NBG (von Neumann-Bernays-Gödel) Axiomatic Set Theory is a mathematical theory that provides a foundation for modern mathematics. It is based on the work of mathematicians John von Neumann, Paul Bernays, and Kurt Gödel and is an extension of the more well-known ZFC (Zermelo-Fraenkel with Choice) set theory.
The main difference between NBG and ZFC is that NBG allows for the existence of proper classes, which are collections of sets that are too large to be considered sets in ZFC. NBG also includes a stronger form of the Axiom of Replacement, known as the Axiom of Class Replacement.
NBG Axiomatic Set Theory is important because it provides a rigorous and consistent foundation for mathematics. It allows mathematicians to work with both sets and proper classes, allowing for a more comprehensive understanding of mathematical structures and concepts.
NBG Axiomatic Set Theory has applications in various areas of mathematics, including logic, category theory, and algebraic geometry. It is also used in the study of large cardinal numbers and in the development of other axiomatic set theories.
While NBG Axiomatic Set Theory is generally accepted and used by mathematicians, there are some debates and controversies surrounding the theory. These include the philosophical implications of the existence of proper classes and whether NBG is a necessary extension of ZFC. There are also ongoing discussions about the consistency and completeness of NBG.