Linked bibliography for the SEP article "Zermelo’s Axiomatization of Set Theory" by Michael Hallett

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Most of the original sources surrounding Zermelo's work were written in German, and some in French; when translations of these works into English are available, bibliographic information for the translations follows the citation of the original text. Similarly for older, relatively inaccessible texts that have been republished in more current works.

  • Bell, J., 2009, The Axiom of Choice, London: College Publications. (Scholar)
  • Benacerraf, P. and H. Putnam (eds.), 1964, Philosophy of Mathematics: Selected Readings, Oxford: Basil Blackwell. (Scholar)
  • ––– (eds.), 1983, Philosophy of Mathematics: Selected Readings, Second Edition, Cambridge: Cambridge University Press. (Scholar)
  • Bernstein, F., 1905, “Über die Reihe der transfiniten Ordnungszahlen”, Mathematische Annalen 60: 187–193. (Scholar)
  • Borel, E., 1905, “Quelque remarques sur les principes de la théorie des ensembles”, Mathematische Annalen 60: 194–195. (Scholar)
  • Browder, F. (ed.), 1976, Mathematical Developments Arising from the Hilbert Problems, Volume 28 of Proceedings of Symposia in Pure Mathematics, Providence: American Mathematical Society. (Scholar)
  • Cantor, G., 1883a, “Ueber unendliche, lineare Punktmannichfaltigkeiten” Mathematische Annalen 21: 545–591. Reprinted in Cantor 1883b and in Cantor 1932: 165–209. English translation in Ewald 1996, Volume 2. (Scholar)
  • –––, 1883b, Grundlagen einer allegemeinen Mannigfaltichkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen, Leipzig: B. G. Teubner. (Scholar)
  • –––, 1895, “Beiträge zur Begründung der transfiniten Mengenlehre, Erster Artikel”, Mathematische Annalen 46: 481–512. Reprinted in Cantor 1932: 282–311. English translation in Cantor 1915. (Scholar)
  • –––, 1897, “Beiträge zur Begründung der transfiniten Mengenlehre, Zweiter Artikel”, Mathematische Annalen 49: 207–246. Reprinted in Cantor 1932: 312–351. English translation in Cantor 1915. (Scholar)
  • –––, 1915, Contributions to the Founding of the Theory of Transfinite Numbers, La Salle: Open Court. English translation of Cantor 1895, 1897 by Philip E. B. Jourdain. (Scholar)
  • –––, 1932, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, mit eläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind herausgegeben von Ernst Zermelo, Berlin: Springer. (Scholar)
  • –––, 1991, Georg Cantor: Briefe. Herausgegeben von Herbert Meschkowski, Berlin: Springer (Scholar)
  • Dedekind, R., 1888, Was sind und was sollen die Zahlen?, Braunschweig: Vieweg und Sohn. Also reprinted in Dedekind 1932: 335–391; English translation in Ewald 1996: 787–833. (Scholar)
  • –––, 1932, Gesammelte mathematische Werke. Band 3. Herausgegeben von Robert Fricke, Emmy Noether and Öystein Ore, Braunschweig: Friedrich Vieweg und Sohn. Reprinted with some omissions by Chelsea Publishing Co., New York, 1969. (Scholar)
  • Dreben, B. and A. Kanamori, 1997, “Hilbert and set theory”, Synthese, 110: 77–125. (Scholar)
  • Ebbinghaus, H.-D., 2007, Ernst Zermelo: An Approach to His Life and Work, Berlin: Springer. (Scholar)
  • –––, 2010, “Introductory note to Über den Begriff von Definitheit in der Axiomatik [Zermelo 1929]”, in Zermelo 2010: 352–357. (Scholar)
  • Ewald, W. (ed.), 1996, From Kant to Hilbert, Oxford: Oxford University Press. (Scholar)
  • Ewald, W., W. Sieg, and M. Hallett (eds.), 2013, David Hilbert's Lectures on the Foundations of Logic and Arithmetic, 1917–1933, Volume 3 of Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933, Berlin: Springer. (Scholar)
  • Felgner, U., 2010, “Introductory note to Untersuchungen über die Grundlagen der Mengenlehre, I [Zermelo 1908b]”, in Zermelo 2010: 160–188. (Scholar)
  • Ferreiros, J., 1999, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Science Networks Historical Studies, Basel: Birkhäuser. Second Revised Edition, 2007 (Scholar)
  • Fraenkel, A., Y. Bar-Hillel, and A. Levy, 1973, Foundations of Set Theory. Amsterdam: North-Holland Publishing. (Scholar)
  • Fraenkel, A. A., 1922, “Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre”, Mathematische Annalen 86: 230–237. (Scholar)
  • –––, 1927, Zehn Vorlesungen über die Grundlegung der Mengenlehre, Leipzig: B. G. Teubner. (Scholar)
  • Frege, G., 1879, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: Louis Nebert. Reprinted in Frege 1964, English translation in van Heijenoort 1967: 1–82. (Scholar)
  • –––, 1893, Grundgesetze der Arithmetik, Band 1, Jena: Hermann Pohle. English translation by Philip Ebert and Marcus Rossberg, Frege, The Basic Laws of Arithmetic, Derived using Concept-Script, Oxford: Oxford University Press, forthcoming. (Scholar)
  • –––, 1903, Grundgesetze der Arithmetik, Band II, Jena: Hermann Pohle. English translation by Philip Ebert and Marcus Rossberg, Frege, The Basic Laws of Arithmetic, Derived using Concept-Script, Oxford: Oxford University Press, forthcoming. (Scholar)
  • –––, 1964, Begriffsschrift und andere Aufsätze. Mit E. Husserls und H. Scholz’ Anmerkungen herausgegeben von Ignacio Angelelli, Darmstadt: Wissenschaftliche Buchgesellschaft. (Scholar)
  • Gödel, K., 1944, “Russell's mathematical logic”, in P. A. Schillp (ed.), The Philosophy of Bertrand Russell, pp. 125–153, La Salle: Open Court. Reprinted in Benacerraf and Putnam 1964: 211–232; Benacerraf and Putnam 1983: 447–469; and in Gödel 1990: 119–141. (Scholar)
  • –––, 1990, Kurt Gödel: Collected Works, Volume 2, edited by Solomon Feferman et al., Oxford: Oxford University Press. (Scholar)
  • Haaparanta, L. (ed.), 2009, The Development of Modern Logic, Oxford: Oxford University Press. (Scholar)
  • Hadamard, J. et al., 1905, “Cinq letters sur la théorie des ensembles”, Bulletin de la société mathématique de France, 33: 261–273. Letters between Baire, Borel, Lebesgue and Hadamard on objections to, and defense of, Zermelo's 1904 proof of the well-ordering theorem. (Scholar)
  • Hallett, M., 1981, “Russell, Jourdain and ‘limitation of size’”, British Journal for the Philosophy of Science, 32: 381–399. (Scholar)
  • –––, 1984, Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press. (Scholar)
  • –––, 2008, “The ‘purity of method’ in Hilbert's Grundlagen der Geometrie”, in P. Mancosu (ed.), The Philosophy of Mathematical Practice, pp. 198–255, Oxford: Clarendon Press. (Scholar)
  • –––, 2010a, “Frege and Hilbert”, in M. Potter and T. Ricketts (eds.), The Cambridge Companion to Frege, Cambridge: Cambridge University Press. (Scholar)
  • –––, 2010b, “Introductory note to Zermelo's two papers on the well-ordering theorem”, in Zermelo 2010: 80–115. (Scholar)
  • Hallett, M. and U. Majer (eds.), 2004, David Hilbert's Lectures on the Foundations of Geometry, 1891–1902, Volume 1 of Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933, Berlin: Springer. (Scholar)
  • Hardy, G. H., 1904, “A theorem concerning the infinite cardinal numbers”, Quarterly Journal of Pure and Applied Mathematics 35: 87–94.
