# Definition:Von Neumann-Bernays-Gödel Set Theory

## Contents

## Definition

**Von Neumann-Bernays-Gödel set theory** is a system of axiomatic set theory.

Its main feature is that it classifies collections of objects into:

- sets, whose construction is strictly controlled

and:

- classes, which have fewer restrictions on how they may be generated.

All sets are classes, but not all classes are sets.

## Von Neumann-Bernays-Gödel Axioms

### The Axiom of Extension

Let $A$ and $B$ be classes.

Then:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

### The Axiom of Specification

Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a function of propositional logic such that:

- $A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes
- $x$ is a free variable whose domain ranges over all sets.

Then the **axiom of specification** gives that:

- $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$

where each of $B$ ranges over arbitrary classes.

## Also known as

**Von Neumann-Bernays-Gödel set theory** is usually seen abbreviated either as **NBG** or **VNB**.

## Source of Name

This entry was named for John von Neumann, Paul Isaac Bernays and Kurt Friedrich Gödel.

## Historical Note

Von Neumann-Bernays-Gödel set theory was devised by John von Neumann, and later revised by Raphael Mitchel Robinson, Paul Isaac Bernays and Kurt Friedrich Gödel.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**von Neumann set theory** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**von Neumann set theory** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 10$ Sets and classes