A logic for category theory

Abstract

MacLane and Feferman have argued that the traditional set theories of Zermelo—Fraenkel and Gödel—Bernays are not suitable foundations for category theory because of the requirement for self-referencing abstractions. The necessity for distinguishing between small and large categories reflects this unsuitability. The purpose of this paper is to demonstrate that a formalization within a natural-deduction-based logic and set theory called NaDSet avoids the difficulties that arise with the use of the traditional set theories. Definitions are provided within NaDSet for most of the fundamental concepts and constructs in category theory. The main result of the paper is a proof that the set of all functors forms a category under appropriate definitions of composition and equivalence. Additional definitions and discussions on products, comma categories, universals, limits and adjoints are presented. They provide evidence to support the claim that any construct, not only in categories, but also in toposes, sheaves, triples and similar theories can be formalized within NaDSet. NaDSet succeeds as a logic and set theory for category theory because the resolution of the paradoxes provided for it is based on an extension of Tarski's reductionist semantics for first-order logic. Self-membership and self-reference is not explicitly excluded. The reductionist semantics is most simply presented as a natural-deduction logic. A sketch of the elementary and logical syntax or proof theory of the logic is provided. A consistency proof for NaDSet is provided elsewhere.