The rise of general-purpose artificial intelligence (AI) systems, particularly large language models (LLMs), has raised pressing moral questions about how to reduce bias and ensure fairness at scale. Researchers have documented a sort of “bias” in the significant correlations between demographics (e.g., race, gender) in LLM prompts and responses, but it remains unclear how LLM fairness could be evaluated with more rigorous definitions, such as group fairness or fair representations. We analyze a variety of technical fairness frameworks and find inherent challenges in each that make the development of a fair LLM intractable. We show that each framework either does not logically extend to the general-purpose AI context or is infeasible in practice, primarily due to the large amounts of unstructured training data and the many potential combinations of human populations, use cases, and sensitive attributes. These inherent challenges would persist for general-purpose AI, including LLMs, even if empirical challenges, such as limited participatory input and limited measurement methods, were overcome. Nonetheless, fairness will remain an important type of model evaluation, and there are still promising research directions, particularly the development of standards for the responsibility of LLM developers, context-specific evaluations, and methods of iterative, participatory, and AI-assisted evaluation that could scale fairness across the diverse contexts of modern human-AI interaction.