4. Computability on the Real Numbers

Abstract

Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3●14159 .... Mathematicians prefer to define the real numbers axiomatically as follows: (ℝ, +,·,0,1, <) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set ℝ with all the properties which are proved in analysis. We will denote the real line topology, that is, the set of all open subsets of ℝ, by _τℝ.