Structures associated with real closed fields and the axiom of choice
Let \(K\) be a real closed field (RCF). A discretely ordered subring \(I\) of \(K\) is an integer part of \(K\) if and only if 1 is the minimal positive element of \(I\) and for every \(x\in K\) there is an \(i\in I\) with \(i\leq x<i+1\). \textit{M. H. Mourgues} and \textit{J. P. Ressayre} [J. Symb. Log. 58, No. 2, 641--647 (1993; Zbl 0786.12005)] used a Zorn's lemma argument to show that every RCF has an integer part. They also used the same non-constructive methods to produce other structures associated with RCFs: namely, value group sections and residue field sections.NEWLINENEWLINEThe author of this article shows that the three structures mentioned above all require non-constructive methods. In particular, he constructs transitive models of ZF containing an RCF \(K\), but no integer part. He also constructs models containing RCFs without value group sections and residue field sections.NEWLINENEWLINEWith regard to integer parts, the author gives extra conditions on RCFs that guarantee the existence of integer parts in ZF. The author strengthens his results and obtains others by proving the following theorem: Let \(T\) be a countable, consistent o-minimal theory. Then there is a transitive model \(N\) of ZF such that \(N\) contains a model \(\hat{M}\models T\) with no unbounded discrete subset.
scientific article; zbMATH DE number 6643012