Studia Logica

, Volume 82, Issue 2, pp 211–244 | Cite as

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

  • Siegfried GottwaldEmail author


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 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 forth-coming Part II will be devoted to category theoretic approaches.


fuzzy sets higher level fuzzy sets set theoretic universes axiomatic set theories categories of fuzzy sets M-valued sets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    ATANASSOV, K., Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, 1999.Google Scholar
  2. [2]
    BAAZ, M., ‘Infinite-valued Gödel logics with 0-1-projections and relativizations’, in P. Hájek, (ed.), Gödel’96, Lect. Notes Logic 6, Springer, Berlin, 1996, pp. 23–33.Google Scholar
  3. [3]
    BĚHOUNEK, L., AND P. CINTULA, ‘Fuzzy class theory’, Fuzzy Sets Syst., 154:34–55, 2005.Google Scholar
  4. [4]
    BELL, J. L., Boolean-valued models and independence proofs in set theory. Oxford Logic Guides 12. Oxford University Press (Clarendon Press), Oxford, 2nd edition, 1985.Google Scholar
  5. [5]
    BLIZARD, W. D., Real-valued multisets and fuzzy sets, Fuzzy Sets Syst., 33:77–97, 1989.CrossRefGoogle Scholar
  6. [6]
    BOTTA, O., AND M. DELORME, An f-sets universe \({\tilde V}\), in M. M. Gupta and E. Sanchez, (eds.), Fuzzy Information and Decision Processes, North-Holland, Amsterdam, 1982, pp. 133–141.Google Scholar
  7. [7]
    BROWN, J. G., ‘A note on fuzzy sets’, Information and Control, 18:32–39, 1971.CrossRefGoogle Scholar
  8. [8]
    CHAPIN, Jr, E. W., ‘Set-valued set theory. I’, Notre Dame J. Formal Logic, 15:619–634, 1974.CrossRefGoogle Scholar
  9. [9]
    CHAPIN, Jr, E. W., ‘Set-valued set theory. II’, Notre Dame J. Formal Logic, 16:255–267, 1975.CrossRefGoogle Scholar
  10. [10]
    CINTULA, P., AND P. HÁJEK, ‘Triangular norm based predicate fuzzy logics’, in Proc. Linz Seminar Fuzzy Set Theory 2005. (to appear).Google Scholar
  11. [11]
    DEMIRCI, M., AND D. ÇOKER, ‘On the axiomatic theory of fuzzy sets’, Fuzzy Sets Syst., 60:181–198, 1993.CrossRefGoogle Scholar
  12. [12]
    DUBOIS D., et al., ‘Terminological dificulties in fuzzy set theory— The case of “Intuitionistic Fuzzy Sets”’, Fuzzy Sets Syst., 156:485–491, 2005.Google Scholar
  13. [13]
    GOTTWALD, S., ‘A cumulative system of fuzzy sets’, in A. Zarach, (ed.), ‘Set Theory Hierarchy Theory’, Mem. Tribute A. Mostowski, Bierutowice 1975, Lect. Notes Math. 537:109–119, Springer, Berlin, 1976.Google Scholar
  14. [14]
    GOTTWALD, S., ‘Untersuchungen zur mehrwertigen Mengenlehre. I, Math. Nachr., 72:297–303, 1976.Google Scholar
  15. [15]
    GOTTWALD, S., ‘Set theory for fuzzy sets of higher level’, Fuzzy Sets Syst., 2:125–151, 1979.CrossRefGoogle Scholar
  16. [16]
    GOTTWALD, S., ‘Fuzzy uniqueness of fuzzy mappings’, Fuzzy Sets Syst., 3:49–74, 1980.Google Scholar
  17. [17]
    GOTTWALD, S., ‘A generalized Lukasiewicz-style identity logic’, in L. P. de Alcantara, (ed.), Mathematical Logic and Formal Systems, Lecture Notes Pure Appl. Math. 94:183–195, Marcel Dekker, New York, 1985. Coll. Pap. Hon. N. C. A. da Costa.Google Scholar
  18. [18]
    GOTTWALD, S., Fuzzy sets and fuzzy logic. The foundations of application – from a mathematical point of view. Artificial Intelligence. Vieweg, Braunschweig/Wiesbaden, 1993.Google Scholar
  19. [19]
