Definition:Class (Class Theory)

Definition
A class is a collection of all sets such that a particular condition holds.

In class builder notation, this is written as:


 * $\left\{{x : p \left({x}\right)}\right\}$

where $p \left({x}\right)$ is a statement containing $x$ as a free variable.

This is read:
 * All $x$ such that $p \left({x}\right)$ holds.

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.