Discontinuities of provably correct operators on the provably recursive real numbers

Abstract

AbstractIn this paper we continue, from [2], the development of provably recursive analysis, that is, the study of real numbers defined by programs which can be proven to be correct in some fixed axiom systemS. In particular we develop the provable analogue of an effective operator on the setof recursive real numbers, namely, a provably correct operator on the setof provably recursive real numbers. In Theorems 1 and 2 we exhibit a provably correct operator onwhich is discontinuous at 0; we thus disprove the analogue of the Ceitin-Moschovakis theorem of recursive analysis, which states that every effective operator onis (effectively) continuous. Our final theorems show, however, that no provably correct operator oncan be proven (inS) to be discontinuous.