Skip to main content
SpringerLink
Log in

The origins of Zermelo's axiomatization of set theory

  • Article
  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

What gave rise to Ernst Zermelo's axiomatization of set theory in 1908? According to the usual interpretation, Zermelo was motivated by the set-theoretic paradoxes. This paper argues that Zermelo was primarily motivated, not by the paradoxes, but by the controversy surrounding his 1904 proof that every set can be well-ordered, and especially by a desire to preserve his Axiom of Choice from its numerous critics. Here Zermelo's concern for the foundations of mathematics diverged from Bertrand Russell's on the one hand and from Felix Hausdorff's on the other.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliography

  1. Baire, R., Borel, E., Hadamard, J. and Lebesgue, H., ‘Cinq lettres sur la théorie des ensembles’, Bull. Soc. Math. de France 33 (1905), 261–273.

    Google Scholar 

  2. Bernstein, F., ‘Ueber die Reihe der transfiniten Ordnungszahlen’, Math. Annalen, 60 (1905), 187–193.

    Google Scholar 

  3. Beth, E., The Foundations of Mathematics, North Holland, Amsterdam, 1964.

    Google Scholar 

  4. Borel, E., Lecons sur la théorie des fonctions ..., Gauthier-Villars, Paris, 1898.

    Google Scholar 

  5. Borel, E., ‘Quelques remarques sur les principes de la théorie des ensembles’, Math. Annalen 60 (1905), 194–195.

    Google Scholar 

  6. Bourbaki, N., Eléments d'histoire des mathématiques, Hermann: Paris, 1969 (second edition).

    Google Scholar 

  7. Brouwer, L. E. J., Over de Grondlagen der Wiskunde, Maas & van Suchtelen, Amsterdam, 1907.

    Google Scholar 

  8. Burali-Forti, C., ‘Una questione sui numeri transfiniti’, Rendiconti del Circolo mat. di Palermo 11 (1897), 154–164.

    Google Scholar 

  9. Cantor, G., ‘Beiträge zur Begründung der transfiniten Mengenlehre’, Math. Annalen 49 (1897), 207–246.

    Google Scholar 

  10. Cantor, G. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by E. Zermelo, Springer, Berlin, 1932.

    Google Scholar 

  11. Cohen, P., ‘The Independence of the Continuum Hypothesis’, Proceedings of the Nat. Acad. of Sciences (USA) 50 (1963), 1143–1148.

    Google Scholar 

  12. Fraenkel, A., ‘Der Begriff “definit” und die Unabhängigkeit des Auswahlsaxioms’, Sitzungsberichte der Preuss. Akad. der Wissen., Phys. Math. Klasse (1922), 253–257.

  13. Fraenkel, A.,et al., Foundations of Set Theory, North Holland, Amsterdam, 1973 (second edition).

    Google Scholar 

  14. Gödel, K., ‘The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis’, Proc. Nat. Acad. of Sciences (USA) 24 (1938), 556–557.

    Google Scholar 

  15. Grattan-Guinness, I., ‘The Correspondence between Georg Cantor and Philip Jourdain’, Jahresbericht Deut. Math.-Verein. 73 (1971), 111–130.

    Google Scholar 

  16. Grattan-Guinness, I., Dear Russell-Dear Jourdain, Duckworth, London, 1977.

    Google Scholar 

  17. Hardy, G., ‘The Continuum and the Second Number Class’, Proc. London Math. Soc. (2)4 (1906), 10–17.

    Google Scholar 

  18. Hausdorff, F., ‘Grundzüge einer Theorie der geordneten Mengen’, Math. Annalen 65 (1908), 435–505.

    Google Scholar 

  19. Hausdorff, F., Grundzüge der Mengenlehre, de Gruyter, Leipzig, 1914.

    Google Scholar 

  20. Heyting, A., Mathematische Grundlagenforschung..., Springer, Berlin, 1934.

    Google Scholar 

  21. Heijenoort, J.van, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard U. Press, Cambridge, 1971 (second edition).

    Google Scholar 

  22. Hilbert, D., Grundlagen der Geometrie, Leipzig, 1899.

  23. Hilbert, D., ‘Mathematische Probleme ...’, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen (1900), 253–297.

  24. Hobson, E., ‘On the Arithmetic Continuum’, Proc. London Math. Soc. (2)4 (1906), 21–28.

    Google Scholar 

  25. Jourdain, P., ‘On the Transfinite Cardinal Numbers of Well-ordered Aggregates’, Phil. Mag. (6)7 (1904), 61–75.

    Google Scholar 

  26. Jourdain, P., ‘On a Proof that Every Aggregate Can Be Well-Ordered’, Math. Annalen 60 (1905), 465–470.

    Google Scholar 

  27. Jourdain, P., ‘On the Question of the Existence of Transfinite Numbers’, Proc. London Math. Soc. (2)4 (1906), 266–283.

