Programmable Real Numbers and Functions

Abstract

In this paper we study the notion of programmability of functions and relations in the sense of algorithmic logic. We introduce a notion of acceptable structure. In acceptable structures a programmable function is a function which is defined by an infinite recursive sequence of cases which happens to be the so called Friedman’s schema. The last notion has been introduced as a generalization of a notion of a recursive function for structures which admitt pairing system. We apply the technique of Friedman’s schemata to study programmability in fields which as we shall prove do not admitt pairing systems. We also study some notions of effectiveness of real numbers.