This paper argues that ensuring AI safety is not simply a difficult problem, but is based on a fundamental logical contradiction. The authors present the “enumeration paradox,” pointing out that the reason we use machine learning is because we cannot enumerate all the necessary safety rules, but making ML safe requires examples that can only be generated from the very enumerations that we acknowledge as impossible. This paradox is confirmed by five independent mathematical proofs (the Five Pillars of Impossibility). The main results are: (1) Geometric impossibility: the set of safe policies has measure 0. (2) Computational impossibility: verifying the safety of a policy is coNP-complete even for non-zero error tolerances. (3) Statistical impossibility: training data for safety (rich examples of rare disasters) is logically contradictory and therefore unobtainable. (4) Information-theoretic impossibility: safety rules contain more uncompressed random information than any feasible network can store. (5) Dynamic impossibility: optimization processes for improving AI performance are hostile to safety, and the gradients of the two objectives are typically in opposite directions. These results show that the pursuit of safe and high-performance AI is not a matter of overcoming technical barriers, but rather a matter of facing fundamental and interrelated barriers. The paper concludes by presenting the strategic tripartite dilemma that this impossibility imposes on the field. Formal validation of the core theorem in Lean4 is currently in progress.
