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Zermelo and Set Theory

Published online by Cambridge University Press:  15 January 2014

Akihiro Kanamori
Akihiro Kanamori*
Affiliation:
Department of Mathematics, Boston University, Boston, MA 02215, E-mail: , [email protected]
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Extract

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

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Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 4 , December 2004 , pp. 487 - 553
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1975] Barwise, Jon, Admissible sets and structures, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[1995] Bell, John L., Type reducing correspondences and well-orderings: Frege's and Zermelo's constructions re-examined, The Journal of Symbolic Logic, vol. 60 (1995), pp. 209–221.CrossRefGoogle Scholar
[1901] Bernstein, Felix, Untersuchungen aus der Mengenlehre, 1901, inaugural dissertation, Göttingen, printed in Halle, reprinted in Mathematische Annalen, vol. 61 (1905), pp. 111–155.Google Scholar
[1997] Boolos, George, Constructing Cantorian counterexamples, The Journal of Philosophical Logic, vol. 26 (1997), pp. 237–239, reprinted in [1998] below, pp. 339–341.CrossRefGoogle Scholar
[1998] Boolos, George, Logic, logic, and logic (Jeffrey, Richard, editor), Harvard University Press, Cambridge, 1998.Google Scholar
[1898] Borel, Emile, Lecons sur la théorie des fonctions, Gauthier-Villars, Paris, 1898.Google Scholar
[1921] Borel, Emile, La théorie du jeu et les équations intégrales á noyau symétrique, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 173 (1921), pp. 1304–1308, reprinted in [1972] below, pp. 901–904, translated in Econometrica , vol. 21 (1953), pp. 97–100.Google Scholar
[1972] Borel, Emile, Oeuvres de Emile Borel, Editions de C.N.R.S., Paris, 1972.Google Scholar
[1956] Bourbaki, Nicolas, Eléments de mathématique. I. Théorie des Ensembles. Chapter III: Ensembles ordonnés, cardinaux, nombres entiers, Actualites Scientifiques et Industrielles #1243, Hermann, Paris, 1956.Google Scholar
[1891] Cantor, Georg, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 1 (1891), pp. 75–78, reprinted in [1932] below, pp. 278–280, translated in Ewald [1996], vol. 2, pp. 920–922.Google Scholar
[1895] Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre. I, Mathematische Annalen, vol. 46 (1895), pp. 481–512, translated in [1915] below, reprinted in [1932] below, pp. 282–311.CrossRefGoogle Scholar
[1897] Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre. II, Mathematische Annalen, vol. 49 (1897), pp. 207–246, translated in [1915] below, reprinted in [1932] below, pp. 312–351.CrossRefGoogle Scholar
[1915] Cantor, Georg, Contributions to the founding of the theory of transfinite numbers, including translations of [1895] and [1897] above with introduction and notes by Jourdain, Philip E. B., Open Court, Chicago, 1915, reprinted by Dover, New York, 1965.Google Scholar
[1932] Cantor, Georg, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, (Zermelo, Ernst, editor), Julius Springer, Berlin, 1932, reprinted by Springer-Verlag, Berlin, 1980.Google Scholar
[1934] Church, Alonzo, The Richard paradox, American Mathematical Monthly, vol. 41 (1934), pp. 356–361.CrossRefGoogle Scholar
[1997] Dawson, John W. Jr., Logical dilemmas. The life and work of Kurt Gödel, A.K. Peters, Cambridge, 1997.Google Scholar
[1888] Dedekind, Richard, Was sind und was sollen die Zahlen?, F. Vieweg, Braunschweig, 1888, sixth, 1930 edition reprinted in [1932] below, pp. 335–390, second, 1893 edition translated in [1963] below, pp. 29–115, third edition translated with commentary in Ewald [1996], vol. 2, pp. 787–833.Google Scholar
[1932] Dedekind, Richard, Gesammelte mathematische Werke, (Fricke, Robert, Noether, Emmy, and Ore, Öystein, editors), vol. 3, F. Vieweg, Braunschweig, 1932, reprinted by Chelsea, New York, 1969.Google Scholar
[1963] Dedekind, Richard, Essays on the theory of numbers, translations by Beman, Wooster W., Dover, New York, 1963, (reprint of original edition, Open Court, Chicago, 1901).Google Scholar
[1997] Dreben, Burton and Kanamori, Akihiro, Hilbert and set theory, Synthèse, vol. 110 (1997), pp. 77–125.CrossRefGoogle Scholar
[2003] Ebbinghaus, Heinz-Dieter, Zermelo: Definiteness and the universe of definable sets, History and Philosophy of Logic, vol. 24 (2003), pp. 197–219.CrossRefGoogle Scholar
[1996] Ewald, William (editor), From Kant to Hilbert: A source book in the foundations of mathematics, Clarendon Press, Oxford, 1996, in two volumes.Google Scholar
