Abstract
This paper presents an axiomatic set theory FST (‘Fuzzy Set Theory’), as a first-order theory within the framework of fuzzy logic in the style of [4]. In the classical ZFC, we use a construction similar to that of a Boolean-valued universe—over an algebra of truth values of the logic we use—to show the nontriviality of FST. We give the axioms of FST. Finally we show that FST interprets ZF.
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
Preview
Unable to display preview. Download preview PDF.
References
Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic Fuzzy Logic is the logic of continuous t-norms and their residua, Soft Computing 4 (2000) 106–112
Grayson, R.J.: Heyting-valued models for intuitionistic set theory, Application of Sheaves: Lect. Notes Math. 753 (1979) 402–414
Grishin, V.N.: O vese aksiomy svertyvania v teorii, osnovannoj na logike bez sokrashchenij, Matematicheskije zametki 66, no. 5 (1999) 643–652
Hâjek, P.: Metamathematics of Fuzzy Logic, Kluwer Academic Publishers (1998)
Hâjek, P.: Function symbols in fuzzy predicate logic, Proc. East-West Fuzzy Colloq. Zittau (2000)
Hâjek, P., Hanikovâ, Z.: A set theory within fuzzy logic, Proc. 31st IEEE ISMVL Warsaw (2001) 319–324
Powell, W. C.: Extending Gödel’s negative interpretation to ZF, J. Symb. Logic 40, no. 2 (1975) 221–229
Shirahata, M.: Phase-valued models of linear set theory, preprint (1999)
Takeuti, G., Titani, S.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, J. Symb. Logic 49 (1984) 851–866
Takeuti, G., Titani, S.: Fuzzy logic and fuzzy set theory, Arch. Math. Logic 32 (1992) 1–32
Titani, S.: A lattice-valued set theory, Arch. Math. Logic 38 (1999) 395–421
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hájek, P., Haniková, Z. (2003). A Development of Set Theory in Fuzzy Logic. In: Fitting, M., Orłowska, E. (eds) Beyond Two: Theory and Applications of Multiple-Valued Logic. Studies in Fuzziness and Soft Computing, vol 114. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1769-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-7908-1769-0_12
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-2522-0
Online ISBN: 978-3-7908-1769-0
eBook Packages: Springer Book Archive