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Categoricity Theorems and Conceptions of Set

Gabriel Uzquiano
Journal of Philosophical Logic
Vol. 31, No. 2 (Apr., 2002), pp. 181-196
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226753
Page Count: 16
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Categoricity Theorems and Conceptions of Set
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Abstract

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine.

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