[1984] Feferman, Solomon, Kurt Gödel: Conviction and caution, Philosophia Naturalis, vol. 21 (1984), pp. 546–562, reprinted in [1998] below, pp. 150–164.Google Scholar
[1987] Feferman, Solomon, Infinity in mathematics: Is Cantor necessary?, L'infinito nella Scienza (di Francia, G. Toraldo, editor), Instituto della Enciclopedia Italiana, Roma, 1987, pp. 151–209, reprinted in [1998] below, particularly pp. 229–248.Google Scholar
[1998] Feferman, Solomon, In light of logic, Oxford University Press, New York, 1998.CrossRefGoogle Scholar
[1999] Ferreirós, José, Labyrinth of thought: A history of set theory and its role in modern mathematics, Birkhauser Verlag, Basel, 1999.CrossRefGoogle Scholar
[1921] Fraenkel, Abraham, Über die Zermelosche Begründung der Mengenlehre, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 30 (1921), pp. 97–98.Google Scholar
[1922] Fraenkel, Abraham, Zu den Grundlagen der Cantor-Zermelosehen Mengenlehre, Mathematische Annalen, vol. 86 (1922), pp. 230–237.CrossRefGoogle Scholar
[1922a] Fraenkel, Abraham, Über den Begriff “definit” und die Unabhängigkeit des Auswahlaxioms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1922, pp. 253–257, translated in van Heijenoort, [1967], pp. 284–289.Google Scholar
[1925] Fraenkel, Abraham, Untersuchungen Über die Grundlagen der Mengenlehre, Mathematische Zeitschrift, vol. 22 (1925), pp. 250–273.CrossRefGoogle Scholar
[1926] Fraenkel, Abraham, Axiomatische Theorie der geordneten Mengen (Untersuchungen über die Grundlagen der Mengenlehre II), Journal für die reine und angewandte Mathematik, vol. 155 (1926), pp. 129–158.CrossRefGoogle Scholar
[1953] Fraenkel, Abraham, Abstract set theory, North Holland, Amsterdam, 1953.Google Scholar
[1958] Fraenkel, Abraham and Bar-Hillel, Yehoshua, Foundations of set theory, North Holland, Amsterdam, 1958.Google Scholar
[1906] Fréchet, Maurice, Sur quelques points du calcul fonctionnel, Rendiconti Circolo Matematico di Palermo, vol. 22 (1906), pp. 1–74.CrossRefGoogle Scholar
[1879] Frege, Gottlob, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Nebert, Halle, 1879, reprinted by Olms, Hildesheim 1964, translated in van Heijenoort [1967], pp. 1–82.Google Scholar
[1884] Frege, Gottlob, Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl, Wilhelm Köbner, Breslau, 1884, translated with German text by John L. Austin, as The Foundations of Arithmetic, A logico-mathematical enquiry into the concept of number , Blackwell, Oxford, 1950, later editions without German text, Harper, New York.Google Scholar
[1893] Frege, Gottlob, Grundgesetze der Arithmetik, Begriffsschriftlich abgeleitet, vol. 1, Hermann Pohle, Jena, 1893, reprinted by Olms, Hildesheim 1962.Google Scholar
[1903] Frege, Gottlob, Grundgesetze der arithmetik, Begriffsschriftlich abgeleitet, vol. 2, Hermann Pohle, Jena, 1903, reprinted by Olms, Hildesheim 1962.Google Scholar
[1976] Frege, Gottlob, Wissenschaftlicher Briefwechsel (Gabriel, Gottfried et alii, editors), Felix Meiner Verlag, Hamburg, 1976.CrossRefGoogle Scholar
[1980] Frege, Gottlob, Philosophical and mathematical correspondence (Gabriel, Gottfried et alii, editors), The University of Chicago Press, Chicago, 1980, abridged from [1976] above by Brian McGuinnes and translated by Hans Kaal.Google Scholar
[1953] Gale, David and Stewart, Frank M., Infinite games with perfect information, Contributions to the theory of games (Kuhn, Harold W. and Tucker, Alan W., editors), vol. 2, Annals of Mathematics Studies, no. 28, Princeton University Press, Princeton, 1953, pp. 245–266.Google Scholar
[1998] George, Alexander and Velleman, Daniel J., Two conceptions of natural number, Truth in mathematics (Dales, H. G. and Oliveri, G., editors), Clarendon Press, Oxford, 1998, pp. 311–327.CrossRefGoogle Scholar
[1905] Gibbs, J. Willard, Elementare Grundlagen der statistischen Mechanik, Barth, Leipzig, 1905, Ernst Zermelo's German translation of Gibbs's Elementary Principles of Statistical Mechanics , Charles Scribner, New York, 1902.Google Scholar
[1930] Gödel, Kürt, Die Vollständigkeit der Axiome des logischen FunktionenkalkÜls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360, reprinted and translated in [1986] below, pp. 102–123.CrossRefGoogle Scholar
[1931] Gödel, Kürt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198, reprinted and translated with minor emendations by the author in [1986] below, pp. 144–195.CrossRefGoogle Scholar
[1933] Gödel, Kürt, The present situation in the foundations of mathematics , 1933, in [1995] below, pp. 45–53, and the page references are to these.CrossRefGoogle Scholar
