Neumann-Bernays-Gödel set theory

The Neumann-Bernays-Gödel (NBG) axioms are a system for axiomatic set theory, developed by John von Neumann, Paul Bernays, and Kurt Gödel. Unlike naive set theory, NBG focuses on the properties of sets and their relationships rather than defining what a "set" fundamentally is.1

NBG distinguishes between "sets" and "proper classes." Proper classes are collections too large to be considered sets, such as the collection of all sets. This distinction helps avoid the paradoxes found in naive set theory by proving that collections leading to paradoxes are proper classes, not sets.1

NBG is closely related to Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While ZFC has an infinite number of axioms, NBG has a finite set of axioms, making its logical study simpler. Every theorem in ZFC is also a theorem in NBG, and NBG's theorems that only concern sets are also theorems in ZFC. Kurt Gödel's theorems show that if NBG (or ZFC) is consistent, its consistency cannot be proven within the system itself, and there will always be undecidable statements.12