In ZFC there is no set that is the set of all sets, for this we introduce the notion of class. But then what is the 'class' of all classes, is it actually a class? Do we apply the same idea again? But then at what stage do we stop? Does this show that classes are not the right notion to go beyond sets, but more of an ad-hoc solution?


Further, within foundational category theory, we have the notion of grothendieck universes, if i recall rightly, this is equivalent to introducing an axiom that an inaccessible cardinal exists. Does this subsume, or is equivalent to the notion of classes?


Finally, what is the formalism that uses classes to extend ZFC, is the NBG?



What notion captures the 'class' of all classes?
Since the question is rather philosophical (e.g., "right notion"), I'll use it as an excuse to record my philosophical opinions on this topic. The intuition underlying ZFC, i.e., the intuition of the cumulative hierarchy of sets, contains two quite vague notions, (1) the notion of "arbitrary subset" of an infinite set, used at successor stages of the hierarchy, and (2) the notion of iterating "forever", beyond any imaginable bound. Although these ideas are vague, they have consequences that can be expressed precisely, and the point of the ZFC axioms is to express enough of the consequences to serve as a foundation for what mathematician ordinarily do. 


To add proper classes to the picture, as in the von Neumann-Bernays-Gödel or Morse-Kelley theories, is to add one more level to the cumulative hierarchy, after all the sets. This is technically useful for some purposes (including some aspects of category theory), but it is incoherent with aspect (2) of the intuition of sets. If it's possible to add one more level, then the hierarchy of sets should have been continued to include that level and many more beyond it. 


For this reason, I view ZFC, possibly augmented with (mild) large-cardinal axioms or reflection principles as an intuitively more acceptable foundation than a class theory. I might well use the terminology of proper classes as a convenient abbreviation for statements about sets (e.g., "$V=L$" abbreviates "all sets are constructible", which can be defined in hte purely set-theoretic context of ZFC). But when people make serious use of proper classes, my picture of what they're doing is that their sets are really just sets of rank below some inaccessible cardinal $\kappa$ and their proper classes are really sets of rank $\kappa$. If they need super-classes of classes and even higher-rank collections, that's no problem as far as I'm concerned; the universe of sets stretches way beyond $\kappa$.