    Google Scholar 

  28. Kennedy, H., ‘Nine Letters from Giuseppe Peano to Bertrand Russell’, Journal of the History of Philosophy 13 (1975), 205–220.

    Google Scholar 

  29. Kline, M., Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, 1972.

    Google Scholar 

  30. König, J., ‘Zum Kontinuum-Problem’, Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg... (Teubner: Leipzig, 1905), 144–147.

    Google Scholar 

  31. König, J., ‘Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem’, Math. Annalen 61 (1905), 156–160.

    Google Scholar 

  32. König, J., ‘Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem (Zweite Mitteilung)’, Math. Annalen 63 (1906), 217–221.

    Google Scholar 

  33. König, J., Neue Grundlagen der Logik, Arithmetik, und Mengenlehre, Viet, Leipzig, 1914.

  34. Kowalewski, G., Bestand und Wandel..., Oldenbourg, Munich, 1950.

    Google Scholar 

  35. Kramer, E., The Nature and Growth of Modern Mathematics, Hawthorn, New York, 1970.

    Google Scholar 

  36. Murata, T., ‘French Empiricism ...’, Commentarii math. Univ. Sancti Pauli 6 (1958), 93–114.

    Google Scholar 

  37. Peano, G., ‘Démonstration de l'intégrabilité des équations différentielles ordinaires’, Math. Annalen 37 (1890), 182–228.

    Google Scholar 

  38. Peano, G., Formulaire de mathématiques, Turin, 1895–1901 (vols. 1–3).

  39. Peano, G., ‘Additione’, Revista de mathematica 8 (1906), 136–143.

    Google Scholar 

  40. Poincaré, H., ‘Les mathématiques et la logique’, Revue de métaphysique et morale 13 (1905), 815–835.

    Google Scholar 

  41. Poincaré, H., ‘Les mathématiques et la logique’, Revue de métaphysique et morale 14 (1906), 294–317.

    Google Scholar 

  42. Quine, W. V., The Ways of Paradox, Random House, New York, 1966.

    Google Scholar 

  43. Russell, B., ‘On Some Difficulties in the Theory of Transfinite Numbers and Order Types’, Proc. London Math. Soc. (2)4 (1906), 29–53.

    Google Scholar 

  44. Russell, B., ‘Mathematical Logic as Based on the Theory of Types’, Amer. Jour. of Math. 30 (1908), 222–262.

    Google Scholar 

  45. Russell, B. and Whitehead, A., Principia Mathematica, Cambridge U. Press: Cambridge, 1910–1913 (vols. 1–3).

    Google Scholar 

  46. Schoenflies, A., ‘Die Entwicklung der Lehre von den Punktmannigfaltigkeiten’, Jahreshericht Deut. Math.-Verein, 8 (1990), pt. 2, 1–251.

    Google Scholar 

  47. Schoenflies, A., ‘Ueber wohlgeordnete Mengen’, Math. Annalen 60 (1905), 181–186.

    Google Scholar 

  48. Sierpinski, W., Cardinal and Ordinal Numbers, Polish Scientific Publishers, Warsaw, 1965 (second edition).

    Google Scholar 

  49. Zermelo, E., ‘Addition transfiniter Cardinalzahlen’, Nachrichten Königl. Gesell. Wissen. zu Göttingen (1901), 34–38.

  50. Zermelo, E., ‘Beweis, dass jede Menge wohlgeordnet werden kann’, Math. Annalen 59 (1904), 514–516.

    Google Scholar 

  51. Zermelo, E., ‘Neuer Beweis für die Möglichkeit einer Wohlordnung’, Math. Annalen 65 (1908), 107–128.

    Google Scholar 

  52. Zermelo, E., ‘Untersuchungen über die Grundlagen der Mengenlehre, I’, Math. Annalen 65 (1908), 261–281.

    Google Scholar 

  53. Zlot, W., ‘The Principle of Choice in Pre-axiomatic Set Theory’, Scripta Math. 25 (1960), 105–123.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for the History and Philosophy of Science and Technology, University of Toronto, Toronto, Canada

    Gregory H. Moore

Authors
  1. Gregory H. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moore, G.H. The origins of Zermelo's axiomatization of set theory. J Philos Logic 7, 307–329 (1978). https://doi.org/10.1007/BF00245932

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00245932

Keywords

Search

Navigation