Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-16T18:02:34.224Z Has data issue: false hasContentIssue false
  • English
  • Français

The Mathematical Development of Set Theory from Cantor to Cohen

Published online by Cambridge University Press:  15 January 2014

Akihiro Kanamori
Akihiro Kanamori*
Affiliation:
Department of Mathematics, Boston University, Boston, Ma 02215, USA.E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf.

What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 2 , Issue 1 , March 1996 , pp. 1 - 71
Copyright
Copyright © Association for Symbolic Logic 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1965] Addison, John W. Jr., Henkin, Leon, and Tarski, Alfred (editors), The theory of models, Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam.Google Scholar
[1990] Albers, Donald J., Alexanderson, Gerald L., and Reid, Constance (editors), More mathematical people, Harcourt Brace Jovanovich, Boston.Google Scholar
[1916] Aleksandrov, Pavel S., Sur la puissance des ensembles mesurables B, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 162, pp. 323–325.Google Scholar
[1929] Aleksandrov, Pavel S. and Urysohn, Paul, Mémoire sur les espaces topologiques compacts, Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen Tweed Reeks, Afdeling Natuurkunde, vol. 14, pp. 1–96.Google Scholar
[1898] Baire, René, Sur les fonctions discontinues qui se rattachent aux fonctions continues, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 129, pp. 1621–1623.Google Scholar
[1899] Baire, René, Sur les fonctions de variables réelles, Annali di Matematica Pura ed Applicata, vol. (3)3, pp. 1–123.Google Scholar
[1967] Banach, Stefan, Oeuvres, vol. 1, Państwowe Wydawnictwo Naukowe, Warsaw, edited by Hartman, Stanislaw and Marczewski, Edward.Google Scholar
[1924] Banach, Stefan and Tarski, Alfred, Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae, vol. 6, pp. 244–277, reprinted in Banach [1967], pp. 118–148, and in Tarski [1986] vol. 1, pp. 121–154.Google Scholar
[1961] Bar-Hillel, Yehoshua, Poznanski, E. I. J., Rabin, Michael O., and Robinson, Abraham (editors), Essays on the foundations of mathematics, Magnes Press, Jerusalem.Google Scholar
[1995] Bartoszyński, Tomek and Judah, Haim, Set theory. On the structure of the real line, A. K. Peters, Wellesley.Google Scholar
[1985] Bell, John L., Boolean-valued models and independence proofs in set theory, second ed., Oxford Logic Guides #12, Oxford University Press, Oxford.Google Scholar
[1964] Benacerraf, Paul and Putnam, Hilary (editors), Philosophy of mathematics. Selected readings, Prentice Hall, Englewood Cliffs, N.J. Google Scholar
[1983] Benacerraf, Paul and Putnam, Hilary (editors), Philosophy of mathematics. Selected readings, second ed., Cambridge University Press, Cambridge.Google Scholar
[1883] Bendixson, Ivar, Quelques théorèmes de la théorie des ensembles de points, Acta Mathematica, vol. 2, pp. 415–429.CrossRefGoogle Scholar
[1937] Bernays, Paul, A system of axiomatic set theory. Part I, The Journal of Symbolic Logic, vol. 2, pp. 65–77.Google Scholar
[1976] Bernays, Paul, A system of axiomatic set theory, Sets and Classes. On the work by Paul Bernays (Müller, Gert H., editor), North-Holland, Amsterdam, pp. 1–119, individually appeared as A system of axiomatic set theory in The Journal of Symbolic Logic , vol. 2 (1937), pp. 65–77; vol. 6 (1941), pp. 1–17; vol. 7 (1942), pp. 65–89, 133–145; vol. 8 (1943), pp. 89–106; vol. 13 (1948), pp. 65–79; and vol. 19 (1954), pp. 81–96.Google Scholar
[1908] Bernstein, Felix, Zur theorie der trigonometrischen Reihen, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, vol. 60, pp. 325–338.Google Scholar
[1984] Blass, Andreas R., Existence of bases implies the Axiom of Choice, Axiomatic set theory (Baumgartner, James E., Martin, Donald A., and Shelah, Saharon, editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, pp. 31–33.CrossRefGoogle Scholar
[1971] Boolos, George S., The iterative concept of set, Journal of Philosophy, vol. 68, pp. 215–231, reprinted in Benacerraf-Putnam [1983], pp. 486–502.CrossRefGoogle Scholar
[1898] Borel, Emile, Leçons sur la théorie des fonctions, Gauthier-Villars, Paris.Google Scholar
[1919] Borel, Emile, Sur la classification des ensembles de mesure nulle, Bulletin de la Société Máthematique de France, vol. 47, pp. 97–125.Google Scholar
[1911] Brouwer, Luitzene. J., Beweis der Invarianz der Dimensionenzahl, Mathematische Annalen, vol. 70, pp. 161–165, reprinted in [1976] below, pp. 430–434.Google Scholar
[1976] Brouwer, Luitzene. J., Collected works, vol. 2, North-Holland, Amsterdam, edited by Freudenthal, Hans.Google Scholar
[1897] Burali-Forti, Cesare, Una questione sui numeri transfini, Rendiconti del Circolo Matematico di Palermo, vol. 11, pp. 154–164, translated in van Heijenoort, [1967], pp. 104–111.CrossRefGoogle Scholar
[1872] Cantor, Georg, Über die Ausdehnung eines Satzes aus der Theorie der trignometrischen Reihen, Mathematische Annalen, vol. 5, pp. 123–132, reprinted in [1932] below, pp. 92–102.Google Scholar
[1874] Cantor, Georg, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 77, pp. 258–262, reprinted in [1932] below, pp. 115–118.Google Scholar
[1878] Cantor, Georg, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 84, pp. 242–258, reprinted in [1932] below, pp. 119–133.Google Scholar
[1879] Cantor, Georg, Über einen Satz aus der Theorie der stetigen Mannigfaltigkeiten, Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, pp. 127–135, reprinted in [1932] below, pp. 134–138.Google Scholar
[1880] Cantor, Georg, Über unendliche, lineare punktmannigfaltigkeiten. II, Mathematische Annalen, vol. 17, pp. 355–358, reprinted in [1932]below, pp. 145–148, but excluding the referenced remark.CrossRefGoogle Scholar
[1883] Cantor, Georg, Über unendliche, lineare punktmannigfaltigkeiten. V, Mathematische Annalen, vol. 21, pp. 545–591, published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen , B. G. Teubner, Leipzig, 1883; reprinted in [1932] below, pp. 165–209.Google Scholar
[1884] Cantor, Georg, Über unendliche, lineare punktmannigfaltigkeiten. VI, Mathematische Annalen, vol. 23, pp. 453–488, reprinted in [1932] below, pp. 210–246.Google Scholar
[1884a] Cantor, Georg, De la puissance des ensembles parfaits de points, Acta Mathematica, vol. 4, pp. 381–392, reprinted in [1932] below, pp. 252–260.Google Scholar
[1887] Cantor, Georg, Mitteilungen zur Lehre vom Transfiniten, Zeitschrift für Philosophic und philosophische Kritik, vol. 91, pp. 81–125, 252–270, reprinted in [1932] below, pp. 378–439.Google Scholar
