# Definition:Class/Zermelo-Fraenkel

## Definition

Denote with $\textrm{ZF}$ the language of set theory endowed with the Zermelo-Fraenkel axioms.

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:

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

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.

This definition "overloads" the membership symbol $\in$ since its operands could now be either sets or classes.

That such does not lead to ambiguity is proved on Class Membership Extension of Set Membership.

### Class Variables

In deriving general results about $\textrm{ZF}$ which mention classes, it is often convenient to have class variables, which denote an arbitrary class.

By convention, these variables are taken on $\mathsf{Pr} \infty \mathsf{fWiki}$ to be the (start of) the capital Latin alphabet, i.e. $A, B, C$, and so on.

## Caution

Unlike in von Neumann-Bernays-Gödel set theory, it is prohibited to quantify over classes.

That is, expressions like:

$\forall A: P \left({A}\right)$