on the nature of undecidability within computing

The problem of undecidability is a defining problem within the theory of computing, presenting certain limits to machine computation that were identified in Alan Turing's seminal paper On Computable Numbers, that still stands as consensus today. This problem takes many forms within computing, from the circle-free problem found within a diagonal computing machine of Turing’s paper, to the canonical halting problem commonly used to demonstrate undecidability in undergrad classrooms, to the more generalized form of Rice’s Theorem that applies to all extensional semantics of a turing machine.

Despite being a foundational issue brought up in the very first paper on the math of mechanical computation, there still persists some confusion on its nature, and even whether Turing specifically presented the undecidability of the halting problem or not. In a recent article by Hamkins and Nenu, the authors attempt to clarify the accuracy of the attribution. They ultimately end with a “nuanced” conclusion: claiming that while Turing laid the framework for the halting problem:

Strictly speaking, Turing did not prove nor even state the undecidability of the halting problem in his 1936 paper, and it is incorrect to suggest that this result or any discussion of it can be found there. It is especially incorrect to attribute to Turing the common self-referential proof of the undecidability of the halting problem, since nothing like that argument appears in Turing’s paper.

This conclusion is a fundamental misunderstanding on the nature of undecidability within computing. Turing didn’t just provide a scaffold used to produce the halting problem, he’s the first to produce a machine form that these sorts of computably undecidable decision problems fundamentally follow: the self-referential set-classification paradox. To fully explore the totality of why this is a misunderstanding, what starts off as merely a correction to this paper, will expound into a more fundamental discussion on the nature of undecidability within computing: we will define and generalize the problem of undecidability within computing; question the merit of recursive undecidability; fix a few algorithmic flaws in Turing’s attempt at computing a diagonal across all sequences; discuss a fallacy in Turing’s second proof undecidability; propose a limit to the undecidability within computing; discuss a fallacy in Turing’s first proof of undecidability; all to end up at the rather surprising result that undercuts and refutes the overwhelmingly accepted Church-Turing thesis, that the turing machine model encapsulates all of computing. It does not.