Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC). This set theory is often referred to by the abbreviation NBG or NGB. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes proper classes, objects having members but that cannot be members of other entities. NBG's principle of class comprehension is predicative; quantified variables in the defining formula can range only over sets. Allowing impredicative comprehension turns NBG into Morse-Kelley set theory (MK). NBG, unlike ZFC and MK, can be finitely axiomatized.

The defining aspect of NBG is the distinction between proper class and set. Let a and s be two individuals. Then the atomic sentence ${\displaystyle a\in s}$ is defined if a is a set and s is a class. In other words, ${\displaystyle a\in s}$ is defined unless a is a proper class. A proper class is very large; NBG even admits of "the class of all sets", the universal class called V. However, NBG does not admit "the class of all classes" (which fails because proper classes are not "objects" that can be put into classes in NBG) or "the set of all sets" (whose existence cannot be justified with NBG axioms).

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