Limit computable integer parts

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every r ∈ R, there exists an i ∈ I so that i ≤ r < i+ 1. Mourgues and Ressayre [11] showed that every real closed field has an integer part. In [6], it is shown that for a countable real closed field R, the integer part obtained by the procedure of Mourgues and Ressayre is ∆ωω (R). We would like to know whether there is a much simpler procedure, yielding an integer part that is ∆2(R)—limit computable relative to R. We show that there is a maximal Z-ring I ⊆ R which is ∆2(R). However, this I may not be an integer part for R. By a result of Wilkie [14], 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 ⊆ 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 ∆2. We know that certain subclasses of ∆ 0 2 are not sufficient. We show that for each n ∈ ω, 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.