A note reviewing Turings 1936

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.

💡 Research Summary

The paper undertakes a meticulous rereading of Alan Turing’s seminal 1936 article “On Computable Numbers, with an Application to the Entscheidungsproblem,” focusing on the section where Turing introduces a non‑computable real number via a diagonal construction. The author, Paola Cattabriga, argues that Turing’s proof does not constitute a conclusive demonstration of the existence of an uncomputable number, because it rests on an untenable assumption about the existence of a decision‑procedure (machine D) that can, in a finite number of steps, determine for any given Turing machine whether it is circular (halts) or circle‑free (produces an infinite binary output).

The paper first reconstructs Turing’s formal framework: computing machines, circular versus circle‑free machines, the notion of a standard description (S.D.) of a machine, and the associated description number (D.N.). It clarifies that a circle‑free machine’s output is an infinite sequence of 0s and 1s, encoded as a “complete configuration” table (C₂), while a circular machine yields only a finite prefix. The author emphasizes that Turing’s universal machine U can simulate any machine M when supplied with M’s S.D., thereby generating M’s output sequence.

In Section 8 of Turing’s original work, a diagonal argument is presented. Turing first defines a sequence β whose n‑th digit is the complement of the n‑th digit of the n‑th computable sequence (ϕₙ(n)). He notes that assuming β is computable leads to a contradiction, and therefore treats the argument as “fallacious” because it presupposes β’s computability. To avoid this circularity, Turing constructs a second diagonal sequence β′ whose n‑th digit is exactly ϕₙ(n). He then posits a hypothetical machine D that, given the S.D. of any machine M, decides in finitely many steps whether M is circle‑free (label “s”) or circular (label “u”). Using D, Turing builds a machine H that, for each natural number N, runs D on N, counts how many numbers up to N correspond to circle‑free machines (R(N)), and writes the R(N)-th digit of the corresponding machine’s output into β′.

The author’s central critique is that the existence of D is impossible. If D must output “s” for a circle‑free machine, it would have to examine the entire infinite output of that machine to confirm that it never halts, which cannot be done in a finite number of steps. Turing’s own text asserts that the decision must be reached in finitely many steps, creating a direct contradiction. Consequently, the whole diagonal construction collapses because it relies on a non‑existent decision procedure.

The paper further analyses the self‑reference that arises when H encounters its own description number K. At that point H would need to compute the first K digits of its own output in order to determine the K‑th digit to write, an operation that leads to an infinite regress. This mirrors the classic liar‑paradox style of Gödel’s incompleteness proof, but here the paradox originates not from logical inconsistency of arithmetic but from the infeasibility of the assumed machine D.

By exposing the hidden assumption, the author concludes that Turing’s diagonal argument does not, by itself, prove the existence of a non‑computable real number. The claim that “the computable numbers are not enumerable, therefore there exists a non‑computable number” lacks a rigorous foundation unless supplemented by an independent undecidability result (such as the halting problem). The paper therefore calls for a reinterpretation of Turing’s original proof, suggesting that the celebrated diagonal argument should be presented as a demonstration of the limits of a certain hypothetical decision procedure rather than as a definitive existence proof of an uncomputable real. This nuanced reading aligns with modern understandings of computability theory, where non‑computable objects are established via reductions to known undecidable problems rather than by a sole diagonal construction.