Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T20:58:29.808Z Has data issue: false hasContentIssue false
  • English
  • Français

The Empty Set, The Singleton, and the Ordered Pair

Published online by Cambridge University Press:  15 January 2014

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

Extract

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 3 , September 2003 , pp. 273 - 298
Copyright
Copyright © Association for Symbolic Logic 2003

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

[1937] Bernays, Paul, A system of axiomatic set theory – Part I, The Journal of Symbolic Logic, vol. 2, pp. 65–77.Google Scholar
[1941] Bernays, Paul, A system of axiomatic set theory – Part II, The Journal of Symbolic Logic, vol. 6, pp. 1–17.CrossRefGoogle Scholar
[1954] Bernays, Paul, A system of axiomatic set theory – Part VII, The Journal of Symbolic Logic, vol. 19, pp. 81–96.Google Scholar
[1847] Boole, George, The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning, Macmillan, Barclay, and Macmillan, Cambridge, reprinted in Ewald [1996], vol. 1, pp. 451–509.Google Scholar
[1854] Boole, George, An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities, Macmillan, London.CrossRefGoogle Scholar
[1954] Bourbaki, Nicolas, Eléments de mathématique. I. Théorie des ensembles, Hermann, Paris.Google Scholar
[1970] Boole, George, Eléments de mathématique. I. Théorie des ensembles, combined ed., Hermann, Paris.Google Scholar
[1874] Cantor, Georg, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal für die reine und angewandte Mathematik, vol. 77, pp. 258–262; reprinted in Cantor [1932] below, pp. 115–118; translated in Ewald [1996], vol. 2, pp. 839–843.Google Scholar
[1880] Boole, George, Über unendliche, lineare Punktmannigfaltigkeiten. II, Mathematische Annalen, vol. 17, pp. 355–358; reprinted in Cantor [1932] below, pp. 145–148.Google Scholar
[1891] Boole, George, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 1, pp. 75–78; reprinted in Cantor [1932] below, pp. 278–280; translated in Ewald [1996], vol. 2, pp. 920–922.Google Scholar
[1895] Boole, George, Beiträge zur Begründung der transfiniten Mengenlehre. I, Mathematische Annalen, vol. 46, pp. 481–512; translated in Cantor [1915] below; reprinted in Cantor [1932] below, pp. 282–311.Google Scholar
[1915] Boole, George, Contributions to the founding of the theory of transfinite numbers, including translations of Cantor [1895] and Cantor [1897] above with introduction and notes by Philip E. B. Jourdain, Open Court, Chicago; reprinted Dover, New York, 1965.Google Scholar
[1932] Boole, George, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (Zermelo, Ernst, editor), Julius Springer, Berlin; reprinted Springer-Verlag, Berlin, 1980.Google Scholar
[1943] Carmichael, Peter A., The null class nullified, Philosophical Review, vol. 52, pp. 61–68.CrossRefGoogle Scholar
[1943a] Carmichael, Peter A., Animadversion on the null class, Philosophy of Science, vol. 10, pp. 90–94.CrossRefGoogle Scholar
[1888] Dedekind, Richard, Was sind und was sollen die Zahlen?, F. Vieweg, Braunschweig; sixth, 1930 edition reprinted in Dedekind [1932] below, pp. 335–390; second, 1893 edition translated in Dedekind [1963] below, pp. 29–115 and also translated in Ewald [1996], vol. 2, pp. 795–833.Google Scholar
[1900] Dedekind, Richard, Über die von drei Moduln erzeugte Dualgruppe, Mathematische Annalen, vol. 53, pp. 371–403; reprinted in Dedekind [1932], vol. 2, pp. 236–271.Google Scholar
[1932] Dedekind, Richard, Gesammelte mathematische Werke (Fricke, Robert, Noether, Emmy, and Ore, Oystein, editors), vol. 3, F. Vieweg, Braunschweig; reprinted Chelsea, New York, 1969.Google Scholar
[1963] Dedekind, Richard, Essays on the theory of numbers (Beman, Wooster W., translator), Dover, New York; reprint of original edition, Open Court, Chicago, 1901.Google Scholar
[1989] Dreben, Burton S., Quine, in Quine in perspective (Barrett, Robert and Gibson, Roger, editors), Blackwell, Oxford.Google Scholar
