{"slug": "ai-proves-medvedev-logic-is-undecidable", "title": "AI proves Medvedev logic is undecidable", "summary": "A paper submitted to arXiv on 11 September 2026 by Rodrigo Nicolau Almeida proves that Medvedev's logic of finite problems is undecidable, settling a longstanding open problem. The proof reduces the periodic tiling problem to non-theoremhood in Medvedev's logic, and the same techniques establish undecidability of Skvortsov's logic of infinite problems and show the two logics are distinct, separated by any aperiodic tiling of the plane. The paper states 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.", "body_md": "# Mathematics > Logic\n\n  [Submitted on 11 Sep 2026]\n\n# Title:Medvedev logic is undecidable\n\n[View PDF](https://arxiv.org/pdf/2609.13359)\n\n[HTML (experimental)](https://arxiv.org/html/2609.13359v1)\n\nAbstract:We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct -- in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields -- for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes.\n\nThe 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. A detailed methodology section outlines how such results were obtained.\n\n## Submission history\n\nFrom: Rodrigo Nicolau Almeida [\n[view email](https://arxiv.org/show-email/7ee5e864/2609.13359)]\n\n**[v1]** Fri, 11 Sep 2026 17:49:49 UTC (44 KB)\n\n### Current browse context:\n\nmath.LO\n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer \n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers \n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps \n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations \n\n*(*[What are Smart Citations?](https://www.scite.ai/))\n# Code, Data and Media Associated with this Article\n\nalphaXiv \n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers \n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub \n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub \n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face \n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast \n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))\n# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower \n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender \n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))\n# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.\n\nHave an idea for a project that will add value for arXiv's community? [**Learn more about arXivLabs**](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/ai-proves-medvedev-logic-is-undecidable", "canonical_source": "https://arxiv.org/abs/2609.13359", "published_at": "2026-09-21 17:01:15+00:00", "updated_at": "2026-09-21 17:24:44.360572+00:00", "lang": "en", "topics": ["ai-research", "artificial-intelligence", "machine-learning"], "entities": ["Rodrigo Nicolau Almeida", "arXiv", "Medvedev's logic of finite problems", "Skvortsov's logic of infinite problems", "ChatGPT Sol 5.6", "Claude Opus 5", "Lean"], "alternates": {"html": "https://wpnews.pro/news/ai-proves-medvedev-logic-is-undecidable", "markdown": "https://wpnews.pro/news/ai-proves-medvedev-logic-is-undecidable.md", "text": "https://wpnews.pro/news/ai-proves-medvedev-logic-is-undecidable.txt", "jsonld": "https://wpnews.pro/news/ai-proves-medvedev-logic-is-undecidable.jsonld"}}