From Gödel to Turing: The Illusion of a Self-Reference That Never Was

The purpose of this article is to challenge some of the foundational pillars of modern mathematical logic, in particular Gödel’s incompleteness theorems and the halting problem as formulated by Turing. Both have traditionally been interpreted as instances of self-reference, although such an interpretation proves problematic when examined with technical and conceptual rigour.

In Gödel’s case, it is argued that the supposedly self-referential statement has been historically misinterpreted, and that, in its precise formulation, it is technically unfeasible within the formal system. In the case of the halting problem, it is shown that the machine which generates the contradiction is defined ambiguously.

To develop this thesis, the article begins with an introduction to Gödel’s first incompleteness theorem, highlighting the key elements of its proof and identifying the points at which a confusion between interpretative levels arises. A detailed refutation of the theorem is then presented, focusing on the structural impossibility of self-reference within the formal system.

Subsequently, the halting problem is examined from both a technical and logical perspective, demonstrating that the classical paradox relies on the conceptual use of a machine that cannot be constructed—or that, if constructed in the only coherent way possible, would dissolve the supposed paradox.

Other theorems that fall into similar errors, based on false self-references and interpretative level confusions, are also briefly analysed.

Finally, some general reflections are offered on the limits of self-reference, the possibilities of evaluation based on multiple identities, the existence of diverse interpretations of a single reality, and the role of language—not only in the cases of Gödel and Turing, but also in other logical and mathematical contexts. These reflections point to the need to revisit certain traditional assumptions and to explore new ways of organising thought.