Definition:Class (Class Theory)

Definition
A class is a collection of sets.

In ZF set theory only restricted comprehension of classes is facilitated.

One approach is to hypothesise a universal set $\mathfrak U$ (e.g. a Grothendieck universe), and conduct one's business locally within $\mathfrak U$.

The Gödel–Bernays axioms allow less restrictive class comprehension.

In particular the collection of all sets form a class.

We aren't freed from such problems, however: there is no axiom schema for the comprehension of a "class of all classes".

Small and Proper Classes
A class is proper if it is not a set.

A class is small if it is a set.