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

0523434 - ÚI 2023 RIV GB eng J - Journal Article
Badia, G. - Cintula, Petr - Hájek, Petr - Tedder, Andrew
How Much Propositional Logic Suffices for Rosser's Undecidability Theorem?
Review of Symbolic Logic. Roč. 15, č. 2 (2022), s. 487-504. ISSN 1755-0203. E-ISSN 1755-0211
R&D Projects: GA ČR GA17-04630S
Institutional support: RVO:67985807
Keywords : undecidability * substructural logic * Robinson arithmetic
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Limited access
http://dx.doi.org/10.1017/S175502032000012X

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. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional 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.
Permanent Link: http://hdl.handle.net/11104/0307787