Hallucinations are a mathematical necessity of powerful AI rather than just a bug that can be patched out.
Developers often assume that better data or more training will eventually stop AI from making things up. This paper proves a fundamental computability-theoretic limit that makes hallucinations inevitable in complex domains. You cannot have a system that is both highly expressive and guaranteed to be error-free. This means that as AI gets more capable of solving hard problems, the risk of plausible-sounding errors will always remain. We must build our infrastructure around the assumption that AI can never be one hundred percent reliable. Hallucination is the price we pay for intelligence.
Hallucination, Abstention, and Recursive Inseparability
arXiv · 2604.28067
The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit that forced-answer obstruction as its main claim. Instead, it attempts to formally describe the corresponding limitation for abstaining systems. Abstention can trivially avoid hallucination if the system is allowed to abstain on every input; the substantive qu