Journal of Symbolic Logic 52 (3):636-650 (1987)

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]
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DOI 10.2307/2274353
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Mathematics in Philosophy, Selected Essays.Stewart Shapiro - 1983 - Journal of Symbolic Logic 53 (1):320.

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Broadening the Iterative Conception of Set.Mark F. Sharlow - 2001 - Notre Dame Journal of Formal Logic 42 (3):149-170.
The Intuitionistic Alternative Set Theory.K. Lano - 1993 - Annals of Pure and Applied Logic 59 (2):141-156.

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