Contested
The Partition Principle does not imply Choice (family 244)
The gist
On October 6, 2026, OpenAI announced a proof that the Partition Principle, in set theory, does not imply the axiom of choice.
DiscoverIA status on October 11, 2026: contested, meaning sourced public objection, with no established refutation.
Text written by an AI, reviewed by Antoine Olivier,
Cite this entry
A single status snapshot since Oct 11, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 11, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-244-partition-principle-r1
@misc{discoveria-oai-fam-244-partition-principle-r1,
author = {{DiscoverIA}},
title = {The Partition Principle does not imply Choice (family 244)},
howpublished = {DiscoverIA registry, entry oai-fam-244-partition-principle},
note = {Status “Contested”, snapshot 1, recorded on October 11, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-244-partition-principle#releve-1},
urldate = {2026-10-11}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
The Partition Principle says that a set split into non-empty parts has at least as many elements as there are parts. The axiom of choice allows one to pick an element from each set in a collection of non-empty sets. If the proof held, the first would not allow one to prove the second.
What remains unclear
The set theorist Asaf Karagila disputes, on his blog, the quality of the writing and the method; he does not show that the proof is wrong, and he does not read the Lean (a language in which a computer checks every step of a proof). That Lean comes from OpenAI alone: no third party has checked the substance.
For experts
Sources, method, key facts, status history, report an error
Status
Contested
Recorded on · karagila.org · openai-pp
Outside the verification path
- Announced
- Recordedcontested
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
Sourced public objection, with no established refutation.
It stays “Contested” until a refutation is established. It would become “Withdrawn” if its author withdrew it or the reference source marked it incorrect.
Sources
- Main sourcegithub.comlink only, reading not recorded
- Disputekaragila.org · openai-pppage read word for word
- Disputekaragila.org · leanlink only, reading not recorded
Method
For family 244, OpenAI publishes a manuscript (preprint) and a Lean formalisation (configuration PartitionPrinciple) in its openai/math repository. Karagila, in a post dated October 8, 2026, writes: "it should be issued a desk rejection for the quality". A second post, "Why I don't trust Lean code from AI" (October 10, 2026), was not read in detail. DiscoverIA status: contested (a sourced public objection exists, with no established refutation). Set on October 11, 2026 from karagila.org (page read, quotation checked word for word).
Limits
The claimant supplies a Lean proof; DiscoverIA has found no independent verification of it.
Key facts
- Claimed by
- OpenAI
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (logic / set theory)
- Checked by
- claimant; no third party on the substance
- Sources
- 3 : 1 read word for word, 2 links only
- This entry
- Reviewed by Antoine Olivier, Oct 18, 2026
Status history
Only one observation so far.
- 01
First observation:ContestedCurrent status
contested on the quality of the writing and the method, not shown to be wrong: the author does not read the Lean
Source: karagila.orgread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 238 of 372, by family number
- Not evaluated : Rigidity of the Turing degrees (family 241)
- Not evaluated : Single-fold Diophantine representations (family 242)
- Not evaluated : Choiceless counting does not capture polynomial time (family 243)
- Contested : The Partition Principle does not imply Choice (family 244)
- Not evaluated : The β-Barendregt–Geuvers–Klop conjecture (family 245)
- Not evaluated : Cannon's conjecture (family 246)
- Not evaluated : An infinite finitely presented residually finite 2-group (family 247)
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