The Predicative Comprehension in NBG is $\exists X\forall y(y\in X \iff \phi)$, where $\phi$ is a formula where no bound class variables occur. A possible quantification over classes (I know this is not the case in FOL) would raise problems like Russell's paradox over sets even about classes?

-

I don't think there is any real problem with allowing quantification over classes in a prototypical "Class Comprehension Scheme". In fact, Morse-Kelley set theory does just this.

But what does happen is that unlike NBG, MK is not a conservative extension of ZFC.

-

NBG is a first order theory. It just has two types of elements, classes and sets. We define all sets to be classes, but not all classes are sets, in fact a class is a set if and only if it is an element of another class.

We can therefore quantify over classes in NBG because classes are objects of the universe, not "definable subsets" like in ZFC.

However, Russell's paradox defines a class, and not a set. If we allow unrestricted comprehension in ZFC then we can define a Russell's class which will then be a set and derive contradiction. In NBG we can define classes, so we can allow this sort of axiom schema.

The reason we would like to limit the formulas in the schema is that the resulting theory is a conservative extension of ZFC, that means that it doesn't prove more about sets than what ZFC does. Much like Arthur Fischer (and Wikipedia) said, if we remove this restriction we find ourselves in the Morse-Kelly set theory which is not a conservative extension of ZFC.

-
Impredicative comprehension for classes would still not allow us to make the Russell class into a set; for example, as you say, MK set theory does allow that sort of comprehension. I don't follow what you're saying in the third paragraph. – Carl Mummert Mar 14 '12 at 11:08
@Carl: I'm not saying that MK allows Russell classes to be sets. I'm just saying that we want a conservative extension of ZFC so we want to limit how much we allow classes to play an active role in the comprehension. – Asaf Karagila Mar 14 '12 at 11:12
"We want a conservative extension of ZFC" if we're really interested in ZFC results and are using proper classes just to avoid circumlocutions. On the other hand, if we want to axiomatize what we intuitively know about the world of classes (a cumulative hierarchy of sets as for ZFC, with one more level at the top, containing all collections of things from lower levels), then MK is much more natural than NBG. (I'm not claiming that adding one level to the hierarchy is a natural thing to do, only that if you decide to do it then MK is a natural description of what you get.) – Andreas Blass Aug 9 '13 at 15:33
@Andreas: Yes, I agree. And this answer was written a long time ago. But upon reflecting it and the comments below it, I think that if one agrees that $\sf ZFC$ is a "good enough theory for sets", then an extension to classes should probably be conservative over that. I do agree that if we want to formalize the universe of classes as we think about it, then $\sf MK$ is much more natural. We can continue in this fashion and really just reach to $\sf ZFC$ with universes (i.e. proper class of inaccessible cardinals). – Asaf Karagila Aug 9 '13 at 15:36