[1934] Gödel, Kürt, On undecidable propositions of formal mathematical systems, (mimeographed lecture notes, taken by Kleene, Stephen C. and Rosser, J. Barkley), 1934, reprinted with revisions in [1986] below, pp. 346–371.Google Scholar
[1938] Gödel, Kürt, The consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences U.S.A., vol. 24 (1938), pp. 556–557, reprinted in [1990] below, pp. 26–27.CrossRefGoogle ScholarPubMed
[1939] Gödel, Kürt, Consistency-proof for the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences U.S.A., vol. 25 (1939), pp. 220–224, reprinted in [1990] below, pp. 28–32.CrossRefGoogle ScholarPubMed
[1939a] Gödel, Kürt, Vortrag Göttingen, 1939, text and translation in [1995] below, pp. 126–155, and the page references are to these.Google Scholar
[1940] Gödel, Kürt, The consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory, Annals of Mathematics Studies, #3, Princeton University Press, Princeton, 1940, reprinted in [1990] below, pp. 33–101.Google Scholar
[1940a] Gödel, Kürt, Lecture [on the] consistency [of the] continuum hypothesis, Brown University, 1940, in [1995] below, pp. 175–185, and the page references are to these.CrossRefGoogle Scholar
[1986] Gödel, Kürt, Collected works (Feferman, Solomon, editor), vol. 1, Oxford University Press, New York, 1986.Google Scholar
[1990] Gödel, Kürt, Collected works (Feferman, Solomon, editor), vol. 2, Oxford University Press, New York, 1990.Google Scholar
[1995] Gödel, Kürt, Collected works (Feferman, Solomon, editor), vol. 3, Oxford University Press, New York, 1995.Google Scholar
[2003] Gödel, Kürt, Collected works (Feferman, Solomon and Dawson, John W. Jr., editors), vol. 4, Oxford University Press, Oxford, 2003.Google Scholar
[2003a] Gödel, Kürt, Collected works (Feferman, Solomon and Dawson, John W. Jr., editors), vol. 5, Oxford University Press, Oxford, 2003.Google Scholar
[1988] Göldfarb, Warren, Poincaré against thelogicists, History and philosophy of modern mathematics (Aspray, William and Kitcher, Philip, editors), Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, Minneapolis, 1988, pp. 61–81.Google Scholar
[1977] Grattan-Güinness, Ivor, Dear Russell – Dear Jourdain, 1977, Duckworth & Co., London, and Columbia University Press, New York.Google Scholar
[2000] Grattan-Güinness, Ivor, The search for mathematical roots 1870-1940: Logics, set theories and the foundations of mathematics from Cantor through Russell and Gödel, Princeton University Press, Princeton, 2000.Google Scholar
[1910] Grelling, Kurt, Die Axiome der Arithmetik mit besonderer Berücksichtigung der Beziehungen zur Mengenlehre, Ph.D. thesis , Göttingen, 1910.Google Scholar
[1984] Hallett, Michael, Cantorian set theory and limitation of size, Logic Guides #10, Clarendon Press, Oxford, 1984.Google Scholar
[1903] Hardy, Godfrey H., A theorem concerning the infinite cardinal numbers, The Quarterly Journal of Pure and Applied Mathematics, vol. 35 (1903), pp. 87–94, reprinted in [1979] below, vol. 7, pp. 427–434.Google Scholar
[1979] Hardy, Godfrey H., Collected papers of G. H. Hardy (Busbridge, I. W. and Rankin, R. A., editors), Clarendon, Oxford, 1979.Google Scholar
[1915] Hartogs, Friedrich, Über das Problem der Wohlordnung, Mathematische Annalen, vol. 76 (1915), pp. 436–443.CrossRefGoogle Scholar
[1966] hauschild, K., Bemerkungen, das Fundierungsaxiom betreffend, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 51–56.CrossRefGoogle Scholar
[1904] Hausdorff, Felix, Der Potenzbegriff in der Mengenlehre, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 13 (1904), pp. 569–571.Google Scholar
[1908] Hausdorff, Felix, GrundzÜge einer Theorie der geordneten Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435–505.CrossRefGoogle Scholar
[1914] Hausdorff, Felix, Grundzüge der Mengenlehre, de Gruyter, Leipzig, 1914, reprinted by Chelsea, New York 1965.Google Scholar
[1995] Heck, Richard G. Jr., Definition by induction in Frege's Grundgesetze der Arithmetik, Frege's philosophy of mathematics (Demopoulos, William, editor), Harvard University Press, Cambridge, 1995, pp. 295–333.Google Scholar
[1906] Hessenberg, Gerhard, Grundbegriffe der Mengenlehre, Vandenhoeck & Ruprecht, Göttingen, 1906, reprinted from Abhandlungen der Fries'schen Schule, Neue Folge 1 (1906), pp. 479–706.Google Scholar
[1899] Hilbert, David, Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen, Teubner, Leipzig, 1899, reprinted as Grundlagen der Geometrie , Leipzig, Teubner, 1899.Google Scholar
