How much could we cover a set by c.e sets?

TL;DR

This paper explores the extent to which arbitrary sets can be covered by computably enumerable (c.e.) sets, introducing a new perspective and definitions in computability theory to analyze set coverage.

Contribution

It proposes a novel approach to measure how much an arbitrary set can be covered by c.e. sets, offering new definitions and insights in computability theory.

Findings

Introduced a new framework for set coverage by c.e. sets

Provided initial definitions and methods for measuring coverage

Explored the relationship between arbitrary sets and c.e. sets in computability

Abstract

"How much c.e. sets could cover a given set?" in this paper we are going to answer this question. Also, in this approach some old concepts come into a new arrangement. The major goal of this article is to introduce an appropriate definition for this purpose. Introduction In Computability Theory (Recursion Theory) in the first step we wish to recognize the sets which could be enumerated by Turing machines (equivalently, algorithms) and in the next step we will compare these sets by some reasonable order (Like Turing degree). Also sometimes with some extra information (Oracles) a class of non c.e. sets show the same behavior as c.e. sets (Post hierarchy and related theorems). Here we try another approach: "Let A be an arbitrary set and we wish to recognize how much this set might be covered by a c.e. set?" Although in some sense this approach could be seen in some definitions of Recursion Theory, but at the best of our knowledge it didn't considered as an approach yet, even though it is able to shed a light on some subjects of Computability of sets. Defining this approach is not quite straightforward and there are some obstacles to define them. To overcome these difficulties we modify the definitions. We have an alternative problem here when we consider recursive sets and not c.e. sets. In this case, the problem would be: "Let A be an arbitrary set and we wish to know that how much this set might be covered by a recursive Set?" Here, we try the first definition and the first problem.