Countable Real Closed Fields and Developments of Bounded Length

Let R be a real closed field. An integer part for R is a subring that sits inside R the way the integers sit inside the reals. Mourgues and Ressayre [10] showed that every real closed field has an integer part. We may associate with R a value group G that is a subgroup of (R+, ·), a residue field k that is a subfield of R, and an integer part I. The Mourgues and Ressayre construction involves mapping the elements of R to generalized power series, called “developments,” with terms corresponding to elements of a wellordered subset of G and with coefficients in k. We consider the case where R is countable. We fix a residue field k and a list r1, r2, . . . of elements that form a transcendence basis for R over k. Once these objects are fixed, the Mourgues and Ressayre (MR) procedure is canonical. Let Rn be the real closure of k(r1, . . . , rn). We show that the elements of Rn have developments of length at most ωω (n−1) , so all elements of R = ∪nRn have developments of length less than ωω ω . We give an example of a real closed field R, with residue field k and transcendence basis r1, r2, . . . for R over k, such that these bounds are sharp: i.e., for each n, the field Rn has an element with a development of length ωω (n−1) . We are interested in effectiveness. We show that for any countable real closed field R, there is a value group G that is ∆2(R) and that this result is sharp. Next, we show that for any countable real closed field R, there is a residue field k that is Π2(R), and this result is the best possible. Finally, we use our upper bounds on the lengths of developments, together with the results on the value group and the residue field, to show that there is a ∆ωω (R) integer part, obtained by the MR-procedure.