[1900] Hilbert, David, Mathematische probleme, Vortrag, gehalten auf dem internationalem Mathematiker-Kongress zu Paris, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, (1900), pp. 253–297, reprinted in Archiv fÜr Mathematik und Physik vol. 3 (1901), no. 1, pp. 44–63 and pp. 213–237, translated in Bulletin of the American Mathematical Society , vol. 8 (1902), pp. 437–479, that translation reprinted in Felix E. Browder (editor), Mathematical Developments Arising from Hilbert's Problems, Proceedings of Symposia in Pure Mathematics , vol. 28, American Mathematical Society, Providence, 1976, pp. 1–34.Google Scholar
[1900a] Hilbert, David, Über den Zahlbegriff, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 8 (1900), pp. 180–184, translated in Ewald [1996], vol. 2, pp. 1092–1095.Google Scholar
[1903] Hilbert, David, Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen , 1903, second edition of [1899] above.Google Scholar
[1905] Hilbert, David, Über die Grundlagen der Logik und der Arithmetik, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 (Leipzig) (Krazer, A., editor), B. G. Teubner, 1905, pp. 174–184.Google Scholar
[1928] Hilbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Julius Springer, Berlin, 1928, second edition, 1938, third edition, 1949. Second edition was translated by Lewis M. Hammond, George G. Leckie, and F. Steinhardt as Principles of Mathematical Logic , New York, Chelsea 1950.Google Scholar
[1985] Hollinger, Henry B. and Zenzen, Michael J., The nature of irreversibility, D. Reidel, Dordrecht, 1985.CrossRefGoogle Scholar
[1891] Husserl, Edmund, Besprechung von E. Schroder, Vorlesungen über die Algebra der Logik (exakte Logik), vol. I, Leipzig, 1890, Göttingische Gelehrte Anzeigen, (1891), pp. 243–278, reprinted in Edmund Husserl, Aufsätze und Rezensionen (1890-1910), Husserliana Vol. XXII , Nijhoff, The Hague 1978, pp. 3–43, translated by Dallas Willard in Edmund Husserl, Early Writings in the Philosophy of Logic and Mathematics. Collected Works V , Kluwer, Dordrecht 1994, pp. 52–91.Google Scholar
[1995] Jané, Ignacio, The role of the absolute infinite in Cantor's conception of set, Erkenntnis, vol. 42 (1995), pp. 375–402.CrossRefGoogle Scholar
[1904] Jourdain, Philip E. B., On the transfinite cardinal numbers of well-ordered aggregates, Philosophical Magazine, vol. 7 (1904), pp. 61–75.Google Scholar
[1905] Jourdain, Philip E. B., On a proof that every aggregate can be well-ordered, Mathematische Annalen, vol. 60 (1905), pp. 465–470.CrossRefGoogle Scholar
[1928] Kalmár, Lázló, Zur Theorie der abstrakten Spiele, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum, vol. 4 (1928), pp. 65–85, (from 1946: Acta Scientiarum Mathematicarum, Szeged).Google Scholar
[1996] Kanamori, Akihiro, The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2 (1996), pp. 1–71.Google Scholar
[1997] Kanamori, Akihiro, The mathematical import of Zermelo's Well-Ordering Theorem, this Bulletin, vol. 3 (1997), pp. 281–311.Google Scholar
[2003] Kanamori, Akihiro, The higher infinite, second ed., Springer-Verlag, Heidelberg, 2003.Google Scholar
[1927] Kőnig, Dénes, Über eine Schlussweise aus dem Endlichen ins Unendliche, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae (Szeged), sectio scientiarum mathematicarum, vol. 3 (1927), pp. 121–130.Google Scholar
[1905] Kőnig, Julius Gyuia, Zum Kontinuum-Problem, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 (Krazer, A., editor), B. G. Teubner, Leipzig, 1905, pp. 144–147, reprinted in Mathematische Annalen , vol. 60 (1905), pp. 177–180 with Berichtigung, Mathematische Annalen , vol. 60 (1905), p. 462.Google Scholar
[1911] Korselt, Alwin, Über einen Beweis des äquivalenzsatzes, Mathematische Annalen, vol. 70 (1911), pp. 294–296.CrossRefGoogle Scholar
[1950] Kowalewski, Gerhard, Bestand und Wandel. Meine Lebenserinnerungen. Zugleich ein Beitrag zur nueuren Geschichte der Mathematik, R. Oldenbourg, München, 1950.CrossRefGoogle Scholar
[2004] Krajewski, Stanisław, Gödel on Tarski, Annals of Pure and Applied Logic, vol. 127 (2004), pp. 303–323.CrossRefGoogle Scholar
[1980] Kreisel, Georg, Kurt Gödel, 28 April 1906-14 January 1978, Memoirs of the Fellows of the Royal Society, vol. 26 (1980), pp. 149–224, Corrections vol. 27 (1981), p. 697 and vol. 28 (1982), p. 718.Google Scholar
[1922] Kuratowski, Kazimierz, Une méthode d'élimination des nombres transfinis des raisonnements mathématiques, Fundamenta Mathematicae, vol. 3 (1922), pp. 76–108.CrossRefGoogle Scholar
