Is automated hallucination detection possible? In this work, we introduce a theoretical framework to analyze the feasibility of automatically detecting hallucinations produced by large language models (LLMs). Inspired by the classical Gold-Angluin framework for language identification and its recent adaptation to language generation by Kleinberg and Mullainathan, we investigate whether an algorithm, trained on examples drawn from an unknown target language $K$ (selected from a countable collection) and given access to an LLM, can reliably determine whether the LLM's outputs are correct or constitute hallucinations. First, we establish an equivalence between hallucination detection and the classical task of language identification. We prove that any hallucination detection method can be converted into a language identification method, and conversely, algorithms solving language identification can be adapted for hallucination detection. Given the inherent difficulty of language identification, this implies that hallucination detection is fundamentally impossible for most language collections if the detector is trained using only correct examples from the target language. Second, we show that the use of expert-labeled feedback, i.e., training the detector with both positive examples (correct statements) and negative examples (explicitly labeled incorrect statements), dramatically changes this conclusion. Under this enriched training regime, automated hallucination detection becomes possible for all countable language collections. These results highlight the essential role of expert-labeled examples in training hallucination detectors and provide theoretical support for feedback-based methods, such as reinforcement learning with human feedback (RLHF), which have proven critical for reliable LLM deployment.