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A note reviewing Turing's 1936

Paola Cattabriga

TL;DR

This note critically reexamines Turing's 1936 proof that there exist noncomputable numbers, arguing that the Section 8 diagonal argument is not a conclusive demonstration and that no evidence supports the existence of a definable noncomputable number. It analyzes the diagonalization against the computable sequences produced by circle-free machines, highlighting the reliance on an effective enumeration and on an abstract D-machine that decides circularity. The authors contend that the purported contradiction arising from self-reference hinges on a conflation of abstract diagonalization with feasible computation, and they propose ways to avoid self-reference that undermine the claimed proof. Overall, the paper questions the solidity of Turing's original argument, emphasizes definability constraints, and calls for clearer separation between abstract concepts and effective construction in discussions of computability.

Abstract

By closely rereading the original Turing's 1936 article, we can gain insight about that it is based on the claim to have defined a number which is not computable, arguing that there can be no machine computing the diagonal on the enumeration of the computable sequences. This article provides a careful analysis of Turing's original argument, demonstrating that it cannot be regarded as a conclusive proof. Furthermore, it shows that there is no evidence supporting the existence of a defined number that is not computable.

A note reviewing Turing's 1936

TL;DR

This note critically reexamines Turing's 1936 proof that there exist noncomputable numbers, arguing that the Section 8 diagonal argument is not a conclusive demonstration and that no evidence supports the existence of a definable noncomputable number. It analyzes the diagonalization against the computable sequences produced by circle-free machines, highlighting the reliance on an effective enumeration and on an abstract D-machine that decides circularity. The authors contend that the purported contradiction arising from self-reference hinges on a conflation of abstract diagonalization with feasible computation, and they propose ways to avoid self-reference that undermine the claimed proof. Overall, the paper questions the solidity of Turing's original argument, emphasizes definability constraints, and calls for clearer separation between abstract concepts and effective construction in discussions of computability.

Abstract

By closely rereading the original Turing's 1936 article, we can gain insight about that it is based on the claim to have defined a number which is not computable, arguing that there can be no machine computing the diagonal on the enumeration of the computable sequences. This article provides a careful analysis of Turing's original argument, demonstrating that it cannot be regarded as a conclusive proof. Furthermore, it shows that there is no evidence supporting the existence of a defined number that is not computable.

Paper Structure

This paper contains 5 sections, 10 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The diagonal process
  3. The main argument
  4. Reviewing Turing's argument
  5. Definable numbers

Figures (7)

  • Figure 1: A representation of $\mathcal{U}$
  • Figure 2: a representation of $\mathcal{H}$ computing the diagonal $\beta ^{\prime}$
  • Figure 3: A representation of $\mathcal{D}$
  • Figure 4: A representation of $\mathcal{H}$
  • Figure 5: $\mathcal{H}$ encounters its own D.N.$K$
  • ...and 2 more figures

Theorems & Definitions (1)

  • remark 1