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

Logic and Logical Philosophy (Online First), Aug 2016

18 Pages Posted: 15 Aug 2016

Date Written: August 5, 2016

Abstract

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}∈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.

Keywords: Universals, Category Theory, Plato's Theory of Forms, Set Theoretic Antinomies, Universal Mapping Properties

Suggested Citation

Ellerman, David, Category Theory and Set Theory as Theories about Complementary Types of Universals (August 5, 2016). Logic and Logical Philosophy (Online First), Aug 2016. Available at SSRN: https://ssrn.com/abstract=2821924

David Ellerman (Contact Author)

Phil. Dept. UC at Riverside ( email )

4044 Mt. Vernon Ave.
Riverside, CA 92507
United States

HOME PAGE: http://www.ellerman.org

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