[1931] Kuratowski, Kazimierz and Tarski, Alfred, Les opérations logiques et les ensembles projectifs, Fundamenta Mathematicae, vol. 17 (1931), pp. 240–248, reprinted in Tarski [1986] vol. 1, pp. 551–559, and translated in Tarski [1983], pp. 143–151.CrossRefGoogle Scholar
[1915] Löwenheim, Leopold, Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447–470, translated in van Heijenoort [1967], pp. 228–251.CrossRefGoogle Scholar
[1927] Luzin, Nikolai, Sur les ensembles analytiques, Fundamenta Mathematicae, vol. 10 (1927), pp. 1–95.CrossRefGoogle Scholar
[1918] Luzin, Nikolai and Sierpiński, Wacław, Sur quelques propriétés des ensembles (A), Bulletin de l'Académie des Sciences Cracovie, Classe des Sciences Mathématiques, Série A, (1918), pp. 35–48.Google Scholar
[1923] Luzin, Nikolai and Sierpiński, Wacław, Sur un ensemble non mesurable B, Journal de Mathématiques Pures et Appliquées, vol. 2 (1923), no. 9, pp. 53–72.Google Scholar
[2001] Mathias, Adrian R. D., Slim models of Zermelo set theory, The Journal of Symbolic Logic, vol. 66 (2001), pp. 487–496.CrossRefGoogle Scholar
[1997] McLarty, Colin, Poincaré: Mathematics & Logic & Intuition, Philosophia Mathematica, vol. 5 (1997), no. 3, pp. 97–115.CrossRefGoogle Scholar
[1917] Mirimanoff, Dimitry, Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles, L'enseignment mathématique, vol. 19 (1917), pp. 3752.Google Scholar
[1917a] Mirimanoff, Dimitry, Remarques sur la théorie des ensembles et les antinomies Cantoriennes. I, L'enseignment mathématique, vol. 19 (1917), pp. 209–217.Google Scholar
[1980] Moore, Gregory H., Beyond first-order logic: The historical interplay between mathematical logic and axiomatic set theory, History and Philosophy of Logic, vol. 1 (1980), pp. 95–137.CrossRefGoogle Scholar
[1982] Moore, Gregory H., Zermelo's Axiom of Choice. Its origins, development, and influence, Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
[1988] Moore, Gregory H., The emergence of first-order logic, History and philosophy of modern mathematics (Aspray, William and Kitcher, Philip, editors), Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, Minneapolis, 1988, pp. 95–135.Google Scholar
[2002] Moore, Gregory H., Die Kontroverse zwischen Gödel und Zermelo, Kurt Gödel, Wahrheit und Beweisbarkeit (Buldt, Bernd et alii, editors), vol. 2, obv-a-hpt, Wien, 2002, pp. 55–64.Google Scholar
[2002a] Moore, Gregory H., Hilbert on the infinite: The role of set theory in the evolution of Hilbert's thought, Historia Mathematica, vol. 29 (2002), pp. 40–64.CrossRefGoogle Scholar
[1939] Mostowski, Andrzej, Über die Unabhangigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32 (1939), pp. 201–252, translated in [1979] below, vol. 1, pp. 290–338.CrossRefGoogle Scholar
[1979] Mostowski, Andrzej, Foundational studies. Selected works (Kuratowski, Kazimierz et alii, editors), North-Holland, Amsterdam, 1979.Google Scholar
[1998] Murawski, Roman, Undefinability of truth. The problem of priority: Tarski vs Gödel, History and Philosophy of Logic, vol. 19 (1998), pp. 153–160.CrossRefGoogle Scholar
[1916] Noether, Emmy, Die allgemeinsten Bereiche aus ganzen transzendenten Zahlen, Mathematische Annalen, vol. 77 (1916), pp. 103–128.CrossRefGoogle Scholar
[1987] Parsons, Charles, Developing arithmetic in set theory without infinity: Some historical remarks, History and Philosophy of Logic, vol. 8 (1987), pp. 201–213.CrossRefGoogle Scholar
[1906] Peano, Giuseppe, Super theorema de Cantor-Bernstein, Rendiconti del Circolo Matematico di Palermo, vol. 21 (1906), pp. 360–366, reprinted in [1957–9] below, vol. 1, pp. 337–344.CrossRefGoogle Scholar
[1957–1959] Peano, Giuseppe, Opera scelte (Cassina, U., editor), Edizioni Cremonese, Rome, 1957–9, in three volumes.Google Scholar
[1990] Peckhaus, Volker, Hilbertprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie, Vanden-hoeck & Ruprecht, Göttingen, 1990.Google Scholar
[1992] Peckhaus, Volker, Hilbert, Zermelo und die Institutionalisierung der mathematischen Logik in Deutschland, Berichte zur Wissenschaftsgeschichte, vol. 15 (1992), pp. 27–38.CrossRefGoogle Scholar
[1994] Peckhaus, Volker, Von Nelson zu Reichenbach. Kurt Greiling in Göttingen und Berlin, Bibliographie der Werke Kurt Grellings, Hans Reichenbach und die Berliner Gruppe (Danneberg, Lutz et alii, editors), Vieweg & Sohn, Braunschwieg, 1994, pp. 53–86.Google Scholar
[2004] Peckhaus, Volker, Paradoxes in Göttingen, One hundred years of Russell's paradox. Papers from the 2001 Munich Russell conference (Link, Godehard, editor), de Gruyter, Berlin, 2004.Google Scholar
[2002] Peckhaus, Volker and Kahle, Reinhard, >Hilbert's paradox, Historia Mathematica, vol. 29 (2002), pp. 157–175.CrossRefGoogle Scholar
