A note reviewing Turing's 1936

TL;DR

This paper critically reexamines Turing's 1936 work, arguing that his proof of non-computability is not conclusive and that there is no evidence for a non-computable number, challenging traditional interpretations.

Contribution

It provides a detailed analysis of Turing's original argument, questioning its validity and the existence of non-computable numbers.

Findings

Turing's 1936 proof is not conclusive.

No evidence supports the existence of non-computable numbers.

The original argument cannot be regarded as definitive.

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.