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A Non-Formulable Theorem: A Fundamental Limit of Finite Syntactic Systems and Its Consequences for Security and AI

arXiv · AI, language, vision and robotics · article · Sep 3, 2026 · UTC

For every coherent and sufficiently expressive finite syntactic system S, we prove the existence of at least one theorem that S cannot produce autonomously. The result is a metatheorem: it proves the existence of a theorem, and applies to every finite syntactic system - security mechanisms, AI systems, formal verifiers, legal systems, economic models, and the formal system in which it is itself proved.

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