The need for fair AI is increasingly clear in the era of general-purpose systems such as ChatGPT, Gemini, and other large language models (LLMs). However, the increasing complexity of human-AI interaction and its social impacts have raised questions of how fairness standards could be applied. Here, we review the technical frameworks that machine learning researchers have used to evaluate fairness, such as group fairness and fair representations, and find that their application to LLMs faces inherent limitations. We show that each framework either does not logically extend to LLMs or presents a notion of fairness that is intractable for LLMs, primarily due to the multitudes of populations affected, sensitive attributes, and use cases. To address these challenges, we develop guidelines for the more realistic goal of achieving fairness in particular use cases: the criticality of context, the responsibility of LLM developers, and the need for stakeholder participation in an iterative process of design and evaluation. Moreover, it may eventually be possible and even necessary to use the general-purpose capabilities of AI systems to address fairness challenges as a form of scalable AI-assisted alignment.