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While reading an introductory text on model theory, I found it interesting that one can reformulate the famous conjectures about twin primes and Mersenne primes in terms of non-standard models of arithmetic:

The twin prime conjecture is true if and only if there is some non-standard model of arithmetic $\mathcal{M}$ and at least one pair of non-standard twin primes in $\mathcal{M}$.

There are infinitely many Mersenne primes if and only if there is a non-standard Mersenne prime.

My question is this: Are there any examples of longstanding open questions in number theory whose eventual solution took the above form, that is, as an implication of some feature of a non-standard model of arithmetic? I would be particularly interested to know of any instances where the only known proof of a certain result makes use of this strategy. Thanks in advance.

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  • $\begingroup$ What introductory text have you read? $\endgroup$
    – Joel Adler
    Commented Mar 23, 2022 at 13:20
  • $\begingroup$ @JoelAdler "An Invitation to Model Theory" by Jonathan Kirby. It's an undergraduate-level introduction and I am by no means an expert on the subject. The statements appear as Proposition 13.11 and Exercise 13.5 in the text. $\endgroup$
    – Menander I
    Commented Mar 23, 2022 at 15:02
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    $\begingroup$ Even if you include all of mathematics and not just number theory, there are not very many examples of conjectures being proved for the first time using nonstandard methods; see this MO question for more info. Your examples above are basically straightforward applications of a transfer principle; the transfer principle is not very "deep," and so cannot be expected to help too much with solving a very hard problem. Any proof using a transfer principle can be rewritten without it, so in some sense it can't play an "essential" role in a proof. $\endgroup$
    – Timothy Chow
    Commented Mar 23, 2022 at 20:21
  • $\begingroup$ For “questions which can be stated arithmetically” rather than just “questions in number theory”, the answer might be yes. Eg: One might use non-standard models to prove that any sentence in $S$ which follows from theory $T$ also follows from theory $U$. Proof theorists who know about models of arithmetic might be able to provide examples. $\endgroup$
    – user44143
    Commented Apr 28, 2022 at 14:24
  • $\begingroup$ @Menander I Seen only now, thanks! $\endgroup$
    – Joel Adler
    Commented Jun 1, 2022 at 15:47

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