Journal of Symbolic Logic

Proper Classes via the Iterative Conception of Set

Mark F. Sharlow

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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 $\mathrm{ZFC} + (\exists x) \lbrack x \text{is a strongly inaccessible cardinal}\rbrack$.

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J. Symbolic Logic, Volume 52, Issue 3 (1987), 636-650.

First available in Project Euclid: 6 July 2007

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Sharlow, Mark F. Proper Classes via the Iterative Conception of Set. J. Symbolic Logic 52 (1987), no. 3, 636--650.

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