Structures Associated with Real Closed Fields and the Axiom of Choice

Abstract

An integer part $I$ of a real closed field $K$ is a discretely ordered subring of $K$ with minimal positive element $1$ such that, for every $x\in K$, there is $i\in I$ with $i\leq x<i+1$. Mourgues and Ressayre showed in [MR] that every real closed field has an integer part. Their construction implicitly uses the Axiom of Choice. We show that $AC$ is actually necessary to obtain the result by constructing a transitive model of $ZF$ which contains a real closed field without an integer part. Then we analyze some cases where the Axiom of Choice is not necessary for obtaining an integer part. On the way, we demonstrate that a class of questions containing the question whether the Axiom of Choice is necessary for the proof of a certain $ZFC$-theorem is algorithmically undecidable. We further apply the methods to show that it is independent of $ZF$ whether every real closed field has a value group section and a residue field section. This also sheds some light on the possibility to effectivize constructions of integer parts and value group sections which was considered e.g. in [DKKL] and [KL]