Definition:Von Neumann-Bernays-Gödel Set Theory
Definition
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
and:
- 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.
Then:
- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$
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.
Symbolically:
- $\forall Y: \map {\text{Fnc}} Y \implies \forall x: \exists y: \forall u: u \in y \iff \exists v: \tuple {v, u} \in Y \land v \in x$
where:
- $\map {\text{Fnc}} X := \forall x, y, z: \tuple {x, y} \in X \land \tuple {x, z} \in X \implies y = z$
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:
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:
- $\forall x \in V: \map f x \in x$
Symbolically:
- $\exists A: \map {\text{Fnc}} A \land \forall x: x \ne \O \implies \exists y: y \in x \land \tuple {x, y} \in A$
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 Neumann, Paul 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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): von Neumann set theory
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): von Neumann set theory
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 10$ Sets and classes