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CANTORIAN SET THEORY

Published online by Cambridge University Press:  02 January 2019

ALEX OLIVER  and
TIMOTHY SMILEY
ALEX OLIVER
Affiliation:
GONVILLE AND CAIUS COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE CB2 1TA, UKE-mail: ado10@cam.ac.uk
TIMOTHY SMILEY
Affiliation:
CLARE COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE CB2 1TL, UKE-mail: tjs1002@cam.ac.uk
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Abstract

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

Keywords

03A0503B2003E3003E7003E75arithmetization of analysisAxiom of Pluralityempty setlevelordered pairsingletonSingular LogicCantorScott
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 4 , December 2018 , pp. 393 - 451
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Barrow, J. D., The Book of Nothing, Jonathan Cape, London, 2000.Google Scholar
Boolos, G., Burgess, J., and Jeffrey, R., Computability and Logic, fourth ed., Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
Cantor, G., Gesammelte Abhandlungen Mathematischen undPhilosophischen Inhalts (Zermelo, E., editor), Springer, Berlin, 1932.CrossRefGoogle Scholar
Cohen, P. J., Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.Google Scholar
Devlin, K. J., The Axiom of Constructibility, Springer, Berlin, 1977.CrossRefGoogle Scholar
Ebbinghaus, H.-D., Ernst Zermelo: An Approach to His Life and Work, Springer, Berlin, 2007.Google Scholar
Enderton, H. B., Elements of Set Theory, Academic Press, New York, 1977.Google Scholar
Ewald, W. (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol II, Clarendon Press, Oxford, 1996.Google Scholar
Fraenkel, A. A., Abstract Set Theory, North-Holland, Amsterdam, 1953.Google Scholar
Frege, G., Grundgesetze der Arithmetik, vol. I, transl. (in part) by Furth, M. as The Basic Laws of Arithmetic, University of California Press, Berkeley, 1967.Google Scholar
Frege, G., A critical elucidation of some points in E. Schröder, Vorlesungen überdie Algebra der Logik (1895), Collected Papers on Mathematics, Logic andPhilosophy (McGuinness, B., editor), Blackwell, Oxford, 1984, pp. 210–228.Google Scholar
Gödel, K., What is Cantor’s continuum problem? (1947), Collected Works, vol. II (Feferman, S., editor in chief), Oxford University Press, New York, 1990, pp. 119–141.Google Scholar
Halmos, P. R., Naive Set Theory, Van Nostrand, Princeton, 1960. The page reference is to the 1974 reprint, Springer, New York.Google Scholar
Hausdorff, F., Grundzüge der Mengenlehre, Veit, Leipzig, 1914.Google Scholar
Hazen, A. P., Small sets. Philosophical Studies, vol. 63 (1991), pp. 119–123.CrossRefGoogle Scholar
Landau, E., Grundlagen der Analysis (1930), transl. by Steinhardt, F. as Foundations of Analysis, AMS Chelsea, Providence, RI, 2009.Google Scholar
Lewis, D., Parts of Classes, Blackwell, Oxford, 1991.Google Scholar
Mates, B., Elementary Logic, Oxford University Press, New York, 1965.Google Scholar
Oliver, A. and Smiley, T., Plural Logic, Oxford University Press, Oxford, 2013.CrossRefGoogle Scholar
Oliver, A. and Smiley, T., Plural Logic, second ed., revised and enlarged, Oxford University Press, Oxford, 2016.CrossRefGoogle Scholar
Potter, M., Set Theory and its Philosophy, Oxford University Press, Oxford, 2004.CrossRefGoogle Scholar
Scott, D., Definitions by abstraction in axiomatic set theory. Bulletin of theAmerican Mathematical Society, vol. 61 (1955), p. 442.Google Scholar
Scott, D., Axiomatizing set theory, Axiomatic Set Theory (Jech, T. J., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, part 2, American Mathematical Society, Providence, RI, 1974, pp. 207–214.CrossRefGoogle Scholar
Stenius, E., Das Problem der Logischen Antinomien, Societas Scientiarum Fennica, Helsinki, 1949.Google Scholar
Stewart, I., Concepts of Modern Mathematics, Penguin, Harmondsworth, 1981.Google Scholar
Tarski, A., On well-ordered subsets of any set. Fundamenta Mathematicae, vol. 32 (1939), pp. 176–183.CrossRefGoogle Scholar
von Neumann, J., An axiomatization of set theory (1925), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (van Heijenoort, J, editor), Harvard University Press, Cambridge, MA, 1967, pp. 393–413.Google Scholar
Zermelo, E., Investigations in the foundations of set theory I (1908), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (van Heijenoort, J, editor), Harvard University Press, Cambridge, MA, 1967, pp. 199–215.Google Scholar
Zermelo, E., On boundary numbers and domains of sets: new investigations in thefoundations of set theory (1930), Ernst Zermelo: Collected Works, vol. I. (Ebbinghaus, H.-D. and Kanamori, A., editors), Springer, Berlin, 2010, pp. 400–430.CrossRefGoogle Scholar
Zermelo, E., On the set-theoretic model (1930), Ernst Zermelo: Collected Works, vol. I (Ebbinghaus, H.-D. and Kanamori, A., editors), Springer, Berlin, 2010, pp. 446–453.CrossRefGoogle Scholar
Zermelo, E., Report to the Emergency Association of German Science about myresearch concerning the foundations of mathematics (1930), Ernst Zermelo:Collected Works, vol. I (Ebbinghaus, H.-D. and Kanamori, A., editors), Springer, Berlin, 2010, pp. 434–443.CrossRefGoogle Scholar