The Axiom of Choice as Paradigm Shift: The Case for the Distinction Between the Ontological and the Methodological Crisis in the Foundations of Mathematics

  • Valérie Lynn TherrienEmail author
Conference paper
Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (PCSHPM)


Seldom has a mathematical axiom engendered the kind of criticism and controversy as did Zermelo’s (1904) Axiom of Choice (henceforth, AC). In this paper, we intend to place the development of the Axiom of Choice in its proper historical context relative to the period often called “the crisis in the foundations of mathematics.” To this end, we propose that the nature of the controversy surrounding AC warrants a division of the Grundlagenkrise der Mathematik into two separate horns: (1) an ontological crisis related to the nature and status of mathematics itself (viz., the nature of its foundation and the logical paradoxes that surrounded early attempts to logically formalize mathematics); and (2) a methodological branch concerned rather with the nature of mathematical practice (viz., the nature of mathematical proofs). These two strands are inexorably intertwined and, though it is not new to suggest that the controversy surrounding AC was related either to the foundational crisis or to a polemic about the nature of mathematical demonstration, it is perhaps new to state that the question of the validity of AC not only was a central question of this period but also, furthermore, was one of its primary drivers—one which led to a profound paradigm shift in the way we construe mathematical reasoning, whether it has led us down a path of embracing realism/Platonism or intuitionism/constructivism.


Axiom of Choice Axiomatic Systems Set Theory Foundations of Mathematics Philosophy of Mathematics History of Mathematics 


