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Equivalence of Countable and Computable

Hantao Zhang

TL;DR

The paper investigates the relationship between Cantor's countability and computability by defining enumerable sets as those with a computable counting bijection from $\mathcal{N}$ and proving this notion is equivalent to computability. It shows that, for formal languages, enumerability coincides with recognizability and develops tools (enumerators, a counting order, and the deleteMin procedure) to characterize enumerable sets and their complexity. It establishes unenumerability for several natural classes (e.g., total functions, decidable languages, deciders, algorithms) via diagonal arguments and extends the framework to infinite-input computability. It also challenges conventional claims about subsets of $\mathcal{N}$, proposing the Counting Bijection Thesis (every countable set has a computable counting bijection) and outlining its profound implications for the relationship between countability and computability, as well as for foundational questions in set theory and computability.

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.

Equivalence of Countable and Computable

TL;DR

The paper investigates the relationship between Cantor's countability and computability by defining enumerable sets as those with a computable counting bijection from and proving this notion is equivalent to computability. It shows that, for formal languages, enumerability coincides with recognizability and develops tools (enumerators, a counting order, and the deleteMin procedure) to characterize enumerable sets and their complexity. It establishes unenumerability for several natural classes (e.g., total functions, decidable languages, deciders, algorithms) via diagonal arguments and extends the framework to infinite-input computability. It also challenges conventional claims about subsets of , proposing the Counting Bijection Thesis (every countable set has a computable counting bijection) and outlining its profound implications for the relationship between countability and computability, as well as for foundational questions in set theory and computability.

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 is a bijection from the set of natural numbers to . We say is {\em enumerable} if either is finite or has a computable counting bijection. We prove that is enumerable if and only if 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.
Paper Structure (7 sections, 21 theorems, 2 equations)

This paper contains 7 sections, 21 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Equivalence of Recognizable and Enumerable
  3. Some Unenumerable Sets
  4. A Criterion for Being Enumerable
  5. Unproven Claims about Countable Sets
  6. Conclusion
  7. Appendix: Basic Concepts

Key Result

Theorem 2.1

$(a)$ If $L=E(M)$, where $M$ is an enumerator, then $L$ is recognizable. $(b)$ If $L$ is a recognizable formal language, there exists an enumerator $M$ such that $L=E(M)$ and every printed string by $M$ is unique.

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Corollary 2.9
  • Definition 3.1
  • ...and 20 more