Limit computable integer parts

Abstract

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every ${r \in R}$ , there exists an ${i \in I}$ so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is ${\Delta^0_2(R)}$ —limit computable relative to R. We show that there is a maximal Z-ring ${I \subseteq R}$ which is ${\Delta^0_2(R)}$ . However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ’77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie’s ideas, we produce a real closed field R with a Z-ring ${I \subseteq R}$ such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class ${\Delta^0_2}$ . We know that certain subclasses of ${\Delta^0_2}$ are not sufficient. We show that for each ${n \in \omega}$ , there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n.