AI proves Medvedev logic is undecidable

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.

Mathematics Logic Submitted on 11 Sep 2026 Title:Medvedev logic is undecidable View PDF https://arxiv.org/pdf/2609.13359 HTML experimental https://arxiv.org/html/2609.13359v1 Abstract: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. 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. A detailed methodology section outlines how such results were obtained. Submission history From: Rodrigo Nicolau Almeida view email https://arxiv.org/show-email/7ee5e864/2609.13359 v1 Fri, 11 Sep 2026 17:49:49 UTC 44 KB Current browse context: math.LO References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both 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. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .