Equivalence of Countable and Computable

TL;DR

This paper establishes that countability and computability are equivalent concepts for sets, linking set theory with computational complexity, and challenges some common assumptions taught in mathematics and computer science courses.

Contribution

It proves the equivalence of enumerable and computable sets, introduces the concept of counting order, and critiques traditional beliefs about countable sets in education.

Findings

Enumerable sets are exactly the computable sets.

The set of all total computable functions is unenumerable.

Some widely accepted statements about countable sets lack formal proof.

Abstract

The concept of ``countable set'' is attributed to Georg Cantor, who set the boundary between countable and uncountable sets in 1874. The concept of ``computable set'' arose in the study of computing models in the 1930s by the founders of computer science, including \Godel, Church, and Turing. However, the connection between countability and computability has not been thoroughly studied in the past eight decades. A {\em counting bijection} of a set $S$ is a bijection from the set of natural numbers to $S$. We say $S$ is {\em enumerable} if either $S$ is finite or $S$ has a computable counting bijection. We prove that $S$ is enumerable if and only if $S$ is computable. This equivalence provides insights to the study of set theory and computability theory and reveals that the concept of countability is related to space and time complexity. We show that the set of total computable functions (or deciders, algorithms) is unenumerable. We also provide a sufficient and necessary condition for a set to be enumerable and use the concept of {\em counting order} to explain why some sets are unenumerable. We show that some popular statements about countable sets, such as ``every subset of a countable set is countable'' or ``every formal language is countable,'' lack a valid proof. These unproven statements are widely taught in college courses on discrete mathematics, set theory, and computability theory.