Books on NBG Axiomatic Set Theory

• gotjrgkr
In summary, NBG Axiomatic Set Theory is a mathematical theory developed by John von Neumann, Paul Bernays, and Kurt Gödel that serves as a foundation for modern mathematics. It extends the well-known ZFC set theory and allows for the existence of proper classes, which are collections of sets that are too large to be considered sets in ZFC. NBG is important because it provides a rigorous and consistent foundation for mathematics and has applications in various areas such as logic, category theory, and algebraic geometry. There are some debates and controversies surrounding NBG, including its philosophical implications and discussions about its consistency and completeness.
gotjrgkr
Hi!
I am looking for a text concerned with NBG axiomatic set theory( Neumann-Bernays-Godel).
Could you recommend some books related with it?

Hi gotjrgkr!

The wonderful text "Set theory" by Jech has a chapter on NBG. It is one of the best books on set theory that I know of.

Thank you!

1. What is NBG Axiomatic Set Theory?

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.

2. What are the main differences between NBG and ZFC?

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.

3. Why is NBG Axiomatic Set Theory important?

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.

4. What are some common applications of NBG Axiomatic Set Theory?

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.

5. Are there any controversies or debates surrounding NBG Axiomatic Set Theory?

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.

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