foundations

Contents

Idea

In the foundations of mathematics, a class theory or class-set theory is a theory of classes and sets. Class theories are usually formulated in the language of logic over type theory. Class theories are important because the allow for the formal definition of universes, a universal class or a class of all sets, allowing for size issues in mathematics to be handled where needed, such as in category theory in defining large categories and locally small categories.

Similar to the case with set theory, there are material and structural versions of class theory as well:

Material approaches to class theories include Morse-Kelley class theory and von Neumann–Bernays–Gödel class theory, and are two-sorted theories with a sort of sets and a sort of classes.

Structural approaches to class theories include category with class structure and the general field of algebraic set theory. Here, classes are defined as objects in a suitably presented well-pointed category, and sets are defined as a class with some smallness structure.

In contrast to material set theories like ZFC and structural set theories like ETCS or SEAR, the notion of set is not a primitive in the theory. It is instead a defined notion from the primitive notion of class.

Semantics and models

Let $U$ be a universe, and let $\mathrm{Set}_U$ be the universe of sets in $U$, and let $\Omega$ be a collection of propositions. A model of set theory is $\mathrm{Set}_U$, while a model of class theory is the collection of predicates $\mathrm{Set}_U \to \Omega$.