[1906] Poincaré, Henri, Les mathématiques de la logique, Revue de métaphysique et de morale, vol. 14 (1906), pp. 17–34, translated in Ewald [1996], vol. 2, pp. 1038–1052.Google Scholar
[1906a] Poincaré, Henri, Les mathématiques et la logique, Revue de métaphysique et de morale, vol. 14 (1906), pp. 294–317, translated in Ewald [1996], vol. 2, pp. 1052–1071.Google Scholar
[1909] Poincaré, Henri, Réflexions sure les deux notes précédentes, Acta Mathematica, vol. 32 (1909), pp. 195–200.CrossRefGoogle Scholar
[1909a] Poincaré, Henri, La logique de l'infini, Revue de métaphysique et de morale, vol. 17 (1909), pp. 461–482.Google Scholar
[2002] Purkert, Walter, GrundzÜge der Mengenlehre – Historische EinfÜhrung, Felix Hausdorff, Gesammelte Werke, vol. 2, 2002, pp. 1–89.Google Scholar
[1981] Rang, Bernhard and Thomas, Wolfgang, Zermelo's discovery of the “Russell paradox”, Historia Mathematica, vol. 8 (1981), pp. 15–22.CrossRefGoogle Scholar
[1905] Richard, Jules, Les principes des mathématiques et le problème des ensembles, Revue générale des sciences pures et appliquées, vol. 16 (1905), p. 541, translated in van Heijenoort [1967], pp. 142–144.Google Scholar
[1963] Rubin, Herman and Rubin, Jean, Equivalents of the Axiom of Choice, North-Holland, Amsterdam, 1963.Google Scholar
[1903] Russell, Bertrand A. W., The principles of mathematics, Cambridge University Press, Cambridge, 1903, later editions, George Allen & Unwin, London.Google Scholar
[1906] Russell, Bertrand A. W., On some difficulties in the theory of transfinite numbers and order types, Proceedings of the London Mathematical Society, vol. 4 (1906), no. 2, pp. 29–53.Google Scholar
[1890] Schröder, Ernst, Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, B. G. Teubner, Leipzig, 1890, reprinted in [1966] below.Google Scholar
[1898] Schröder, Ernst, Über zwei Definitionen der Endlichkeit und G. Cantor'sehe Sätze, Deutsche Akademie der Naturforscher, Nova Acta Academiae Caesareae Leopodino-Carolinae Germanicae Naturae Curiosorum, vol. 71 (1898), pp. 303–362.Google Scholar
[1966] Schröder, Ernst, Vorlesungen über die Algebra der Logik, Chelsea, New York, 1966, three volumes.Google Scholar
[2001] Schwalbe, Ulrich and Walker, Paul, Zermelo and the early history of Game Theory, Games and Economic Behavior, vol. 34 (2001), pp. 123–137.CrossRefGoogle Scholar
[1952] Shepherdson, John C., Inner modelsfor set theory – Part II, The Journal of Symbolic Logic, vol. 17 (1952), pp. 225–237.CrossRefGoogle Scholar
[1999] Sieg, Wilfried, Hilberts programs: 1917-1922, this Bulletin, vol. 5 (1999), pp. 1–44.Google Scholar
[1930] Sierpiński, Wacław and Tarski, Alfred, Sur une propriété caractéristique des nombres inaccessibles, Fundamenta Mathematicae, vol. 15 (1930), pp. 292–300, reprinted in Tarski [1986], vol. 1, pp. 289–297.CrossRefGoogle Scholar
[1920] Skolem, Thoralf, Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Videnskaps-selskapets Skrifter, I, Matematisk-Naturvidenskabelig Klass (#4), (1920), pp. 1–36, reprinted in [1970] below, pp. 103–136, partially translated in van Heijenoort [1967], pp. 252–263.Google Scholar
[1923] Skolem, Thoralf, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingfors den 4-7 Juli 1922, Den femte skandinaviska matematik-erkongressen, Redogörelse, Akademiska-Bokhandeln, Helsinki, 1923, pp. 217–232, reprinted in [1970] below, pp. 137–152, translated in van Heijenoort [1967], pp. 290–301.Google Scholar
[1930] Skolem, Thoralf, Einige Bemerkungen zu der Abhandlung von E. Zermelo: “Über die Definitheit in der Axiomatik”, Fundamenta Mathematicae, vol. 15 (1930), pp. 337–341, reprinted in [1970] below, pp. 275–279.CrossRefGoogle Scholar
[1962] Skolem, Thoralf, Abstract set theory, Notre Dame Mathematical Lecture Notes #8, Notre Dame, 1962.Google Scholar
[1970] Skolem, Thoralf, Selected works in logic (Fenstad, Jens E., editor), Universitetsforlaget, Oslo, 1970.Google Scholar
[1979] Specker, Ernst, Paul Bernays, Logic Colloquium '78 (Boffa, Maurice, van Dalen, Dirk, and McAloon, Kenneth, editors), North-Holland, Amsterdam, 1979.Google Scholar
[1907] Stäckel, Paul, Zu H. Webers Elementarer Mengenlehre, Jahresbericht der deutschen Mathematiker-Vereinigung, vol. 16 (1907), pp. 425–428.Google Scholar
[1998] Tait, William W., Zermelo's conception of set theory and reflection principles, The philosophy of mathematics today (Schirn, Matthias, editor), Oxford University Press, Oxford, 1998, pp. 469–483.CrossRefGoogle Scholar
[1924] Tarski, Alfred, Sur les ensembles finis, Fundamenta Mathematicae, vol. 6 (1924), pp. 45–95, reprinted in [1986] below, vol. 1, pp. 67–117.CrossRefGoogle Scholar
