By the time I got out of bed yesterday, three people had asked me if I'd seen the OpenAI announcement that the Partition Principle does not imply the Axiom of Choice. The number kept increasing throughout the day, and I can see why. I am well known for my interest in this problem. Indeed, at least two people asked whether I plan to send Sam Altman a bottle of whisky, as promised on my Problems page. The answer is no, and I want to explain why you should be very angry at OpenAI.
First, let me clarify. I am not entirely against the use of AI. I understand it is a tool, and many people find it useful. I think we need to better understand how this technology changes our field and how we want to use it before we rush to commit to it. This is why I generally avoid using AI for mathematics, though I am happy to use LLMs to consolidate information, generate an infographic, or proofread an email. The framework for using it in research is still missing. arXiv has introduced rate limiting in response to these tools, and almost all papers should now include an AI statement. We also need to decide how to judge an author's contribution. Should chat logs be made public, or at least available to reviewers and editors?
But back to the Partition Principle. I took a brief look at the preprint released by OpenAI, though not the Lean code, since that was enormous and I know little about Lean. The preprint was unclear, muddled, and oddly structured. The terminology was "a bit off," and it contained theorems I would not expect to be proved, stated in a strange manner. Lemma 7.4 and Lemma 8.1 are not something you would expect in a paper like this.
The references were also problematic. A paper of mine on the Bristol model is cited for a basic introduction to symmetric extensions, which is not the reference I would have chosen. Three unpublished, unrefereed lecture notes that are not on arXiv are also used, including one of mine that refers to a proposition I am certain appears in published work. That is not what you would expect from a serious preprint claiming to solve a problem.
A competent set theory paper outside my specialty can usually be skimmed to grasp its strategy. In my own area of expertise, the Axiom of Choice, I can quickly work out what the authors intended. With OpenAI's paper, this was impossible. If it were submitted to a journal, it should be desk rejected for quality. The onus is on the author to meet the standards of clear communication, and OpenAI has dropped hundreds of "solutions" that are incomprehensibly written.
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