Fuzzy Sets and Fuzzy Classes in Universes of Sets Michal Holcapekˇ National Supercomputing Center IT4Innovations division of the University of Ostrava Institute for Research and Applications of Fuzzy Modeling web: irafm.osu.cz Prague seminar on Non–Classical Mathematics, June 13, 2015 M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 1 / 54 Outline 1 Motivation 2 Universes of sets over L 3 Fuzzy sets and fuzzy classes in U Concept of fuzzy sets in U Basic relations and operations in F(U) Functions between fuzzy sets Fuzzy power set and exponentiation Concept of fuzzy class in U Basic graded relations between fuzzy sets Functions between fuzzy sets in a certain degree 4 Graded equipollence of fuzzy sets in F(U) Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence 5 Conclusion M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 2 / 54 A poor interest about cardinal theory of fuzzy sets S. Gottwald. Fuzzy uniqueness of fuzzy mappings. Fuzzy Sets and Systems, 3:49–74, 1980. M. Wygralak. Vaguely defined objects. Representations, fuzzy sets and nonclassical cardinality theory. Theory and Decision Library. Series B: Mathematical and Statistical Methods, Kluwer Academic Publisher, 1996. M. Wygralak. Cardinalities of Fuzzy Sets. Kluwer Academic Publisher, Berlin, 2003. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 3 / 54 Set and fuzzy set theories Zermelo–Fraenkel set theory with the axiom of choice (ZFC) - sets are introduced formally, classes are introduced informally; von Neumann–Bernays–Gödel axiomatic set theory (NBG) - classes are introduced formally, sets are special classes (difference between sets and proper classes is essential) type theory Gotwald cumulative system of fuzzy sets Novak axiomatic fuzzy type theory (FTT) Behounek–Cintulaˇ axiomatic fuzzy class theory (FCT) M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 4 / 54 Set and fuzzy set theories Zermelo–Fraenkel set theory with the axiom of choice (ZFC) - sets are introduced formally, classes are introduced informally; von Neumann–Bernays–Gödel axiomatic set theory (NBG) - classes are introduced formally, sets are special classes (difference between sets and proper classes is essential) type theory Gotwald cumulative system of fuzzy sets Novak axiomatic fuzzy type theory (FTT) Behounek–Cintulaˇ axiomatic fuzzy class theory (FCT) M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 4 / 54 Outline 1 Motivation 2 Universes of sets over L 3 Fuzzy sets and fuzzy classes in U Concept of fuzzy sets in U Basic relations and operations in F(U) Functions between fuzzy sets Fuzzy power set and exponentiation Concept of fuzzy class in U Basic graded relations between fuzzy sets Functions between fuzzy sets in a certain degree 4 Graded equipollence of fuzzy sets in F(U) Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence 5 Conclusion M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 5 / 54 Degrees of membership Definition A residuated lattice is an algebra L = hL; ^; _; !; ⊗; ?; >i with four binary operations and two constants such that 1 hL; ^; _; ?; >i is a bounded lattice, 2 hL; ⊗; >i is a commutative monoid and 3 the adjointness property is satisfied, i.e., a ≤ b ! c iff a ⊗ b ≤ c holds for each a; b; c 2 L. Our prerequisite In our theory, we assume that each residuated lattice is complete and linearly ordered. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 6 / 54 Degrees of membership Definition A residuated lattice is an algebra L = hL; ^; _; !; ⊗; ?; >i with four binary operations and two constants such that 1 hL; ^; _; ?; >i is a bounded lattice, 2 hL; ⊗; >i is a commutative monoid and 3 the adjointness property is satisfied, i.e., a ≤ b ! c iff a ⊗ b ≤ c holds for each a; b; c 2 L. Our prerequisite In our theory, we assume that each residuated lattice is complete and linearly ordered. