The AI Incompleteness Theorem

In this paper, we propose a conceptual framework and prove "The AI Incompleteness Theorem" within that framework. The theorem presented here states that it is not possible to guarantee an AI/ML inferencing system that is hallucination-free, even with an infinite amount of model training data and with an infinite amount of compute resources to train AI/ML models with that data. We prove this theorem using the properties of infinite sets, Cantor's theorem and Hilbert's Grand Hotel paradox. We also provide a new definition of AGI (artificial general intelligence) within this conceptual framework. We define a system having attained AGI as a system that does not produce infinitely many hallucinations when presented with infinitely many input datasets. We present a corollary that states AGI is not possible even with an infinite amount of training data and an infinite amount of compute resources.