# 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:

- $\set {x: \map p x}$

where $\map p x$ is a statement containing $x$ as a free variable.

This is read:

**All $x$ such that $\map p x$ 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:

- $\set {x : \map P x}$

where $\map P x$ is a statement containing $x$ as a free variable.

More formally, a **class** $\set {x: \map P x}$ serves to define the following definitional abbreviations involving the membership symbol:

\(\displaystyle y \in \set {x: \map P x}\) | \(\quad \text{for} \quad\) | \(\displaystyle \map P y\) | |||||||||||

\(\displaystyle \set {x: \map P x} \in y\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z \in y: \forall x: \paren {x \in z \iff \map P x}\) | |||||||||||

\(\displaystyle \set {x: \map P x} \in \set {y: \map Q y}\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z: \paren {\map Q z \land \forall x: \paren {x \in z \iff \map P x} }\) |

where:

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

Through these "rules", every statement involving $\set {x: \map P x}$ 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** if and only if:

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

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 - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory