Degree-Theoretic Aspects of Computably Enumerable Reals

Abstract

A real α is computable if its left cut, L(α); is computable. If (qi)i is a computable sequence of rationals computably converging to α, then {qi}, the corresponding set, is always computable. A computably enumerable (c.e.) real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to α. For example, every representation A of α is Turing reducible to L(α). Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. re- als are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below degT L(α) necessarily contains a representation of α.