Limit computable integer parts

作者：D'Aquino Paola; Knight Julia; Lange Karen^*

来源：Archive for Mathematical Logic, 2011, 50(7-8): 681-695.

DOI：10.1007/s00153-011-0241-z

• Integer part
• Real closed field
• Z-ring
• Computable structure

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every r is an element of R, there exists an i is an element of 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 subset of 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 subset of 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 is an element of 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

• 出版日期2011-11