Theorems not Formulable in Set Theory Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. 
What is an example of a theorem that can't be cast in terms of set theory? Is there any reason we know of so far to doubt set theory's ability to describe all of mathematics?
 A: There is actually a precise sense in which any foundation of math will fail to formalize certain statements which we should believe are true.
Suppose I view the mathematical universe as a structure $V$ in the language of set theory (this is the point of view taken by ZF, ZFC, NF, etc.). Then - informally - I can talk about the theory of $V$ in this language. This theory can be coded by a single real, and - assuming $V$ really is "everything" - this real should exist. So 

There is a real coding the true $\{\in\}$-theory of the universe

is a "true" statement. However, this sentence cannot be expressed in the language of set theory in any satisfactory way, by Tarski's theorem: https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem. We could expand our language to make this sentence expressible, but then truth in that expanded language would not be expressible, so we would need to further expand the language . . .
Exactly how problematic you find this sort of thing varies from person to person; but certainly I find it a good argument for caution - I wouldn't want to commit myself to never using a more expressive language, even if I don't think I'm likely to need it.

Note that this has nothing to do with classes (although there is some overlap), and is really an essential feature of how first-order logic works.
