Category theory and set theory as theories about complementary types of universals
 Citation data:

Logic and Logical Philosophy, ISSN: 14253305, Vol: 26, Issue: 2, Page: 145162
 Publication Year:
 2017
 Repository URL:
 http://philsciarchive.pitt.edu/id/eprint/12353
 DOI:
 10.12775/llp.2016.022
 Author(s):
 Publisher(s):
 Tags:
 Arts and Humanities
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article description
Instead of the halfcentury old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The settheoretic 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 selfpredicative in the sense of uF ∈ uF . But the mathematical theory of categories, dating from the midtwentieth century, includes a theory of alwaysselfpredicative universals which can be seen as forming the “other bookend” to the neverselfpredicative universals of set theory. The selfpredicative universals of category theory show that the problem in the antinomies was not selfpredication per se, but negated selfpredication. They also provide a model (in the Platonic Heaven of mathematics) for the selfpredicative 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.