    GOTTWALD, S., A treatise on many-valued logics. Studies in Logic and Computation. 9. Research Studies Press, Baldock, 2001.Google Scholar
  20. [20]
    GOTTWALD, S., and P. Hájek, ‘T-norm based mathematical fuzzy logics’, in E. P. Klement and R. Mesiar, (eds), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, Elsevier, Dordrecht, 2005, pp. 275–299.Google Scholar
  21. [21]
    HÁJEK, P., Metamathematics of Fuzzy Logic. Trends in Logic. 4. Kluwer Acad. Publ., Dordrecht, 1998.Google Scholar
  22. [22]
    HÁJEK, P., ‘On arithmetic in the Cantor-Lukasiewicz fuzzy set theory’, Arch. Math. Logic, 44:763–782, 2005.Google Scholar
  23. [23]
    HÁJEK, P., AND Z. HANIKOVÁ, ‘A set theory within fuzzy logic’, in 31st IEEE Internat. Symp. Multiple-Valued Logic, Warsaw, 2001, pp. 319–323. IEEE Computer Soc., Los Alamitos/Ca, 2001.Google Scholar
  24. [24]
    HÁJEK, P., AND Z. HANIKOVÁ, ‘A development of set theory in fuzzy logic’, in M. Fitting and E. Orlowska, (eds), Beyond Two, Studies in Soft Computing. Physica-Verlag, Heidelberg 2003, pp. 273–285.Google Scholar
  25. [25]
    JAHN, K. U., Über einen Ansatz zur mehrwertigen Mengenlehre unter Zulassung kontinuumvieler Wahrheitswerte. Master's thesis, Universität Leipzig, 1969.Google Scholar
  26. [26]
    KLAUA, D., ‘Über einen Ansatz zur mehrwertigen Mengenlehre’, Monatsber. Deutsch. Akad. Wiss. Berlin, 7:859–867, 1965.Google Scholar
  27. [27]
    KLAUA, D., ‘Grundbegrifie einer mehrwertigen Mengenlehre’, Monatsber. Deutsch. Akad. Wiss. Berlin, 8:782–802, 1966.Google Scholar
  28. [28]
    KLAUA, D., ‘Über einen zweiten Ansatz zur mehrwertigen Mengenlehre’, Monatsber. Deutsch. Akad. Wiss. Berlin, 8:161–177, 1966.Google Scholar
  29. [29]
    KLAUA, D., ‘Ein Ansatz zur mehrwertigen Mengenlehre’, Math. Nachr., 33:273–296, 1967.Google Scholar
  30. [30]
    KLAUA, D., ‘Stetige Gleichmächtigkeiten kontinuierlich-wertiger Mengen’, Monatsber. Deutsch. Akad. Wiss. Berlin, 12:749–758, 1970.Google Scholar
  31. [31]
    KLAUA, D., ‘Zum Kardinalzahlbegrifiin der mehrwertigen Mengenlehre’, in G. Asser, J. Flachsmeyer, and W. Rinow, (eds), Theory of Sets and Topology. Deutscher Verlag Wissensch., Berlin, 1972, pp. 313–325. Collection Papers Honour Felix Hausdorfi.Google Scholar
  32. [32]
    KLAUA, D., ‘Zur Arithmetik mehrwertiger Zahlen’, Math. Nachr., 57:275–306, 1973.Google Scholar
  33. [33]
    KLEENE, S. C., Introduction to Metamathematics, North-Holland Publ. Comp./van Nostrand, Amsterdam/New York, 1952.Google Scholar
  34. [34]
    LAKE, J., ‘Sets, fuzzy sets, multisets and functions’, J. London Math. Soc., II. Ser., 12:323–326, 1976.Google Scholar
  35. [35]
    LANO, K., Mathematical frameworks for vagueness and approximate reasoning. PhD thesis, Bristol University, 1988.Google Scholar
  36. [36]
    LANO, K., ‘Formal frameworks for approximate reasoning’, Fuzzy Sets Syst., 51:131–146, 1992.CrossRefGoogle Scholar
  37. [37]
    LANO, K., ‘Fuzzy sets and residuated logic’, Fuzzy Sets Syst., 47:203–220, 1992.CrossRefGoogle Scholar
  38. [38]
    NOVÁK, V., ‘An attempt at Gödel-Bernays-like axiomatization of fuzzy sets’, Fuzzy Sets Syst., 3:323–325, 1980.Google Scholar
  39. [39]
    NovÁK, V., ‘Fuzzy type theory’, Fuzzy Sets Syst., 149:235–273, 2004.Google Scholar
  40. [40]
    PRATI, N., ‘An axiomatization of fuzzy classes’, Stochastica, 12:65–78, 1988.Google Scholar
  41. [41]
    PRATI, N., ‘About the axiomatizations of fuzzy set theory’, Fuzzy Sets Syst., 39:101–109, 1991.CrossRefGoogle Scholar
