Skip to main content
×
×
Home

On Arbitrary sets and ZFC

  • José Ferreirós (a1)
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.

Copyright
References
Hide All
Bernays, Paul [1935], Sur le platonisme dans les mathématiques, L'Enseignement Mathématique, vol. 34, pp. 5269, References to the English version in (P. Benacerraf and H. Putnam, editors), Philosophy of Mathematics: selected readings , Cambridge University Press, 1983, pp. 258-271.
Cantor, Georg [1874], Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal für die reine und angewandte Mathematik, vol. 77, pp. 258262, also in Cantor's Gesammelte Abhandlungen , Berlin, Springer, 1932, pp. 115-118. English translation in Ewald [1996].
Cantor, Georg [1892], Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinung, vol. 1, pp. 7578, also in Cantor's Gesammelte Abhandlungen , Berlin, Springer, 1932, pp. 278-280. English translation in Ewald [1996].
Cavaillès, J. [1938], Remarques sur la formation de la theorie abstraite des ensembles, dissertation, in Cavaillès, Philosophie mathématique . Paris, Hermann, 1962.
Cooke, Roger [1993], Uniqueness of trigonometric series and descriptive set theory, 1870-1985, Archive for History of Exact Sciences, vol. 45, pp. 281334.
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].
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.
Devlin, Keith [1984], Constructibility, Springer-Verlag, Berlin.
Ewald, William B. (editor) [1996], From Kant to Hilbert, vol. 2, Oxford University Press.
Feferman, Solomon [1965], Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56, pp. 325345.
Feferman, Solomon (editor) [1990], Kurt Gödel, collected works, vol. II, Oxford University Press.
Feferman, Solomon [1998], In the light of logic, Oxford University Press.
Ferreirós, José [2001], The road to modern logic, this Bulletin, vol. 7, pp. 441484.
Ferreirós, José [2007], Labyrinth of thought. A history of set theory and its role in modern mathematics, Birkhäuser, Basel, (first edition 1999).
Fraenkel, Abraham, Bar-Hillel, Yehoshua, and Levy, Azriel [1973], Foundations of set theory, North-Holland, Amsterdam.
Frege, Gottlob [1893], Grundgesetze der Arithmetik, vol. 1, Pohl, Jena, reprinted Olms, Hildesheim, 1966.
Goldstein, Catherine, Schappacher, N., and Schwermer, J. (editors) [2007], The shaping of arithmetic after C. F. Gauss's Disquisitiones Arithmeticae, Berlin.
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.
Hilbert, David [1925], Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161–90, references to the English translation in Heijenoort [1967], 367-392.
Hintikka, Jaako [1999], Is the axiom of choice a logical or set-theoretical principle?, Dialectica, vol. 53, pp. 283290.
Jané, Ignasi [2001], Reflections on Skolem's relativity of set-theoretical concepts, Philosophia Mathematica, vol. 9, pp. 129153.
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. 373393.
Jané, Ignasi [2005b], Higher-order logic reconsidered, The Oxford handbook of philosophy of mathematics and logic (Shapiro, S., editor), Oxford University Press.
Jensen, R. [1995], Inner models and large cardinals, this Bulletin, vol. 1, pp. 393407.
Jourdain, P. E. B. [19061914], 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, 121.
Kanamori, A. and Foreman, M. (editors) [2010], Handbook of set theory, Springer, Berlin, 3 volumes. ISBN: 978-1-4020-4843-2.
Kanamori, Akihiro [1994], The higher infinite, Springer, Berlin.
Kanamori, Akihiro [1995], The emergence of descriptive set theory, From Dedekind to Gödel (Hintikka, Jaakko, editor), Kluwer, Dordrecht, pp. 241262.
Kanamori, Akihiro [1996], The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2, pp. 171.
Krömer, Ralf [2007], Tool and object: A history and philosophy of category theory, Birkhäuser, Basel/Boston.
Laugwitz, Detlef [1999], Bernhard Riemann 1826–1866: Turning points in the conception of mathematics, Birkhäuser, Basel.
Lavine, Shaugan [1994], Understanding the infinite, Harvard University Press.
Maddy, Penelope [1988], Believing the axioms, Part I & II, The Journal of Symbolic Logic, vol. 53, pp. 481-511, 736764.
Maddy, Penelope [1997], Naturalism in mathematics, Oxford University Press.
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. 8192.
Martin, Donald A. [1998], Mathematical evidence, Truth in mathematics (Dales, H. G. and Oliveri, G., editors), Oxford University Press.
Meschkowski, H. and Nilson, W. [1991], Georg Cantor: Briefe, Springer, Berlin.
Moore, Gregory H. [1982], Zermelo's Axiom of Choice. Its origins, development and influence, Springer, Berlin.
Moschovakis, Yiannis N. [1994], Notes on set theory, Springer-Verlag, New York.
Mostowski, Andrzej [1967], Recent results in set theory, The philosophy of mathematics (Lakatos, I., editor), North-Holland, Amsterdam.
Parsons, Charles [2008], Mathematical thought and its objects, Cambridge University Press.
Peano, Giuseppe [1889], Arithmetices principia, nova methodo exposita, Bocca, Torino, partial English translation in Heijenoort [1967].
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. 130, Part II: 20th Century Metalogic to 21st Century Semantics, History and Philosophy of Logic, vol. 23 (2), pp. 77-94.
Russell, Bertrand [1910], Principia mathematica, vol. I, Cambridge University Press, with A. N. Whitehead.
Russell, Bertrand [1920], Introduction to mathematical philosophy, 2nd ed., Allen & Unwin, London, reprinted in New York, Dover, 1993.
Shapiro, Stewart [1991], Foundations without foundationalism: A case for second-order logic, Oxford University Press.
Shapiro, Stewart [1997], Philosophy of mathematics: Structure and ontology, Oxford University Press.
Simpson, Stephen G. [1999], Subsystems of second-order arithmetic, Springer, Berlin.
Skolem, Thoralf [1923], Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Skolem's selected works in logic, Universitetsforlaget, Oslo, 1970. English translation in Heijenoort [1967].
Tait, William [2005], The provenance of pure reason: Essays in the philosophy of mathematics and its history, Oxford University Press.
Väänänen, Jouko [2001], Second-order logic and foundations of mathematics, this Bulletin, vol. 7, pp. 504520.
Van Heijenoort, Jean [1967], From Frege to Gödel: A source book in mathematical logic, Harvard University Press.
Weston, Thomas [1976], Kreisel, the Continuum Hypothesis and second order set theory, Journal of Philosophical Logic, vol. 5, pp. 281298.
Weyl, Hermann [1918], Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, references to the reprint New York, AMS Chelsea, 1973.
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. 213.
Weyl, Hermann [1949], Philosophy of mathematics and natural science, Princeton University Press, translated with additions from the original German, published in 1927.
Wittgenstein, Ludwig [1976], Lectures on the foundations of mathematics, Cambridge, 1939, edited by Diamond, Cora, Cornell University Press, Ithaca, N.Y.
Woodin, Hugh [2001], The Continuum Hypothesis, Parts I and II, Notices of the American Mathematical Society, vol. 48, pp. 567-576, 681690.
Zermelo, Ernst [1908], Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65, pp. 261281, English translation in Heijenoort [1967], 199-215.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed