Set theorist Asaf Karagila refuses to review OpenAI’s claimed solution of the Partition Principle problem (does PP imply AC?): ‘It sucked’; calls the 722-paper release a ‘Denial of Service’
Event confirmedAwaiting review
The takeaway
OpenAI’s catalogue claims the Partition Principle does not imply the Axiom of Choice, one of set theory’s oldest open problems (Russell, 1906), with a partial Lean formalization. The best-known expert on the problem, who had promised a bottle of whisky for a solution, called the paper unreadable and declined to check it.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Unverified claim
- Importance
- 3 of 5
- Last verified
- 9 October 2026
Your AI and this story
- GPT-6 Astra161 days after its cutoff
- Claude Opus 5.5100 days after its cutoff
- Gemini 3.8 Flash191 days after its cutoff
- Grok 4.7130 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 100 days before it.
Key facts
- Claim (OpenAI family 244): assuming Con(ZF), there is a model of ZF in which every surjective image of a set injects into it (Partition Principle) while AC fails; choice for ordinal-indexed families holds. A second construction gives a transitive symmetric extension with no new countable sequences of ground-model elements
- Lean scope (lean/docs/244.md): formalizes the relative-consistency implication and a model construction from a countable transitive ground model with an internal strongly inaccessible cardinal; ‘The paper’s stronger transitive-model preservation assertions are outside that selected construction’
- History: Russell (1906) conjectured PP is equivalent to AC; AC ⇒ PP is easy; the converse is often called the oldest open problem in set theory. Frank Gilson’s 2026 claim (arXiv 2601.01855) was withdrawn in v8 (10 Sep 2026): ‘the previously claimed final PP+not-AC construction is withdrawn’
- Karagila (8 Oct): read the preprint, not the Lean code (‘that code was enormous’); found odd lemmas (7.4, 8.1), ‘terminology a bit off’, and references to unpublished lecture notes, including his own; ‘if this was an academic paper submitted to a journal, it should be issued a desk rejection for the quality’
- Karagila on the release: ‘OpenAI drops some hundreds of “solutions”, incomprehensibly written “solutions”, and we are all expected to jump on them’; ‘It seems to me, that the AI tech companies would like us to conform to their standards, rather than spend the time and energy to conform to ours’
- Karagila is not against AI as a tool and uses LLMs for minor tasks, but avoids them for mathematics until the community has a framework (e.g. whether chat logs should go to referees)
- Correctness of OpenAI’s construction: not assessed by any expert as of 9 Oct
What happened
Within a day of OpenAI’s release, people kept asking Asaf Karagila whether he had seen the claimed solution and whether he would send Sam Altman the whisky promised on his Problems page. His answer, in a blog post on 8 October, was no on both counts. He judged the writing unfit for a journal and said that checking it would mean dropping his own research and students for a paper whose authors “don’t bother to communicate better”. He argued that the release lets OpenAI suggest “progress” while the public hears “solutions”, and that funders and policymakers may conclude mathematicians can be replaced.
The construction itself has a Lean formalization of its main relative-consistency statement, which Karagila did not examine. A human claim to the same result (Gilson, January 2026) had been withdrawn in September.
Why it matters
It shows the gap between a machine-checkable claim and acceptance by the field. The problem’s main expert refuses the cost of reading AI output written below the field’s standards, so a possibly correct proof of a century-old problem may stay in limbo until someone checks the Lean statement against the intended theorem or rewrites the argument.
What is disputed or not yet verified
| Verification | Partial Lean formalization by OpenAI (review status ‘unchecked’); no expert check |
|---|
Sources
5 sources from 4 sites. Numbers match the chips in the text.
5 sources: 3 primary, 2 reactions
Primary
- OpenAI math catalogue (CONTENTS.md, family 244)github.com, paper
- OpenAI math: Lean scope note for family 244github.com, code
- Gilson: A countable-support symmetric iteration separating PP from AC (arXiv 2601.01855, claim withdrawn in v8)arxiv.org, paper
Reactions
- Asaf Karagila: OpenAI, the Partition Principle, and mathematics (8 Oct 2026)karagila.org, discussion
- Hacker News discussion (172 points)news.ycombinator.com, discussion
Changes
- Filed from Karagila’s post and OpenAI’s repo (HN front page)