Scholar
[1997] Mazur, B., Introduction to the deformation theory of Galois representations, Modular forms and Fermat's Last Theorem (Cornell, G., Silverman, J., and Stevens, S., editors), Springer-Verlag, 1997, pp. 243–312.CrossRefGoogle Scholar
[2007] McLarty, C., The rising sea: Grothendieck on simplicity and generality I, Episodes in the history of recent algebra (Gray, J. and Parshall, K., editors), American Mathematical Society, 2007, pp. 301–326.Google Scholar
[2008] McLarty, C., “There is no ontology here”: visual and structural geometry in arithmetic, The philosophy of mathematical practice (Mancosu, P., editor), Oxford University Press, 2008, pp. 370–406.CrossRefGoogle Scholar
[1950] Mostowski, A., Some impredicative definitions in the axiomatic set theory, Fundamenta Mathematicae, vol. 37 (1950), pp. 111–124.CrossRefGoogle Scholar
[1978] Mumford, D. and Tate, J., Fields Medals IV. An instinct for the key idea, Science, vol. 202 (1978), pp. 737–739.CrossRefGoogle ScholarPubMed
[2008] Osserman, B., The Weil conjectures, Princeton companion to mathematics (Gowers, T., Barrow-Green, J., and Leader, I., editors), Princeton University Press, 2008, pp. 729–732.Google Scholar
[1978] Takeuti, G., A conservative extension of Peano Arithmetic, Two applications of logic to mathematics, Princeton University Press, 1978, pp. 77–135.Google Scholar
[2008] Totaro, B., Algebraic topology, Princeton companion to mathematics (Gowers, T., Barrow-Green, J., and Leader, I., editors), Princeton University Press, 2008, pp. 383–396.Google Scholar
[1997] Washington, L., Galois cohomology, Modular forms and Fermat's Last Theorem (Cornell, G., Silverman, J., and Stevens, S., editors), Springer-Verlag, 1997, pp. 101–120.CrossRefGoogle Scholar
[1995] Wiles, A., Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, vol. 141 (1995), pp. 443–551.CrossRefGoogle Scholar