Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T10:55:12.379Z Has data issue: false hasContentIssue false
  • English
  • Français

On Arbitrary sets and ZFC

Published online by Cambridge University Press:  15 January 2014

José Ferreirós
José Ferreirós*
Affiliation:
Instituto De Filosofia, CCHS–CSIC, Albasanz, 26–28, 28037 Madrid, SpainE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Set theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 17 , Issue 3 , September 2011 , pp. 361 - 393
Copyright
Copyright © Association for Symbolic Logic 2011

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

Bernays, Paul [1935], Sur le platonisme dans les mathématiques, L'Enseignement Mathématique, vol. 34, pp. 52–69, References to the English version in (P. Benacerraf and H. Putnam, editors), Philosophy of Mathematics: selected readings , Cambridge University Press, 1983, pp. 258-271.Google Scholar
Cantor, Georg [1874], Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal für die reine und angewandte Mathematik, vol. 77, pp. 258–262, also in Cantor's Gesammelte Abhandlungen , Berlin, Springer, 1932, pp. 115-118. English translation in Ewald [1996].Google Scholar
Cantor, Georg [1892], Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinung, vol. 1, pp. 75–78, also in Cantor's Gesammelte Abhandlungen , Berlin, Springer, 1932, pp. 278-280. English translation in Ewald [1996].Google Scholar
Cavaillès, J. [1938], Remarques sur la formation de la theorie abstraite des ensembles, dissertation, in Cavaillès, Philosophie mathématique . Paris, Hermann, 1962.Google Scholar
Cooke, Roger [1993], Uniqueness of trigonometric series and descriptive set theory, 1870-1985, Archive for History of Exact Sciences, vol. 45, pp. 281–334.CrossRefGoogle Scholar
Dedekind, Richard [1888], Was sind und was sollen die Zahlen?, reprinted in Gesammelte mathematische Werke , vol. 3, New York, Chelsea, 1969. References to the English translation in Ewald [1996].Google Scholar
Dedekind, Richard [1932], Gesammelte mathematische Werke,(Fricke, R., Noether, E., and Ore, Ö., editors), Braunschweig, 3 vols. Reprint in 2 vols. New York, Chelsea, 1969.Google Scholar
Devlin, Keith [1984], Constructibility, Springer-Verlag, Berlin.Google Scholar
Ewald, William B. (editor) [1996], From Kant to Hilbert, vol. 2, Oxford University Press.Google Scholar
Feferman, Solomon [1965], Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56, pp. 325–345.Google Scholar
Feferman, Solomon (editor) [1990], Kurt Gödel, collected works, vol. II, Oxford University Press.Google Scholar
Feferman, Solomon [1998], In the light of logic, Oxford University Press.CrossRefGoogle Scholar
Ferreirós, José [2001], The road to modern logic, this Bulletin, vol. 7, pp. 441–484.Google Scholar
Ferreirós, José [2007], Labyrinth of thought. A history of set theory and its role in modern mathematics, Birkhäuser, Basel, (first edition 1999).Google Scholar
Fraenkel, Abraham, Bar-Hillel, Yehoshua, and Levy, Azriel [1973], Foundations of set theory, North-Holland, Amsterdam.Google Scholar
Frege, Gottlob [1893], Grundgesetze der Arithmetik, vol. 1, Pohl, Jena, reprinted Olms, Hildesheim, 1966.Google Scholar
Goldstein, Catherine, Schappacher, N., and Schwermer, J. (editors) [2007], The shaping of arithmetic after C. F. Gauss's Disquisitiones Arithmeticae, Berlin.Google Scholar
Hilbert, David [1900], Mathematische Probleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1900), reprint in Gesammelte Abhandlungen , vol. 3, Springer, 1935, 146–56, References to the partial translation in Ewald [1996], pp. 253-297.Google Scholar
Hilbert, David [1925], Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161–90, references to the English translation in Heijenoort [1967], 367-392.Google Scholar
Hintikka, Jaako [1999], Is the axiom of choice a logical or set-theoretical principle?, Dialectica, vol. 53, pp. 283–290.Google Scholar
Jané, Ignasi [2001], Reflections on Skolem's relativity of set-theoretical concepts, Philosophia Mathematica, vol. 9, pp. 129–153.Google Scholar
Jané, Ignasi [2005a], The iterative conception of sets from a Cantorian perspective, Logic, Method logy and Philosophy of Science. Proceedings of the twelfth international congress, King's College Publications, London, pp. 373–393.Google Scholar
Jané, Ignasi [2005b], Higher-order logic reconsidered, The Oxford handbook of philosophy of mathematics and logic (Shapiro, S., editor), Oxford University Press.Google Scholar
Jensen, R. [1995], Inner models and large cardinals, this Bulletin, vol. 1, pp. 393–407.Google Scholar
Jourdain, P. E. B. [1906–1914], The development of the theory of transfinite numbers, Archiv für Mathematik und Physik, vol. 10, pp. 254-281, vol. 14, 287-311, vol. 16, 21-43, vol. 22, 1–21.Google Scholar
