The Wikipedia article on [Hahn Series](https://en.wikipedia.org/wiki/Hahn_series) ~~mentions~~ mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".



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> Is this correct? If so, what was this approach, and where can I read about it?
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I have read most of Hahn's paper [über die nichtarchimedischen Größensysteme](https://zbmath.org/?q=an%3A38.0501.01), 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.



Hahn's approach to Hilbert's 17th problem?
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.


