Foundations of a set theory for fuzzy sets. 40 years of development

International Journal of General Systems

Volumn 37, Issue 1, 2008, Pages 69-81

  Gottwald, Siegfried ^a  

^a UNIVERSITY OF LEIPZIG   (Germany)

Author keywords

Axiom systems for fuzzy sets; Categories of fuzzy sets; Fuzzy set theory; Universes of fuzzy sets

Indexed keywords

PROBLEM SOLVING;

AXIOM SYSTEMS; CATEGORIES OF FUZZY SETS; FUZZY SET THEORY; UNIVERSES OF FUZZY SETS;

FUZZY SETS;

EID: 36448937526     PISSN: 03081079     EISSN: 15635104     Source Type: Journal    
DOI: 10.1080/03081070701499922     Document Type: Article

References (48)

1. Běhounek, L. and Cintula, P., 2005. Fuzzy class theory. Fuzzy Sets and Systems, 154 (1), 34-55.
2. Bell, J.L., 1977. Boolean-Valued Models and Independence Proofs in Set Theory, Oxford: Oxford University Press.
3. Blizard, W.D., 1989. Real-valued multisets and fuzzy sets. Fuzzy Sets and Systems, 33 (1), 77-97.
4. Brown, J.G., 1971. A note on fuzzy sets. Information and Control, 18 (1), 32-39.
5. Chapin, E.W. Jr., 1974. Set-valued set theory. I. Notre Dame Journal of Formal Logic, 15, 619-634.
6. Chapin, E.W. Jr., 1975. Set-valued set theory. II. Notre Dame Journal of Formal Logic, 16, 255-267.
7. Cintula, P., 2005, From Fuzzy Logic to Fuzzy Mathematics. PhD Thesis. Prague Technical University.
8. Demirci, M. and Çoker, D., 1993. On the axiomatic theory of fuzzy sets. Fuzzy Sets and Systems, 60, 181-198.
9. Eytan, M., 1981. Fuzzy sets: a topos-logical point of view. Fuzzy Sets and Systems, 5, 47-67.
10. Fourman, M.P. and Scott, D.S., 1979. Sheaves and logic. Applications of Sheaves. Berlin: Springer, 302-401.
11. Fraenkel, A.A., Bar-Hillel, Y. and Levy, A., 1973. Foundations of Set Theory. Amsterdam.: North Holland Publ. Comp.
12. Goguen, J.A., 1967. L-fuzzy sets. Journal of Mathematical Analysis and Applications, 18, 145-174.
13. Goguen, I.A., 1974. Concept respresentation in natural and artificial languages: axioms, extensions and applications for fuzzy sets. International Journal of Man-Machine Studies, 6, 513-561.
14. Gottwald, S., 1979. Set theory for fuzzy sets of higher level. Fuzzy Sets and Systems, 2, 125-151.
15. Gottwald, S., 1980. Fuzzy uniqueness of fuzzy mappings. Fuzzy Sets and Systems, 3 (1980), 49-74.
16. Gottwald, S., 1993. Fuzzy Sets and Fuzzy Logic. Vieweg: Braunschweig/Wiesbaden and Tecnea: Toulouse.
17. Gottwald, S., 2001. A. Treatise on Many-Valued Logics. Baldock, GB: Research Studies Press.
18. Gottwald, S., 2003. Universes of fuzzy sets - a short survey. Proceedings 33rd Internat. Symp. Multiple-Valued Logic. Tokyo 2003, Los Alamitos: IEEE Comp. Soc., 71-76.
19. Hájek, P., 2005. On arithmetic in the Cantor-Lukasiewicz fuzzy set theory. Archive Mathematical Logic, 44, 763-782.
20. Hájek, P. and Haniková, Z., 2001. A set theory within fuzzy logic. Proc. 31st ISMVL., Warsaw 2001, Los Alamitos/CA: IEEE Comp. Soc, 319-323.
21. Higgs, D., 1973, A category approach to Boolean-valued set theory, Preprint, University of Waterloo.
22. Höhle, U., 1992. M-valued sets and sheaves over integral commutative CL-monoids. Applications of Category Theory to Fuzzy Subsets. Dordrecht: Kluwer, 33-72.