[1891] Cantor, Georg, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. I, pp. 75–78, reprinted in [1932] below, pp. 278–280.Google Scholar
[1895] Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre. I, Mathematische Annalen, vol. 46, pp. 481–512, translated in [1915] below; reprinted in [1932] below, pp. 282–311.Google Scholar
[1897] Cantor, Georg, Beiträge zur Begründung der transfiniten Mengenlehre. II, Mathematische Annalen, vol. 49, pp. 207–246, translated in [1915] below; reprinted in [1932] below, pp. 312–351.Google Scholar
[1915] Cantor, Georg, Contributions to the founding of the theory of transfinite numbers, Open Court, Chicago, translation of [1895] and [1897] above with introduction and notes by Jourdain, Philip E. B.; reprinted Dover, New York, 1965.Google Scholar
[1932] Cantor, Georg, Gesammelte Abhandlungen mathematicschen und philosophischen Inhalts, Julius Springer, Berlin, edited by Zermelo, Ernst, reprinted Springer-Verlag, Berlin, 1980.Google Scholar
[1962] Cavaillès, Jean, Philosophie mathématique, Hermann, Paris, includes French translation of Noether-Cavaillès [1937].Google Scholar
[1990] Chang, Chen-Chung and Keisler, H. Jerome, Model theory, third ed., North-Holland, Amsterdam.Google Scholar
[1979] Coffa, J. Alberto, The humble origins of Russell's paradox, Russell, vol. 33-34, pp. 31–37.Google Scholar
[1963] Cohen, Paul J., The independence of the Continuum Hypothesis. I, Proceedings of the National Academy of Sciences U.S.A., vol. 50, pp. 1143–1148.Google Scholar
[1964] Cohen, Paul J., The independence of the Continuum Hypothesis. II, Proceedings of the National Academy of Sciences U.S.A., vol. 51, pp. 105–110.CrossRefGoogle ScholarPubMed
[1965] Cohen, Paul J., Independence results in set theory, in Addison-Henkin-Tarski [1965], pp. 39–54.CrossRefGoogle Scholar
[1979] Dauben, Joseph W., Georg Cantor. His mathematics and philosophy of the infinite, Harvard University Press, Cambridge, paperback edition 1990.Google Scholar
[1871] Dedekind, Richard, Supplement to the second edition of Dirichlet's Vorlesungen über Zahlentheorie, F. Vieweg, Braunschweig, reprinted in [1932] below, vol. 3, pp. 399–407.Google Scholar
[1872] Dedekind, Richard, Stetigkeit und irrationale Zahlen, F. Vieweg, Braunschweig, fifth, 1927 edition reprinted in [1932] below, vol. 3, pp. 315–334; translated in [1963] below, pp. 1–27.Google Scholar
[1888] Dedekind, Richard, Was sind und was sollen die Zahlen?, F. Vieweg, Braunschweig, sixth, 1930 edition reprinted in [1932] below, vol. 3, pp. 335–390; second, 1893 edition translated in [1963] below, pp. 29–115.Google Scholar
[1900] Dedekind, Richard, Über die von drei Moduln erzeugte Dualgruppe, Mathematische Annalen, vol. 53, pp. 371–403, reprinted in [1932] below, vol. 2, pp. 236–271.CrossRefGoogle Scholar
[1932] Dedekind, Richard, Gesammelte mathematische Werke, F. Vieweg, Baunschweig, edited by Robert Fricke, Emmy Noether and Öystein Ore, reprinted Chelsea, New York, 1969.Google Scholar
[1963] Dedekind, Richard, Essays on the theory of numbers, Dover, New York, translation by Wooster W. Beman (reprint of original edition, Open Court, Chicago, 1901).Google Scholar
[1979] Dreben, Burton and Goldfarb, Warren D., The decision problem: Solvable classes of quantificational formulas, Addison-Wesley, Reading.Google Scholar
[1869] du Bois-Reymond, Paul, Bemerkungen über die verschidenen Werthe, welche eine Function zweier reellen Variabeln erhält, wenn man diese Variabeln entweder nach einander oder gewissen Beziehungen gemäss gleichzeitig verschwinden lässt, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 70, pp. 10–45.Google Scholar
[1875] du Bois-Reymond, Paul, Über asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen, Mathematische Annalen, vol. 8, pp. 363–414.Google Scholar
[1976] Dugac, Pierre, Richard Dedekind et les fondements des mathématiques, Collection des travaux de l'Académie Internationale d'Histoire des Sciences #24, J. Vrin, Paris.Google Scholar
[1964] Easton, William B., Powers of regular cardinals, Ph.D. thesis , Princeton University, abstracted as: Proper classes of generic sets, Notices of the American Mathematical Society , vol. 11 (1964), p. 205; published in abridged form as [1970] below.Google Scholar
[1970] Easton, William B., Powers of regular cardinals, Annals of Mathematical Logic, vol. 1, pp. 139–178.Google Scholar
[1956] Ehrenfeucht, Andrzej and Mostowski, Andrzej M., Models of axiomatic theories admitting automorphisms, Fundamenta Mathematicae, vol. 43, pp. 50–68, reprinted in Mostowski [1979], pp. 494–512.CrossRefGoogle Scholar
[1962] Erdös, Paul and Hajnal, András, Some remarks concerning our paper “On the structure of set mappings”, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 13, pp. 223–226.Google Scholar
[1984] Erdös, Paul, Hajnal, András, Máté, Attila, and Rado, Richard, Combinatorial set theory: Partition relations for cardinals, North-Holland, Amsterdam.Google Scholar
[1965] Erdös, Paul, Hajnal, András, and Rado, Richard, Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 93–196.CrossRefGoogle Scholar
[1956] Erdös, Paul and Rado, Richard, A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62, pp. 427–489, reprinted in Gessel-Rota [1987], pp. 179–241.CrossRefGoogle Scholar
[1943] Erdös, Paul and Tarski, Alfred, On families of mutually exclusive sets, Annals of Mathematics, vol. 44, pp. 315–329, reprinted in Tarski [1986], vol. 2, pp. 591–605.CrossRefGoogle Scholar
[1961] Erdös, Paul and Tarski, Alfred, On some problems involving inaccessible cardinals, in Bar-Hillel–Poznanski–Rabin–Robinson [1961], pp. 50–82; reprinted in Tarski [1986], vol. 4, pp. 79–111.Google Scholar
[1963] Feferman, Solomon and Levy, Azriel, Independence results in set theory by Cohen's method. II (abstract), Notices of the American Mathematical Society, vol. 10, p. 593.Google Scholar
[1971] Felgner, Ulrich, Models of ZF-set theory, Lecture Notes in Mathematics #223, Springer-Verlag, Berlin.Google Scholar
[1995] Ferreirós, José, “What fermented in me for years”: Cantor's discovery of transfinite numbers, Historia Mathematica, vol. 22, pp. 33–42.Google Scholar
[1956] Fodor, Géza, Eine Bemerking zur Theorie der regressiven Funktionen, Acta Scientiarum Mathematicarum, Szeged, vol. 17, pp. 139–142.Google Scholar
[1921] Fraenkel, Abraham A., Über die Zermelosche Begründung der Mengenlehre (abstract), Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 30, no. II, pp. 97–98.Google Scholar
[1922] Fraenkel, Abraham A., Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86, pp. 230–237.Google Scholar
