Jourdain’s 1904 Generalization of Grundlagen

  • Arie HinkisEmail author
Part of the Science Networks. Historical Studies book series (SNHS, volume 45)


In 1904, Jourdain (see Grattan-Guinness 1977 prologue) published two papers. The first, in January (1904), was titled “On the transfinite cardinal numbers of well-ordered aggregates”; the second, in March (1904a), was titled “On the transfinite cardinal numbers of number-classes in general”. The papers are remarkable because they matched Cantor’s theory of inconsistent sets and offered a general construction of Cantor’s scale of number-classes, both unpublished at the time (see ‎ Chap. 4).


Union Theorem Comparability Theorem Ordinal Number Cardinal Number Transfinite Induction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Bernstein F. Bemerkung zur Mengenlehre, Göttingen Nachrichten. 1904;557–60.Google Scholar
  2. Bernstein F. Untersuchungen aus der Mengenlehre. Mathematische Annalen. 1905;61:117–55.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Cantor G. Ein Beitrag zur Mannigfeltigkeitslehre, (‘1878 Beitrag’). Cantor 1932;119–33.Google Scholar
  4. Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1895 Beiträge’). Cantor 1932;282–311. English translation: Cantor 1915.Google Scholar
  5. Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1897 Beiträge’). Cantor 1932;312–56. English translation: Cantor 1915.Google Scholar
  6. Ebbinghaus HD. Ernst Zermelo. An approach to his life and work. New York: Springer; 2007.zbMATHGoogle Scholar
  7. Ewald W. editor. From Kant to Hilbert: a source book in the foundations of mathematics. 2 vols. Oxford: Clarendon Press; 1996.Google Scholar
  8. Fraenkel AA. Abstract set theory. 3rd ed. Amsterdam: North Holland; 1966.Google Scholar
  9. Grattan-Guinness I. The correspondence between Georg Cantor and Philip Jourdain. Jahresbericht der Deutschen Mathematiker-Vereinigung. 1971a;73(Part 1):111–39.MathSciNetzbMATHGoogle Scholar
  10. Grattan-Guinness I. Dear Russell - dear Jourdain. A commentary on Russell’s logic, based on his correspondence with Philip Jourdain. New York: Columbia University Press; 1977.zbMATHGoogle Scholar
  11. Hardy GH. A theorem concerning the infinite cardinal numbers. The Quarterly Journal of Pure and Applied Mathematics. 1903;35:87–94.zbMATHGoogle Scholar
  12. Harward AE. On the transfinite numbers. Philosophical Magazine (6). 1905;10(58):439–60.zbMATHCrossRefGoogle Scholar
  13. Hessenberg G. Grundbegriffe der Mengenlehre. Abhandlungen der Friesschen Schule. 1906;2(1):479–706. reprinted Göttingen, Vardenhoeck & Ruprecht 1906.Google Scholar
  14. Jourdain PEB. On the transfinite cardinal numbers of well-ordered aggregates. Philosophical Magazine (6). 1904a;7(37):61–75.zbMATHCrossRefGoogle Scholar
  15. Jourdain PEB. On the transfinite cardinal numbers of number-classes in general. Philosophical Magazine (6). 1904b;7(39):294–303.zbMATHCrossRefGoogle Scholar
  16. Jourdain PEB. The definition of a series similarly ordered to the series of all ordinal numbers. Messenger of Mathematics. 1905b;35(2):56–8.Google Scholar
  17. Jourdain PEB. On the comparison of aggregates. Quarterly Journal of Pure and Applied Mathematics. 1907b;38:352–67.Google Scholar
  18. Jourdain PEB. The multiplication of alephs. Mathematische Annalen. 1908a;65:506–12.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Kuratowski K, Mostowski A. Set theory. Amsterdam: North Holland; 1968.zbMATHGoogle Scholar
  20. Lakatos I. What does a mathematical proof proves?, In: Worral J, Curry G, editors. Mathematics, science and epistemology, Cambridge University Press 1978a;61–9. Also in Tymoczko T editor. New directions in the philosophy of mathematics: an anthology, Birkhäuser 1986;153–62.Google Scholar
  21. Levy A. Basic set theory, Dover Publications Inc 2002. Originally published by Springer in 1979.Google Scholar
  22. Mańka R, Wojciechowska A. On two Cantorian theorems. Annals of the Polish Mathematical Society, Series II: Mathematical News. 1984;25:191–8.zbMATHGoogle Scholar
  23. Moore GH. The origins of Zermelo’s axiomatisation of set theory. J Philosophical Logic. 1978;7:307–29.zbMATHCrossRefGoogle Scholar
  24. Moore GH. Zermelo’s axiom of choice: its origin, development and influence. Berlin: Springer; 1982.CrossRefGoogle Scholar
  25. Peano G, Super Theorema de Cantor-Bernstein, Rendiconti del Circulo Mathmatico di Palermo, 21:360–6. This printing is from May 1906. The paper was reprinted in Revista de Mathematica, 1906;8:136–57. The second printing is dated August 1906. The second printing contains an additional part titled “additione”.Google Scholar
  26. Russell B. The collected papers of Bertrand Russell: foundations of logic, 1903–1905, vol 4, edited by Urquart A, London: Routledge; 1994.Google Scholar
  27. Schoenflies A. Die Entwicklung der Lehre von den Punktmannigfeltigkeiten, I, Jahresbericht der Deutschen Mathematiker-Vereinigung 1900;8.Google Scholar
  28. Schröder E. Über Zwei Defitionen der Endlichkeit und G. Cantorsche Sätze, Nova Acta. Abhandlungen der Kaiserlichen Leopold-Carolinschen deutchen Akademie der Naturfoscher. 1898;71:301–62.Google Scholar
  29. Whitehead AN. On Cardinal Numbers. Am J Math. 1902;24:365–94.MathSciNetCrossRefGoogle Scholar
  30. Zermelo E. Beweiss dass jede Menge wholgeordnet werden kann, Mathematiche Annalen 1904;59:514–6. English translation: van Heijenoort 1967;139–41.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.The Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityTel AvivIsrael

Personalised recommendations