  1. Baire, René; Borel, Émile; Hadamard, Jacques; & Lebesgue, Henri. (1905), “Cinq lettres sur la théorie des ensembles”, Bulletin de la Société Mathématique de France (tome 33), pp. 261-73.Google Scholar
  2. Bell, John L.. (2009). Axiom of Choice (Vol. 22 of Studies in Logic). College Publications: London, 248pp.Google Scholar
  3. Bell, John L.. (2011). “The Axiom of Choice in the Foundations of Mathematics”, in Sommaruga, Giovanni (Ed.), Foundational Theories of Classical and Constructive Mathematics. Springer: Dordrecht, pp. 157-170.Google Scholar
  4. Borel, Émile. (1904), “Quelques remarques sur les principes de la théorie des ensembles”, Mathematische Annalen (tome 50), pp. 194-5.MathSciNetCrossRefGoogle Scholar
  5. Borel, Émile. (1946). “L’axiome du choix et la mesure des ensembles”, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Tome Deux-Cent-Vingt-Deuxième, Gauthier-Villars : Paris, p. 309-10.Google Scholar
  6. Borel, Émile. (1947). “Les paradoxes de l’axiome du choix”, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Tome Deux-Cent-Vingt-Quatrième, Gauthier-Villars : Paris, pp. 1537-8.Google Scholar
  7. Cantor, Georg. (1883), “Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen”. Leipzig, Teubner.Google Scholar
  8. Gauthier, Yvon. (2010). Logique arithmétique: L’arithmétisation de la logique. Presses de l’Université Laval: Québec, 205pp.Google Scholar
  9. Hilbert, David. (1925). “Über das Unendliche”, Mathematische Annalen (vol. 96), pp. 161-90.MathSciNetCrossRefGoogle Scholar
  10. Jech, Thomas J.. (1973). The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, Volume 75. North-Holland Publishing Company: Amsterdam, 202pp.Google Scholar
  11. Jech, Thomas J.. (1982), “About the Axiom of Choice”, in Barwise J. (Ed.), Handbook of Mathematical Logic (Vol. 90 of Studies in Logic and the Foundations of Mathematics). Elsevier: Amsterdam, pp. 346-70.CrossRefGoogle Scholar
  12. Kanamori, Akihiro (1997), “The Mathematical Import of Zermelo’s Well-Ordering Theorem”, The Bulletin of Symbolic Logic (vol. 3, no. 3), pp. 281-311.MathSciNetCrossRefGoogle Scholar
  13. Kanamori, Akihiro. (2012), “Set Theory From Cantor to Cohen”, in Handbook of the History of Logic (vol. 6: Sets and Extensions in the Twentieth Century, Gabbay, Dov M., Kanamori, Akihiro & Woods, John (eds.),. Elsevier: Amsterdam, pp. 1-71.Google Scholar
  14. Kuratowski, Kazimierz. (1922). « Une méthode d’élimination des nombres transfinis des raisonnements mathématiques », Fundamenta Matematica (vol. 3), pp. 76-108.Google Scholar
  15. Lebesgue, Henri (1941). “Les controverses de la théorie des ensembles et la question des fondements”, in Les entretiens de Zurich sur les fondements et les méthodes des sciences mathématiques, 8-9 décembre 1938. Gonseth : Zurich, pp. 109-122.Google Scholar
  16. Martin-Löf, Per. (2006). « 100 years of Zermelo’s Axiom of choice : what was the problem with it? », The Computer Journal, (vol. 49, no. 3), pp. 345-50.Google Scholar
  17. Mancosu, Paolo; Zach, Richard & Badesa, Calixto. (2009), “The Development of Mathematical Logic from Russell to Tarski, 1900-1935”, in The Development of Modern Logic, Haaparantha, Leila (ed.). Oxford University Press: New York, pp. 318-470.CrossRefGoogle Scholar
  18. Moore, Gregory H.. (1978), “The Origins of Zermelo’s Axiomatization of Set Theory”, Journal of Philosophical Logic (vol. 7, no. 1), pp. 307-329.Google Scholar
  19. Moore, Gregory H.. (1982), Zermelo’s Axiom of Choice: Its Origins, Development and Influence, Vol. 8 Studies in the History of Mathematics and Physical Sciences. Springer-Verlag: New York, 410pp.Google Scholar
  20. Moore, Gregory H.. (1983). “Lebesgue’s Measure Problem and Zermelo’s Axiom of Choice: the Mathematical Effects of a Philosophical Dispute”, Annals of the New York Academy of Sciences (vol. 412, iss. 1), pp. 129-54.CrossRefGoogle Scholar
  21. Putnam, Hilary. (1975), “What is Mathematical Truth”, Historia Mathematica (vol. 2), pp. 529-33.MathSciNetCrossRefGoogle Scholar
  22. Russell, Bertrand. (1903). The Principles of Mathematics. Cambridge University Press: Cambridge.Google Scholar
  23. Sierpiński, Wacław. (1918). “L’axiome de M. Zermelo et son rôle dans la Théorie des Ensembles et l’Analyse“, Bulletin de l’Académie des Sciences de Cracovie, série A, pp. 97-152.Google Scholar
  24. Sierpiński, Wacław. (1958), Cardinal and Ordinal Numbers (Monografje Matematyczne Tom 34). Państwowe Wydawnictwo Naukowe : Warsaw, 487pp.Google Scholar
  25. Sierpiński, Wacław. (1967), « L’axiome du Choix », Notre Dame Journal of Logic (vol. VIII, no. 4), pp. 257-66.Google Scholar
  26. Strepāns, Juris. (2012), “History of the Continuum in the 20th Century”, in Handbook of the History of Logic (vol. 6: Sets and Extensions in the Twentieth Century, Gabbay, Dov M., Kanamori, Akihiro & Woods, John (eds.),. Elsevier: Amsterdam, pp. 1-71.Google Scholar
  27. Therrien, Valérie Lynn. (2012). “Wittgenstein and the Labyrinth of ‘Actual’ Infinity: The Critique of Transfinite Set Theory“, Ithaque (vol. 10), pp. 43-65.Google Scholar
  28. Zermelo, Ernst. (1904). “Bewis, dass jede Menge wohlgeordnet warden kann”, in From Frege to Gödel. A Source Book in Mathematical Logic, 1897-1931, van Heijenoort, Jean (ed.), (1967). Harvard University Press: Cambridge, pp. 139-41.Google Scholar
  29. Zermelo, Ernst. (1908). “Neuer Beweis für die Möglichkeit einer Wohlordung ”, in Ernst Zermelo: Collected Works/Gesammelte Werke, Volume I/Band I, Ebbinghaus, Heinz-Dieter and Kanamori, Akihiro (eds.), (2010). Springer-Verlag : Berlin, pp. 120-59.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Western UniversityLondonCanada

Personalised recommendations