A definable number which cannot be approximated algorithmically

TL;DR

This paper introduces the concept of approachable numbers by Turing machines, shows that many non-computable numbers are approachable, and provides an example of a number that cannot be approached by any TM.

Contribution

It formalizes the notion of approachability by Turing machines and identifies a specific number that cannot be approximated algorithmically.

Findings

Many non-computable numbers are approachable by TMs

An example of a non-approachable number is provided

The concept of approachability extends understanding of computability

Abstract

The Turing machine (TM) and the Church thesis have formalized the concept of computable number, this allowed to display non-computable numbers. This paper defines the concept of number "approachable" by a TM and shows that some (if not all) known non-computable numbers are approachable by TMs. Then an example of a number not approachable by a TM is given.