[1922a] Fraenkel, Abraham A., Über den Begriff ‘definit’ und die Unabhängigkeit des Auswahlaxioms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 253–257, translated in van Heijenoort, [1967], pp. 284–289.Google Scholar
[1930] Fraenkel, Abraham A., Georg Cantor, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 39, pp. 189–266, published separately as Georg Cantor , B. G. Teubner, Leipzig, 1930; published in abridged form in Cantor [1932], pp. 452-483.Google Scholar
[1953] Fraenkel, Abraham A., Abstract set theory, North-Holland, Amsterdam.Google Scholar
[1879] Frege, Gottlob, Begriffsschrift, eine der arithmetischen nachgëbildete Formelsprache des reinen Denkens, Nebert, Halle, reprinted Hildesheim, Olms, 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, translated with German text by John L. Austin, as The foundations of arithmetic, a logico-mathematical enquiry into the concept of number , Oxford, Blackwell, 1950; later editions without German text, Harper, New York.Google Scholar
[1891] Frege, Gottlob, Function und Begriff, Hermann Pohle, Jena, translated in [1952] below, pp. 21–41.Google Scholar
[1893] Frege, Gottlob, Grundgesetze der Arithmetik, Begriffsschriftlich abgeleitet, vol. 1, Hermann Pohle, Jena, reprinted Hildesheim, Olms, 1962; partially translated in [1964] below.Google Scholar
[1895] Frege, Gottlob, Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik, Archiv für systematische Philosophie, vol. 1, pp. 433–456, translated in [1952] below, pp. 86-106.Google Scholar
[1903] Frege, Gottlob, Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik, Archiv für systematische Philosophie, vol. 2 of [1893]; reprinted Hildesheim, Olms, 1962.Google Scholar
[1952] Frege, Gottlob, Translations from the philosophical writings of Gottlob Frege, Blackwell, Oxford, translated and edited by Geach, Peter and Black, Max, second, revised edition 1960; latest edition, Rowland & Littlewood, Totowa, 1980.Google Scholar
[1964] Frege, Gottlob, The basic laws of arithmetic. Exposition of the system, California Press, Berkeley, California, edited by Furth, Montgomery.Google Scholar
[1992] Garciadiego, Alejandro, Bertrand Russell and the origins of the set-theoretic “paradoxes”, Birkhäuser, Boston.Google Scholar
[1936] Gentzen, Gerhard, Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematicshe Annalen, vol. 112, pp. 493–565, translated in [1969] below, pp. 132–213.CrossRefGoogle Scholar
[1943] Gentzen, Gerhard, Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Mathematicshe Annalen, vol. 119, pp. 140–161, translated in [1969] below, pp. 287–308.Google Scholar
[1969] Gentzen, Gerhard, The collected papers of Gerhard Gentzen, North-Holland, Amsterdam, edited by Szabo, M. E..Google Scholar
[1987] Gessel, Ira and Rota, Gian-Carlo (editors), Classic papers in combinatorics, Birkhäuser, Boston.CrossRefGoogle Scholar
[1992] Gitik, Moti and Magidor, Menachem, The singular cardinals hypothesis revisited, Set theory of the Continuum (Judah, Haim, Just, Winfried, and Woodin, W. Hugh, editors), Mathematical Sciences Research Institute Publication #26, Springer-Verlag, Berlin, pp. 243–279.Google Scholar
[1930] Gödel, Kurt F., Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37, pp. 349–360, reprinted and translated in [1986] below, pp. 102–123.CrossRefGoogle Scholar
[1931] Gödel, Kurt F., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , Monatshefte für Mathematik und Physik, vol. 38, pp. 173–198, reprinted and translated with minor emendations by the author in [1986] below, pp. 144–195.Google Scholar
[1938] Gödel, Kurt F., 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, pp. 556–557, reprinted in [1990] below, pp. 26–27.Google Scholar
[1939] Gödel, Kurt F., Consistency-proof for the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences U.S.A., vol. 25, pp. 220–224, reprinted in [1990] below, pp. 28–32.Google Scholar
[1940] Gödel, Kurt F., 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, reprinted in [1990] below, pp. 33–101.Google Scholar
[1947] Gödel, Kurt F., What is Cantor's Continuum problem?, American Mathematical Monthly, vol. 54, pp. 515–525, Errata, vol. 55 (1948), p. 151; reprinted in [1990] below, pp. 176–187; revised and expanded version in Benacerraf-Putnam [1964], pp. 258–273; this version reprinted with minor emendations by the author in [1990] below, pp. 254–270.Google Scholar
[1986] Gödel, Kurt F., Collected works, vol. 1, Oxford University Press, New York, edited by Feferman, Solomon and others.Google Scholar
[1990] Gödel, Kurt F., Collected works, vol. 2, Oxford University Press, New York, edited by Feferman, Solomon and others.Google Scholar
[1979] Goldfarb, Warren D., Logic in the twenties: the nature of the quantifier, The Journal of Symbolic Logic, vol. 44, pp. 351–368.Google Scholar
[1990] Graham, Ronald L., Rothschild, Bruce L., and Spencer, Joel H., Ramsey theory, second ed., Wiley & Sons, New York.Google Scholar
[1974] Grattan-Guinness, Ivor, The rediscovery of the Cantor-Dedekind correspondence, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 76, pp. 104–139.Google Scholar
[1978] Grattan-Guinness, Ivor, How Bertrand Russell discovered his paradox, Historia Mathematica, vol. 5, pp. 127–137.Google Scholar
[1994] Gray, Robert, Georg Cantor and transcendental numbers, American Mathematical Monthly, vol. 101, pp. 819–832.Google Scholar
[1971] Hájek, Petr, Sets, semisets, models, in Scott [1971], pp. 67–81.Google Scholar
[1956] Hajnal, András, On a consistency theorem connected with the generalized continuum problem, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 2, pp. 131–136.CrossRefGoogle Scholar
[1961] Hajnal, András, On a consistency theorem connected with the generalized continuum problem, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 12, pp. 321–376.Google Scholar
[1984] Hallett, Michael, Cantorian set theory and limitation of size, Logic Guides #10, Clarendon Press, Oxford.Google Scholar
[1961] Halpern, James D., The independence of the Axiom of Choice from the boolean prime ideal theorem (abstract), Notices of the American Mathematical Society, vol. 8, pp. 279–280.Google Scholar
[1964] Halpern, James D., The independence of the Axiom of Choice from the boolean prime ideal theorem, Fundamenta Mathematicae, vol. 55, pp. 57–66.CrossRefGoogle Scholar
[1966] Halpern, James D. and Läuchli, Hans, A partition theorem, Transactions of the American Mathematical Society, vol. 124, pp. 360–367.Google Scholar
[1971] Halpern, James D. and Levy, Azriel, The Boolean prime ideal theorem does not imply the axiom of choice, in Scott [1971], pp. 83–134.Google Scholar
