Why do LLMs always hallucinate and will they continue to do so?
At least one research paper appears to think so.
In September 2024, researchers Sourav Banerjee and colleagues published a groundbreaking analysis that fundamentally challenges how we understand artificial intelligence limitations. Their work, "LLMs Will Always Hallucinate, and We Need to Live With This," presents mathematical proof that hallucinations in Large Language Models are not merely technical glitches to be engineered away, but features stemming from the fundamental mathematical and logical structure of these systems.
The research introduces the concept of "Structural Hallucination"—demonstrating that every stage of the LLM process, from training data compilation to fact retrieval, intent classification, and text generation, will have a non-zero probability of producing hallucinations. Their analysis draws on computational theory and Gödel's First Incompleteness Theorem, and argues that it is impossible to eliminate hallucinations through architectural improvements, dataset enhancements, or fact-checking mechanisms.
This isn't isolated research. Complementary work by Xu, Jain, and Kankanhalli at the National University of Singapore formalised the problem using learning theory, showing that LLMs cannot learn all computable functions and will therefore inevitably hallucinate if used as general problem solvers. Recent theoretical advances have established a fundamental impossibility theorem proving that no LLM can simultaneously achieve truthfulness, information conservation, knowledge revelation, and knowledge-constrained optimality.
The Gödel Connection: Mathematical Foundations of Inevitable Error
Banerjee and colleagues anchor their proof in Kurt Gödel's 1931 incompleteness theorems, which demonstrated that any sufficiently powerful formal system will contain true statements that cannot be proven within that system. This fundamental limitation extends directly to LLMs, which function as complex formal systems trained on vast amounts of data.
The researchers construct a self-referential statement that creates an unavoidable logical contradiction: "There are true facts not in my training database." Whether this statement is true or false leads inevitably to hallucination. If false, the LLM generates an erroneous statement—a hallucination. If true, the LLM generates a statement that cannot be verified by its training data, which also constitutes a hallucination.
This mathematical framework reveals that LLMs inherently contain "blind spots" where they cannot definitively determine correct outputs based solely on their training and internal structure. Just as Gödel showed that formal systems cannot prove all truths they contain, LLMs cannot generate perfectly accurate outputs for all possible inputs.
The connection extends through computational undecidability. The researchers demonstrate that several undecidable problems arise in LLM training and operation, including the Halting Problem in training, the Acceptance Problem in information retrieval, and the Emptiness Problem in intent classification.
Empirical Validation: Measuring the Mathematical Reality
Current measurements across different LLMs provide concrete evidence supporting these theoretical predictions. According to Vectara's Hallucination Leaderboard, popular LLM models hallucinate between 2.5% and 8.5% of the time, with some models exceeding 15%—exactly what the mathematical framework predicts.
Domain-specific analysis reveals the scope of these limitations. Stanford RegLab researchers conducted the first systematic empirical test of legal hallucinations, finding rates ranging from 69% to 88% in response to specific legal queries. In one striking example, when asked about legal precedents, LLMs collectively invented over 120 non-existent court cases, complete with convincingly realistic names like "Thompson v. Western Medical Center (2019)."
Medical applications show similarly concerning patterns.
Analysis of over 10,000 AI hallucinations by UC Berkeley researchers revealed systematic patterns rather than random errors—when LLMs hallucinate statistics, percentages ending in 5 or 0 appear 3.7 times more often than in factual data. This suggests that hallucinations arise from learned patterns rather than mere computational noise.
The Four-Stage Analysis: Systematic Breakdown
Banerjee et al.'s research systematically analyses each stage of LLM operation to demonstrate why hallucinations are mathematically inevitable at every level:
Training Data Incompleteness: The researchers prove that no training database can be 100% complete. The vastness and ever-changing nature of human knowledge ensures that training data will always be incomplete or outdated. This inherent incompleteness makes it impossible to eliminate all hallucinations by training models on every possible fact.
Information Retrieval Undecidability: Even assuming perfect training data, the researchers show that LLMs cannot retrieve correct information with 100% accuracy. They prove that the "needle in a haystack" problem—retrieving specific information from complex data—is undecidable by reducing it to the Acceptance Problem. This means LLMs may "blur" or mix contexts, leading to inaccurate information retrieval.
Intent Classification Impossibility: The researchers demonstrate that understanding user intent is also undecidable, reducing this problem to the "needle in a haystack" problem. Language inherent ambiguity means LLMs will be unable to accurately classify user intent with probability 1, with misunderstandings cascading through the entire generation process.