[1996] Ewald, William, From Kant to Hilbert: A source book in the foundations of mathematics, Clarendon Press, Oxford.Google Scholar
[1999] Ferreirós, José, Labyrinth of thought: A history of set theory and its role in modern mathematics, Birkhauser Verlag, Basel.CrossRefGoogle Scholar
[1995] Forster, Thomas E., Set theory with a universal set, second ed., Oxford Logic Guides, vol. 31, Clarendon Press, Oxford.CrossRefGoogle Scholar
[1921] Fraenkel, Abraham, Über die Zermelosche Begründung der Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 30, pp. 97–98.Google Scholar
[1922] Fraenkel, Abraham, Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86, pp. 230–237.Google Scholar
[1879] Frege, Gottlob, Begriffsschrift, eine der arithmetischen nachgebildete Formel-sprache 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, Blackwell, Oxford, 1950; later editions without German text, Harper, New York.Google Scholar
[1891] Frege, Gottlob, Function und Begriff, Hermann Pohle, Jena; translated in Frege [1952] below, pp. 21–41.Google Scholar
[1893] Frege, Gottlob, Grundgesetze der Arithmetik, Begriffsschriftlich abgeleitet, vol. 1, Hermann Pohle, Jena; reprinted Olms, Hildesheim, 1962.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 Frege [1952] below, pp. 86–106.Google Scholar
[1952] Frege, Gottlob, Translations from the philosophical writings of Gottlob Frege (Geach, Peter and Black, Max, translators and editors), Blackwell, Oxford; second, revised edition 1960; latest edition, Rowland & Littlewood, Totowa, 1980.Google Scholar
[1992] Garciadiego, Alejandro R., Bertrand Russell and the origins of the set-theoretic “paradoxes”, Birkhäuser, Boston.Google Scholar
[1931] Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38, pp. 173–198; reprinted, together with a translation incorporating minor emendations by Gödel, in Gödel [1986] below, pp. 144–195.CrossRefGoogle Scholar
[1986] Gödel, Kurt, Collected works (Feferman, Solomon et al., editors), vol. 1, Oxford University Press, New York.Google Scholar
[1975] Grattan-Guinness, Ivor, Wiener on the logics of Russell and Schröder. An account of his doctoral thesis, and of his subsequent discussion of it with Russell, Annals of Science, vol. 32, pp. 103–132.Google Scholar
[1978] Grattan-Guinness, Ivor, How Bertrand Russell discovered his paradox, Historia Mathematica, vol. 5, pp. 127–137.Google Scholar
[1837] Hamilton, William R., Theory of conjugate functions, or algebraic couples: with a preliminary and elementary essay on algebra as the science of pure time, Transactions of the Royal Irish Academy, vol. 17, pp. 293–422; reprinted in The mathematical papers of Sir William Rowan Hamilton (H. Halberstam and R. E. Ingram, editors), Cambridge University Press, Cambridge, pp. 1931–1967.Google Scholar
[1914] Hausdorff, Felix, Grundzüge der Mengenlehre, de Gruyter, Leipzig; reprinted Chelsea, New York, 1965.Google Scholar
[1937] Hausdorff, Felix, Mengenlehre; third, revised edition of Hausdorff [1914]; translated by Auman, John R. as Set theory, Chelsea, New York, 1962.Google Scholar
[1995] Heck, Richard G. Jr., Definition by induction in Frege's Grundgesetze der Arithmetik, in Frege's philosophy of mathematics (Demopoulos, William, editor), Harvard University Press, Cambridge, pp. 295–333.Google Scholar
[1891] Husserl, Edmund, Besprechung von E. Schröder, Vorlesungen über die Algebra der Logik (exakte Logik), Leipzig, 1890, vol. I, Göttingische Gelehrte Anzeigen; 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, pp. 243–278.Google Scholar
[1997] Kanamori, Akihiro, The mathematical import of Zermelo's well-ordering theorem, this Bulletin, vol. 3, pp. 281–311.Google Scholar
[1975] Kennedy, Hubert C., Nine letters from Giuseppe Peano to Bertrand Russell, Journal of the History of Philosophy, vol. 13, pp. 205–220.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 Kuratowski [1988] below, pp. 1–11.Google Scholar