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 6 / 54 Example of linearly ordered residuated lattice Example Let T be a left continuous t-norm. Then L = h[0; 1]; min; max; T; !T ; 0; 1i; W where α !T β = fγ 2 [0; 1] j T(α; γ) ≤ βg, is a complete linearly ordered residuated lattice. E.g., Łukasiewicz algebra is determined by TL(a; b) = max(0; a + b − 1): The residuum is then given by a !TL b = min(1 − a + b; 1): M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 7 / 54 Universe of sets motivated by Grothendieck Definition A universe of sets over L is a non-empty class U of sets in ZFC satisfying the following properties: (U1) x 2 y and y 2 U, then x 2 U, (U2) x; y 2 U, then fx; yg 2 U, (U3) x 2 U, then P(x) 2 U, S (U4) x 2 U and yi 2 U for any i 2 x, then i2x yi 2 U, (U5) x 2 U and f : x ! L, then R(f ) 2 U, where L is the support of the residuated lattice L. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 8 / 54 Examples Universes of sets over L class of all sets, class of all finite sets, Grothendieck universes (suitable sets of sets). M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 9 / 54 Sets and classes in U In ZFC, we have sets (introduced by axioms) classes (introduced informally as collections of sets) Definition Let U be a universe of sets over L. We say that a set x in ZFC is a set in U if x 2 U, a class x in ZFC is a class in U if x ⊆ U, a class x in U is a proper class in U if x 62 U. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 10 / 54 Sets and classes in U In ZFC, we have sets (introduced by axioms) classes (introduced informally as collections of sets) Definition Let U be a universe of sets over L. We say that a set x in ZFC is a set in U if x 2 U, a class x in ZFC is a class in U if x ⊆ U, a class x in U is a proper class in U if x 62 U. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 10 / 54 Sets and classes in U In ZFC, we have sets (introduced by axioms) classes (introduced informally as collections of sets) Definition Let U be a universe of sets over L. We say that a set x in ZFC is a set in U if x 2 U, a class x in ZFC is a class in U if x ⊆ U, a class x in U is a proper class in U if x 62 U. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 10 / 54 Sets and classes in U In ZFC, we have sets (introduced by axioms) classes (introduced informally as collections of sets) Definition Let U be a universe of sets over L. We say that a set x in ZFC is a set in U if x 2 U, a class x in ZFC is a class in U if x ⊆ U, a class x in U is a proper class in U if x 62 U. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 10 / 54 Theorem (Extensibility of sets in U) Let x 2 U. Then there exists y 2 U such that jxj ≤ jy n xj. Basic properties Theorem Let x; y 2 U and yi 2 U for any i 2 x. Then we have 1 ; and fxg belong to U, 2 x × y, x t y, x \ y and yx belong to U, 3 if z 2 U [ fLg and f : x ! z, then f and R(f ) belong to U, 4 if z ⊆ U and jzj ≤ jxj, then z belongs to U, 5 Q F T i2x yi, i2x yi and i2x yi belong to U. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 11 / 54 Basic properties Theorem Let x; y 2 U and yi 2 U for any i 2 x. Then we have 1 ; and fxg belong to U, 2 x × y, x t y, x \ y and yx belong to U, 3 if z 2 U [ fLg and f : x ! z, then f and R(f ) belong to U, 4 if z ⊆ U and jzj ≤ jxj, then z belongs to U, 5 Q F T i2x yi, i2x yi and i2x yi belong to U. Theorem (Extensibility of sets in U) Let x 2 U. Then there exists y 2 U such that jxj ≤ jy n xj. M. Holcapekˇ (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 11 / 54 Outline 1 Motivation 2 Universes of sets over L 3 Fuzzy sets and fuzzy classes in U Concept of fuzzy sets in U Basic relations and operations in F(U) Functions between fuzzy sets Fuzzy power set and exponentiation Concept of fuzzy class in U Basic graded relations between fuzzy sets Functions between fuzzy sets in a certain degree 4 Graded equipollence of fuzzy sets in F(U) Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence 5 Conclusion M.