23. Höhle, U., 1995a. Commutative, residuated 1-monoids. Non-Classical Logics and Their Applications to Fuzzy Subsets. Dordrecht: Kluwer, 53-106.
24. Höhle, U., 1995b. Presheaves over GL-monoids. Non-Classical Logics and Their Applications to Fuzzy Subsets. Dordrecht: Kluwer, 127-157.
25. Höhle, U., 2000. Classification of subsheaves over GL-algebras. Logic Colloquium '98. Natick, MA: K.A. Peters, 238-261.
26. Horčik, R. and Cintula, P., 2004. Product Lukasiewicz logic. Archive Mathe-matical Logic, 43, 477-503.
27. Klaua, D., 1965. Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, 7, 859-867.
28. Klaua, D., 1966a. Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, 8, 161-177.
29. Klaua, D., 1966b. Grundbegriffe einer mehrwertigen Mengenlehre. Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, 8, 782-802.
30. Lake, J., 1976. Sets, fuzzy sets, multisets and functions. Journal of London Mathematical Society, II. Serial, 12, 323-326.
31. Lawvere, F.W., 1964. An elementary theory of the category of sets. Proceedings of National Academy of Sciences USA, 52, 1506-1511.
32. Novák, V., 1980. An attempt at Godel-Bernays-like axiomatization of fuzzy sets. Fuzzy Sets Systems, 3, 323-325.
33. Prati, N., 1988. An axiomatization of fuzzy classes. Stochastica, 12, 65-78.
34. Prati, N., 1991. About the axiomatizations of fuzzy set theory. Fuzzy Sets and Systems, 39, 101-109.
35. Prati, N., 1992. On the comparison between fuzzy set axiomatizations. Fuzzy Sets and Systems, 46, 167-175.
36. Scott, D.S., 1979. Identity and existence in intuitionistic logic Applications of Sheaves. New York: Springer, 660-696.
37. Skolem, Th., 1957. Bemerkungen zum komprehensionsaxiom. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 1-17.
38. Takeuti, G. and Titani, S., 1984. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic, 49, 851-866.
39. Takeuti, G. and Titani, S., 1986. The Mathematics of Fuzzy Systems. Köln: TÜV Rheinland, 291-301.
40. Takeuti, G. and Titani, S., 1992. Fuzzy logic and fuzzy set theory. Archive Mathematical Logic, 32, 1-32.
41. Tamir, D.E., Cao, Z.-Q., Kandel, A. and Mott, J.L., 1990. An axiomatic approach to fuzzy set theory. Information Sciences, 52, 75-83.
42. Toth, H., 1993. Axiomatic f-set theory I: basic concepts. Journal of Fuzzy Mathematics, 1, 109-135.
43. Weidner, A.J., 1974, An axiomatization of fuzzy set theory. PhD Thesis. Univ. Notre Dame.
44. Weidner, A.J., 1981. Fuzzy sets and Boolean-valued universes. Fuzzy Sets and Systems, 6, 61-72.
45. Wyler, O., 1995. Fuzzy logic and categories of fuzzy sets. Non-Classical Logics and Their Applications to Fuzzy Subsets. Dordrecht: Kluwer, 235-268.
46. Zadeh, L.A., 1965a. Fuzzy sets. Information and Control, 8, 338-353.
47. Zadeh, L.A., 1965b. Fuzzy sets and systems. In: J.Fox, ed. System Theory. New York: Polytechnic Press, 29-39.
48. Zhang, J.-W., 1980. A unified treatment of fuzzy set theory and Boolean-valued set theory - fuzzy set structures and normal fuzzy set structures. Journal of Mathematical Analysis and Applications, 76, 297-301.