Sets vs Classes: Is Anything Beyond A Set?

[dodo]

The stupid question of the day:

The existence of proper classes is often proven by contradiction: assume that some object is a set, you'll find a contradiction, therefore it is not a set. We baptized those as "classes".

Will (can) this even happen to classes? To find some object, assume it is a class, and get a contradiction, proving it is something else?

 

[Hurkyl]

Except for the question of internal vs external in nonstandard analysis, I can't think of any situation in ordinary mathematics where such a thing could come up. The only reason we see it with sets and classes is because one of the main applications of sets is to serve as a simplified yet extensive version of higher-order logic.