# Definition:Von Neumann-Bernays-Gödel Set Theory

## 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 NeumannPaul 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.