# Definition:Von Neumann-Bernays-Gödel Axioms

## Definition

The Von Neumann-Bernays-Gödel axioms are a basis for axiomatic set theory.

The axioms are as follows:

### 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 propositional function 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:

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

where each of $B$ ranges over arbitrary classes.

## Source of Name

This entry was named for John von NeumannPaul Isaac Bernays and Kurt Friedrich Gödel.