[1931] Tarski, Alfred, Sur les ensembles définissables de nombres réels, Fundamenta Mathematicae, vol. 17 (1931), pp. 210–239, translated in [1983] below, pp. 110–142, and reprinted in [1986] below, vol. 1, pp. 517–548.CrossRefGoogle Scholar
[1933] Tarski, Alfred, Pojȩcie prawdy w jȩzykach nauk dedukcyjnych (The concept of truth in the languages of deductive sciences), Prace Towarzystwa Naukowego Warszawskiego, Wydział III, Nauk Matematyczno-fizycznych (Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III, Sciences Mathématiques et Physiques) #34, 1933, see also [1935] below.Google Scholar
[1935] Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen, German translation of [1933] with a postscript, Studia Philosophica, vol. 1 (1935), pp. 261–405, reprinted in [1986] below, vol. 2, pp. 51–198, translated in [1983] below, pp. 152–278ε.Google Scholar
[1938] Tarski, Alfred, Über unerreichbare Kardinalzahlen, Fundamenta Mathematicae, vol. 30 (1938), pp. 68–89, reprinted in [1986] below, vol. 2, pp. 359–380.CrossRefGoogle Scholar
[1939] Tarski, Alfred, On well-ordered subsets of any set, Fundamenta Mathematicae, vol. 32 (1939) , pp. 176–183, reprinted in [1986] below, vol. 2, pp. 551–558.CrossRefGoogle Scholar
[1983] Tarski, Alfred, Logic, semantics, metamathematics. Papers from 1923 to 1938, second ed., Hackett, Indianapolis, 1983, translations by Woodger, J. H..Google Scholar
[1986] Tarski, Alfred, Collected papers (Givant, Steven R. and McKenzie, Ralph N., editors), Birkhauser, Basel, 1986.Google Scholar
[1987] Taussky-Todd, Olga, Remembrances of Kurt Gödel, Gödel remembered (Wein-gartner, Paul and Schmetterer, Leopold, editors), Bibliopolis, Naples, 1987.Google Scholar
[1993] Taylor, R. Gregory, Zermelo, reductionism, and the philosophy of mathematics, Notre Dame Journal of Formal Logic, vol. 34 (1993), pp. 539–563.CrossRefGoogle Scholar
[2002] Taylor, R. Gregory, Zermelo's Cantonan theory of systems of infinitely long propositions, this Bulletin, vol. 8 (2002), pp. 478–513.Google Scholar
[1958] Ulam, Stanisław M., John von Neumann, 1903–1957, Bulletin of the American Mathematical Society, vol. 64 (1958), pp. 1–49.CrossRefGoogle Scholar
[1999] Uzquiano, Gabriel, Models of second-order Zermelo set theory, this Bulletin, vol. 5 (1999), pp. 289–302.Google Scholar
[2000] Dalen, Dirk van and Ebbinghaus, Heinz-Diter, Zermelo and the Skolem paradox, this Bulletin, vol. 6 (2000), pp. 145–161.Google Scholar
[1967] van Heijenoort, Jean (editor), From Frege to Gödel: A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, 1967, reprinted 2002.Google Scholar
[1923] von Neumann, John, Zur Einführung der transfiniten Zahlen, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae (Szeged), sectio scientiarum mathematicarum, vol. 1 (1923), pp. 199–208, reprinted in [1961] below, vol. 1, pp. 24–33, translated in van Heijenoort [1967], pp. 346–354.Google Scholar
[1925] von Neumann, John, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240, Berichtigung, ibd. vol. 155, p. 128, reprintedin [1961] below, vol. 1, pp. 34–56, translated in van Heijenoort [1967], pp. 393–413.CrossRefGoogle Scholar
[1928] von Neumann, John, Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99 (1928), pp. 373–391, reprinted in [1961] below, vol. 1, pp. 320–338.CrossRefGoogle Scholar
[1928a] von Neumann, John, Die Axiomatisierung der Mengenlehre, Mathematische Zeitschrift, vol. 27 (1928), pp. 669–752, reprinted in [1961] below, vol. 1, pp. 339–422.CrossRefGoogle Scholar
[1928b] von Neumann, John, Zur Theorie der Gesellschaftsspiele, Mathematisches Annalen, vol. 100 (1928), pp. 295–320, reprinted in [1961] below, vol. 6, pp. 1–26.CrossRefGoogle Scholar
[1929] von Neumann, John, Über eine Widerspruchsfreitheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 227–241, reprinted in [1961] below, vol. 1, pp. 494–503.CrossRefGoogle Scholar
[1961] von Neumann, John, John von Neumann. Collected works (Taub, Abraham H., editor), Pergamon Press, New York, 1961.Google Scholar
[1944] von Neumann, John and Morgenstern, Oskar, Theory of games and economic behavior, Princeton University Press, Princeton, 1944.Google Scholar
[1963] Vopěnka, Petr and Hajek, Petr, Über die Gültigkeit des Fundierungsaxioms in speziellen Systemen der Mengentheorie, Zeitschrift für mathematische Logic und Grundlagen der Mathematik, vol. 9 (1963), pp. 235–241.CrossRefGoogle Scholar
[1974] Wang, Hao, From mathematics to philosophy, Humanities Press, New York, 1974.Google Scholar
