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

• Authors: Ellerman David
• Languages of publication: EN

Abstracts

EN

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.

Keywords

EN

universals; category theory; Plato’s Theory of Forms; set theoretic antinomies; universal mapping properties

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2017
• Volume: 26
• Issue: 2
• Pages: 145–162

Dates

issued

2016-06-15

Contributors

author

Ellerman David

• Department of Philosophy, University of California at Riverside, 4044 Mt. Vernon Ave, Riverside, CA 92507 USA

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Identifiers

DOI

10.12775/LLP.2016.022