Where do Set Theory and Number Theory meet together? As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of natural numbers. But on one hand we have Forcing Methods and Theory of Core Model to investigate about reals and the real line, and on the other hand for some statements in Number Theory we have, equivalent statements expressed by real or complex numbers, using Analytic Number Theory. Therefore, it seems it's possible to reconcile two hands!!
Now my question is:
Is there any theorem in Number Theory that can be proved by tools of Set Theory, especially by methods of consistency results?
Any reference is appreciated.
 A: Recently, I saw the abstract of a talk by Matteo Viale, that sounds 
very interesting to me. See USING FORCING TO PROVE THEOREMS: AN EXAMPLE AROUND
SCHANUEL’S CONJECTURE. In it the following is claimed:
Let $SC(K, \mathbb{C})$, for a subfield $K$ of complex numbers $\mathbb{C},$ denote the following version of Schanuel's conjecture: For $a_1, \dots, a_n$ in $\mathbb{C}$ which are linearly independent over $K,$
$\hspace{4.cm}$ $trdg_K(a_1, \dots a_n, exp(a_1), \dots, exp(a_n)) \geq n.$
Using forcing, a new proof of the following is given: There is $K$, a countable subfield of the
complex numbers, such that $SC(K, \mathbb{C})$ holds. See the abstract for more details.

I posted this as a new answer, as my previous answer was somehow old, and I didn't want to make this answer into that one.


Update:
The paper can be found here.
A: The following example gives a connection between descriptive set theory and the theory of approximation by algebraic numbers:
There exists a  classification, due to Mahler, of real (and complex) numbers into four classes $A, S, T$ and $U$ according
to their properties of approximation by algebraic numbers. 
In the paper The Borel Classes of Mahler's $A$, $S$, $T$, and $U$ Numbers, the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.
--
A: One example, which was quite striking for me is the following theorem by Goodstein:
http://en.wikipedia.org/wiki/Goodstein%27s_theorem
A big surprise is not only the fact that it cannot be proven in Peano arithmetic but also the fact that people figured out that each Goodstein sequence eventually terminates: since these sequences may take huge values!
A: How about this: Inaccessible cardinals and Andrew Wiles's proof
I guess turns out to be false that the Wiles proof requires inaccessible cardinals.
But Grothendieck cohomology theories do, so could you perhaps consider them to be "number theory"?
