{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-11-medvedev-logic-undecidable/","md":"https://postcutoff.com/e/2026-09-11-medvedev-logic-undecidable/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-11-medvedev-logic-undecidable","date":"2026-09-11","date_precision":"day","short_title":"Medvedev's logic of finite problems (1962) shown undecidable","deck":"Key idea from ChatGPT Sol 5.6, central argument checked in Lean by Claude Opus 5","takeaway":"Rodrigo Nicolau Almeida and Søren Brinck Knudstorp proved that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable, which settles what they call a longstanding open problem (arXiv 2609.13359, 11 Sep 2026).","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.13359: Medvedev logic is undecidable (Almeida, Knudstorp)","url":"https://arxiv.org/abs/2609.13359","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2609.39577: Non-finite axiomatizability and undecidability of Cheq (Han Xiao)","url":"https://arxiv.org/abs/2609.39577","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-09-30","updated":"2026-10-29","orgs":["Rodrigo Nicolau Almeida","Søren Brinck Knudstorp","OpenAI","Anthropic"],"title":"Medvedev's logic of finite problems (1962) shown undecidable; key idea from ChatGPT Sol 5.6, central argument checked in Lean by Claude Opus 5","summary":"Rodrigo Nicolau Almeida and Søren Brinck Knudstorp proved that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable, which settles what they call a longstanding open problem (arXiv 2609.13359, 11 Sep 2026). They reduce the periodic tiling problem to it. The authors 'do not take any credit' for the key ideas: ChatGPT Sol 5.6 obtained them on 4 Sep 2026, and Claude Opus 5 produced a Lean verification of the central argument. All prompts and model outputs are published.","key_facts":["Medvedev introduced the logic of finite problems in 1962 (Dokl. Akad. Nauk 142); whether it is decidable was a longstanding open question","Main theorem: Medvedev's logic ML is undecidable, via a reduction from periodic tiling to non-theoremhood; similarly Skvortsov's logic of infinite problems is undecidable, and the two logics are separated by any aperiodic tiling of the plane","Methodology §7: 'the key idea for the undecidability of ML – namely the construction of the tiling posets P_W, and the encoding of the reduction – was obtained by prompting ChatGPT Sol 5.6 … whilst the authors assume responsibility for the correctness of the results, they do not take any credit in such ideas'","The authors rewrote the text themselves and invoke the Leiden Declaration; Claude Opus 5 was used for proofreading and for a Lean verification of the main central argument; prompts, preliminary documents and verification ledgers are public on the first author's website","Status: preprint (32 pages)","Follow-up: Han Xiao (arXiv 2609.39577, 30 Sep 2026) proves the logic Cheq is not finitely axiomatizable (a human result 'established before the rise of AI') and that Medvedev's undecidability implies Cheq's; the paper notes Theorem 6.2 'does use the AI-generated proof that Medvedev logic is undecidable'","AI disclosure (abstract, arXiv 2609.13359): 'The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5.' arXiv comment: 'main result obtained via LLM (see methodology section)'"],"key_numbers":[],"tags":["math","logic","decidability","tiling","lean","gpt-5-6","claude-opus-5"],"science":{"field":"mathematics","subfield":"mathematical logic / superintuitionistic logics","problem":"Decidability of Medvedev's logic of finite problems","result":"Medvedev's logic (and Skvortsov's logic) is undecidable.","open_since":"1962","ai_system":["GPT-5.6 Sol (ChatGPT)","Claude Opus 5"],"human_role":"Humans chose the problem and wrote structured prompts; ChatGPT found the core construction; the humans checked and rewrote it","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; central argument verified in Lean by Claude Opus 5 (per the authors), not the whole paper; proofs checked and rewritten by the authors"},"status":"pending","shock":""},"body_md":"## What happened\n\nTwo logicians, one of whom wrote his PhD on undecidability, suspected a targeted search could settle Medvedev's logic. They prompted ChatGPT Sol 5.6, which returned a tiling-based reduction on 4 September. Claude Opus 5 formalised the central argument in Lean, and the authors wrote the paper and published the whole prompt trail.