Unlimited Category Theories for Mathematics are Inconsistent: A Discussion of Michael Ernst's "The Prospects for Unlimited Category Theory: Doing What Remains to be Done"

TL;DR

This paper discusses Ernst's proof that any axiomatic theory of categories fulfilling certain natural requirements for an 'unlimited' foundation of mathematics is inherently inconsistent, highlighting fundamental limitations in category-based foundations.

Contribution

It presents Ernst's proof demonstrating the inconsistency of proposed axiomatic theories of unlimited categories that meet specific natural criteria.

Findings

Ernst proved the inconsistency of certain axiomatic category theories.

The paper clarifies the limitations of unlimited category theories.

It discusses the historical context and implications for mathematical foundations.

Abstract

Proponents of category theory long hoped to escape the limits of set theory by founding mathematics on an unlimited category theory in which large categories, such as the category Grp of all groups, the category Top of all topological spaces, and the category Cat of all categories, would be (first-class) entities rather than just classes. Several proposals were put forward by Lawvere, MacLane, and Feferman, but none were successful. Feferman, in 1969 and 2013, proposed three requirements which an axiomatic theory of categories should fulfull in order to be an "unlimited category theory for mathematics". But in 2015 Michael Ernst proved that if an axiomatic theory of categories satisfied Fefermann's three requirements, then it is inconsistent. This paper, a presentation to the Indiana University, Bloomington, Logic Seminar, exposits Ernst's paper.