We may use the word “class” generically to mean all those mathematical objects which satisfy a certain predicate, reified as a single conceptual entity. Like groups or topological spaces, for example. Most foundational systems simply do not have a good way to formalize classes. Let's call this “the class problem”. The problem becomes particularly serious when we try to formalize concepts which implicitly involve classes of classes, such as categories. Universe hierarchies do not solve the class problem, because equivocating about the size of the universe undermines the conceptual integrity of the system as a whole.
In this talk I will present and defend my own solution to the class problem, a new foundational system called OE. (The name comes from “Operations and Equalities”.) In OE we have a specific object, Class, representing the class of classes in the system. The behavior of Class is illustrated by the following basic facts, provable in OE:
Class ∈ Class
(→) ∈ Class → Class → Class
(×) ∈ Class → Class → Class
∀C∈Class. C × C → C ≅ C → C → C
Similarly, there is a category of categories which is (literally) an object in itself. I will explain how this all works, what it might mean for formal and informal foundations of math, and limitations of the method. I will also sketch a proof that OE is consistent relative to ZFC.