---
tag: W473FH7Z0P
kind: definition
name: Class
msc: ["03E30", "03E70"]
defines: Y48FA69563
uses: [0HE5WDQ9FJ, 0SJ3RX9TXA, 086QKPF0A6, 0HESMMSYEG, 0RDJK7EQYY]
sources:
  - url: https://en.wikipedia.org/wiki/Class_(set_theory)
---

## Definition

Work in [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
Let $\varphi(x, p_1, \dots, p_n)$ be a formula of [[first-order logic|086QKPF0A6]] in the language $\set{\in}$, and let $p_1, \dots, p_n$ be sets, the parameters.
The *class* defined by $\varphi$ with these parameters is the collection
$$C = \set{x : \varphi(x, p_1, \dots, p_n)},$$
and $x \in C$ is an abbreviation for $\varphi(x, p_1, \dots, p_n)$.
A class $C$ is a *set* if there is a [[set|0SJ3RX9TXA]] $a$ with $\forall x\, (x \in a \leftrightarrow x \in C)$.
Otherwise $C$ is a *proper class*.

In class theories such as von Neumann–Bernays–Gödel set theory, classes are instead objects of the theory.
There, a set is a class that is an element of some class.

## Remarks

Example: the class $\set{x : x \subseteq \omega}$ is a set, namely the power set of $\omega$.
Non-example: the class $V = \set{x : x = x}$ of all sets is not a set, so it is a proper class.
If $V$ were a set $a$, then separation would give the set $\set{x \in a : x \notin x}$, and [[Russell's paradox|0HESMMSYEG]] would follow.
The class of all [[ordinals|0RDJK7EQYY]] is also a proper class, by the Burali-Forti argument.
Every set $a$ is a class, defined by the formula $x \in p$ with parameter $p = a$.