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  • Cited by 4
  • Print publication year: 1999
  • Online publication date: May 2010

Degree-Theoretic Aspects of Computably Enumerable Reals

    • By Cristian S. Calude, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, {cristian,coles,hertling,bmk}, Richard Coles, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, {cristian,coles,hertling,bmk}, Peter H. Hertling, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, {cristian,coles,hertling,bmk}, Bakhadyr Khoussainov, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand, {cristian,coles,hertling,bmk}
  • Edited by S. Barry Cooper, University of Leeds, John K. Truss, University of Leeds
  • Publisher: Cambridge University Press
  • DOI: https://doi.org/10.1017/CBO9780511565670.003
  • pp 23-40

Summary

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 α is a real which is the limit of an increasing computable sequence of rationals. It 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 representation and a noncomputable representation of degree degTL(α). In fact, the representations of any noncomputable c.e. real are downwards dense, and yet not every c.e. Turing degree below degTL(α) necessarily contains a representation of α.

Introduction

Computability theory essentially studies the relative computability of sets of natural numbers. Since Gödel introduced a method for coding structures using natural numbers, computability has been applied to many areas of mathematics, for example, to the theory of linear orders, to group theory and to real analysis. In this paper we will consider an application of computability theory to the real numbers.

The real numbers ℝ may be defiend in several different ways. In classical analysis, the reals are those entities which are the limit of a Cauchy sequence. In mathematical logic the real numbers are defined as Dedekind cuts of sets.