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

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"