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The Wikipedia article on Hahn Series mentions mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".

Is this correct? If so, what was this approach, and where can I read about it?

I have read most of Hahn's paper über die nichtarchimedischen Größensysteme, where Hahn series were introduced, but I have not seen Hilbert's 17th problem on positive polynomials being mentioned there. I have also skimmed his list of publications and not found anything else that looks relevant.

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    $\begingroup$ Like you, I'm confident that Hahn does not discuss Hilbert's problem in the paper you cited, and, like you, I am not aware of any of his publications that apply that work to Hilbert's 17th problem. However, in the paper you mention, Hahn does discuss Hilbert's (arithmetic) completeness condition, which is discussed by Hilbert in his famous list of problems. $\endgroup$ May 8, 2022 at 15:55
  • $\begingroup$ Very interesting, @PhilipEhrlich! So what you're suggesting is that the Wikipedia reference to Hilbert's 17th problem should rather be pointing to Hilbert's 2nd problem, right? I would like to check but can't find a pdf of Hahn's paper right now. But I see that your paper talks about this! $\endgroup$ May 8, 2022 at 16:46
  • $\begingroup$ Yes, that is precisely what I am suggesting. $\endgroup$ May 8, 2022 at 17:01
  • $\begingroup$ Thank you! I've corrected the statement on the Wiki page. And in case that you want to make this into an answer to the question, I'll be happy to accept it. $\endgroup$ May 8, 2022 at 17:05
  • $\begingroup$ I will do as you suggest later. For the time being, let me simply clarify that Hahn did not see his work as directly applicable to Hilbert's second problem, but rather to Hilbert's completeness condition that Hilbert mentions in his discussion. $\endgroup$ May 8, 2022 at 17:17

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As you suspect, Hahn does not discuss Hilbert's 17th problem in the paper you cite, and, like you, I am not aware of any of Hahn's publications that applies his work on non-Archimedean ordered systems to Hilbert's 17th problem.

However, in the paper you mention Hahn does discuss Hilbert's (arithmetic) completeness condition, which is discussed by Hilbert in the second problem of his famous list of problems. There, however, Hahn is not concerned with the consistency of the real number system, which is the subject of the second problem, but rather with generalizing Hilbert's completeness condition so as to be applicable to non-Archimedean as well as Archimedean ordered Abelian groups and ordered fields.

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