\n\n## Why it matters\n\nIt answers a well-known question in non-classical logic, and other logicians already build on it: on 30 September Han Xiao derived the undecidability of the logic Cheq from it, citing \"the AI-generated proof\". It is also a model of transparent disclosure: full prompt and output archives are published alongside the paper and treated like a dataset.","disputed":[{"label":"Verification","text":"Unrefereed preprint; central argument verified in Lean by Claude Opus 5 (per the authors), not the whole paper; proofs checked and rewritten by the authors","sources":[]}],"related":[{"id":"2026-10-13-ai-assisted-results-october-2026","url":"https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/","date":"2026-10-13","date_precision":"day","short_title":"October 2026: the arXiv stream of conjectures settled with disclosed AI help continues","deck":null,"takeaway":"In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":17,"official":17,"filed":"2026-10-13","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-06-openai-periodic-tiling-conjecture-dimension-three","url":"https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/","date":"2026-10-06","date_precision":"day","short_title":"Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 155, dated 23 Sept) claims a finite set of cubes in ℤ³ whose translates tile space, while no tiling by them is periodic. The same holds for its solid version in ℝ³. Greenfeld and Tao had disproved the periodic tiling conjecture only in some very high dimension. In one and two dimensions the conjecture is true, so three would be the lowest dimension where it fails. The Lean proof, which also covers the lower dimensions, was accepted in a community Comparator run. Erik Demaine and Stefan Langerman reused the paper's gadgets to prove that deciding whether one connected polycube tiles 3D space is undecidable. Chaim Goodman-Strauss said the paper has novel ideas but is 'unnecessarily difficult to read'. No published expert check of the whole proof was found as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-07-24-claude-opus-5","url":"https://postcutoff.com/e/2026-07-24-claude-opus-5/","date":"2026-07-24","date_precision":"day","short_title":"Anthropic releases Claude Opus 5","deck":"Near-Fable-5 intelligence at half the price","takeaway":"Opus 5 brought most of Fable 5's capability to half the price.","category":"model-release","category_label":"Model releases","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":9,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Anthropic"]},{"id":"2026-07-09-gpt-5-6-sol-terra-luna","url":"https://postcutoff.com/e/2026-07-09-gpt-5-6-sol-terra-luna/","date":"2026-07-09","date_precision":"day","short_title":"OpenAI broadly releases GPT-5.6 after government-gated preview","deck":null,"takeaway":"First frontier model whose public release was explicitly gated by US government review, and the model family involved in the July 2026 sandbox-escape incident.","category":"model-release","category_label":"Model releases","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":7,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2026-06-02-leiden-declaration-ai-mathematics","url":"https://postcutoff.com/e/2026-06-02-leiden-declaration-ai-mathematics/","date":"2026-06-02","date_precision":"day","short_title":"Leiden Declaration on Artificial Intelligence and Mathematics sets community norms for AI in maths (4,000+ signatories)","deck":null,"takeaway":"It is the broadest grassroots statement of mathematicians' norms on AI: disclosure, attribution, integrity of proof, and independence from industry.","category":"policy-safety","category_label":"Policy & safety","importance":3,"confidence":"medium","status":{"key":"partly","labels":["Partly confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Lorentz Center","International Mathematical Union"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"edit","text":"Related: added [Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)](https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/) (OpenAI's claimed 3D translational tile with no periodic tiling)"},{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (unrefereed; Lean covers the central argument only); added the abstract's AI disclosure verbatim"},{"date":"2026-10-13","type":"edit","text":"Related: added [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/) (October catalogue, split out of the summer catalogue)"},{"date":"2026-10-07","type":"edit","text":"Added Han Xiao's Cheq follow-up (arXiv 2609.39577), which builds on the AI-generated undecidability proof"},{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":134,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":73,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":164,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":103,"in_training_data":false}],"short_url":null}