# 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.**

## Definition in ZF Set Theory

A **class** in $\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds.

In $\textrm{ZF}$, **classes** are written using class builder notation:

- $\left\{{x : P \left({x}\right)}\right\}$

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

More formally, a **class** $\left\{ {x : P \left({x}\right)}\right\}$ serves to define the following definitional abbreviations involving the membership symbol:

\(\displaystyle y \in \left\{ {x: P \left({x}\right)}\right\}\) | \(\quad \text{for} \quad\) | \(\displaystyle P \left({y}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in y\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z \in y: \forall x: \left({x \in z \iff P \left({x}\right)}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in \left\{ {y: Q \left({y}\right)}\right\}\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z: \left({Q \left({z}\right) \land \forall x: \left({x \in z \iff P \left({x}\right)}\right)}\right)\) | $\quad$ | $\quad$ |

where:

- $x, y ,z$ are variables of $\textrm{ZF}$
- $P, Q$ are propositional functions.

Through these "rules", every statement involving $\left\{{x : P \left({x}\right) }\right\}$ can be reduced to a simpler statement involving only the basic language of set theory.

## Proper Class

A **proper class** is a class which is not a set.

That is, $A$ is a **proper class** iff:

- $\neg \exists x: x = A$

where $x$ is a set.

A class which is not a proper class is a small class.

## Also see

- Results about
**classes**can be found here.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 4.2$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.1$: Sets