# 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?

• I would ask more broadly than ZFC, so as to get a better handle on proper classes. In that case, I some theories, no. But see discussion at en.m.wikipedia.org/wiki/Axiom_of_limitation_of_size, where it mentions a model of NBG without Choice such that $Ord$ is smaller than $V$, both proper classes Apr 4 at 21:42
• It's consistent with $\mathsf{ZFC}$ that every proper is in bijection with $V$ (more precisely, the scheme saying that every $\Sigma_n$-definable proper class is in bijection with $V$ for each $n$ is consistent with $\mathsf{ZFC}$), but it's also consistent that this fails. $\mathsf{ZFC}$ does prove, appropriately phrased, that every proper class surjects onto $\mathsf{Ord}$, but again there can be proper classes into which $\mathsf{Ord}$ does not inject. Apr 4 at 23:10
• Does this answer your question? Bijective-equivalent collections of proper classes in set theory Apr 4 at 23:10

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.
• >Proper classes are so large, that they don't really have a cardinality  Fun fact 1: In $ZF$, this statement (when state correctly) is equivalent to AC.  Fun fact 2: In $ZF$, even tho classes can be "not big", there exists a (class) function $f:V→Ord$ such that a class $C$ is proper if and only if $f''C$ is a proper class (An example of such $f$: the rank function)  Fun fact 3: It is consistent with $ZF$-regularity that there is no function like the above $f$