Monotonically Computable Real Numbers

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A real number x is called semi-computable if it is a limit of an increasing or decreasing computable sequence (x n ), n∈ℕ of ratio­nal numbers. In this case, a later member of the sequence is always a better approximation to x in the sense that |x − x n | ≥ |x − x m |, if n ≤ m. As a natural generalization, we call a real number x k-monotone computable (k-mc, for short), for any real number k >0,, if there is a computable sequence (xn),n∈ℕ of rational numbers which converges to x k-monotonically in the sense that k · |x − x n | ≥ |x − x m | for any n ≤ m and x is monotonically computable (mc, for short) if it is k-mc for some k >0. Various properties of k-mc real numbers are discussed in this pa­per. Among others we show that a real number is computable if it is k-mc for some k <1; the 1-mc real numbers are just the semi-computable real numbers and the set of all mc real numbers is contained properly in the set of weakly computable real numbers, where x is weakly computable if there are semi-computable real numbers y, z such that x = y + z. Furthermore, we show also an infinite hierarchy of mc real numbers.