Dear Mike: One reference is the work of M-F Coste-Roy and H. Lombardi in Real Algebraic Geometry; a quick Google search yielded: http://hlombardi.free.fr/publis/AMega90-1.html Also they have some papers in the J.ACM, apparently. In fact, it seems to be a huge area of model theory: see http://name.math.univ-rennes1.fr/michel.coste/Borel/w1abs.html Cheers, Phil On Thu, 4 May 2006, Michael Barr wrote:
Is there a reference for the fact that a countable decidably ordered field has a constructable (and decidably ordered) real closure?
Maybe 20 years ago, an undergraduate did a project under my supervision proving exactly that (although the proofs are not constructive). While I don't think it would prove feasible for actual computation, the fact that it exists is interesting. Finally, I have looked more closely at it and if it is not known, I think it publishable. Unfortunately, the student (who was a Commonwealth Fellow at Cambridge) lost interest in math and is doing other things. But I think I know how to get in touch with him.
Michael