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Published online by Cambridge University Press:  01 August 2008

The Ohio State University and University of St Andrews
University of Oxford


1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

Research Article
Copyright © Association for Symbolic Logic 2008

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Antonelli, A., & May, R. (2005). Frege's other program. Notre Dame Journal of Formal Logic, 46, 117.CrossRefGoogle Scholar
Bar-Hillel, Y., Fraenkel, A., & Lévy, A. (1973). Foundations of Set Theory. Amsterdam: North Holland.Google Scholar
Bernays, P. (1961). Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehere. In Bar-Hillel, Y., Poznanski, E. I. J., Robinson, A., and Rabin, M. O., editors. Essays on the Foundations of Mathematics. Jerusalem: Magnes Press, pp. 349.Google Scholar
Bernays, P. (1976). On the problem of schemata of infinity in axiomatic set theory. Sets and Classes: On the Work of Paul Bernays. Amsterdam: North Holland, pp. 121172.CrossRefGoogle Scholar
Boolos, G. (1971). The iterative conception of set. Journal of Philosophy, 68, 215231. Reprinted in Boolos (1998a), 1329.CrossRefGoogle Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 521. Reprinted in Boolos (1998a), 88104.CrossRefGoogle Scholar
Boolos, G. (1993). Whence the contradiction?. Aristotelian Society, Supplementary Volume, 67, 213233. Reprinted in Boolos (1998a), 220236.CrossRefGoogle Scholar
Boolos, G. (1998a). Logic, logic, and logic. Jeffrey, Richard, editor. Cambridge: Harvard University Press.Google Scholar
Boolos, G. (1998b). Must we believe in set theory? See Boolos, G. (1998a), 120–132. Also in (2000) Between Logic and Intuition: Essays in Honor of Charles Parsons. Sher, G., and Tieszen, R., editors. Cambridge: Cambridge University Press, 257268.Google Scholar
Burgess, J. (2004). E pluribus unum: plural logic and set theory. Philosophia Mathematica, 12(3), 13221.CrossRefGoogle Scholar
Burgess, J. (2005). Fixing Frege. Princeton University Press.Google Scholar
Dummett, M. (1991). Frege: Philosophy of Mathematics. London: Duckworth.Google Scholar
Ebbinghaus, H.-D. (2003). Zermelo: definiteness and the universe of definable sets. History and Philosophy of Logic, 24, 197219.CrossRefGoogle Scholar
Friedman, H. (2003). Similar Subclasses. Unpublished typescript. Available from: Scholar
Hale, B. (2000). Abstraction and set theory. Notre Dame Journal of Formal Logic, 41(4), 379398.Google Scholar
Hale, B., & Wright, C. (2001). The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press.CrossRefGoogle Scholar
Hilbert, D., & Ackermann, W. (1928). Die Grundlagen der theoretischen Logik. Berlin: Springer.Google Scholar
Jané, I., & Uzquiano, G. (2004). Well-and non-well-founded Fregean extensions. Journal of Philosophical Logic, 33(5), 437465.CrossRefGoogle Scholar
Koellner, P. (2003). The search for new axioms, PhD Thesis, Massachusetts Institute of Technology.Google Scholar
Lévy, A. (1960a). Axiom schemata of strong infinity in axiomatic set theory. Pacific Journal of Mathematics, 10, 223238.CrossRefGoogle Scholar
Lévy, A. (1960b). Principles of reflection in axiomatic set theory. Fundamenta Mathematicae, 49, 110.CrossRefGoogle Scholar
Lévy, A. (1968). On von neumann's axiom system for set theory. The American Mathematical Monthly, 75(7), 762763.CrossRefGoogle Scholar
Linnebo, Ø. (2007). Burgess on plural logic and set theory. Philosophia Mathematica, 15(1), 7993.CrossRefGoogle Scholar
Moore, G. H. (1982). Zermelo's Axiom of Choice: Its Origins, Development and Influence. New York: Springer-Verlag.CrossRefGoogle Scholar
Rayo, A., & Uzquiano, G., editors. (2006). Absolute Generality. Oxford: Oxford University Press.Google Scholar
Shapiro, S. (1987). Principles of reflection and second-order logic. Journal of Philosophical Logic, 16, 309333.CrossRefGoogle Scholar
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic. Oxford: Clarendon Press.Google Scholar
Shapiro, S. (2003). Prolegomenon to any future neo-logicist set theory: abstraction and indefinite extensibility. British Journal for the Philosophy of Science, 54, 5991.CrossRefGoogle Scholar
Shapiro, S., & Wright, C. (2006). All things indefinitely extensible. In Rayo, A. & Uzquiano, G., editors. Absolute Generality. Oxford: Oxford University Press, pp. 255304.Google Scholar
Tait, W. (1990). The iterative hierarchy of sets. Ivyum, 39, 6579.Google Scholar
Tait, W. (1998). Zermelo's conception of set theory and reflection principles. In Schrin, M., editor. The Philosophy of Mathematics Today. Oxford University Press, pp. 469483.Google Scholar
Tait, W. (2000). Cantor's grundlagen and the paradoxes of set theory. In Sher, G. & Tieszen, R., editors. Between Logic and Intuition: Essays in Honour of Charles Parsons. Cambridge: Cambridge University Press, pp. 269290.CrossRefGoogle Scholar
Van Dalen, D., & Ebbinghaus, H.-D. (2000). Zermelo and the skolem paradox. The Bulletin of Symbolic Logic, 6, 145161.CrossRefGoogle Scholar
von Neumann, J. (1925). Eine Axiomatisierung der Mengenlehre. Journal für die Reine und Angewandte Mathematik, 154, 219240.Google Scholar
von Neumann, J. (1928). Die Axiomatizierung der Mengenlehre. Mathematische Zeitschrift, 27, 339422.Google Scholar
Wang, H. (1974). From Mathematics to Philosophy. London: Routledge & Kegan Paul.Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16, 2947.CrossRefGoogle Scholar