Category theory and set theory as theories about complementary types of universals

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Logic and Logical Philosophy, ISSN: 1425-3305, Vol: 26, Issue: 2, Page: 145-162

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David P. Ellerman
Uniwersytet Mikolaja Kopernika/Nicolaus Copernicus University, Nicolaus Copernicus University
Arts and Humanities
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Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal uF = {x | F (x)} for a property F (.) could never be self-predicative in the sense of uF ∈ uF . But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals which can be seen as forming the “other bookend” to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.

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