{"id":"openai-math/The-Partition-Principle-does-not-imply-Choice","pdf":"https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026","tags":["set-theory","axiom-of-choice","logic"],"score":0,"title":"The Partition Principle does not imply Choice","domain":"Mathematical logic","family":"244","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.LO","summary":"We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://karagila.org/2026/openai-pp/","kind":"web","site":"karagila.org","text":"By the time I got out of bed yesterday, I had three people ask me if I'd seen the OpenAI announcement that the Partition Principle does not (...)","added":"2026-10-24","title":"OpenAI, the Partition Principle, and mathematics","author":"Asaf Karagila"}],"formalized":true,"family_title":"The Partition Principle does not imply Choice"}