[Math] 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.