Von Neumann-Bernays-Gödel set theory

Von Neumann-Bernays-Gödel set theory or $N B G$ ¹ is a Material set theory and conservative extension of $Z F C$ , in which the notion of a Class is treated directly. Notably, there exists a universal class $𝑉$ of all elements.

Setup

We consider a universe $W$ of objects. We modify the setup of material set theory so that sethood is replaced with classhood, namely we have the primitive notions

$𝑥 = 𝑦$ iff $𝑥$ is the same object as $𝑦$
$ℭ 𝔩 𝔰 (𝑥)$ iff $𝑥$ is a Class;
$𝑥 \in 𝑦$ iff $ℭ 𝔩 𝔰 (𝑦)$ and $𝑥$ is a member of $𝑦$

We also have a sethood predicate $𝔐 (𝑥)$ , however it is no longer primitive, since one can define

𝔐 (𝑥) d e f ⟺ ℭ 𝔩 𝔰 (𝑥) \land (\exists ℭ 𝔩 𝔰 𝑦) [𝑥 \in 𝑦]

and any class $𝑥$ which is not a set is a proper class, denoted $\neg 𝔈 (𝑥)$ . As with my treatment of other material set theories, we will allow for the existence of an Urelement.

Axioms

We take the following axioms²: #m/def/set/nbg

Fundamentals

Axioms of Class Existence

whence follows the important Class existence theorem schema, which generalizes these axioms into instances of a single schema.

Set axioms

Lemmata

Class existence theorem schema

#state/develop | #lang/en | #SemBr

Paul Halmos, who was not a fan, joked that this should stand for “No Bloody Good”. ↩
2015. Introduction to Mathematical Logic, §4.1, pp. 231ff. ↩