Definition:Class (Class Theory)

Informal 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.

Small and Proper Classes
A class is proper if it is not a set. That is, $A$ is a proper class iff $\neg \exists x: x = A$ where $x$ is a set.

A class is small if it is equal to some set. See the definition of small class.

Notation
Classes are sometimes denoted by variables $A, B, C,\dots$