[1905] Hamel, Georg, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f(x + y) = f(x) + f (y), Mathematische Annalen, vol. 60, pp. 459–462.CrossRefGoogle Scholar
[1964] Hanf, William P., Incompactness in languages with infinitely long expressions, Fundamenta Mathematicae, vol. 53, pp. 309–324.Google Scholar
[1915] Hartogs, Friedrich, Über das Problem der Wohlordnung, Mathematische Annalen, vol. 76, pp. 436–443.Google Scholar
[1908] Hausdorff, Felix, Grundzüge einer Theorie der geordneten Mengen, Mathematische Annalen, vol. 65, pp. 435–505.CrossRefGoogle Scholar
[1914] Hausdorff, Felix, Grundzüge der Mengenlehre, de Gruyter, Leipzig, reprinted Chelsea, New York, 1965.Google Scholar
[1914a] Hausdorff, Felix, Bemerkung über den Inhalt von Punktmengen, Mathematische Annalen, vol. 75, pp. 428–433.CrossRefGoogle Scholar
[1916] Hausdorff, Felix, Die Mächtigkeit der Borelschen Mengen, Mathematische Annalen, vol. 77, pp. 430–437.CrossRefGoogle Scholar
[1932] Hausdorff, Felix, Zur Theorie der linearen metrischen Räume, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 167, pp. 294–311.Google Scholar
[1937] Hausdorff, Felix, Mengenlehre , third, revised edition of [1914]; translated by John R. Auman as Set Theory , Chelsea, New York, 1962.Google Scholar
[1975] Hawkins, Thomas W., Lebesgue's theory of integration. Its origins and development, second ed., Chelsea, New York.Google Scholar
[1949] Henkin, Leon, The completeness of the first-order functional calculus, The Journal of Symbolic Logic, vol. 14, pp. 159–166.Google Scholar
[1906] Hessenberg, Gerhard, Grundbegriffe der mengenlehre, Vandenhoeck & Ruprecht, Göttingen, reprinted from Abhandlungen der Fries'schen Schule, Neue Folge , vol. 1 (1906), pp. 479–706.Google Scholar
[1900] Hilbert, David, Mathematische Probleme, Vortrag, gehalten auf dem internationalem Mathematiker-Kongress zu Paris. 1900. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp. 253–297, translated in the Bulletin of the American Mathematical Society , vol. 8 (1902), pp. 437–479.Google Scholar
[1926] Hilbert, David, Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161–190, translated in van Heijenoort [1967], pp. 367–392; partially translated in Benacerraf-Putnam [1983], pp. 183–201.Google Scholar
[1928] Hilbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Julius Springer, Berlin, 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 , Chelsea, New York, 1950.Google Scholar
[1993] Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[1973] Jech, Thomas J., The Axiom of Choice, North-Holland, Amsterdam.Google Scholar
[1978] Jech, Thomas J., Set theory, Academic Press, New York.Google Scholar
[1995] Jech, Thomas J., Singular cardinals and the PCF theory, this Bulletin, vol. 1, pp. 408–424.Google Scholar
[1966] Jech, Thomas J. and Sochor, Antonin, Applications of the θ-model, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14, pp. 351–355.Google Scholar
[1972] Jensen, Ronald B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4, pp. 229–308.Google Scholar
[1968] Kac, Mark and Ulam, Stanislaw M., Mathematics and logic, Praeger, New York.Google Scholar
[1994] Kanamori, Akihiro, The higher infinite, Springer-Verlag, Berlin.Google Scholar
[1995] Kanamori, Akihiro, The emergence of descriptive set theory, From Dedekind to Gödel. Essays on the development of the foundations of mathematics (Hintikka, Jaakko, editor), Synthese Library No. 251, Kluwer Publishing, Dordrecht, pp. 241–262.Google Scholar
[∞] Kanamori, Akihiro, The higher infinite II, Springer-Verlag, Berlin, to appear.Google Scholar
[1987] Kechris, Alexander S. and Louveau, Alain, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series #128, Cambridge University Press, Cambridge.Google Scholar
[1962] Keisler, H. Jerome, Some applications of the theory of models to set theory, Logic, methodology and philosophy of science (Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors), Stanford University Press, Stanford, Proceedings of the 1960 [and first] International Congress (Stanford, California), pp. 80–86.Google Scholar
[1962a] Keisler, H. Jerome, The equivalence of certain problems in set theory with problems in the theory of models (abstract), Notices of the American Mathematical Society, vol. 9, pp. 339–340.Google Scholar
[1964] Keisler, H. Jerome and Tarski, Alfred, From accessible to inaccessible cardinals, Fundamenta Mathematicae, vol. 53, pp. 225–308, corrections vol. 57 (1965), p. 119; reprinted in Tarski [1986] vol. 4, pp. 129–213.Google Scholar
[1927] könig, Dénes, Über eine Schlussweise aus dem Endlichen ins Unendliche: Punktmengen. Kartenfärben. Verwandtschaftsbeziehungen. Schachspiel, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, sectio scientiarum mathematicarum, vol. 3, pp. 121–130.Google Scholar
[1905] König, Julius Gyula, Zum Kontinuum-Problem, Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 (Krazer, A., editor), B. G. Teubner, Leipzig, reprinted in Mathematische Annalen , vol. 60 (1905), pp. 177–180 and Berichtigung, p. 462, pp. 144–147.Google Scholar
[1971] Kreisel, Georg, Observations on popular discussions of foundations, in Scott [1971], pp. 189–198.Google Scholar
[1980] Kunen, Kenneth, Set theory. An introduction to independence proofs, North-Holland, Amsterdam.Google Scholar
[1984] Kunen, Kenneth and Vaughan, Jerry E. (editors), Handbook of set-theoretic topology, North-Holland, Amsterdam.Google Scholar
[1921] Kuratowski, Kazimierz, Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta Mathematicae, vol. 2, pp. 161–171, reprinted in [1988] below, pp. 1–11.Google Scholar
[1922] Kuratowski, Kazimierz, Üne méthode d'élimination des nombres transfinis des raisonnements mathématiques, Fundamenta Mathematicae, vol. 3, pp. 76–108.Google Scholar
[1988] Kuratowski, Kazimierz, Selected papers, Państwowe Wydawnictwo Naukowe, Warsaw, edited by Borsuk, Karol and others.Google Scholar
[1931] Kuratowski, Kazimierz and Tarski, Alfred, Les opérations logiques et les ensembles projectifs, Fundamenta Mathematicae, vol. 17, pp. 240–248, reprinted in Tarski [1986] vol. 1, pp. 551–559, and in Kuratowski [1988], pp. 367–375; translated in Tarski [1983], pp. 143–151.Google Scholar
[1935] Kurepa, Djuro R., Ensembles ordonnés et ramifiés. Thèse, Paris, published as Publications mathématiques de l'Université de Belgrade, vol. 4, pp. 1–138.Google Scholar
[1942] Kuratowski, Kazimierz, A propos d'une généralisation de la notion d'ensembles bien ordonnés, Acta Mathematica, vol. 75, pp. 139–150.Google Scholar
