Definition:Von Neumann-Bernays-Gödel Axioms

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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 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.




Source of Name

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