Fuzzy Sets and Fuzzy Classes in Universes of Sets Michal Holˇ capek 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. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 1 / 54
Outline Motivation 1 Universes of sets over L 2 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 Graded equipollence of fuzzy sets in F ( U ) 4 Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence Conclusion 5 M. Holˇ capek (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. Holˇ capek (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) Bˇ ehounek–Cintula axiomatic fuzzy class theory (FCT) M. Holˇ capek (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) Bˇ ehounek–Cintula axiomatic fuzzy class theory (FCT) M. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 4 / 54
Outline Motivation 1 Universes of sets over L 2 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 Graded equipollence of fuzzy sets in F ( U ) 4 Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence Conclusion 5 M. Holˇ capek (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 = � L , ∧ , ∨ , → , ⊗ , ⊥ , ⊤� with four binary operations and two constants such that � L , ∧ , ∨ , ⊥ , ⊤� is a bounded lattice, 1 � L , ⊗ , ⊤� is a commutative monoid and 2 the adjointness property is satisfied, i.e., 3 a ≤ b → c iff a ⊗ b ≤ c holds for each a , b , c ∈ L . Our prerequisite In our theory, we assume that each residuated lattice is complete and linearly ordered. M. Holˇ capek (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 = � L , ∧ , ∨ , → , ⊗ , ⊥ , ⊤� with four binary operations and two constants such that � L , ∧ , ∨ , ⊥ , ⊤� is a bounded lattice, 1 � L , ⊗ , ⊤� is a commutative monoid and 2 the adjointness property is satisfied, i.e., 3 a ≤ b → c iff a ⊗ b ≤ c holds for each a , b , c ∈ L . Our prerequisite In our theory, we assume that each residuated lattice is complete and linearly ordered. M. Holˇ capek (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 = � [ 0 , 1 ] , min , max , T , → T , 0 , 1 � , where α → T β = � { γ ∈ [ 0 , 1 ] | T ( α, γ ) ≤ β } , is a complete linearly ordered residuated lattice. E.g., Łukasiewicz algebra is determined by T L ( a , b ) = max ( 0 , a + b − 1 ) . The residuum is then given by a → T L b = min ( 1 − a + b , 1 ) . M. Holˇ capek (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 ∈ y and y ∈ U , then x ∈ U , (U2) x , y ∈ U , then { x , y } ∈ U , (U3) x ∈ U , then P ( x ) ∈ U , (U4) x ∈ U and y i ∈ U for any i ∈ x , then � i ∈ x y i ∈ U , (U5) x ∈ U and f : x → L , then R ( f ) ∈ U , where L is the support of the residuated lattice L . M. Holˇ capek (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. Holˇ capek (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 ∈ 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 �∈ U . M. