Degree-Theoretic Aspects of Computably Enumerable Reals [chapter]

Cristian S. Calude, Richard Coles, Peter H. Hertling, Bakhadyr Khoussainov, S. Barry Cooper, John K. Truss
Models and Computability  
A real is computable if its left cut, L ; is computable. If q i i is a computable sequence of rationals computably converging to ; then fq i g; 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 : For example, every
more » ... on A of is Turing reducible to L : Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L necessarily contains a representation of :
doi:10.1017/cbo9780511565670.003 fatcat:3amdwueza5d6vpytitwjgdsg5y