×

zbMATH — the first resource for mathematics

Broadening the iterative conception of set. (English) Zbl 1034.03052
Summary: The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for the theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine’s set theory NF.
MSC:
03E70 Nonclassical and second-order set theories
PDF BibTeX XML Cite
Full Text: DOI
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. 215–31. 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., Logic, Logic, and Logic , edited by R. Jeffrey, Harvard University Press, Cambridge, 1998. · Zbl 0955.03008
[5] Forster, T. E., Set Theory with a Universal Set , vol. 20 of Oxford Logic Guides , The Clarendon Press, Oxford, 1992. · Zbl 0755.03029
[6] Fraenkel, A. A., Y. Bar-Hillel, and A. Levy, Foundations of Set Theory , revised edition, North-Holland Publishing Co., Amsterdam, 1973. · Zbl 0248.02071
[7] Gödel, K., ”Russell’s mathematical logic”, pp. 447–69 in Philosophy of Mathematics: Selected Readings , edited by P. Benacerraf and H. Putnam, Cambridge University Press, Cambridge, 2d edition, 1983.
[8] Hailperin, T., ”A set of axioms for logic”, The Journal of Symbolic Logic , vol. 9 (1944), pp. 1–19. · Zbl 0060.02201
[9] Hallett, M., Cantorian Set Theory and Limitation of Size , vol. 10 of Oxford Logic Guides , The Clarendon Press, New York, 1984. · Zbl 0656.03030
[10] Hinnion, R., ”Stratified and positive comprehension seen as superclass rules over ordinary set theory”, Zeitschrift für mathematische Logik und Grundlagen der Mathematik , vol. 36 (1990), pp. 519–34. · Zbl 0696.03026
[11] Holmes, M. R., ”The set-theoretical program of Quine succeeded, but nobody noticed”, Modern Logic , vol. 4 (1994), pp. 1–47. · Zbl 0804.03040
[12] Prati, N., ”A partial model of NF with ZF”, Mathematical Logic Quarterly , vol. 39 (1993), pp. 274–78. · Zbl 0807.03035
[13] Quine, W., ”New foundations for mathematical logic”, American Mathematical Monthly , vol. 44 (1937), pp. 70–80. JSTOR: · Zbl 0016.19301
[14] Sharlow, M. F., ”Proper classes via the iterative conception of set”, The Journal of Symbolic Logic , vol. 52 (1987), pp. 636–50. JSTOR: · Zbl 0646.03044
[15] Wang, H., From Mathematics to Philosophy , Routledge and Kegan Paul, London, 1974. · Zbl 0554.03002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.