Some properties of D.C.E. reals and their degrees

Abstract

In effective analysis, various sub-classes of real numbers are discussed, and effective versions of classical results on the analysis of real numbers are studied. The computable reals, and the computably enumerable (c.e.) reals are perhaps the more fundamental classes, and are identified by their (weak) computable properties. It is known that the c.e. reals are not closed under the arithmetic operations, and when we take the field generated by the c.e. reals (which are the d.c.e. reals), we get a new sub-class of reals with interesting properties. We will show that the d.c.e. reals not only forms an algebraic field, but is also a real closed field, hence has a decidable theory. Some properties of Turing degrees are also examined. In particular, we show that every jump class contains a degree free of d.c.e. reals, and hence contains a degree that is not \omega-r.e.