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Proper classes via the iterative conception of set1

Published online by Cambridge University Press:  12 March 2014

Mark F. Sharlow
Mark F. Sharlow*
Affiliation:
Department of Philosophy, U.C.L.A., Los Angeles, California 90024
*
P. O. Box 2039, Santa Monica, California 90406
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Abstract

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃x) [x is a strongly inaccessible cardinal].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 636 - 650
Copyright
Copyright © Association for Symbolic Logic 1987

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Footnotes

1

The author wishes to thank A. J. Macintyre and an anonymous referee for their valuable suggestions on the original version of this paper.

References

REFERENCES

[Dr] Drake, Frank R., Set theory: an introduction to large cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, North-Holland, Amsterdam, 1974.Google Scholar
[Ha] Hallett, Michael, Cantorian set theory and limitation of size, Oxford Logic Guides, vol. 10, Clarendon Press, Oxford, 1984.Google Scholar
[No] Novák, Vilém, An attempt at Gödel-Bernays-like axiomatization of fuzzy sets, Fuzzy Sets and Systems, vol. 3 (1980), pp. 323–325.CrossRefGoogle Scholar
[Pa] Parsons, Charles, Mathematics in philosophy: selected essays, Cornell University Press, Ithaca, New York, 1983.Google Scholar
[TZ] Takeuti, Gaisi and Zaring, Wilson M., Introduction to axiomatic set theory, 2nd ed., Graduate Texts in Mathematics, vol. 1, Springer-Verlag, Berlin and New York, 1982.CrossRefGoogle Scholar