Nearly Computable Real Numbers
Peter Hertling, Philip Janicki

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 of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function the sequence converges computably to . 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 nearly computable if there exists a computable sequence of rational numbers that converges to 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…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Benford’s Law and Fraud Detection · Advanced Topology and Set Theory
