TITLE: What in Mathematics cannot be described within set theory?
QUESTION [27 upvotes]: I have begun reading Patrick Suppes' book Axiomatic Set Theory. The first sentence in chapter 1 reads: "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
What are these rare exceptions? What in mathematics cannot be described within set theory?

REPLY [4 votes]: I think there is an argument to be made that nothing in mathematics can not be formalized in set theory unless you are deliberately going out of your way to push the limits of set theory in unnatural ways. I will explain what this means later. 
First, you should ask what it means to be an object of mathematics. Except for finite or recursive objects, it is difficult to make an argument for in what sense certain objects exist. For example, is the first uncountable ordinal $\omega_1$ really just an object described by set theory or that its existence is entirely dependent on the underlying set theory. Or if you are familiar with algebra, most of the interesting theorem are about subrings, subgroups, etc and rarely about the multiplication of the ring or group elements. You are not exactly proving these theorems in the theory of rings or groups, so where exactly are the familiar theorems of algebra being proved. Anyway these examples are to show that ordinary mathematics is already using some of the set construction principles perhaps formalized in something like $\mathsf{ZFC}$. Upon some ontological inspection, several particular parts of mathematics have very strict rules, how come one take it for granted that one is allowed to take unions of sets, pairs elements together, define subsets using certain properties, etc. If one tries to apply mathematical rigor to this area, then one may realizes that one was already using certain rules for the manipulation of diverse mathematical objects. Of the many possible alternative, $\mathsf{ZFC}$ set theory (and its extensions) has been a particularly use and natural such foundation. 
Finally, I am usually very skeptical when someone asserts that something can not be handled in first order set theory and asserts that one should work in class set theory or some foundational category theory. A common argument is that the object of study is a proper class (not a set) and hence totally beyond the capabilities of set theory. Generally saying something is a proper class means it is first order describable in the language of set theory. 
For the most part, if someone asserts that a theorem of a algebra, number theory, etc (anything not about set theory or category theory) and he or she states that the problem can not be handled in set theory, I generally think it is an elaborate example of: "I can not classify the finite groups of order 2 in set theory because the class of groups of order 2 is a proper class".