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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Boolos, G.: Iteration again, Philos. Topics 17 (1989), 15-21.
Crossley, J. and Dummett, M. (eds): Formal Systems and Recursive Functions, North-Holland, 1962.
McGee, V.: How we learn mathematical language, Philos. Rev. 106 (1997), 35-68.
Montague, R.: Set Theory and Higher-Order Logic, in Crossley and Dummett [2], 1962, pp. 131-148.
Potter, M.: Sets: An Introduction, Clarendon Press, 1990.
Scott, D.: Axiomatizing set theory, in T. Jech (ed.), Axiomatic Set Theory: Proceedings of Symposia on Pure Mathematics, vol. 13, Part 2, 1974, pp. 204-214.
Shapiro, S. and Weir, A.: New V, ZF, and Abstraction, Philos. Math. 3 (1999).
Smullyan, R. and Fitting, M.: Set Theory and the Continuum Problem, Oxford Logic Guides 64, Oxford University Press, 1996.
Uzquiano, G.: Models of second-order Zermelo set theory, Bull. Symbolic Logic 5 (1999), 289-302.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Uzquiano, G. Categoricity Theorems and Conceptions of Set. Journal of Philosophical Logic 31, 181–196 (2002). https://doi.org/10.1023/A:1015276715899
Issue Date:
DOI: https://doi.org/10.1023/A:1015276715899