The Importance of Developing a Foundation for Naive Category Theory

TL;DR

This paper argues that naive category theory requires a foundational basis to avoid paradoxes, demonstrating that current informal methods are inconsistent and resemble Russell's paradox, thus emphasizing the need for a formal foundation.

Contribution

It provides a formal argument showing that naive category theory without a proper foundation is inconsistent and contains paradoxes similar to Russell's paradox.

Findings

Naive category theory uses informal 'cookbook' methods for constructing categories.

Current naive approaches are inconsistent and contain paradoxes.

A formal foundation is necessary to replace set theory as a basis for mathematics.

Abstract

Recently Feferman (Rev. Symb. Logic 6: 6-15, 2013) has outlined a program for the development of a foundation for naive category theory. While Ernst (ibid. 8: 306-327, 2015) has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a 'cookbook recipe' is used for constructing categories, and it is explicitly shown with a formalized argument that this 'foundationless' naive category theory therefore contains a paradox similar to the Russell paradox of naive set theory.