---
tag: Y48FA69563
kind: concept
name: Class
msc: ["03E30", "03E70"]
uses: [0BSJ7RQ9BW, 0HE5WDQ9FJ, 0HESMMSYEG, 0YSCSX5YZC]
sources:
  - url: https://en.wikipedia.org/wiki/Class_(set_theory)
same_as:
  wikidata: Q217594
  wikipedia: Class (set theory)
---

## Summary

In set theory, a class is a collection of sets defined by a property that all its members share.
Every [[set|0BSJ7RQ9BW]] is a class, but some classes are too large to be sets; these are the proper classes.
Examples of proper classes are the class of all sets and the class of all [[ordinals|0YSCSX5YZC]].
Classes let one speak about such large collections without the contradictions of [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

A class is the collection of all sets with some property, such as "being a set" or "being a group".
Small collections, such as the even numbers, are sets.
Some collections are too big to be a set: if the collection of all sets were a set, one could form Russell's paradoxical set from it.
So one calls them proper classes, and does not allow them as elements of other collections.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], a class is just a way of talking about a formula; other theories treat classes as objects.