[1959] Kuratowski, Kazimierz, On the cardinal number of ordered sets and of symmetrical structures in dependence on the cardinal numbers of its chains and antichains, Glasnik Matematičko-fizički i astronomski, Periodicum mathematico-physicum et astronomicum, vol. 14, pp. 183–203.Google Scholar
[1962] Läuchli, Hans, Auswahlaxiom in der Algebra, Commentarii Mathematici Helvetici, vol. 37, pp. 1–18.Google Scholar
[1994] Lavine, Shaughan M., Understanding the infinite, Harvard University Press, Cambridge.Google Scholar
[1902] Lebesgue, Henri, Intégrale, longueur, aire, Annali di Matematica Pura ed Applicata, vol. (3) 7, pp. 231–359, reprinted in [1972] below, vol. 1, pp. 203–331.Google Scholar
[1905] Lebesgue, Henri, Sur les fonctions représentables analytiquement, Journal de Mathématiques Pures et Appliquées, vol. (6)1, pp. 139–216, reprinted in [1972] below, vol. 3, p. 103–180.Google Scholar
[1972] Lebesgue, Henri, Oeuvres scientifiques, Kundig, Geneva.Google Scholar
[1957] Levy, Azriel, Indépendance conditionelle de V = L et d'axiomes qui se rattachent au système de M. Gödel, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 245, pp. 1582–1583.Google Scholar
[1960] Levy, Azriel, Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10, pp. 223–238.Google Scholar
[1960a] Levy, Azriel, Principles of reflection in axiomatic set theory, Fundamenta Mathematicae, vol. 49, pp. 1–10.Google Scholar
[1963] Levy, Azriel, Independence results in set theory by Cohen's method. I, III, IV (abstracts), Notices of the American Mathematical Society, vol. 10, pp. 592–593.Google Scholar
[1964] Levy, Azriel, Measurable cardinals and the continuum hypothesis (abstract), Notices of the American Mathematical Society, vol. 11, pp. 769–770.Google Scholar
[1965] Levy, Azriel, Definability in axiomatic set theory I, Logic, methodology and philosophy of science (Bar-Hillel, Yehoshua, editor), Proceedings of the 1964 International Congress, Jerusalem, North-Holland, Amsterdam, pp. 127–151.Google Scholar
[1970] Levy, Azriel, Definability in axiomatic set theory II, Mathematical logic and foundations of set theory (Bar-Hillel, Yehoshua, editor), North-Holland, Amsterdam, pp. 129–145.Google Scholar
[1967] Levy, Azriel and Solovay, Robert M., Measurable cardinals and the Continuum Hypothesis, Israel Journal of Mathematics, vol. 5, pp. 234–248.Google Scholar
[1905] Lindelöf, Ernst, Remarques sur un théorème fondamental de la théorie des ensembles, Acta Mathematica, vol. 29, pp. 183–190.Google Scholar
[1926] Lindenbaum, Adolf and Tarski, Alfred, Communication sur les recherches de la théorie des ensembles, Sprawozdana z Posiedzen Towarzystwa Naukowego Warszawskiego, Wydzial III, Nauk Matematyczno-fizycznych (Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, Sciences Mathematiques et Physiques), vol. 19, pp. 299–330, reprinted in Tarski [1986] vol. 1, pp. 171–204.Google Scholar
[1844] Liouville, Joseph, Des remarques relatives 1° à des classes très-étendues de quantités dont la valeur n'est ni rationelle ni même réductible à des irrationnelles algébriques; 2° à un passage du livre des Principes où Newton calcule l'action exercée par une sphère sur un point extérieur, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 18, pp. 883–885.Google Scholar
[1851] Liouville, Joseph, Sur des classes très-étendues de quantités dont la valeur nest ni algébrique ni même réductible à des irrationnelles algébriques, Journal de Mathématiques Pures et Appliquées, vol. 16, pp. 133–142.Google Scholar
[1955] łoś, Jerzy, Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems (Skolem, Thoralf et al., editors), North-Holland, Amsterdam, pp. 98–113.Google Scholar
[1915] Löwenheim, Leopold, Über Möglichkeiten im Relativkalkul, Mathematische Annalen, vol. 76, pp. 447–470, translated in van Heijenoort [1967], pp. 228–251.Google Scholar
[1914] Luzin, Nikolai N., Sur un problème de M. Baire, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 158, pp. 1258–1261.Google Scholar
[1917] Luzin, Nikolai N., Sur la classification de M. Baire, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 164, pp. 91–94.Google Scholar
[1925] Luzin, Nikolai N., Sur un problème de M. Emile Borel et les ensembles projectifs de M. Henri Lebesgue; les ensembles analytiques, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 180, pp. 1318–1320.Google Scholar
[1925a] Luzin, Nikolai N., Sur les ensembles projectifs de M. Henri Lebesgue, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 180, pp. 1572–1574.Google Scholar
[1925b] Luzin, Nikolai N., Les propriétés des ensembles projectifs, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Paris, vol. 180, pp. 1817–1819.Google Scholar
[1918] Luzin, Nikolai N. and Sierpiński, Wacław, Sur quelques propriétés des ensembles (A), Bulletin de l'Académie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles, Série A, pp. 35–48, reprinted in Sierpiński, [1975], pp. 192–204.Google Scholar
[1923] Luzin, Nikolai N., Sur un ensemble non mesurable B, Journal de Mathématiques Pures et Appliquées, vol. (9)2, pp. 53–72, reprinted in Sierpinski [1975], pp. 504–519.Google Scholar
[1911] Mahlo, Paul, Über lineare transfinite Mengen, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig Mathematische-Physische Klasse, vol. 63, pp. 187–225.Google Scholar
[1912] Mahlo, Paul, Zur Theorie und Anwendung der ρ0-Zahlen, Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig Mathematische-Physische Klasse, vol. 64, pp. 108–112.Google Scholar
[1913] Mahlo, Paul, Zur Theorie und Anwendung der ρ0-Zahlen II, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, vol. 65, pp. 268–282.Google Scholar
[1913a] Mahlo, Paul, Über Teilmengen des Kontinuums von dessen Mächtigkeit, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, vol. 65, pp. 283–315.Google Scholar
[1971] Mansfield, Richard B., A Souslin operation for , Israel Journal of Mathematics, vol. 9, pp. 367–379.Google Scholar
[1969] Martin, Donald A. and Solovay, Robert M., A basis theorem for sets of reals, Annals of Mathematics, vol. 89, pp. 138–159.Google Scholar
[1983] Meschkowski, Herbert, Georg Cantor. Leben. Werk und Wirkung, Bibliographisches Institut, Mannheim.Google Scholar
[1991] Meschkowski, Herbert and Nilson, Winfried (editors), Briefe / Georg Cantor, Springer-Verlag, Berlin.Google Scholar
[1984] Miller, Arnold W., Special subsets of the real line, in Kunen-Vaughan [1984], pp. 201–233.Google 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, pp. 37–52.Google Scholar
[1917a] Mirimanoff, Dimitry, Remarques sur la théorie des ensembles et les antinomies Cantoriennes. I, L'Enseignment Mathématique, vol. 19, pp. 209–217.Google Scholar
