Discontinuities of provably correct operators on the provably recursive real numbers

Journal of Symbolic Logic 48 (4):913-920 (1983)
In 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 system S. In particular we develop the provable analogue of an effective operator on the set C of recursive real numbers, namely, a provably correct operator on the set P of provably recursive real numbers. In Theorems 1 and 2 we exhibit a provably correct operator on P which is discontinuous at 0; we thus disprove the analogue of the Ceitin-Moschovakis theorem of recursive analysis, which states that every effective operator on C is (effectively) continuous. Our final theorems show, however, that no provably correct operator on P can be proven (in S) to be discontinuous
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