Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

Siegfried Gottwald
2006 Studia Logica: An International Journal for Symbolic Logic  
For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical) survey of quite a lot
more » ... of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forthcoming Part II will be devoted to category theoretic approaches. Keywords: fuzzy sets, higher level fuzzy sets, set theoretic universes, axiomatic set theories, categories of fuzzy sets, M -valued sets * This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. Mostowski, H. Rasiowa and C. Rauszer, and held in Warsaw and Ruciane-Nida from 23rd to
doi:10.1007/s11225-006-7197-8 fatcat:vxo5u3lvy5g53ewkj5bdddfa6y