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Category Theory and Set Theory as Theories about Complementary Types of Universals

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Ellerman D.

Logic and Logical Philosophy

2017

vol. 26

issue 2

145–162

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 u_F = {x | F(x)} for a property F(.) could never be self-predicative in the sense of u_F \in u_F . 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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1 Logic and Logical Philosophy

1 Ellerman D.

1 2016

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Category Theory and Set Theory as Theories about Complementary Types of Universals