[1956] Montague, Richard M., Zermelo-Fraenkel set theory is not a finite extension of Zermelo set theory (abstract), Bulletin of the American Mathematical Society, vol. 62, p. 260.Google Scholar
[1961] Montague, Richard M., Fraenkel's addition to the axioms of Zermelo, in Bar-Hillel-Poznanski-Rabin-Robinson [1961], pp. 91–114.Google Scholar
[1982] Moore, Gregory H., Zermelo's Axiom of Choice. Its origins, development and influence, Springer-Verlag, New York.Google Scholar
[1988] Moore, Gregory H., The roots of Russell's paradox, Russell, vol. 8, pp. 46–56, (new series).Google Scholar
[1988a] Moore, Gregory H., The origins of forcing, Logic colloquium '86 (Drake, Frank R. and Truss, John K., editors), North-Holland, Amsterdam, pp. 143–173.Google Scholar
[1988b] 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, pp. 95–135.Google Scholar
[1989] Moore, Gregory H., Towards a history of Cantor's Continuum Problem, The history of modern mathematics. Vol. 1: Ideas and their reception (Rowe, David E. and McCleary, John, editors), Academic Press, Boston, pp. 79–121.Google Scholar
[1981] Moore, Gregory H. and Garciadiego, Alejandro R., Burali-Forti's paradox: A reappraisal of its origins, Historia Mathematica, vol. 8, pp. 319–350.Google Scholar
[1980] Moschovakis, Yiannis, Descriptive set theory, North-Holland, Amsterdam.Google Scholar
[1939] Mostowski, Andrzej M., Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32, pp. 201–252, translated in [1979] below, vol. 1, pp. 290–338.Google Scholar
[1949] Mostowski, Andrzej M., An undecidable arithmetical statement, Fundamenta Mathematicae, vol. 36, pp. 143–164, reprinted in [1979] below, vol. 1, pp. 531–552.Google Scholar
[1979] Mostowski, Andrzej M., Foundational studies. Selected works, North-Holland, Amsterdam, edited by Kuratowski, Kazimierz and others.Google Scholar
[1971] Myhill, John R. and Scott, Dana S., Ordinal definability, in Scott [1971], pp. 271–278.Google Scholar
[1990] Nešetřil, Jaroslav and Rödl, Vojtěch, Mathematics of Ramsey theory, Springer-Verlag, Berlin.Google Scholar
[1937] Noether, Emmy and Cavaillès, Jean (editors), Briefwechsel Cantor-Dedekind, Hermann, Paris, translated into French in Cavaillès [1962].Google Scholar
[1971] Oxtoby, John C., Measure and category. A survey of the analogies between topological and measure spaces, Springer-Verlag, New York.Google Scholar
[1977] Parsons, Charles, What is the iterative concept of set?, Logic, foundations of mathematics and computability theory (Butts, Robert E. and Hintikka, Jaakko, editors), Proceedings of the Fifth International Congress of Logic, Methodology and the Philosophy of Science (London, Ontario 1975), The University of Western Ontario Series in Philosophy and Science, vol. 9, D. Reidel, Dordrecht, pp. 335–367, reprinted in Benacerraf-Putnam [1983], pp. 503–529.Google Scholar
[1897] Peano, Guiseppe, Studii di logica matematica, Atti della Accademia delle scienze di Torino, Classe di scienze fisiche, matematiche e naturali, vol. 32, pp. 565–583, reprinted in [1958] below, pp. 201–217.Google Scholar
[1905–1908] Peano, Guiseppe, Formulario mathematico, Bocca, Torino, reprinted Edizioni Cremonese, Rome, 1960.Google Scholar
[1911] Peano, Guiseppe, Sulla definizione di funzione, Atti della Accademia nazionale dei Lincei, Rendiconti, Classe di scienze fisiche, matematiche e naturali, vol. 20–I, pp. 3–5.Google Scholar
[1913] Peano, Guiseppe, Review of: A.N. Whitehead and B. Russell, Principia Mathematica, vols. I, II, Bollettino di bibliografia e storia delle scienze matematiche (Loria), vol. 15, pp. 47–53, 75–81, reprinted in [1958] below, pp. 389–401.Google Scholar
[1958] Peano, Guiseppe, Opere scelte, vol. 2, Edizioni Cremonese, Rome.Google Scholar
[1990] Peckhaus, Volker, “Ich habe mich wohl gehütet alle Patronen auf einmal zu verschiessen.” Ernst Zermelo in Göttingen, History and Philosophy of Logic, vol. 11, pp. 19–58.Google Scholar
[1883] Peirce, Charles S., A theory of probable inference. Note B. The logic of relatives, Studies in Logic by Members of the John Hopkins University, pp. 187–203, reprinted in Charles Hartshorne and Paul Weiss (editors), Collected Papers of Charles Sanders Peirce , vol. 3, Harvard University Press, Cambridge, pp. 195–209.Google Scholar
[1972] Pincus, David, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, The Journal of Symbolic Logic, vol. 37, pp. 721–743.Google Scholar
[1969] Pinl, M., Kollegen in einer dunklen Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 71, pp. 167–228.Google Scholar
[1990] Potter, Michael D., Sets. An introduction, Clarendon Press, Oxford.Google Scholar
[1989] Purkert, Walter, Cantor's views on the foundations of mathematics, The history of modern mathematics. Vol. 1: Ideas and their reception (Rowe, David E. and McCleary, John, editors), Academic Press, Boston, pp. 49–65.Google Scholar
[1987] Purkert, Walter and Ilgauds, Hans J., Georg Cantor: 1845–1918, Birkhäuser Verlag, Basel.Google Scholar
[1960] Quine, Willard V. O., Word and object, MIT Press, Cambridge.Google Scholar
[1984] Raisonnier, Jean, A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel Journal of Mathematics, vol. 48, pp. 48–56.Google Scholar
[1930] Ramsey, Frank P., On a problem of formal logic, Proceedings of the London Mathematical Society, vol. (2)30, pp. 264–286, reprinted in Gessel-Rota [1987], pp. 2–24.Google Scholar
[1981] Rang, Bernhard and Thomas, Wolfgang, Zermelo's discovery of the “Russell paradox“, Historia Mathematica, vol. 8, pp. 15–22.Google Scholar
[1951] Robinson, Abraham, On the metamathematics of algebra, North Holland, Amsterdam.Google Scholar
[1947] Robinson, Raphael M., On the decomposition of spheres, Fundamenta Mathematicae, vol. 34, pp. 246–270.Google Scholar
[1938] Rothberger, Fritz, Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen, Fundamenta Mathematicae, vol. 30, pp. 215–217.Google Scholar
[1939] Rothberger, Fritz, Sur un ensemble toujours de première categorie qui est depourvu de la properté λ, Fundamenta Mathematicae, vol. 32, pp. 294–300.Google Scholar
[1948] Rothberger, Fritz, On some problems of Hausdorff and Sierpiński, Fundamenta Mathematicae, vol. 35, pp. 29–46.Google Scholar
[1985] Rubin, Herman and Rubin, Jean E., Equivalents of the Axiom of Choice, II, North-Holland, Amsterdam, revised, expanded version of their 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, 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. (2)4, pp. 29–53, reprinted in [1973] below, pp. 135–164.Google Scholar
[1959] Russell, Bertrand A. W., My philosophical development, George Allen & Unwin, London.Google Scholar
