Category:Axioms/Von Neumann-Bernays-Gödel Axioms

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.

This needs considerable tedious hard slog to complete it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.