Axiomatic set theory

Von Neumann-Bernays-Gödel set theory

Von Neumann-Bernays-Gödel set theory or NBG¹ is a Material set theory and conservative extension of ZFC, 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
• ℭ𝔩𝔰⁡(𝑥) iff 𝑥 is a Class;
• 𝑥 ∈𝑦 iff ℭ𝔩𝔰⁡(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)def⟺ℭ𝔩𝔰⁡(𝑥)∧(∃ℭ𝔩𝔰⁡𝑦)[𝑥∈𝑦]

and any class 𝑥 which is not a set is a proper class, denoted ¬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: nbg

Fundamentals

1. Axiom of Extensionality for Classes
2. Emptyset Axiom
3. Axiom of Pairing for Classes

Axioms of Class Existence

1. Elementhood Relation Class Axiom
2. Axiom of Intersection for Classes
3. Complement Axiom for classes
4. Domain Axiom for classes
5. Universal Relation Axiom
6. Axioms of Permutation for classes

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

Set axioms

7. Axiom of Union
8. Powerset Axiom
9. Axiom of Subsets
10. Axiom of Replacement for classes
11. Axiom of Infinity

Lemmata

• Class existence theorem schema

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Footnotes

1. Paul Halmos, who was not a fan, joked that this should stand for “No Bloody Good”. ↩

2. 2015. Introduction to Mathematical Logic, §4.1, pp. 231ff. ↩