AI proves Medvedev logic is undecidable
Researchers Rodrigo Nicolau Almeida and Søren Brinck Knudstorp have proven that Medvedev's logic of finite problems is undecidable, resolving a longstanding open problem in mathematics. The proof utilizes a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic, with similar techniques applied to show the undecidability of Skvortsov's logic of infinite problems. The core ideas and technical work were generated using ChatGPT Sol 5.6 and formally verified in Lean by Claude Opus 5.
- ▪Medvedev's logic of finite problems has been proven to be undecidable via a reduction from the periodic tiling problem.
- ▪Skvortsov's logic of infinite problems is also undecidable, and it is distinct from Medvedev's logic as separated by any aperiodic tiling of the plane.
- ▪The core idea and technical work of the proof were obtained using the AI model ChatGPT Sol 5.6.
- ▪The results were formally verified in the Lean programming language by the AI model Claude Opus 5.
- ▪The paper was submitted to arXiv on September 11, 2026, and has implications for fields such as propositional dependence logic and toposes.
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| Original publisher | arXiv.org |
| Canonical URL | https://arxiv.org/abs/2609.13359 |
| Publication time | Mon, 21 Sep 2026 17:01:15 +0000 |
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Mathematics > Logic arXiv:2609.13359 (math) [Submitted on 11 Sep 2026] Title:Medvedev logic is undecidable Authors:Rodrigo Nicolau Almeida, Søren Brinck Knudstorp View a PDF of the paper titled Medvedev logic is undecidable, by Rodrigo Nicolau Almeida and S{\o}ren Brinck Knudstorp View PDF HTML (experimental) 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.
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