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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.

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    $\begingroup$ related question, but not the same: How would one even begin to try to prove that a simple number-theoretic statement is undecidable? $\endgroup$
    – Mohammad Golshani
    Commented Dec 20, 2014 at 8:29
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    $\begingroup$ The implications Con(ZF)->Con(ZFC), due to Goedel, and Con(ZF)->Con(ZF+ not AC), due to Cohen, are both arithmetical statements, whose truth was first established using set-theoretical methods (inner models and forcing, respectively). $\endgroup$
    – Ali Enayat
    Commented Dec 20, 2014 at 14:18
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    $\begingroup$ @AliEnayat, that's right, but I mean Is there any attempt or general approach to attack to some conjectures like Schanual's conjecture and etc.? $\endgroup$
    – Rahman. M
    Commented Dec 20, 2014 at 14:35
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    $\begingroup$ You probably have heard that there are model theorists (such as Zil'ber) who have studied Schanuel's conjecture, but not particularly using set theory to the best of my knowledge. $\endgroup$
    – Todd Trimble
    Commented Dec 20, 2014 at 17:45

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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.

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  • $\begingroup$ How does one define arithmetical set in the context of sets of sets? $\endgroup$
    – Wojowu
    Commented Dec 20, 2014 at 8:46
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    $\begingroup$ Is this really a theorem in number theory? Not every result about the integers, or the positive integers, is something of concern to people in number theory. Does it have an application to a more conventional theorem in number theory? $\endgroup$
    – KConrad
    Commented Dec 20, 2014 at 17:37
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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.

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    $\begingroup$ @Rahman. M, I hope this answer fits more into what you were looking for. Maybe someone asks Viale to come and gives more information, then I will delete my answer. $\endgroup$
    – Mohammad Golshani
    Commented Dec 6, 2015 at 7:25
  • $\begingroup$ Matteo gave this talk at the Newton Institute, and it should be available here at some point. $\endgroup$
    – user3462
    Commented Dec 8, 2015 at 16:51
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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!

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    $\begingroup$ Why would a number theorist be interested in these sequences? $\endgroup$
    – Douglas Zare
    Commented Dec 21, 2014 at 7:49
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    $\begingroup$ This is not number theory. $\endgroup$
    – KConrad
    Commented Dec 6, 2015 at 9:08
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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"?

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    $\begingroup$ Grothendieck's cohomology aka derived functors do not, in cases of number theoretic interest, require inaccessibles. Colin McLarty has written about this, as referenced at my answer there. $\endgroup$
    – David Roberts ♦
    Commented Dec 22, 2014 at 9:07

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