Ellerman D. Category Theory and Set Theory as Theories about Complementary Types of Universals

Published online. — Logic and Logical Philosophy. — (August 5, 2016). — 18 p. Interactive menu. English. (OCR-слой).[David Ellerman. Department of Philosophy.
University of California. Riverside. USA].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 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.Criteria for a Theory of Universals.
Set Theory as the Theory of Non-self-predicative Universals.
Self-Predicative or Concrete Universals in Philosophy.
Self-predicative Universals in Partial Orders.
Self-predicative Universals in General Categories.
Self-Predicative Universals and the Antinomies.
The Third Man Argument.References (37 publ).