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.

Classes can be introduced into ZF set theory by defining $y \in \{ x : P(x) \}$, $\{ x : P(x) \} \in y$, and $\{ x : P(x) \} \in \{ y : Q(y) \}$.

Once defined, every statement involving $\{ x : P(x) \}$ can be reduced to a simpler statement involving only the language of set theory:


 * $y \in \{ x : P\left({x}\right) \} \dashv \vdash P\left({y}\right)$


 * $\{ x : P\left({x}\right) \} \in y \dashv \vdash \exists z \in y: \forall x: \left({ x \in z \iff P\left({x}\right) }\right)$


 * $\{ x : P\left({x}\right) \} \in \{ y : Q\left({y}\right) \} \dashv \vdash \exists z: \left({ Q\left({z}\right) \land \forall x: \left({ x \in z \iff P\left({x}\right) }\right) }\right)$

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.