Impredicative Comprehension about classes?

Question

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?

Answer 1

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.

Answer 2

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.