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Idealist and realist elements in Cantor's approach to set theory

Philosophia Mathematica 18 (2):193-226 (2010)
There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I conclude that his account of that distinction is only tenable if the definiteness of the set-theoretical universe is rejected. Partially supported by the Spanish CICYT, grant MTM 2008–03389. Earlier versions of this paper were presented at the VIII International Ontology Congress in San Sebastian, and at the Seminar in Logic and the Philosophy of Mathematics of the University of Bristol. I wish to thank the participants for their comments. I am also grateful to Joan Bertran, Joan Climent, and one anonymous referee for their careful reading of the paper and their helpful remarks and suggestions. CiteULike Connotea What's this?
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