[1973] Russell, Bertrand A. W., Essays in analysis, George Braziller, New York, edited by Lackey, Douglas.Google Scholar
[1890] Schröder, Ernst, Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, B. G. Teubner, Leipzig, reprinted in [1966] below.Google Scholar
[1895] Schröder, Ernst, Vorlesungen über die Algebra der Logik (exakte Logik). Vol. 3: Algebra und Logik der Relative, B. G. Teubner, Leipzig, reprinted in [1966] below.Google Scholar
[1966] Schröder, Ernst, Vorlesungen über die Algebra der Logik, Chelsea, New York, three volumes.Google Scholar
[1961] Scott, Dana S., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences, Serie des Sciences Mathématiques, Astronomiques et Physiques, vol. 9, pp. 521–524.Google Scholar
[1967] Scott, Dana S., A proof of the independence of the Continuum Hypothesis, Mathematical Systems Theory, vol. 1, pp. 89–111.Google Scholar
[1971] Scott, Dana S. (editor), Axiomatic set theory, Proceedings of Symposia in Pure Mathematics vol. 13, part 1, American Mathematical Society, Providence.Google Scholar
[1974] Scott, Dana S., Axiomatizing set theory, Axiomatic set theory (Jech, Thomas J., editor), Proceedings of Symposia in Pure Mathematics vol. 13, part 2, American Mathematical Society, Providence, pp. 207–214.Google Scholar
[1980] Shelah, Saharon, Going to Canossa (abstract), Abstracts of papers presented to the American Mathematical Society, vol. 1, p. 630.Google Scholar
[1984] Shelah, Saharon, Can you take Solovay's inaccessible away?, Israel Journal of Mathematics, vol. 48, pp. 1–47.Google Scholar
[1951] Shepherdson, John C., Innermodels for set theory—Part I, The Journal of Symbolic Logic, vol. 16, pp. 161–190.Google Scholar
[1952] Shepherdson, John C., Inner models for set theory—Part II, The Journal of Symbolic Logic, vol. 17, pp. 225–237.Google Scholar
[1953] Shepherdson, John C., Inner models for set theory—Part III, The Journal of Symbolic Logic, vol. 18, pp. 145–167.Google Scholar
[1959] Shoenfield, Joseph R., On the independence of the axiom of constructibility, American Journal of Mathematics, vol. 81, pp. 537–540.Google Scholar
[1967] Shoenfield, Joseph R., Mathematical logic, Addison-Wesley, Reading.Google Scholar
[1971] Shoenfield, Joseph R., Unramified forcing, in Scott [1971], pp. 357–381.Google Scholar
[1977] Shoenfield, Joseph R., Axioms of set theory, Handbook of mathematical logic (Barwise, K. Jon, editor), North-Holland, Amsterdam, pp. 321–344.Google Scholar
[1918] Sierpiński, Wacław, l'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse, Bulletin de l'Académie des Sciences de Cracovie, Classe des Sciences Mathematiques et Naturelles, Seŕie A, pp. 97–152, reprinted in [1975] below, pp. 208–255.Google Scholar
[1924] Sierpiński, Wacław, Sur l'hypothèse du continu (2ℵ0 = ℵ1), Fundamenta Mathematicae, vol. 5, pp. 177–187, reprinted in [1975] below, pp. 527–536.Google Scholar
[1925] Sierpiński, Wacław, Sur une class d'ensembles, Fundamenta Mathematicae, vol. 7, pp. 237–243, reprinted in [1975] below, pp. 571–576.Google Scholar
[1928] Sierpiński, Wacław, Sur un ensemble non dénombrable, dont toute image continue est de mesure nulle, Fundamenta Mathematicae, vol. 11, pp. 302–304, reprinted in [1975] below, pp. 702–704.Google Scholar
[1934] Sierpiński, Wacław, Hypothese du continu, Monografie Matematyczne, vol. 4, Warsaw, second revised edtion, Chelsea, New York, 1956.Google Scholar
[1975] Sierpiński, Wacław, Oeuvres choisies, vol. 2, Państwowe Wydawnictwo Naukowe, Warsaw, edited by Hartman, Stanislaw and others.Google Scholar
[1976] Sierpiński, Wacław, Ouevres choisies, vol. 3, Państwowe Wydawnictwo Naukowe, Warsaw, edited by Hartman, Stanislaw and others.Google Scholar
[1930] Sierpiński, Wacław and Tarski, Alfred, Sur unepropriété caractéristique des nombres inaccessibles, Fundamenta Mathematicae, vol. 15, pp. 292–300, reprinted in Sierpinski [1976], pp. 29–35 and in Tarski [1986] vol. 1, pp. 289–297.Google 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, no. 4, 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 matematikerkongressen, Redogörelse, Akademiska-Bokhandeln, Helsinki, 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, pp. 337–341, reprinted in [1970] below, pp. 275–279.Google Scholar
[1933] Skolem, Thoralf, Ein kombinatorischer Satz mit Anwendung auf ein logisches Entscheidungsproblem, Fundamenta Mathematicae, vol. 20, pp. 254–261, reprinted in [1970] below, pp. 337–344.Google Scholar
[1933a] Skolem, Thoralf, Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems, Norsk Matematisk Forenings Skrifter, vol. 2, no. 10, pp. 73–82, reprinted in [1970] below, pp. 345–354.Google Scholar
[1934] Skolem, Thoralf, Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Assagen mit ausschliesslich Zahlenvariablen, Fundamenta Mathematicae, vol. 23, pp. 150–161, reprinted in [1970] below, pp. 355–366.Google Scholar
[1970] Skolem, Thoralf, Selected works in logic, Univesitetsforlaget, Oslo, edited by Fenstad, Jens E..Google Scholar
[1963] Solovay, Robert M., Independence results in the theory of cardinals. I, II (abstracts), Notices of the American Mathematical Society, vol. 10, p. 595.Google Scholar
[1965] Solovay, Robert M., 2ℵ0 can be anything it ought to be (abstract), in Addison-Henkin-Tarski [1965], p. 435.Google Scholar
[1965a] Solovay, Robert M., Measurable cardinals and the continuum hypothesis (abstract), Notices of the American Mathematical Society, vol. 12, p. 132.Google Scholar
[1965b] Solovay, Robert M., The measure problem (abstract), Notices of the American Mathematical Society, vol. 12, p. 217.Google Scholar
[1969] Solovay, Robert M., The cardinality of sets of reals, Foundations of mathematics (Bulloff, Jack J., Holyoke, Thomas C., and Hahn, Samuel W., editors), Symposium papers commemorating the sixtieth birthday of Kurt Gödel, Springer-Verlag, Berlin, pp. 58–73.Google Scholar
[1970] Solovay, Robert M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92, pp. 1–56.Google Scholar
[1957] Specker, Ernst, Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom), Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 3, pp. 173–210.Google Scholar
[1910] Steinitz, Ernst, Algebraische Theorie der Korper, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 137, pp. 167–309.Google Scholar
[1917] Suslin, Mikhail Ya., Sur une définition des ensembles mesurables B sans nombres transfinis, Comptes Rendus Hebdomadaires des Seances de l'Académie des Sciences, Paris, vol. 164, pp. 88–91.Google Scholar
[1920] Suslin, Mikhail Ya., Problème 3, Fundamenta Mathematicae, vol. 1, p. 223.Google Scholar
