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.



Where do Set Theory and Number Theory meet together?
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](http://www.jstor.org/stable/2160682?seq=1#page_scan_tab_contents), the author studies these classes from the point of view of Descriptive Set Theory, and determines their complexity in the Borel hierarchy.


--