  • Hartogs, F., 1915, “Über das Problem der Wohlordnung”, Mathematische Annalen, 76: 438–442. (Scholar)
  • Harward, A. E., 1905, “On the transfinite numbers”, Philosophical Magazine 10(6): 439–460. (Scholar)
  • Hausdorff, F., 1914, Grundzüge der Mengenlehre, Leipzig: Von Veit. (Scholar)
  • Hawkins, T., 1970, Lebesgue's Theory of Integration. New York: Blaisdell. Reprinted by the Chelsea Publishing Company, New York, 1979. (Scholar)
  • van Heijenoort, J. (ed.), 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Cambridge, Massachusetts: Harvard University Press. (Scholar)
  • Heinzmann, G., 1986, Poincaré, Russell, Zermelo et Peano. Textes de la discusion (1906–1912) sur les fondements des mathématiques: des antinomies à la prédicativité, Paris: Albert Blanchard. (Scholar)
  • Hessenberg, G., 1906, “Grundbegriffe der Mengenlehre”, Abhandlungen der neuen Fries'schen Schule (Neue Folge) 1: 479–706. (Scholar)
  • Hilbert, D., 1899, “Grundlagen der Geometrie”, in Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Leipzig: B. G. Teubner. Republished as Chapter 5 in Hallett and Majer 2004. (Scholar)
  • –––, 1900a, “Mathematische Probleme”, Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, pp. 253–296. English translation by Mary Winston Newson, 1902, “Mathematical Problems” Bulletin of the American Mathematical Society 8: 437–479. (Scholar)
  • –––, 1900b, “Über den Zahlbegriff”, Jahresbericht der deutschen Mathematiker-Vereinigung 8: 180–185. Reprinted (with small modifications) in Second to Seventh Editions of Hilbert 1899. (Scholar)
  • –––, 1902, “Grundlagen der Geometrie”. Ausarbeitung by August Adler for lectures in the Sommersemester of 1902 at the Georg-August Universität, Göttingen. Library of the Mathematisches Institut. Published as Chapter 6 in Hallett and Majer 2004. (Scholar)
  • –––, 1918, “Axiomatisches Denken”, Mathematische Annalen, 78: 405–415. Reprinted in Hilbert 1935: 146–156; English translation in Ewald 1996: volume 2, pp. 1105–1115. (Scholar)
  • –––, 1920, “Probleme der mathematischen Logik”, Lecture notes for a course held in the Wintersemester of 1920 at the Georg-August Universität, Göttingen, ausgearbeitet by Moses Schönfinkel and Paul Bernays. Library of the Mathematisches Institut, Universität Göttingen. Published in Ewald et al. 2013, Chapter 2. (Scholar)
  • –––, 1935, Gesammelte Abhandlungen, Band 3. Berlin: Julius Springer. (Scholar)
  • Jourdain, P. E. B., 1904, “On the transfinite cardinal numbers of well-ordered aggregates”, Philosophical Magazine 7(6): 61–75. (Scholar)
  • –––, 1905a, “On a proof that every aggregate can be well-ordered”, Mathematische Annalen 60: 465–470. (Scholar)
  • –––, 1905b, “On transfinite numbers of the exponential form”, Philosophical Magazine 9(6): 42–56. (Scholar)
  • Kanamori, A., 1996, “The mathematical development of set theory from Cantor to Cohen”, Bulletin of Symbolic Logic 2: 1–71. (Scholar)
  • –––, 1997, “The mathematical import of Zermelo's well-ordering theorem”, Bulletin of Symbolic Logic 3: 281–311. (Scholar)
  • –––, 2003, “The empty set, the singleton, and the ordered pair”, Bulletin of Symbolic Logic 9: 273–298. (Scholar)
  • –––, 2004, “Zermelo and set theory”, Bulletin of Symbolic Logic 10: 487–553. (Scholar)
  • –––, 2012, “In praise of replacement”, Bulletin of Symbolic Logic, 18: 46–90. (Scholar)
  • Kuratowski, C., 1921, “Sur la notion de l'ordre dans la théorie des ensembles”, Fundamenta Mathematicae 2: 161–171. (Scholar)
  • –––, 1922, “Une méthode d'élimination des nombres transfini des raisonnements mathématiques”, Fundamenta Mathematicae 3: 76–108. (Scholar)