Kanamori, A. and Foreman, M. (editors) [2010], Handbook of set theory, Springer, Berlin, 3 volumes. ISBN: 978-1-4020-4843-2.Google Scholar
Kanamori, Akihiro [1994], The higher infinite, Springer, Berlin.Google Scholar
Kanamori, Akihiro [1995], The emergence of descriptive set theory, From Dedekind to Gödel (Hintikka, Jaakko, editor), Kluwer, Dordrecht, pp. 241–262.Google Scholar
Kanamori, Akihiro [1996], The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2, pp. 1–71.Google Scholar
Krömer, Ralf [2007], Tool and object: A history and philosophy of category theory, Birkhäuser, Basel/Boston.Google Scholar
Laugwitz, Detlef [1999], Bernhard Riemann 1826–1866: Turning points in the conception of mathematics, Birkhäuser, Basel.Google Scholar
Lavine, Shaugan [1994], Understanding the infinite, Harvard University Press.Google Scholar
Maddy, Penelope [1988], Believing the axioms, Part I & II, The Journal of Symbolic Logic, vol. 53, pp. 481-511, 736–764.CrossRefGoogle Scholar
Maddy, Penelope [1997], Naturalism in mathematics, Oxford University Press.Google Scholar
Martin, Donald A. [1976], Hilbert's first problem: The continuum hypothesis, Mathematical developments arising from Hilbert problems (Browder, Felix E., editor), American Mathematical Society, pp. 81–92.CrossRefGoogle Scholar
Martin, Donald A. [1998], Mathematical evidence, Truth in mathematics (Dales, H. G. and Oliveri, G., editors), Oxford University Press.Google Scholar
Meschkowski, H. and Nilson, W. [1991], Georg Cantor: Briefe, Springer, Berlin.Google Scholar
Moore, Gregory H. [1982], Zermelo's Axiom of Choice. Its origins, development and influence, Springer, Berlin.Google Scholar
Moschovakis, Yiannis N. [1994], Notes on set theory, Springer-Verlag, New York.Google Scholar
Mostowski, Andrzej [1967], Recent results in set theory, The philosophy of mathematics (Lakatos, I., editor), North-Holland, Amsterdam.Google Scholar
Parsons, Charles [2008], Mathematical thought and its objects, Cambridge University Press.Google Scholar
Peano, Giuseppe [1889], Arithmetices principia, nova methodo exposita, Bocca, Torino, partial English translation in Heijenoort [1967].Google Scholar
Reck, Erich and Awodey, Steve [2002], Completeness and categoricity, Part I: 19th century axiomatics to 20th century metalogic, History and Philosophy of Logic, vol. 23, no. 1, pp. 1–30, Part II: 20th Century Metalogic to 21st Century Semantics, History and Philosophy of Logic, vol. 23 (2), pp. 77-94.Google Scholar
Russell, Bertrand [1910], Principia mathematica, vol. I, Cambridge University Press, with A. N. Whitehead.Google Scholar
Russell, Bertrand [1920], Introduction to mathematical philosophy, 2nd ed., Allen & Unwin, London, reprinted in New York, Dover, 1993.Google Scholar
Shapiro, Stewart [1991], Foundations without foundationalism: A case for second-order logic, Oxford University Press.Google Scholar
Shapiro, Stewart [1997], Philosophy of mathematics: Structure and ontology, Oxford University Press.Google Scholar
Simpson, Stephen G. [1999], Subsystems of second-order arithmetic, Springer, Berlin.CrossRefGoogle Scholar
Skolem, Thoralf [1923], Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Skolem's selected works in logic, Universitetsforlaget, Oslo, 1970. English translation in Heijenoort [1967].Google Scholar
Tait, William [2005], The provenance of pure reason: Essays in the philosophy of mathematics and its history, Oxford University Press.CrossRefGoogle Scholar
Väänänen, Jouko [2001], Second-order logic and foundations of mathematics, this Bulletin, vol. 7, pp. 504–520.Google Scholar
Van Heijenoort, Jean [1967], From Frege to Gödel: A source book in mathematical logic, Harvard University Press.Google Scholar
Weston, Thomas [1976], Kreisel, the Continuum Hypothesis and second order set theory, Journal of Philosophical Logic, vol. 5, pp. 281–298.Google Scholar
Weyl, Hermann [1918], Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, references to the reprint New York, AMS Chelsea, 1973.Google Scholar
Weyl, Hermann [1944], Mathematics and logic. A brief survey serving as a preface to a review of “The Philosophy of Bertrand Russell,”, American Mathematical Monthly, vol. 53, pp. 2–13.Google Scholar
Weyl, Hermann [1949], Philosophy of mathematics and natural science, Princeton University Press, translated with additions from the original German, published in 1927.Google Scholar
Wittgenstein, Ludwig [1976], Lectures on the foundations of mathematics, Cambridge, 1939, edited by Diamond, Cora, Cornell University Press, Ithaca, N.Y. Google Scholar
Woodin, Hugh [2001], The Continuum Hypothesis, Parts I and II, Notices of the American Mathematical Society, vol. 48, pp. 567-576, 681–690.Google Scholar
Zermelo, Ernst [1908], Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65, pp. 261–281, English translation in Heijenoort [1967], 199-215.Google Scholar