Reference textbook developing NBG set theory

Question

I'm starting Borceux "Handbook of Categorical Algebra". It starts with a brief discussion of the logical foundations of category theory. He describes two approaches: 1.defining universes and 2. With NBG set theory. I'm interested in the second approach, so I want to study the axiomatics of that system. The book that I'm looking for should present the logical construction of the theory and some important results such as proving that NBG is an extension of ZFC, some results on model theory and a profound discussion of its applicability to category theory if possible.

Answer 1

Sadly, there aren't that many sources on this really important topic because Grothendieck universes have largely taken hold as the "solution" to collections that are too large to be sets. I'm not sure why this is.

That being said, there is a wonderful book on NBG set theory by Smullyan abd Fitting, Set Theory and the Continuum Problem.I think you'll find it very helpful. But be careful-many Dover printings came out with missing symbols and the result was a disaster. I got stuck with one. So be sure the book isn't defective before you buy-ask.

As for the relationship with category theory, there are no books per se (although there really should be),but there are several books that do mention the issue.There's a brief but informative discussion in Adamek,et. al.'s The Joy of Cats online. There is also a good but more sophisticated section in the mathematical logic textbook of my old teacher, Elliot Mendelson, where the logical subtleties are detailed in an interesting manner.

It's important also to know that NBG isn't the only form of set theory that's been proposed with proper classes to act as a unified foundation for both category theory and set theory. For example, there's a modified form of Willard Quine's New Foundations that looks quite promising and has somewhat different axioms from NBG.(Note: Quine's original formulation of NF, which initially got a lot of positive response from mathematicians and logicians,runs into a major stumbling block in it's original form: Namely, you can disprove the axiom of choice within any consistent axiom system of it! Mathematicians,primarily Tom Forster at Cambridge,have created modified versions of NF, which are equivalent to ZFC for "small" classes since then that avoid this problem.) A good presentation of the basic theory can be found in the online textbook by Holmes, available here.

Answer 2

Mendelson's Introduction to Mathematical Logic has a good development of the basic facts of set theory in NBG and includes a discussion of its relationship with ZFC (it is a conservative extension of ZF, and hence the model theory of NBG is essentially the same as that of ZF). As for a profound discussion of its applicability to category theory, I don't know of one. The system MK introduced in the appendix to Kelley's book on topology is probably closer to ordinary practice in category theory, since it is common to quantify over all elements of a proper class in categorical constructions. Both systems will let you talk about particular large categories, e.g., topological spaces or groups, and about particular functors between them, e.g., the fundamental group functor. Neither system allows you to talk very satisfactorily about large categories in general.

Answer 3

Two references which list the axioms of NBG are

(1) Chapter one of Godel's 1940 monograph titled "The consistency of the axiom of choice and the generalized continuum hypothesis from the axioms of set theory"

(2) Chapter II article 7 in Foundations of set theory by Fraenkel, Bar-Hillel and Levy.

Remark: NBG is a conservative extension of ZF - See pages 131-32 in reference (2) above for a proof sketch. Although NBG is finitely axiomatizable, this is not a real advantage. Godel used this system in (1) only to have a more economical definition of the constructible universe.

There is also a system called Morse-Kelley (stronger than ZF) about which you can read in the appendix of Kelley's General topology and also on Wikipedia.