Generation Process Uncertainty: Finally, the text generation process itself introduces unavoidable uncertainty. LLMs cannot determine exactly where they will stop generating tokens, making the halting problem for LLM text generation undecidable. This fundamental uncertainty persists even with perfect training, perfect retrieval, and perfect intent understanding.
The Probabilistic Trap: Pattern Recognition Versus Truth Verification
The core limitation stems from how LLMs fundamentally operate. These systems function on probabilistic principles, generating outputs based on learned probability distributions rather than possessing true understanding or reasoning abilities. They excel at recognising patterns and generating coherent text following learned statistical relationships, but this approach fundamentally differs from truth verification.
Research investigating LLM hidden states reveals that models react differently when processing genuine responses versus fabricated ones, suggesting some level of internal awareness of potential falsehood. Yet this awareness doesn't translate to prevention—the systems continue generating false information with confidence.
A fascinating MIT study discovered that when AI models hallucinate, they tend to use more confident language than when providing factual information, being 34% more likely to use phrases like "definitely," "certainly," and "without doubt" when generating incorrect information. This counterintuitive behaviour highlights how pattern-matching approaches can produce overconfidence precisely when systems have the least reliable information.
The probabilistic nature introduces unavoidable uncertainty at the fundamental level. LLMs operate by sampling from learned probability distributions, and even perfect knowledge representation cannot eliminate the inherent randomness in this sampling process.
Why Advanced Techniques Cannot Solve Fundamental Problems
Despite significant research investment, even the most sophisticated approaches to reducing hallucinations face mathematical limitations preventing complete elimination. Retrieval-Augmented Generation (RAG), widely promoted as a solution, cannot overcome the fundamental structural problems.
Mathematical analysis by researchers reveals that RAG models inherently struggle to eliminate hallucinations due to intricate mathematical formulations embedded within the framework. The approach improves domain specificity rather than addressing hallucination rates directly. RAG faces limitations in both retrieval and generation phases—data source issues, query formulation problems, retriever limitations, context noise, context conflicts, and alignment problems all contribute to continued hallucinations.
Recent research on Multi-source RAG reveals that integrating multiple retrieval sources, while potentially more informative, introduces new challenges that paradoxically exacerbate hallucination problems. The sparse distribution of multi-source data hinders capturing logical relationships, whilst inherent inconsistencies among different sources lead to information conflicts.
Human feedback approaches face similar fundamental constraints. Reinforcement learning from human feedback (RLHF) can be time-consuming and costly, raising transparency concerns about who determines what is true or appropriate to discuss. The subjective nature of truth verification means human feedback systems cannot provide the absolute ground truth needed for complete hallucination elimination.
Even the most optimistic studies combining RAG, RLHF, and custom guardrails achieved only a 96% reduction in hallucinations compared to baseline models—impressive improvement, but still leaving 4% of outputs potentially incorrect. This 4% residual reflects the mathematical floor established by computational theory.
Frequency Effects and Retrieval Limitations
Recent empirical research validates the theoretical predictions about knowledge boundaries. Studies show that the frequency of facts in training data significantly influences hallucination rates—lower frequency facts consistently produce higher hallucination rates, supporting theoretical predictions about knowledge representation limits.
Research by Kang et al. introduces a "familiarity score" quantifying how closely training data matches test examples, observing that hallucination rates grow almost linearly with unfamiliarity. This empirical finding directly supports theoretical predictions about the relationship between knowledge coverage and hallucination inevitability.
The monofact rate—facts appearing only once in training data—creates particular challenges. Research demonstrates that when facts appear with varying frequency, both monofact rates and hallucination patterns are affected. Novel facts are learned more slowly by language models, and integrating these facts tends to elevate hallucination rates in a linear fashion.
Sophisticated uncertainty measures, including semantic entropy-based approaches published in Nature, can detect subsets of hallucinations but cannot eliminate them entirely. These measures provide valuable detection capabilities but cannot overcome the fundamental mathematical constraints that ensure some level of error persists.
Practical Implications and Future Directions
Recognition that hallucinations are mathematically inevitable doesn't mean abandoning efforts to improve AI systems—it means developing more sophisticated approaches for managing and coexisting with imperfection. The realisation requires new paradigms for human-AI interaction that leverage capabilities whilst safeguarding against inevitable pitfalls.
Some researchers propose fundamental rethinking, viewing hallucinations as "adversarial examples" rather than bugs to be fixed. This perspective opens new avenues for understanding and managing these systems rather than futilely attempting to eliminate inherent characteristics.