[1988] Kuratowski, Kazimierz, Selected papers (Borsuk, Karol et al., editors), Państwowe Wydawnictwo Naukowe, Warsaw.Google Scholar
[1991] Lewis, David, Parts of classes, Basil Blackwell, Cambridge.Google Scholar
[1992] Mathias, Adrian R. D., The ignorance of Bourbaki, Mathematical Intelligencer, vol. 14, pp. 4–13.Google Scholar
[1982] Moore, Gregory H., Zermelo's axiom of choice: Its origins, development, and influence, Springer-Verlag, New York.Google Scholar
[1995] Moore, Gregory H., The origins of Russell's paradox: Russell, Couturat, and the antinomy of infinite number, in From Dedekind to Gödel (Hintikka, Jaakko, editor), Synthese Library, vol. 251, Kluwer, Dordrecht, pp. 215–239.Google Scholar
[1889] Peano, Giuseppe, Arithmetices principia nova methodo exposita, Bocca, Turin; reprinted in Peano [1957–9] below, vol. 2, pp. 20–55; partially translated in van Heijenoort [1967], pp. 85–97; translated in Peano [1973] below, pp. 101–134.Google Scholar
[1889a] Peano, Giuseppe, I principii di geometrica logicamente espositi, Bocca, Turin; reprinted in Peano [1957–9] below, vol. 2, pp. 56–91.Google Scholar
[1890] Peano, Giuseppe, Démonstration de l'intégrabilité des équations différentielles ordinaires, Mathematische Annalen, vol. 37, pp. 182–228; reprinted in Peano [1957–9] below, vol. 1, pp. 119–170.Google Scholar
[1891] Peano, Giuseppe, Principii di logica matematica, Rivista di matematica, vol. 1, pp. 1–10; reprinted in Peano [1957–9] below, vol. 2, pp. 92–101; translated in Peano [1973] below, pp. 153–161.Google Scholar
[1894] Peano, Giuseppe, Notations de logique mathématique (introduction au formulaire de mathématiques), Guadaguini, Turin.Google Scholar
[1897] Peano, Giuseppe, 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 Peano [195–9] below, vol. 2, pp. 201–217; translated in Peano [1973] below, pp. 190–205.Google Scholar
[1905–1908] Peano, Giuseppe, Formulario mathematico, Bocca, Torin; reprinted Edizioni Cremonese, Rome, 1960.Google Scholar
[1911] Peano, Giuseppe, 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, Giuseppe, Review of: A. N.Whitehead and B. Russell, Principia Mathematica, vol. I, II, Bollettino di bibliografia e storia delle scienze matematiche (Loria), vol. 15, pp. 47–53, 75–81; reprinted in Peano [1957–9] below, vol. 2, pp. 389–401.Google Scholar
[1957–1959] Peano, Giuseppe, Opera scelte, three volumes (Cassina, U., editor), Edizioni Cremonese, Rome.Google Scholar
[1973] Peano, Giuseppe, Selected works of Giuseppe Peano (Kennedy, Hubert C., translator and editor), University of Toronto Press, Toronto.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 (Peirce, Charles S., editor), Little-Brown, Boston; reprinted in Peirce [1931–1958] below, vol. 3, pp. 195–209, pp. 187–203.CrossRefGoogle Scholar
[1885] Peirce, Charles S., On the algebra of logic: A contribution to the philosophy of notation, The American Journal of Mathematics, vol. 7, pp. 180–202; reprinted in Peirce [1931–1958] below, vol. 3, pp. 210–238, and in Ewald [1996], vol. 1, pp. 608–32.Google Scholar
[1931–1958] Peirce, Charles S., Collected papers of Charles Sanders Peirce (Hartshorne, Charles and Weiss, Paul, editors), Harvard University Press, Cambridge.Google Scholar
[1934] Quine, Willard V., A system of logistic, Harvard University Press, Cambridge.CrossRefGoogle Scholar
[1936] Quine, Willard V., Set-theoretic foundations for logic, The Journal of Symbolic Logic, vol. 1, pp. 45–57; reprinted in Quine [1966] below, pp. 83–99.Google Scholar
[1937] Quine, Willard V., New foundations for mathematical logic, American Mathematical Monthly, vol. 44 (1937), pp. 70–80.Google Scholar
[1940] Quine, Willard V., Mathematical logic, Norton, New York.Google Scholar
[1960] Quine, Willard V., Word and object, MIT Press, Cambridge.Google Scholar
[1963] Quine, Willard V., Set theory and its logic, Harvard University Press, Cambridge; revised edition, 1969.Google Scholar
