Completions of countable non-standard models of Q

Abstract

In this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that these completions are real closed, i.e. each positive number is a square, and each polynomial of odd degree has a root. This way, we give a direct proof of a consequence of a theorem of Hauschild. In a previous version of this note, not being aware of these results, we missed to mention this reference. We thank Matthias Aschenbrenner for pointing out this and related work. We also give some information about the set of real parts of the finite elements of such completions -about the more interesting results along this we have been informed by Matthias Aschenbrenner. The main idea to achieve the results relies on a way to describe real zeros of a polynomial in terms of first order logic. This is achieved by carefully using the sign changes of such a polynomial.