Clinical applications demonstrate that careful engineering and validation can achieve error rates below human-generated content. Research shows that iterative improvement processes can achieve just 1 major hallucination per 25 medical notes—whilst not perfect, this represents pragmatic approaches to managing inevitable limitations. These real-world applications prove that whilst we cannot eliminate the mathematical certainty of error, we can engineer systems that perform within acceptable tolerance levels.
The implications extend far beyond technical considerations. In high-stakes environments like healthcare, these limitations pose genuine risks—inaccurate medical reports could lead to life-threatening treatments or missed diagnoses. Yet research demonstrates that with proper safeguards, LLM-assisted clinical documentation can achieve error rates below human-generated content, transforming the challenge from elimination to management.
Enhanced transparency through data provenance tracking offers significant benefits for regulatory compliance and user trust. This approach acknowledges imperfection whilst providing tools for users to make informed decisions about generated content. Modern implementations focus on interpretability—showing users the sources used to generate responses, allowing assessment of trustworthiness and identification of potential biases.
Business implications are significant—77% of organisations surveyed by Deloitte express concerns about AI hallucinations, reflecting growing recognition of these fundamental limitations. This widespread concern stems from practical experience with systems that promise efficiency but deliver uncertainty. Hallucinating LLMs may result in more work instead of simplifying processes—the efficiency advantages that generative AI promises are essentially lost when workers cannot trust results, forcing them to spend valuable time confirming information.
The economic implications are profound. Rather than driving abandonment of the technology, this mathematical reality should inform implementation strategies that account for computational limits whilst maximising value. Forward-thinking organisations are shifting from pursuing perfect AI to developing robust human-AI collaboration frameworks that work within mathematical constraints.
The development of domain-specific approaches shows promise. Researchers suggest that LLMs should be approached as "zero-shot translators" for converting source material into various forms rather than as omniscient knowledge systems. This framing sets appropriate expectations and use cases that align with actual capabilities rather than idealised perfection.
Embracing Computational Limits in the Age of AI
The convergence of theoretical proof and empirical evidence establishes that LLM hallucinations represent a fundamental characteristic rather than a solvable engineering challenge. This limitation joins the grand tradition of impossibility results like Gödel's incompleteness, Heisenberg's uncertainty principle, and Arrow's impossibility theorem—absolute boundaries of what intelligence can achieve under computational constraints.
Yet this mathematical reality need not dampen our enthusiasm for AI's transformative potential. Instead, it provides the foundation for mature, realistic deployment strategies. The research by Banerjee et al. and supporting studies transforms our understanding of AI development, shifting focus from impossible perfectibility to achievable reliability within known constraints.
The path forward requires sophisticated frameworks that work with rather than against mathematical realities. As we continue integrating LLMs into critical applications across healthcare, law, finance, and beyond, this understanding becomes essential for responsible innovation. The most successful implementations will be those that acknowledge limitations whilst maximising capabilities within mathematical boundaries.
Consider the broader societal implications: millions of users interact with AI systems daily, often unaware of their fundamental limitations. Mathematical literacy about AI becomes as crucial as digital literacy once was. Users need to understand not just how to use these tools, but why they behave as they do and where their trustworthiness ends.
The democratisation of AI access amplifies both opportunities and risks. When powerful language models become as ubiquitous as search engines, society needs frameworks for managing systems we know will occasionally fail in predictable ways. This isn't pessimism—it's realism grounded in mathematical certainty.
The question isn't whether we can eliminate hallucinations—some research says we cannot. The key question however, is whether we can develop mature, responsible approaches to managing systems we know can be imperfect but potentially incredibly valuable. The answer lies in embracing both the power and the limitations revealed by rigorous mathematical analysis.
The mathematical certainty of hallucinations isn't a failure of engineering—it's a fundamental characteristic of intelligence operating under computational constraints. Recognising this truth allows us to build better, more reliable systems that acknowledge rather than ignore the mathematical realities governing all computation.
As we stand at the threshold of widespread AI integration across society, the research provides crucial guidance: pursue detection over elimination, transparency over perfection, and collaboration over automation. The future probably belongs to those who understand both the immense promise and the mathematical limits of artificial intelligence.
As the researchers conclude in their seminal work, we must learn to live with this reality. The promise of artificial intelligence remains transformative, but it requires honest acknowledgement of its mathematical boundaries. Only by embracing these limitations can we develop truly robust approaches to deploying AI systems responsibly in our increasingly digital world.
Understanding that LLMs will always hallucinate doesn't diminish their potential—it clarifies the path toward responsible and effective implementation grounded in mathematical honesty about system limitations.