[1924] Tarski, Alfred, Sur quelques théorèmes qui équivalent à l'axiome du choix, Fundamenta Mathematicae, vol. 5, pp. 147–154, reprinted in [1986] below, vol. 1, pp. 41–48.Google Scholar
[1931] Tarski, Alfred, Sur les ensembles définissables de nombres réels, Fundamenta Mathematicae, vol. 17, pp. 210–239, translated in Tarski [1983], pp. 110–142.Google 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), no. 34, see also [1935] below.Google Scholar
[1935] Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1, pp. 261–405, German translation of [1933] with a postscript; reprinted in [1986] below, vol. 2, pp. 51–198; translated in [1983] below, pp. 152–278.Google Scholar
[1951] Tarski, Alfred, A decision method for elementary algebra and geometry, University of California Press, Berkeley, prepared by McKinsey, J. C. C., second revised edition.Google Scholar
[1962] Tarski, Alfred, Some problems and results relevant to the foundations of set theory, Logic, methodology and philosophy of science (Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors), Stanford University Press, Stanford, pp. 125–135, reprinted in [1986] below, vol. 4, pp. 115–125.Google Scholar
[1983] Tarski, Alfred, Logic, semantics, metamathematics. papers from 1923 to 1938, second ed., Hackett, Indianapolis, translations by Woodger, J. H..Google Scholar
[1986] Tarski, Alfred, Collected papers, Birkhäuser, Basel, edited by Givant, Steven R. and McKenzie, Ralph N..Google Scholar
[1984] Todorčević, Stevo, Trees and linearly ordered sets, in Kunen-Vaughan [1984], pp. 235–293.Google Scholar
[1930] Ulam, Stanisław M., Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 140–150, reprinted in [1974] below, pp. 9–19.Google Scholar
[1974] Ulam, Stanisław M., Sets, numbers and universes, Selected Works, MIT Press, Cambridge, edited by Beyer, W. A., Mycielski, Jan and Rota, Gian-Carlo.Google Scholar
[1972] van Dalen, Dirk and Monna, A.F., Sets and integration. An outline of the development, Wolters-Noordhoff, Groningen.Google Scholar
[1984] van Douwen, Eric K., The integers and topology, in Kunen-Vaughan [1984], pp. 111–168.Google Scholar
[1967] van Heijenoort, Jean (editor), From Frege to Gödel: A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge.Google Scholar
[1990] Vaughan, Jerry, Small uncountable cardinals and topology, Open problems in topology (Van Mill, Jan and Reed, George M., editors), North-Holland, Amsterdam, pp. 195–218.Google Scholar
[1905] Vitali, Guiseppe, Sul problema della misura dei gruppi di punti di una retta, Bologna, Tip. Gamberini e Parmeggiani.Google Scholar
[1923] von Neumann, John, Zur Einführung der transfiniten Zahlen, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, sectio scientiarum mathematicarum, vol. 1, pp. 199–208, reprinted in [1961] below, 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 (Crelle's Journal), vol. 154, pp. 219–240, Berichtigung, vol. 155, p. 128; reprinted in [1961] below, pp. 34–56; translated in van Heijenoort [1967], pp. 393–413.Google Scholar
[1928] von Neumann, John, Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99, pp. 373–391, reprinted in [1961]below, pp. 320–338.Google Scholar
[1928a] von Neumann, John, Die Axiomatisierung der Mengenlehre, Mathematische Zeitschrift, vol. 27, pp. 669–752, reprinted in [1961] below, pp. 339–422.Google Scholar
[1929] von Neumann, John, Über eine Widerspruchfreiheitsfrage in der axiomaticschen Mengenlehre, Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 160, pp. 227–241, reprinted in [1961] below, pp. 494–508.Google Scholar
[1961] von Neumann, John, John von Neumann. Collected works, vol. 1, Pergamon Press, New York, edited by Taub, Abraham H..Google Scholar
[1962] Vopěnka, Petr, Construction of models of set theory by the method of ultraproducts (in Russian), Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 8, pp. 293–304.Google Scholar
[1964] Vopěnka, Petr, The independence of the Continuum Hypothesis (in Russian), Commentationes Mathematicae Universitatis Carolinae, vol. 5, pp. 1–48, supplement I; translated in American Mathematical Society Translations ,vol. 57 (1966), pp. 85–112.Google Scholar
[1967] Vopěnka, Petr, The general theory of ∇-models, Commentationes Mathematicae Universitatis Carolinae, vol. 8, pp. 145–170.Google Scholar
[1985] Wagon, Stanley, The Banach-Tarski paradox, Encyclopedia of Mathematics and Its Applications, vol. 24, Cambridge University Press, Cambridge, paperback edition 1993.Google Scholar
[1974] Wang, Hao, From mathematics to philosophy, Humanities Press, New York.Google Scholar
[1974a] Wang, Hao, The concept of set, in Wang [1974], pp. 181–223; reprinted in Benacerraf-Putnam [1983], pp. 530–570.Google Scholar
[1910] Whitehead, Alfred N. and Russell, Bertrand A. W., Principia mathematica, vol. 1, Cambridge University Press, Cambridge.Google Scholar
[1912] Whitehead, Alfred N., Principia mathematica, vol. 2, Cambridge University Press, Cambridge.Google Scholar
[1913] Whitehead, Alfred N., Principia mathematica, vol. 3, Cambridge University Press, Cambridge.Google Scholar
[1914] Wiener, Norbert, A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17, pp. 387–390, reprinted in van Heijenoort [1967], pp. 224–227.Google Scholar
[1956] Wittgenstein, Ludwig, Remarks on the foundations of mathematics, Basil Blackwell, Oxford, edited by von Wright, Georg H., Rhees, Rush and Elizabeth, G. Anscombe, M., second printing, 1967.Google Scholar
[1904] Zermelo, Ernst, Beweis, dass jede Menge wohlgeordnet werden kann (Aus einem an Herrn Hilbert gerichteten Briefe), Mathematische Annalen, vol. 59, pp. 514–516, translated in van Heijenoort [1967], pp. 139–141.Google Scholar
[1908] Zermelo, Ernst, Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65, pp. 107–128, translated in van Heijenoort [1967], pp. 183–198.Google Scholar
[1908a] Zermelo, Ernst, Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65, pp. 261–281, translated in van Heijenoort [1967], pp. 199–215.Google Scholar
[1929] Zermelo, Ernst, Über den Begriff der Definitheit in der Axiomatik, Fundamenta Mathematicae, vol. 14, pp. 339–344.Google Scholar
[1930] Zermelo, Ernst, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47.Google Scholar
[1931] Zermelo, Ernst, Über Stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 41, pp. 85–88.Google Scholar
[1935] Zermelo, Ernst, Grundlagen einer allgemeinen Theorie der mathematicschen Satzsysteme, Fundamenta Mathematicae, vol. 25, pp. 136–146.Google Scholar
[1935] Zorn, Max, A remark on method in transfinite algebra, Bulletin of the American Mathematical Society, vol. 41, pp. 667–670.Google Scholar