Category:Axioms/Von Neumann-Bernays-Gödel Axioms

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This category contains axioms related to 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 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.