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arXiv · 2006.12275

How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?

Guillermo Badia·Petr Cintula·Petr Hajek·Andrew Tedder
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Abstract

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.

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Guillermo Badia, Petr Cintula, Petr Hajek, Andrew Tedder. 2020-06-18. How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?. https://arxiv.org/abs/2006.12275

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