Question

I am trying to define an embedding whose range includes classes. Is there a coherent way of assigning "cardinality" to proper classes?

Answer 1

Erin, there is no need to do this. I do not know of any practical reasons for doing it. And, of course, "cardinality" has to be properly interpreted to make some sense of the word.

In extensions of set theory where classes are allowed (not just formally as in ZFC, but as actual objects as in MK or GB), sometimes it is suggested to add an axiom (due to Von Neumann, I believe) stating that any two classes are in bijection with one another. Under this axiom, the "cardinality" of a proper class would be ORD, the class of all ordinals. (By the way, by class forcing, given any proper class, one can add a bijection between the class and ORD without adding sets, so this assumption bears no implications for set theory proper.)

Without assuming Von Neumann's axiom, or the axiom of choice, I know of no sensible way of making sense of this notion, as now we could have some proper classes that are "thinner" than others, or even incomparable. Of course, we could study models where this happens (for example, work in ZF, assume there is a strong inaccessible $\kappa$, and consider $V_\kappa$ as the universe of sets, and $Def(V_\kappa)$ in Gödel's sense (or even $V_{\kappa+1}$) as the collection of classes).