Noetherian Gödel logics

Abstract

Abstract We introduce Noetherian Gödel logics, Gödel logics where the set of truth values is a closed subset of $[0,1]$ containing $0$ and $1$ and without any infinite ascending sequences. There are infinitely many such logics, including the well-known logic $\textsf {G}^\downarrow $ whose set of truth values is $T_\downarrow = \{0\}\cup \{1/n:n\in \mathbb {N}\setminus \{0\}\}$. We compute the complexity of satisfiability and validity for each Noetherian Gödel logic and, in particular, in the logic $\textsf {G}^\downarrow $. This yields optimal strengthening of the results of Baaz–Leitsch–Zach and Hájek