Medvedev Logic is Not Decidable. It is π01 -complete. Who Would Have Guessed?

2026-09-10

AI Summary

研究旨在证明Medvedev逻辑的可判定性，但最终证明其为π01-完全且不可判定。通过将周期多米诺问题与直觉主义公式关联，揭示了该逻辑无递归可枚举的完备证明系统。

Abstract

This project began as an attempt to prove that Medvedev logic is decidable with the help of generative AI systems. The author (as well as the generative AI systems, or at least they claim to be since I have asked them) was surprised by its eventual conclusion. We prove that Medvedev logic ML, the intermediate logic of finite problems, is Pi-01-complete under computable many-one reductions. Consequently, ML is not recursively enumerable, a fortiori undecidable, and admits no recursively enumerable sound and complete proof calculus. The proof connects the periodic domino problem with intuitionistic formulas through a shared intermediate structure that we call a Wang-Medvedev pair. Such a pair consists of a finite partially ordered set of roles together with demands. Demands define the interaction between roles. A realization labels nonempty subsets of a finite set with these roles, respecting the order and satisfying the demands. We associate a pair with each finite Wang system and show that it has a realization iff the system tiles a finite torus. We then construct an intuitionistic formula that fails on some finite Medvedev frame iff the same pair is realizable. Realizability thus provides the link between periodic tilings and the countermodels.

Problem

• Medvedev logic
• undecidability
• Pi-01-complete
• computable many-one reductions
• recursively enumerable

Innovation

• Pi-01-complete
• undecidable
• Wang-Medvedev pair
• periodic domino problem