Abstract
Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic. But we cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.
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References
Cantini, A.: The undecidability of Grishin’s set theory. Studia Logica 74, 345–368 (2003)
Chang, C.C.: The axiom of comprehension in infinite valued logic. Math. Scand 93, 9–30 (1963)
Fenstad, J.E.: On the consistency of the axiom of comprehensionin the Lukasiewicz infinite valued logic. Math. Scand 14, 65–74 (1964)
Gottwald, S.: Untersuchungen zur mehrwertigen Mengenlehre I. Math. Nachrichten 72, 297–303 (1976)
Gottwald, S.: Untersuchungen zur mehrwertigen Mengenlehre II. Math. Nachrichten 74, 329–336 (1976)
Gottwald, S.: Untersuchungen zur mehrwertigen Mengenlehre III. Math. Nachrichten 79, 207–217 (1977)
Grayson, R.J.: Heyting-valued models for intuitionistic set theory, Application of Sheaves. Lect. Notes Math. 753 402–414 (1979)
Grishin, V.H.: Ob odnoi nestandartnoi logike i ee primenenii k teorii mnozhestv. In: Issledovaniya po formalnym yazykom i neklassicheskim logikam. Nauka Moscow 135–171 (1974)
Grishin, V.H.: Ob algebraicheskoi semantike logiki lez sokrastrchenii. In: Issled. po form. logike i neklass. logikam. Nauka Moscow 1978, pp. 247–265
Grishin, V.H.: Predikatnye i teoretiko-mnozhestvennye ischislenia osnovannye na logike bez sokrashchenia. Izv. AN SSSR 18 (1), 47–68 (1981): English translation, Math. USSR Izvestiya 18, 41–59 (1982)
Grishin, V.N.: O vese aksiomy svertyvania v teorii, osnovannoj na logike bez sokrashchenij. Matematicheskije zametki 66 (5), 643–652 (1999)
Hájek, P.: Metamathematics of fuzzy logic, Kluwer 1998
Hájek, P.: Function symbols in fuzzy predicate logic. In: Proc. East West Fuzzy Logic Catalog. 2000, Zittau, pp. 2–8
Hájek, P.: Mathematical fuzzy logic and set theory developed in it. In: Proc. Takeuti Symposium, Kobe, Japan 2003, pp. 58–69
Hájek, P., Haniková, Z.: A set theory within fuzzy logic, Proc. 31st IEEE ISMVL Warsaw 2001, pp. 319–324
Hájek, P., Haniková, Z.: A Development of Set Theory in Fuzzy Logic. In: Beyond Two: Theory and Applications of Multiple-Valued Logic (Fitting, M., Orlowska, E. (eds.)) - Heidelberg, Physica-Verlag 2003, pp. 273–285
Hájek, P., Paris, J., Shepherdson, J.: The liar paradox and fuzzy logic. Journ. Symb. Logic 65, 339–346 (2000)
Hájek, P., Pudlák, P.: Metamathematics of first-order arithmetic. Springer-Verlag 1993
Klaua, D.: Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–867 (1965)
Klaua, D.: Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 8, 782–802 (1966)
Klaua, D.: Ein Ansatz zur mehrwertigen Mengenlehre. Math. Nachr. 33, 273–295 (1967)
Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, 2000
Ragaz, M. E.: Arithmetische Klassifikation von Formelnmengen der unendlichwertigen Logik. Thesis, ETH Zürich, 1981
Restall, G.: Arithmetic and truth in Lukasiewicz’s infinitely valued logic. Logique et Analyse 36, 25–38 (1993) (published 1995)
Rogers, H. Jr.: Theory of recursive functions and effective computability. McGraw-Hill, 1967
Shirahata, M.: Phase-valued models of linear set theory, preprint, 1999
Shirahata, M.: Linear set theory with strict comprehension. In: Proc. Asian Logic Conf., World Scientific 1996, pp. 223–245
Shirahata, M.: A linear conservative extension of Zermelo-Fraenkel set theory. Studia Logica 56, 361–392 (1996)
Shirahata, M.: A coherence space semantics for linear set theory. Preprint, 1999
Shirahata, M.: Fixpoint theorem in linear set theory. Preprint 1999
Skolem, T.: Bemerkungen zum Komprehensionsaxiom. Zeitchr. f. Math. Logik uns Grundl. der Math. 3, 1–17 (1957)
Takeuti, G., Titani, S.: Globalization of intuitionistic set theory. Annals Pure Appl Logic 33, 195–211 (1987)
Takeuti, G., Titani, S.: Heyting valued universes of intuitionistic set theory. In: (Müller, G. H. et al., (ed.)) Logic Symposia Hakone 1979, 1980, Springer-Verlag 1981, pp. 189–306
Takeuti, G., Titani, S.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Logic 49, 851–866 (1984)
Takeuti, G., Titani, S.: Fuzzy logic and fuzzy set theory. Arch. Math. Logic 32, 1–32 (1992)
Terui, K.: Light affine set theory: a naive set theory of polynomial time. to appear in Studia Logica 2003
Titani, S.: Completeness of global intuitionistic set theory. Journ. Symb. Logic 62, 605–627 (1997)
Titani, S.: A lattice-valued set theory. Arch. Math. Logic 38, 395–421 (1999)
White, R. B.: The consistency of the axiom of comprehension in the infinite valued logic of Łukasiewicz. Journ. Phil. Logic 8, 502–537 (1979)
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Long before them, Klaua and Gottwald studied various forms of iterated fuzzy power set constructions inside classical set theory, see the references.
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Hájek, P. On arithmetic in the Cantor- Łukasiewicz fuzzy set theory. Arch. Math. Logic 44, 763–782 (2005). https://doi.org/10.1007/s00153-005-0284-0
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DOI: https://doi.org/10.1007/s00153-005-0284-0