A Distinction Between Legitimacy and Authority: A Meta-Critique of the Overextension of Gödelian Anti-AI Arguments

Gödel's incompleteness theorem (1931) and Turing's halting problem (1936) are among the most important achievements of twentieth-century mathematics and logic. Since their appearance, however, their conclusions have been widely extended beyond their original mathematical context—as in-principle denials of the possibility of artificial intelligence (Lucas, 1961; Penrose, 1989, 1994), as ultimate verdicts on the boundaries of human cognition, and as fundamental limits on the power of formal systems. This paper challenges none of the mathematical proofs: the encoding, the self-referential construction, the diagonal construction, and the formal derivations are all acknowledged as strictly legitimate. What this paper criticizes is the overextension by which later writers repackaged these mathematical facts as philosophical verdicts: Lucas (1961) concluded on this basis that machines cannot possess minds; Penrose (1989, 1994) pushed the verdict further, to consciousness and non-computable physics; and in the framing layer of the English translation, Meltzer and Braithwaite (1962) entrenched the "incompleteness" label and attributed it to Gödel himself. Through a sentence-by-sentence examination of Gödel's 1931 original and Turing's 1936 original, this paper shows that neither originator ever delivered any such verdict. The overextension was, from beginning to end, the work of later writers. This paper proposes the Overextension Criterion: when the property of an element within a system is extended into a property of the system itself, the resulting generalization lacks authority—the operation is legitimate, but the conclusion has no power to rule on the system. The criterion is validated in a unified manner across the liar paradox, Russell's paradox, the halting problem, the barber paradox, and Tarski's theorem. The final claim of this paper is not that "Lucas's and Penrose's conclusions are false," but that "their chains of inference are invalid": a chain of inference that has incorporated an overextending leap must be set aside and re-examined, whatever its conclusion happens to be.