{"id":"openai-math/Relative-independence-of-the-separable-quotient-problem","pdf":"https://github.com/openai/math/blob/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026","tags":["set-theory","functional-analysis","banach-spaces"],"score":0,"title":"Relative independence of the separable quotient problem","domain":"Functional analysis","family":"323","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ 1 , we obtain independence relative to the consistency of ZFC alone.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=it%20is%20very%20cool%20to%20see%20that%20the%20separable%20quotient%20problem%20is%20independent%20of%20ZFC","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"it is very cool to see that the separable quotient problem is independent of ZFC. These results are truly spectacular, and the fact that we are able to witness their resolutions is extremely exciting. What concerns me much more is the potential misalignment between the objectives of AI companies…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":false,"family_title":"Independence of the separable quotient problem"}