  42. [42]
    PRATI, N., ‘On the comparison between fuzzy set axiomatizations’, Fuzzy Sets Syst., 46:167–175, 1992.CrossRefGoogle Scholar
  43. [43]
    RASIOWA, H., An Algebraic Approach to Non-Classical Logics. North-Holland Publ. Comp./PWN, Amsterdam/Warsaw, 1974.Google Scholar
  44. [44]
    SCHWARTZ, D., ‘Mengenlehre über vorgegebenen algebraischen Systemen’, Math. Nachr., 53:365–370, 1972.Google Scholar
  45. [45]
    SKOLEM, T., ‘Bemerkungen zum Komprehensionsaxiom’, Zeitschr. math. Logik Grundl. Math., 3:1–17, 1957.Google Scholar
  46. [46]
    TAKEUTI, G., AND S. TITANI, ‘Intuitionistic fuzzy logic and intuitionistic fuzzy set theory’, J. Symb. Logic, 49:851–866, 1984.Google Scholar
  47. [47]
    TAKEUTI, G., AND S. TITANI, ‘Global intuitionistic fuzzy set theory’, in The Mathematics of Fuzzy Systems, Interdisciplinary Syst. Res. 88:291–301. TÜV Rheinland, Köln (Cologne), 1986.Google Scholar
  48. [48]
    TAKEUTI, G., AND S. TITANI, ‘Fuzzy logic and fuzzy set theory’, Arch. Math. Logic, 32:1–32, 1992.CrossRefGoogle Scholar
  49. [49]
    TAMIR, D. E., Z.-Q. CAO, A. KANDEL, AND J. L. MOTT, ‘An axiomatic approach to fuzzy set theory’, Information Sci., 52:75–83, 1990.CrossRefGoogle Scholar
  50. [50]
    TITANI, S., ‘Completeness of global intuitionistic set theory’, J. Symb. Logic, 62:506–528, 1997.Google Scholar
  51. [51]
    TITANI, S., ‘Numbers in a lattice valued set theory’, RIMS Kokyuroku, 1010:174–192, 1997.Google Scholar
  52. [52]
    TITANI, S., ‘A lattice-valued set theory’, Arch. Math. Logic, 38:395–421, 1999.CrossRefGoogle Scholar
  53. [53]
    TOTH, H., ‘Axiomatic f-set theory. I: Basic concepts’, J. Fuzzy Math., 1:109–135, 1993.Google Scholar
  54. [54]
    WEIDNER, A. J., An axiomatization of fuzzy set theory, PhD thesis, Univ. Notre Dame, 1974.Google Scholar
  55. [55]
    WEIDNER, A. J., ‘Fuzzy sets and Boolean-valued universes’, Fuzzy Sets Syst., 6:61–72, 1981.CrossRefGoogle Scholar
  56. [56]
    WHITE, R. B., ‘The consistency of the axiom of comprehension in the infinite-valued predicate logic of Lukasiewicz’, J. Philosophical Logic, 8:509–534, 1979.Google Scholar
  57. [57]
    YASUGI, M., ‘Definitive valuation of set theory’, Comment. Math. Univ. St. Pauli, 30:175–191, 1981.Google Scholar
  58. [58]
    YASUGI, M., ‘Continuous valuation and logic’, Nagoya Math. J., 85:175–188, 1982.Google Scholar
  59. [59]
    ZADEH, L. A., ‘Fuzzy sets’, Information and Control, 8:338–353, 1965.CrossRefGoogle Scholar
  60. [60]
    ZHANG, JIN WEN, ‘The normal fuzzy set structures and the Boolean-valued models’, J. Huazhong Inst. Tech. (English Ed.), 2(1):1–9, 1980.Google Scholar
  61. [61]
    ZHANG, JIN WEN, ‘A unified treatment of fuzzy set theory and Boolean-valued set theory–fuzzy set structures and normal fuzzy set structures’, J. Math. Anal. Appl., 76:297–301, 1980.Google Scholar
  62. [62]
    ZHANG, JIN WEN, ‘Between fuzzy set theory and Boolean valued set theory’, in M. M. Gupta and E. Sanchez, (eds), Fuzzy Information and Decision Processes, North-Holland, Amsterdam, 1982, pp. 143–147.Google Scholar
  63. [63]
    ZHANG, JIN WEN, ‘Fuzzy set structure with strong implication’, in P. P. Wang, (ed.), Advances in Fuzzy Sets, Possibility Theory, and Applications, Plenum Press, New York, 1983, pp. 107–136.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute for Logic and Philosophy of ScienceLeipzig UniversityLeipzigGermany

Personalised recommendations