{"schema":"postcutoff/event@1","as_of":"2026-10-09T19:24:00+02:00","url":"https://postcutoff.com/e/2026-10-08-karagila-openai-partition-principle/","md":"https://postcutoff.com/e/2026-10-08-karagila-openai-partition-principle/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-08-karagila-openai-partition-principle","date":"2026-10-08","date_precision":"day","short_title":"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'","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"OpenAI math catalogue (CONTENTS.md, family 244)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI math: Lean scope note for family 244","url":"https://github.com/openai/math/blob/main/lean/docs/244.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Gilson: A countable-support symmetric iteration separating PP from AC (arXiv 2601.01855, claim withdrawn in v8)","url":"https://arxiv.org/abs/2601.01855","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"Asaf Karagila: OpenAI, the Partition Principle, and mathematics (8 Oct 2026)","url":"https://karagila.org/2026/openai-pp/","type":"discussion","group":"reactions","domain":"karagila.org"},{"n":5,"title":"Hacker News discussion (172 points)","url":"https://news.ycombinator.com/item?id=50013902","type":"discussion","group":"reactions","domain":"news.ycombinator.com"}],"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["OpenAI"],"title":"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'","summary":"OpenAI's 6 Oct 2026 math catalogue includes family 244, \"The Partition Principle does not imply Choice\": if ZF is consistent, so is ZF + PP + choice for ordinal-indexed families + ¬AC (manuscript dated 24 Sep; Lean formalization of the relative-consistency statement). On 8 Oct Asaf Karagila, who works on exactly this problem and lists it on his Problems page with a whisky prize, wrote that the preprint \"sucked. It is unclear, muddled, and has a strange structure\", that a journal should desk-reject it for quality, and that he will not spend time reading it or send Sam Altman the whisky. He called the mass release \"the equivalent of a Denial of Service\" for mathematicians. The post reached 172 points on Hacker News.","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"],"key_numbers":[],"tags":["math","set-theory","axiom-of-choice","partition-principle","openai","reactions","unverified","lean"],"science":{"field":"mathematics","subfield":"set theory","problem":"Does the Partition Principle imply the Axiom of Choice? (Russell 1906)","result":"Claimed by OpenAI's internal model: no (relative consistency of ZF + PP + ¬AC), with a partial Lean formalization; the leading expert declined to review it.","open_since":"1906","ai_system":["OpenAI internal model (unreleased)"],"human_role":"Autonomous per OpenAI's fixed procedure","verification":{"key":"none","label":"Unverified claim","text":"Partial Lean formalization by OpenAI (review status 'unchecked'); no expert check"},"status":"pending","shock":"A problem older than ZF itself was claimed in an unreadable machine-written paper, and the expert most likely to check it publicly refused.","short":"Partition Principle vs Axiom of Choice"},"body_md":"## What happened\n\nWithin a day of OpenAI's release, people kept asking Asaf Karagila whether he had seen the claimed solution and whether he\nwould send Sam Altman the whisky promised on his Problems page. His answer, in a blog post on 8 October, was no on both\ncounts. He judged the writing unfit for a journal and said that checking it would mean dropping his own research and\nstudents for a paper whose authors \"don't bother to communicate better\". He argued that the release lets OpenAI suggest\n\"progress\" while the public hears \"solutions\", and that funders and policymakers may conclude mathematicians can be\nreplaced.\n\nThe construction itself has a Lean formalization of its main relative-consistency statement, which Karagila did not\nexamine. A human claim to the same result (Gilson, January 2026) had been withdrawn in September.\n\n## Why it matters\n\nIt shows the gap between a machine-checkable claim and acceptance by the field. The problem's main expert refuses the cost\nof reading AI output written below the field's standards, so a possibly correct proof of a century-old problem may stay in\nlimbo until someone checks the Lean statement against the intended theorem or rewrites the argument.","disputed":[{"label":"Verification","text":"Partial Lean formalization by OpenAI (review status 'unchecked'); no expert check","sources":[]}],"related":[{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":58,"official":12,"filed":"2026-10-07","updated":"2026-10-09","orgs":["OpenAI"]}],"people":[{"id":"sam-altman","name":"Sam Altman","url":"https://postcutoff.com/person/sam-altman/"}],"posts":[{"id":"2026-10-08-karagila-openai-partition-principle","title":"Asaf Karagila: 'OpenAI, the Partition Principle, and mathematics'","url":"https://postcutoff.com/p/2026-10-08-karagila-openai-partition-principle/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-09","type":"filed","text":"Created from Karagila's post and OpenAI's repo (HN front page)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-09","run":null,"sources_read":"Karagila's post and OpenAI's repo (HN front page)","updated":"2026-10-09","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":null}