Set theorist pans OpenAI's Partition Principle proof claim as muddled and unreviewable

OpenAI has released a preprint claiming that the Partition Principle does not imply the Axiom of Choice, a result that would settle a set-theory question the post describes as more than a century old. But a researcher who says he has spent over 15 years working on the Axiom of Choice argues the paper is written so badly that a journal would reject it without review, and that he will not spend his time decoding it.

The critique comes from a post on karagila.org, a set theory blog whose author says he was flooded with messages after OpenAI's announcement. The post went on to reach the front page of Hacker News. Its author had long promised, via his "Problems" page, to send Sam Altman a bottle of whisky if this exact problem were solved. His answer now, he writes, is no.

A harsh review of the preprint

According to the post, the author skimmed the preprint but skipped the Lean code, which he calls enormous and says he is not equipped to assess. The prose alone fails on nearly every axis in his telling: it is unclear and muddled, its structure is strange, and its terminology is, in his words, "a bit off". Lemmas 7.4 and 8.1 are stated in ways he would not expect in a serious paper on the subject.

The references draw particular criticism. One of his own papers on the Bristol model is cited for a basic introduction to symmetric extensions, which he says is not the reference he would have chosen, and at least three cited works are unpublished, unrefereed lecture notes that never appeared on arXiv, including his own notes cited for a proposition he believes appears in published papers by him and others.

He contrasts this with ordinary practice: given a typical set theory paper, even outside his specialty, he can skim it and grasp the strategy and construction. For OpenAI's preprint he says that is impossible. Were it submitted to a journal, he argues it should be desk-rejected, because the burden of communicating clearly sits with the author, much as a paper written in Swedish would be turned away by the Proceedings of the American Mathematical Society.

"Progress" in the press release, "solutions" in the headlines

The post's sharpest accusation concerns incentives. OpenAI's press release carefully frames the work as "progress", he writes, but the surrounding media cycle reports "solutions". He claims OpenAI has dropped "some hundreds" of such incomprehensibly written outputs, and compares the effect on mathematicians to a denial-of-service attack: taking the claims seriously would mean abandoning active research, supervision and teaching to sift through them. If the company genuinely wanted to help, he argues, it would have consulted domain experts on the quality of the output before publishing.

The bigger fight over norms

The author is careful to say he is not against AI in general. He uses LLMs to consolidate information, generate infographics and proofread emails. His complaint is that no framework yet exists for using AI in mathematics, and that norms are being improvised under pressure: arXiv has introduced rate limiting partly in response to these tools, most papers now carry an AI statement, and open questions remain about whether chat logs should be published, or at least made available to reviewers and editors, and how an author's contribution should be judged.

His closing worry is about perception. The public, funding bodies and policymakers generally do not understand how mathematical research works, he writes, and headlines about AI "solutions" feed the idea that mathematicians can be replaced by bots. His analogy: automating the knife does not replace the chef.

Why it matters

The Partition Principle question is genuinely old and hard, and a verified resolution would be a major event in set theory. But this episode suggests the bottleneck is not proof discovery at all. It is verification and communication: mathematics that experts cannot read cannot be absorbed into the literature, however much Lean code ships alongside it. The dispute also previews an accounting problem AI creates for every technical field, since generation is cheap while expert review is expensive, and someone has to pay that cost. Until labs internalize the editorial standards of the fields they aim to serve, claims like this one will produce headlines rather than knowledge, and the promised whisky stays in the cupboard.