Abstract
Let \(\mathsf{{G}}_\downarrow \) be the Gödel logic whose set of truth values is 
. Baaz-Leitsch-Zach have shown that \(\mathsf{{G}}_\downarrow \) is not recursively axiomatizable and Hájek showed that it is not arithmetical. We find the optimal strengthening of their theorems and prove that the set of validities of \(\mathsf{{G}}_\downarrow \) is \(\varPi ^1_1\) complete and the set of satisfiable formulas in \(\mathsf{{G}}_\downarrow \) is \(\varSigma ^1_1\) complete.
J.P. Aguilera—Supported by FWF grant I4513N and FWO grant 3E017319.
J. Bydzovsky—Supported by FWF grant P31955 and I4427.
J.P. Aguilera and D. Fernández-Duque—Supported by FWO-FWF Lead Agency Grant G030620N.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Baaz, M., Leitsch, A., Zach, R.: Incompleteness of a first-order Gödel logic and some temporal logics of programs. In: Proceedings of CSL 1995, vol. 68 (1996)
Baaz, M., Preining, N., Zach, R.: Completeness of a hypersequent calculus for some first-order Godel logics with delta. In: ISMVL, p. 9. IEEE Computer Society (2006)
Barwise, J.: Admissible Sets and Structures. Perspectives in Mathematical Logic. Springer, Heidelberg (1975)
Barwise, K.J., Gandy, R., Moschovakis, Y.N.: The next admissible set. J. Symb. Log. 36, 108–120 (1971)
Dummett, M.: A propositional calculus with denumerable matrix. J. Symb. Log. 24, 97–106 (1959)
Gödel, K.: Zum intuitionistischen aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien 69, 65–66 (1932)
Hájek, P.: Fuzzy logic and the arithmetical hierarchy, III. Studia Log. 68, 129–142 (2001). https://doi.org/10.1023/A:1011906423560
Hájek, P.: A non-arithmetical Gödel logic. Log. J. IGPL 13, 435–441 (2005)
Kröger, F.: Temporal Logic of Programs. Monographs in Theoretical Computer Science (EATCS Series), vol. 8. Springer, Berlin (1987). https://doi.org/10.1007/978-3-642-71549-5
Montagna, F.: Three complexity problems in quantified fuzzy logic. Studia Log. 68, 143–152 (2001). https://doi.org/10.1023/A:1011958407631
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Aguilera, J.P., Bydzovsky, J., Fernández-Duque, D. (2022). A Non-hyperarithmetical Gödel Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2022. Lecture Notes in Computer Science(), vol 13137. Springer, Cham. https://doi.org/10.1007/978-3-030-93100-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-93100-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-93099-8
Online ISBN: 978-3-030-93100-1
eBook Packages: Computer ScienceComputer Science (R0)