Category theory and set theory as theories about complementary types of universals

Logic and Logical Philosophy 26 (2):1-18 (2017)
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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.

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Author's Profile

David Ellerman
University of Ljubljana

Citations of this work

Category theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.

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References found in this work

The iterative conception of set.George Boolos - 1971 - Journal of Philosophy 68 (8):215-231.
Adjointness in Foundations.F. William Lawvere - 1969 - Dialectica 23 (3‐4):281-296.
On Frege's way out.W. V. Quine - 1955 - Mind 64 (254):145-159.
Hegel, british idealism, and the curious case of the concrete universal.Robert Stern - 2007 - British Journal for the History of Philosophy 15 (1):115 – 153.
The third man again.P. T. Geach - 1956 - Philosophical Review 65 (1):72-82.

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