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Peter Laubenheimer, Thomas Schick, and Ulrich Stuhler: Completions of countable
non-standard models of Q
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.
We also give some information about the set of real parts of the finite elements
of such completions.
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.
Thomas Schick