# 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 Pairing

For any two sets, there exists a set to which only those two sets are elements:

$\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

### The Axioms of Class Existence

 $(\text B 1)$ $:$ $\ds \exists X: \forall u, v: \tuple {u, v} \in X \iff u \in v$ $\in$-relation $(\text B 2)$ $:$ $\ds \forall X, Y: \exists Z: \forall u: u \in Z \iff u \in X \land u \in Y$ intersection $(\text B 3)$ $:$ $\ds \forall X: \exists Z: \forall u: u \in Z \iff u \notin X$ complement $(\text B 4)$ $:$ $\ds \forall X: \exists Z: \forall u: u \in Z \iff \exists v: \tuple {u, v} \in X$ domain $(\text B 5)$ $:$ $\ds \forall X: \exists Z: \forall u, v: \tuple {u, v} \in Z \iff u \in X$ $(\text B 6)$ $:$ $\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {v, w}, u} \in X$ $(\text B 7)$ $:$ $\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {u, w}, v} \in X$

### The Axiom of Unions

For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:

$\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

### The Axiom of Powers

For every set, there exists a set of sets whose elements are all the subsets of the given set.

$\forall x: \exists y: \paren {\forall z: \paren {z \in y \iff \paren {w \in z \implies w \in x} } }$

### The Axiom of Replacement

For every mapping $f$ and set $x$ in the domain of $f$, the image $f \sqbrk x$ is a set.

Symbolically:

$\forall Y: \map {\text{Fnc}} Y \implies \forall x: \exists y: \forall u: u \in y \iff \exists v: \tuple {v, u} \in Y \land v \in x$

where:

$\map {\text{Fnc}} X := \forall x, y, z: \tuple {x, y} \in X \land \tuple {x, z} \in X \implies y = z$

and the notation $\tuple {\cdot, \cdot}$ is understood to represent Kuratowski's formalization of ordered pairs.

### The Axiom of Infinity

There exists a set containing:

$(1): \quad$ a set with no elements
$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$

### The Axiom of Foundation

For any non-empty class, there is an element of the class that shares no element with the class.

$\forall X: X \ne \O \implies \exists y: y \in X \land y \cap X = \O$

### The Axiom of Global Choice

There exists a mapping $f : V \setminus \set \O \to V$, where $V$ is the universal class, such that:

$\forall x \in V: \map f x \in x$

Symbolically:

$\exists A: \map {\text{Fnc}} A \land \forall x: x \ne \O \implies \exists y: y \in x \land \tuple {x, y} \in A$

## 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.