Definition:Von Neumann-Bernays-Gödel Set Theory

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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


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.


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

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.


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


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

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:

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


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

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.