Cardinality of proper classes In ZFC set theory, we know that the size of every set can be measured through its cardinality, but what about proper classes? We can view a proper class W which has a 'cardinality' larger than any set, the question is, can we construct another class V which has greater 'cardinality' than W?
 A: First I want to address the main issue of "the cardinality of a proper class": they don't have one.
Proper classes are so large, that they don't really have a cardinality. For example, take the class $\text{Card}$. It obviously doesn't have a cardinality, as it is transitive and well-ordered by $\in$ and such a set cannot contain its cardinality (the existence of some $X$ so that $|X| \in X$, $X$ is transitive and $X$ is well-ordered by $\in$ violates the axiom of regularity).
Similar things trivially hold for $V$ and similar classes. For the rest of proper classes, we can use Limitation of Size to show that $|C| = |V|$ for any proper class, yet $|V|$ is undefined.
Next, regarding "given a proper class $C$, can we find some proper class $D$ larger than $C$?" Well, working in something such as $\textsf{NBG}$, the answer is a strong no due to the aforementioned limitation of size. In other systems which don't have limitation of size, I believe that the statement "given a proper class $C$, we can find some proper class $D$ larger than $C$" is independent.
