TITLE: Is there a way to prove that a theorem has no elementary proof? Or to prove that something may have no proof?
QUESTION [7 upvotes]: Recently I was trying to prove something, more or less elementarily, but eventually started going in circles. My prof said that the proof involves mathematical tools that I've not seen yet, and that it's unlikely there is an elementary proof. 
Which got me wondering: would there be a way to prove that there is no elementary proof to a particular theorem, or even a particular kind of theorem? (other than trivial cases such as theorems already involving non-elementary maths)
So, I guess, to rephrase: a way to prove that theorem $X$ can be proved using mathematical tools that are not higher than those needed to state $X$ in the first place?
Or how about theorems that are true but maybe have no proof; can this be known? Can we prove that a theorem may or may not be true, but would have no proof if it were?

I guess this is more metamathematics than mathematics, as it would involve theorems about the properties of theorems.... is there a branch of math that studies this sort of thing? Is this what "proof theory" is?
If there is such a subject: any book recommendations that would be accessible to someone in his 2nd undergrad year? This subject is interesting to me.

REPLY [5 votes]: There has been successful work towards  formalizations of something that may be not not too far from what you are asking for. The most interesting is the Reverse Mathematics program pioneered by Harvey Friedman.  
It is not clear to what degree (if at all) the scales developed in the Reverse Mathematics program match informal notions of "elementary proof." That notion varies widely between fields. For example, in prime number theory, "elementary proof" means roughly a proof that does not use deep tools from the theory of complex variables. In that sense, there is an elementary proof of the Prime Number Theorem, found independently by Erdős and Selberg. But it is much harder than the non-elementary proof!