Characterization of the Computable Real Numbers by Means of Primitive Recursive Functions

Abstract

One usually defines the notion of a computable real number by using recursive functions. However, there is a simple way due to A. Mostowski to characterize the computable real numbers by using only primitive recursive functions.We prove Mostowski’s result differently and apply it to get other simple characterizations of this kind. For instance, a real number is shown to be computable if and only if it belongs to all members of some primitive recursive sequence of nested intervals with rational end points and with lengths arbitrarily closely approaching 0.

Acknowledgments

The author thanks an anonymous referee for many appropriate and useful suggestions. An immense debt of gratitude is owed also to George Barmpalias - he attracted the authors attention to the fact that some essential results of the paper follow immediately from a theorem in Mostowski’s paper [2].