Where do Set Theory and Number Theory meet together?

Question

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.

related question, but not the same: How would one even begin to try to prove that a simple number-theoretic statement is undecidable? — Mohammad Golshani, Dec 20 '14 at 08:29
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). — Ali Enayat, Dec 20 '14 at 14:18
@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.? — Rahman. M, Dec 20 '14 at 14:35
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. — Todd Trimble, Dec 20 '14 at 17:45

Mohammad Golshani · Answer 1 · 2015-12-17T07:44:43.050

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.

@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. — Mohammad Golshani, Dec 06 '15 at 07:25
Matteo gave this talk at the Newton Institute, and it should be available here at some point. — tci, Dec 08 '15 at 16:51

Mohammad Golshani · Accepted Answer · 2015-12-06T07:21:37.293

3

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.

--

edited Dec 06 '15 at 07:21

answered Dec 20 '14 at 08:35

Mohammad Golshani

31,966

How does one define arithmetical set in the context of sets of sets? – Wojowu Dec 20 '14 at 08:46
6

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? – KConrad Dec 20 '14 at 17:37

score 2 · Answer 3 · answered Dec 20 '14 at 21:14

2

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!

answered Dec 20 '14 at 21:14

truebaran

9,140

7

Why would a number theorist be interested in these sequences? – Douglas Zare Dec 21 '14 at 07:49
2

This is not number theory. – KConrad Dec 06 '15 at 09:08

score 1 · Answer 4 · edited Apr 13 '17 at 12:58

1

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"?

edited Apr 13 '17 at 12:58

Community

1

answered Dec 20 '14 at 16:14

Gerald Edgar

40,238

6

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. – David Roberts Dec 22 '14 at 09:07

Where do Set Theory and Number Theory meet together?

Is there any theorem in Number Theory that can be proved by tools of Set Theory, especially by methods of consistency results?

4 Answers4