  • Mancosu, P., 2009, “Measuring the size of infinite collections of natural numbers: was Cantor's theory of infinite number inevitable?”, Review of Symbolic Logic 2: 612–646. (Scholar)
  • –––, 2010, The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900–1940. Oxford: Oxford University Press. (Scholar)
  • Mancosu, P., R. Zach, and C. Badesa, 2009, “The development of mathematical logic from Russell to Tarski, 1900–1935”, in Haaparanta 2009: 318–470. Reprinted in Mancosu 2010: 5–119. (Scholar)
  • Mirimanoff, D., 1917a, “Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles”, L'enseignement mathématique 19: 37–52. (Scholar)
  • –––, 1917b, “Remarques sur la théorie des ensembles et les antinomies Cantoriennes (I)”, L'enseignement mathématique 19: 208–217. (Scholar)
  • –––, 1921, “Remarques sur la théorie des ensembles et les antinomies Cantoriennes (II)”, L'enseignement mathématique 21: 29–52. (Scholar)
  • Moore, G., 1976, “Ernst Zemelo, A. E. Harward, and the axiomatisation of set theory”, Historia Mathematica 3: 206–209. (Scholar)
  • –––, 1982, Zermelo's Axiom of Choice: Its Origins, Development and Influence. Berlin: Springer. (Scholar)
  • Peano, G., 1906, “Additione”, Revista di mathematica 8: 143–157. Reprinted in Heinzmann 1986: 106–120. (Scholar)
  • Peckhaus, V., 1990, Hilbertprogramm und kritische Philosophie: das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie, Volume 7 of Studien zur Wissenschafts- Sozial- und Bildungsgeschichte, Göttingen: Vandenhoek and Ruprecht. (Scholar)
  • Poincaré, H., 1905, “Les mathématiques et la logique”, Revue de métaphysique et de morale 13: 815–835. Reprinted with alterations in Poincaré 1908: Part II, Chapter 3; and, with these alterations noted, in Heinzmann 1986: 11–34. English translation in Ewald 1996: 1021–1038. (Scholar)
  • –––, 1906a, “Les mathématiques et la logique”, Revue de métaphysique et de morale 14: 17–34. Reprinted with alterations in Poincaré 1908: Part II, Chapter 3; and, with these alterations noted, in Heinzmann 1986: 35–53. English translation in Ewald 1996: 1038–1052. (Scholar)
  • –––, 1906b, “Les mathématiques et la logique”, Revue de métaphysique et de morale 14: 294–317. Reprinted with alterations in Poincaré 1908: Part II, Chapter 5; and, with these alterations noted, in Heinzmann 1986: 35–53. English translation in Ewald 1996: 1052–1071. (Scholar)
  • –––, 1908, Science et méthode, Paris: Ernst Flammarion. English translation in Poincaré 1913b, and retranslated by Francis Maitland as Science and Method, New York: Dover Publications. (Scholar)
  • –––, 1909, “Le logique de l'infini”, Revue de métaphysique et de morale 17: 462–482. Reprinted in Poincaré 1913a: 7–31. (Scholar)
  • –––, 1913a, Dernières Pensées, Paris: Ernest Flammarion. English translation published in 1963 as Mathematics and Science: Last Essays, New York: Dover Publications. (Scholar)
  • –––, 1913b, The Foundations of Science, New York: Science Press. Preface by Poincaré and an Introduction by Josiah Royce. Contains English translation by G. B. Halsted of Poincaré 1908. (Scholar)
  • Ramsey, F. P., 1926, “The foundations of mathematics”, Proceedings of the London Mathematical Society 25 (Second Series): 338–384. Reprinted in Ramsey 1931: 1–61, and Ramsey 1978: 152–212. (Scholar)
  • –––, 1931, The Foundations of Mathematics and Other Logical Essays, R. B. Braithwaite (ed.), London: Routledge and Kegan Paul, London. (Scholar)
  • –––, 1978, Foundations: Essays in Philosophy, Logic, Mathematics and Economics, D. H. Mellor (ed.), London: Routledge and Kegan Paul. (Scholar)
