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Journal of Philosophical Logic

, Volume 42, Issue 5, pp 697–725 | Cite as

The Iterative Conception of Set

A (Bi-)Modal Axiomatisation
  • J. P. Studd
Article

Abstract

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A modal stage theory, \({\textsf{MST}}\), is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that \({\textsf{MST}}\) interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory.

Keywords

Set theory Iterative conception of set Russell’s paradox Indefinite extensibility Modal logic 

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References

  1. 1.
    Boolos, G. (1971). The iterative conception of set. The Journal of Philosophy, 68(8), 215–231. Reprinted in G. Boolos (1998). Logic, logic, and logic. Cambridge, MA.: Harvard Univesity Press. Page references to reprint.Google Scholar
  2. 2.
    Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21. Reprinted in G. Boolos (1998). Logic, logic, and logic. Cambridge, MA.: Harvard Univesity Press. Page references to reprint.Google Scholar
  3. 3.
    Boolos, G. (1995). The logic of provability. Cambridge: Cambridge University Press.Google Scholar
  4. 4.
    Braüner, T., & Ghilardi, S. (2007). First-order modal logic. In: P. Blackburn, J. van Benthem, F. Wolter (Eds.), Handbook of modal logic. Amsterdam: Elsevier.Google Scholar
  5. 5.
    Bull, R., & Segerberg, K. (2001). Basic modal logic. In: D. Gabbay & F. Guenthner (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 3). Dordrecht: Kluwer.Google Scholar
  6. 6.
    Burgess, J.P. (2002). Basic tense logic. In: D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 7). Dordrecht: Kluwer.Google Scholar
  7. 7.
    Burgess, J.P., & Rosen, G. (1997). A subject with no object. Oxford: Clarendon Press.Google Scholar
  8. 8.
    Cartwright, R. (1994). Speaking of everything. Noûs, 28(1), 1–20.CrossRefGoogle Scholar
  9. 9.
    Dummett, M. (1991). Frege: Philosophy of mathematics. London: Duckworth.Google Scholar
  10. 10.
    Fine, K. (1981). First-order modal theories I—sets. Noûs, 15(2), 177–205.CrossRefGoogle Scholar
  11. 11.
    Fine, K. (2005). Our knowledge of mathematical objects. In: T.S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 1). Oxford: Clarendon Press.Google Scholar
  12. 12.
    Fine, K. (2006). Relatively unrestricted quantification. In: A. Rayo & G. Uzquiano (Eds.), Absolute generality. Oxford: Oxford University Press.Google Scholar
  13. 13.
    Gödel, K. (1964). What is Cantor’s continuum problem? In: P. Benacerraf, & H. Putnam (Eds.) (1983). Philosophy of mathematics: selected readings (2nd ed.). Cambridge: Cambridge University Press. Revised and expanded version of Gödel, K. (1947). What is Cantor’s continuum problem? The American Mathematical Monthly, 54(9), 515–525.Google Scholar
  14. 14.
    Goldblatt, R. (1992). Logics of time and computation (2nd ed.). Stanford, CA.: CSLI.Google Scholar
  15. 15.
    Jech, T. (2003). Set theory (3rd, revised ed.). New York, NY.: Springer.Google Scholar
  16. 16.
    Lévy, A. (1960). Axiom schemata of strong infinity in axiomatic set theory. Pacific Journal of Mathematics, 10(1), 223–238.CrossRefGoogle Scholar
  17. 17.
    Lévy, A., & Vaught, R. (1961). Principles of partial reflection in the set theories of Zermelo and Ackermann. Pacific Journal of Mathematics, 11(3), 1045–1062.CrossRefGoogle Scholar
  18. 18.
    Linnebo, Ø. (2010). Pluralities and sets. The Journal of Philosophy, 107(3), 144–164.Google Scholar
  19. 19.
    Mathias, A. (2001). Slim models of Zermelo set theory. Journal of Symbolic Logic, 66(2), 487–496.CrossRefGoogle Scholar
  20. 20.
    Parsons, C. (1977). What is the iterative conception of set?. In: R. Butts & J. Hintikka (Eds.), Proceedings of the 5th international congress of logic, methodology and philosophy of science 1975, part I: Logic, foundations of mathematics, and computability theory. Dordrecht, Reidel. Reprinted in P. Benacerraf, & H. Putnam (1983). Philosophy of mathematics: selected readings (2nd ed.). Cambridge: Cambridge University Press. Pages references to reprint.Google Scholar
  21. 21.
    Parsons, C. (1983) Sets and modality. In: Mathematics in philosophy. Ithaca, NY: Cornell University Press.Google Scholar
  22. 22.
    Paseau, A. (2007). Boolos on the justification of set theory. Philosophia Mathematica, 15(1), 30–53.CrossRefGoogle Scholar
  23. 23.
    Potter, M. (2004). Set theory and its philosophy: A critical introduction. Oxford: Oxford University Press.CrossRefGoogle Scholar
  24. 24.
    Scott, D. (1974). Axiomatizing set theory. In: Axiomatic set theory II: Proceedings of symposia in pure mathematics (Vol. 13). Providence, RI., American Mathematical Society.Google Scholar
  25. 25.
    Shoenfield, J. (1967) Mathematical logic. Reading, MA.: Addison-Wesley.Google Scholar
  26. 26.
    Uzquiano, G. (1999). Models of second-order Zermelo set theory. Bulletin of Symbolic Logic, 5(3), 289–302.CrossRefGoogle Scholar
  27. 27.
    Venema, Y. (2001). Temporal logic. In: L. Goble (Ed.), The Blackwell guide to philosophical logic. Oxford: Blackwell.Google Scholar
  28. 28.
    Wang, H. (1974). From mathematics to philosophy. London: Routledge.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Faculty of Philosophy, Radcliffe Humanities, Radcliffe Observatory QuarterUniversity of OxfordOxfordUK

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