TITLE: Size Of Proper Classes
QUESTION [5 upvotes]: There is a well-known hierarchy of infinite cardinalities for sets. I've heard it said that proper classes are from a certain point of view "too large" to be sets. 
Are some proper classes larger than others, in some set-theoretical system or other? Or are they all the same size or all incomparable in systems extending ZFC to talk about classes? 
I suppose the motivation is: Is it possible to think about larger and larger classes? And have a class of all classes "too large" to be a class, call it a "collection", etc?

REPLY [9 votes]: It is consistent with $\sf ZFC$ that every two proper classes have a bijection between them. That is to say, if $A$ and $B$ are two proper classes then there is a class $C$ which is a class of ordered pairs which is a bijection from $A$ onto $B$.
For example, if we assume that $V=L$, then this is true.
But it is also consistent that this is not the case, and for example there is no bijection between $\sf Ord$, the class of ordinals, and $V$, the class of all sets.
For your motivation, do note that in $\sf ZFC$, or even in class-able set theories like $\sf NBG$ or $\sf MK$ which extend $\sf ZFC$, one cannot talk about classes of proper classes.