[1996] Wang, Hao, A logical journey. From Godei to philosophy, The MIT Press, Cambridge, 1996.Google Scholar
[1910] Weyl, Hermann, Über die Definitionen der mathematischen Grundbegriffe, Mathematisch-naturwissenschaftliche Blätter, vol. 7 (1910), pp. 93–95 and 109–113.Google Scholar
[1918] Weyl, Hermann, Das Kontinuum, Verlag von Veit, Leipzig, 1918.CrossRefGoogle Scholar
[1910–1913] Whitehead, Alfred N. and Russell, Bertrand A. W., Principia mathematica, Cambridge University Press, Cambridge, 1910–13, in three volumes.Google Scholar
[1894] Zermelo, Ernst, Untersuchungen zur Variationsrechnung, Ph.D. thesis , University of Berlin, 1894, 97 pp.Google Scholar
[1896] Whitehead, Alfred N. and Russell, Bertrand A. W., Über einen Satz der Dynamik und die mechanische Wärmetheorie, Annalen der Physik und Chemie, Neue Folge, vol. 57 (1896), pp. 485–494.Google Scholar
[1896a] Whitehead, Alfred N. and Russell, Bertrand A. W., Über mechanische Erklärungen irreversibler Vorgänge. Eine Antwort auf Hrn. Boltzmanns “Entgegnung”, Annalen der Physik und Chemie, Neue Folge, vol. 59 (1896), pp. 793–801.Google Scholar
[1901] Whitehead, Alfred N. and Russell, Bertrand A. W., Über die Addition transfiniter Kardinalzahlen, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, (1901), pp. 34–38.Google Scholar
[1902] Whitehead, Alfred N. and Russell, Bertrand A. W., Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche. Erste Mitteilung, Zeitschrift fÜr Mathematik und Physik, vol. 47 (1902), pp. 201–237.Google Scholar
[1904] Whitehead, Alfred N. and Russell, Bertrand A. W., Beweis, dass jede Menge wohlgeordnet werden kann (Aus einem an Herrn Hilbert gerichteten Briefe), Mathematische Annalen, vol. 59 (1904), pp. 514–516, translated in van Heijenoort [1967], pp. 139–141.Google Scholar
[1908] Whitehead, Alfred N. and Russell, Bertrand A. W., Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), pp. 107–128, translated in van Heijenoort [1967], pp. 183–198.Google Scholar
[1908a] Whitehead, Alfred N. and Russell, Bertrand A. W., Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), pp. 261–281, translated in van Heijenoort [1967], pp. 199–215.Google Scholar
[1909] Whitehead, Alfred N. and Russell, Bertrand A. W., Sur les ensembles finis et le principe de l'induction complète, Acta Mathematica, vol. 32 (1909), pp. 185–193.Google Scholar
[1909a] Whitehead, Alfred N. and Russell, Bertrand A. W., Über die Grundlagen der Arithmetik, Atti delIV Congresso Internazionale dei Matematici, Roma 1908 (Castelnuovo, G., editor), vol. 2, Accademia dei Lincei, Roma, 1909, pp. 8–11.Google Scholar
[1913] Whitehead, Alfred N. and Russell, Bertrand A. W., Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proceedings ofthe Fifth International Congress of Mathematicians, Cambridge 1912 (Hobson, Ernest W. and Love, A. E. H., editors), vol. 2, Cambridge University Press, Cambridge, 1913, pp. 501–504.Google Scholar
[1914] Whitehead, Alfred N. and Russell, Bertrand A. W., Über ganze transzendente Zahlen, Mathematische Annalen, vol. 75 (1914), pp. 434–442, with Nachtrag, Alfred N. Whitehead and Bertrand A. W. Russell, Über ganze transzendente Zahlen, Mathematische Annalen , vol. 75 (1914),, p. 442.Google Scholar
[1928] Whitehead, Alfred N. and Russell, Bertrand A. W., Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, vol. 29 (1928), pp. 436–460.Google Scholar
[1929] Whitehead, Alfred N. and Russell, Bertrand A. W., Über den Begriff'der Definitheit in der Axiomatik, Fundamenta Mathematicae, vol. 14 (1929), pp. 339–344.Google Scholar
[1930] Whitehead, Alfred N. and Russell, Bertrand A. W., Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen Über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47, translated in Ewald [1996], vol 2, pp. 1208–1233.Google Scholar
[1932] Whitehead, Alfred N. and Russell, Bertrand A. W., Über Stufen der Quantification und die Logik des Unendlichen, Jahresbericht der deutschen Mathematiker-Vereinigung (Angelegenheiten), vol. 41 (1932), pp. 85–88.Google Scholar
[1932a] Whitehead, Alfred N. and Russell, Bertrand A. W., Über mathematische Systeme und die Logik des Unendlichen, Forschungen und Fortschritte, vol. 8 (1932), pp. 6–7.Google Scholar
[1935] Whitehead, Alfred N. and Russell, Bertrand A. W., Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme (Erste Mitteilung), Fundamenta Mathematicae, vol. 25 (1935), pp. 136–146.Google Scholar
[1904] Zermelo, Ernst and Hahn, Hans, Weiterentwicklung der Variationsrechnung in den letzten Jahren, Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 2, part 1, Teubner, Leipzig, 1904, pp. 626–641.Google Scholar