OpenAI, the Partition Principle, and Mathematics
OpenAI, the Partition Principle, and the State of Mathematics
By Asaf Karagila
Date: October 08, 2026
Yesterday, before I had even fully woken up, I was bombarded by inquiries. Three different people asked if I had seen the latest announcement from OpenAI: the claim that the Partition Principle does not imply the Axiom of Choice.
As the day progressed, the frequency of these questions only grew. Specifically, at least two individuals wanted to know if I intended to follow through on the promise listed on my "Problems" page—namely, sending a bottle of expensive whisky to Sam Altman.
Let me explain why I am not doing that, and why the mathematical community should be deeply concerned.
The Role of AI in Research
I recognize that AI is a powerful tool. I acknowledge its merits and the utility many find in it. However, I believe we are rushing headlong into its adoption without a proper understanding of how this technology alters our field.
I generally avoid using AI for actual mathematical derivation. While I am perfectly happy using LLMs for the following:
- Consolidating disparate information.
- Generating helpful infographics.
- Proofreading professional correspondence.
...I believe the overarching framework for its integration into mathematics is currently non-existent. We are in a "discovery phase" where we observe the consequences in real-time and must decide as a collective whether to embrace or restrict these tools.
Community Adjustments
We are already seeing the fallout:
- arXiv has implemented rate limiting to combat AI-generated submissions.
- Most papers now require an "AI Statement" (even if just to say none was used).
- Pending: Deciding how to judge author contributions.
- Pending: Determining if chat logs must be provided to editors and reviewers.
Critique of the OpenAI Preprint
I performed a cursory review of the preprint released by OpenAI. I did not dive into the Lean code, as the codebase was gargantuan and my proficiency in Lean is limited. However, the written portion is problematic.
The terminology is imprecise, and the structure is bizarre. There are theorems included that simply wouldn't appear in a standard paper, and the phrasing is often alien. Specifically, Lemma 7.4 and Lemma 8.1 are completely out of place for a work of this nature.
The Referencing Problem
The bibliography is particularly alarming for a "serious" claim to have solved a century-old problem:
- One of my papers on the Bristol model is cited for a basic introduction to symmetric extensions—a poor choice of reference for that specific point.
- The paper relies on at least three sets of lecture notes that are unpublished, unrefereed, and not even on arXiv (including my own notes, used to cite a proposition that already exists in published literature).
Comparison of Standards
| Feature | Standard Set Theory Paper | OpenAI Preprint |
|---|---|---|
| Skimmability | High; experts can grasp the strategy quickly. | Low; nearly impossible to parse. |
| Communication | Adheres to academic norms. | Incomprehensible/Non-standard. |
| Referencing | Peer-reviewed, published sources. | Unpublished lecture notes. |
| Clarity | Clear construction and intent. | Obscure and fragmented. |
The "Denial of Service" Attack
In my 15+ years of working on the Axiom of Choice, I have developed a knack for skimming papers to understand the author's intent. Even if I'm not an expert in a specific sub-technicality, the strategy is usually clear.
With this paper, that is impossible. If this were submitted to a journal, it would be a desk rejection based on quality alone. The burden of communication lies with the author. Just as a paper written in Swedish would be rejected by the Proceedings of the American Society, a paper that is incomprehensible to its target audience fails its primary purpose.
OpenAI has essentially dumped hundreds of incomprehensible solutions on us. This is the mathematical equivalent of a Denial of Service (DoS) attack:
I am currently supervising PhD students and managing my own active research. Why should I sacrifice my time because OpenAI didn't bother to communicate clearly?
The Danger of the "Bot" Narrative
The media sees "solutions." The public, funding agencies, and policymakers—who are not mathematicians and do not understand that math research differs fundamentally from biology or philology—see a narrative where bots replace humans.
"This is the equivalent of replacing the chef with the kitchen tools. Even if the knife is automatic and can dice vegetables on its own, there is still a vast distance between that and a palatable meal you actually want to eat."
The tech giants want us to adapt to their standards of output rather than spending the effort to meet ours.
Final Verdict
The claim is:
While this would be a massive leap forward, the presentation is nowhere near an acceptable professional level.
I will be sending Sam Altman a bottle of whisky. No. I will not be sending any whisky, nor do I plan to spend my limited time deciphering a paper that refuses to speak the language of mathematics.