  • Richard, J., 1905, “Les principes des mathématiques et le problème des ensembles”, Révue général des sciences pures et appliqués 16: 541. English translation in van Heijenoort 1967: 142–144. (Scholar)
  • Russell, B., 1902, Letter to Frege. In Heijenoort 1967: 124–125. (Scholar)
  • –––, 1903, The Principles of Mathematics, Volume 1, Cambridge: Cambridge University Press. (Scholar)
  • Schoenflies, A., 1905, “Über wohlgeordnete Mengen”, Mathematische Annalen 60: 181–186. (Scholar)
  • Skolem, T., 1923, “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre”, Matimatikerkrongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaiska matematikerkongressen, redogörelse, 1923, pp. 217–232. Reprinted in Skolem 1970: 137–152 which also preserves the original pagination. English translation in Heijenoort 1967: 290–301. (Scholar)
  • –––, 1970, Selected Papers in Logic, Oslo: Universitetsforlaget. Edited by Jens Erik Fenstad. (Scholar)
  • von Neumann, J., 1923, “Zur Einführung der transfiniten Zahlen”, Acta Litterarum ac Scientiarum Regiæ Universitatis Hungaricæ Francisco-Josephinæ. Sectio Scientiæ-Mathematicæ 1, pp. 199–208. Reprinted in von Neumann 1961: 24–33. English translation in van Heijenoort 1967: 346–354. (Scholar)
  • –––, 1928, “Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre”, Mathematische Annalen 99: 373–391. Reprinted in von Neumann 1961: 320–338. (Scholar)
  • –––, 1961, John von Neumann: Collected Works, Volume 1, Oxford: Pergamon Press. (Scholar)
  • Weyl, H., 1910, “Über die Definitionen der mathematischen Grundbegriffe”, Mathematisch-naturwissenschaftliche Blätter 7, pp. 93–95, 109–113. Reprinted in Weyl 1968, Volume 1, 298–304. (Scholar)
  • –––, 1968, Gesammelte Abhandlungen, 4 Volumes, Berlin: Springer. (Scholar)
  • Young, W. H. and G. C. Young, 1906, The Theory of Sets of Points, Cambridge: Cambridge University Press. (Scholar)
  • Zermelo, E., 1904, “Beweis, daß jede Menge wohlgeordnet werden kann”, Mathematische Annalen 59: 514–516. Reprinted in Zermelo 2010: 114–119, with a facing-page English translation, and an Introduction by Michael Hallett (2010b). English translation also in van Heijenoort 1967: 139–141. (Scholar)
  • –––, 1908a, “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Mathematische Annalen 65: 107–128. Reprinted in Zermelo 2010: 120–159, with a facing-page English translation, and an Introduction by Michael Hallett (2010b). English translation also in van Heijenoort 1967: 183–198. (Scholar)
  • –––, 1908b, “Untersuchungen über die Grundlagen der Mengenlehre, I”, Mathematische Annalen 65: 261–281. Reprinted in Zermelo 2010: 189–228, with a facing-page English translation, and an Introduction by Ulrich Felgner (2010). English translation also in van Heijenoort 1967: 201–215. (Scholar)
  • –––, 1929, “Über den Begriff von Definitheit in der Axiomatik”, Fundamenta Mathematicae 14: 339–344. Reprinted with facing-page English translation in Zermelo 2010: 358–367, with an Introduction by Heinz-Dieter Ebbinghaus (2010). (Scholar)
  • –––, 1930, “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre”, Fundamenta Mathematicae 16: 29–47. Reprinted with facing-page English translation in Zermelo 2010: 400–431, with an Introduction by Akihiro Kanamori. English translation also in Ewald 1996, Volume 2, pp. 1219–1233. (Scholar)
  • –––, 2010, Collected Works. Volume I: Set Theory, Miscellanea, H.-D. Ebbinghaus and A. Kanamori (eds.), Berlin: Springer. (Scholar)

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