Iteration one more time. (English) Zbl 1071.03038

Cook, Roy T.

Notre Dame J. Formal Logic 44, No. 2, 63-92 (2003).

Summary: A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos’s NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain.

MSC:

03E70	Nonclassical and second-order set theories
03E65	Other set-theoretic hypotheses and axioms
03A05	Philosophical and critical aspects of logic and foundations

Keywords:

logicism; neologicist set theory; abstraction principle; axiom of infinity; axiom of foundation

References:

[1]	Aczel, P., Non-well-founded Sets , vol. 14 of CSLI Lecture Notes , CSLI, Stanford, 1988. · Zbl 0668.04001
[2]	Boolos, G., ”The iterative conception of set”, The Journal of Philosophy , vol. 68 (1971), pp. 215–32. Reprinted in [?], pp. 13–29.
[3]	Boolos, G., ”Iteration again”, Philosophical Topics , vol. 17 (1989), pp. 5–21. Reprinted in [?], pp. 88–104. · Zbl 0972.03511
[4]	Boolos, G., “Is Hume’s Principle analytic?” pp. 245–61 in Language, Thought, and Logic. Essays in Honour of Michael Dummett , edited by R. G. Heck, Jr., Oxford University Press, New York, 1997. Reprinted in [?], pp. 301–14. · Zbl 0930.03005
[5]	Boolos, G., Logic, Logic, and Logic , edited by R. Jeffrey, Harvard University Press, Cambridge, 1998. · Zbl 0955.03008
[6]	Cook, R., and P. Ebert, ”Abstraction and identity”, unpublished manuscript.
[7]	Fine, K., The Limits of Abstraction , edited by R. Jeffrey, Oxford University Press, Oxford, 2002. · Zbl 1043.00005
[8]	Frege, G., Grundgesetze der Arithmetik. Band I, II , Georg Olms Verlagsbuchhandlung, Hildesheim, 1962. · JFM 25.0101.02
[9]	Hale, B., ”Reals by abstraction”, Philosophia Mathematica. Series III , vol. 8 (2000), pp. 100–23. · Zbl 0968.03010
[10]	Hallett, M., Cantorian Set Theory and Limitation of Size , vol. 10 of Oxford Logic Guides , The Clarendon Press, New York, 1984. · Zbl 0656.03030
[11]	Rieger, A., ”An argument for Finsler-Aczel set theory”, Mind , vol. 109 (2000), pp. 241–53.
[12]	Shapiro, S., Foundations without Foundationalism. A Case for Second-order Logic , vol. 17 of Oxford Logic Guides , The Clarendon Press, New York, 1991. · Zbl 0732.03002
[13]	Shapiro, S., and A. Weir, ”New V, ZF” and abstraction, Philosophia Mathematica. Series III , The George Boolos Memorial Symposium (Notre Dame, IN, 1998)vol. 7 (1999), pp. 293–321. · Zbl 0953.03061
[14]	Uzquiano, G., and I. Jané, ”Well and non-well-founded Fregean extensions”, forthcoming in Journal of Philosophical Logic. · Zbl 1058.03054
[15]	Uzquiano, G., ”Models of second-order Zermelo set theory”, The Bulletin of Symbolic Logic , vol. 5 (1999), pp. 289–302. JSTOR: · Zbl 0939.03056
[16]	Weir, A., ”Neo-Fregeanism: An embarrassment of riches”, Notre Dame Journal of Formal Logic , vol. 44 (2003), pp. 13–48. · Zbl 1071.03006