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 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.

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.