on the nature of undecidability within computing

The problem of undecidability is a defining problem within the theory of computing, presenting the appearance of certain limits to computation that were identified in Alan Turing's seminal paper On Computable Numbers, that still stands as consensus today. This form of problem can take many forms within computing, from the circle-free problem found in a diagonal construction 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 Joel Hamkins and Theador 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 error on the nature of undecidability within computing. While the Hamkins and Nenu do go into some great surrounding detail, and present an argument that may even be valid from the conventional perspective... both their paper, and the outstanding consensus on the matter lack a certain technical insight into the problem found within Turing’s diagonal machine 𝓗 to understand why the problem described in his paper is very much of the same form of problem as the more widely known halting variation. 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 error is truly an error: what starts off as merely a correction to this paper, will expound into a more fundamental discussion on the nature of undecidability within computing, to end up at a rather surprising result that undercuts and retutes the overwhelmingly accepted Church-Turing thesis that all computation possible can be done on a turing machine.