---
tag: JTPZ1HZ9KK
kind: concept
name: Von Neumann–Bernays–Gödel set theory
msc: ["03E30", "03E70"]
specializes: [RTHHCX84HN]
uses: [RTHHCX84HN, Y48FA69563, 0HE5WDQ9FJ]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann–Bernays–Gödel_set_theory
same_as:
  wikidata: Q278770
  wikipedia: Von Neumann–Bernays–Gödel set theory
---

## Summary

Von Neumann–Bernays–Gödel set theory, NBG, is an axiomatic [[set theory|RTHHCX84HN]] whose objects are [[classes|Y48FA69563]].
In it, sets are the classes that are members of other classes.
It proves the same statements about sets as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice.
Unlike ZFC, it can be given by finitely many axioms.
It lets one speak directly about large collections, such as the class of all sets or of all groups.

## Explanations

### L1 human

In ZFC, a phrase such as "the collection of all sets" is only a way of talking.
NBG makes such collections into real objects, called classes.
A class that is small enough to belong to another class is a set, and the others are proper classes.
Proper classes cannot be elements, so Russell's paradox does not arise.
Category theory often uses this setting to talk about the category of all sets.