Nearly Computable Real Numbers

TL;DR

This paper introduces nearly computable real numbers, a new class of numbers characterized by a specific convergence property of nearly computably Cauchy sequences, and explores their properties and relationship to classical computability notions.

Contribution

It defines nearly computable real numbers, proves their key properties, and establishes their position within the hierarchy of computability and randomness notions.

Findings

Nearly computable reals form a real closed field.

Noncomputable nearly computable reals are weakly 1-generic and strongly Kurtz random.

No promptly simple set is Turing reducible to a nearly computable real.

Abstract

In this article we call a sequence $(a_n)_n$ of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function $r:\mathbb{N}\to\mathbb{N}$ the sequence $(d(a_{r(n+1)},a_{r(n)}))_n$ converges computably to $0$. We show that there exists a strictly increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number $\alpha$ nearly computable if there exists a computable sequence $(a_n)_n$ of rational numbers that converges to $\alpha$ and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (2002) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (2017) and by Stephan and Wu (2005) by showing that any nearly computable real number that is not computable is weakly $1$-generic (and, therefore, hyperimmune and not Martin-L\"of random) and strongly Kurtz random (and, therefore, not $K$-trivial), and we strengthen a result by Downey and LaForte (2002) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.