Hallucination, abstention, and computable inseparability

Abstract

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 question is how large the domain of guaranteed correct non-abstaining answers can be. We formulate this question using separation in the arithmetical hierarchy. Given disjoint sets $A$ and $B$, any system that answers Yes on all queries indexed by $A$ and No on all queries indexed by $B$ induces a separator of $A$ from $B$. By combining this observation with the classical existence theorem of $\Delta_{n}^{0}$-inseparable pairs of $\Sigma_{n}^{0}$-sets, we yield a computability-theoretic trade-off between avoiding hallucination by abstention and maintaining a large domain of guaranteed coverage.