[1966] Quine, Willard V., Selected logic papers, Harvard University Press, Cambridge; enlarged edition, 1995.Google Scholar
[1992] Quine, Willard V., Pursuit of truth, revised ed., Harvard University Press, Cambridge.Google Scholar
[1981] Rang, Bernhard and Thomas, Wolfgang, Zermelo's discovery of the “Russell paradox”, Historia Mathematica, vol. 8, pp. 15–22.Google Scholar
[1901] Russell, Bertrand A. W., Sur la logique des relations avec des applications à la théorie des séries, Revue de mathématiques (Rivista di matematica), vol. 7, pp. 115–148; partly reprinted in The collected papers of Bertrand Russell, vol. 3 (Gregory H. Moore, editor), Routledge, London, 1993, pp. 613–627; translated in same, pp. 310–349.Google Scholar
[1903] Russell, Bertrand A. W., The principles of mathematics, Cambridge University Press, Cambridge; later editions, George Allen & Unwin, London.Google Scholar
[1908] Russell, Bertrand A. W., Mathematical logic as based on the theory of types, American Journal of Mathematics, vol. 30, pp. 222–262; reprinted in van Heijenoort [1967], pp. 150–182.CrossRefGoogle Scholar
[1918] Russell, Bertrand A. W., Mysticism and logic, and other essays, Longmans, Green & Co., New York.Google Scholar
[1944] Russell, Bertrand A. W., My mental development, The philosophy of Bertrand Russell (Schilpp, Paul A., editor), The Library of Living Philosophers, vol. 5, Northwestern University, Evanston, pp. 3–20.Google Scholar
[1945] Russell, Bertrand A. W., A history of Western philosophy, Simon Schuster, New York.Google Scholar
[1959] Russell, Bertrand A. W., My philosophical development, George Allen & Unwin, London.Google Scholar
[1890] Schröder, Ernst, Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, B. G. Teubner, Leipzig; reprinted in Schröder [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 Schröder [1966] below.Google Scholar
[1966] Schröder, Ernst, Vorlesungen über die Algebra der Logik, three volumes, Chelsea, New York.Google Scholar
[1962] Scott, Dana S., Quine's individuals, Logic, methodology and the philosophy of science (Nagel, Ernst, editor), Stanford University Press, Stanford, pp. 111–115.Google Scholar
[1971] Sinaceur, Mohammed A., Appartenance et inclusion: un inédit de Richard Dedekind, Revue d'Histoire des Sciences et de leurs Applications, vol. 24, pp. 247–255.CrossRefGoogle 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
[1931] Tarski, Alfred, Sur les ensembles définissables de nombres réels, Fundamenta Mathematicae, vol. 17, pp. 210–239; translated in Logic, semantics, metamathematics. Papers from 1923 to 1938 (J. H. Woodger, translator), second ed., Hackett, Indianapolis, 1983, pp. 110–142.Google Scholar
[1967] van Heijenoort, Jean, From Frege to Gödel: A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge; reprinted 2002.Google Scholar
[1925] von Neumann, John, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154, pp. 219–240; Berichtigung in 155, p. 128; reprinted in von Neumann [1961] below, pp. 34–56; translated in van Heijenoort [1967], pp. 393–413.Google Scholar
[1928] von Neumann, John, Über dieDefinition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99, pp. 373–391; reprinted in von Neumann [1961] below, pp. 320–338.Google Scholar
[1929] von Neumann, John, Über eine Widerspruchfreiheitsfrage in der axiomaticschen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160, pp. 227–241; reprinted in von Neumann [1961] below, pp. 494–508.Google Scholar
[1961] von Neumann, John, John von Neumann. Collected works (Taub, Abraham H., editor), vol. 1, Pergamon Press, New York.Google Scholar
[1910–1913] Whitehead, Alfred N. and Russell, Bertrand A. W., Principia Mathematica, three volumes, 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
[1904] Zermelo, Ernst, Beweis, dass jedeMenge 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
[1930] Zermelo, Ernst, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47; translated in Ewald [1996], vol. 2, pp. 1208–1233.Google Scholar