Monotonically Computable Real Numbers
Robert Rettinger, Xizhong Zheng, Romain Gengler, Burchard von Braunmühl · Mathematical logic quarterly · 2002
Area number x is called k-monotonically computable (k-mc), for constant k > 0, if there is a computable sequence (xn)n ∈ ℕ of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k · |x — xn| ≥ |x — xm| for any m > n and x is monotonically computable (mc) if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence (xs)s ∈ ℕ of rational numbers converging to x such that the sum \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\sum _{s \in \mathbb{N}}$\end{document}|xs — xs + 1| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.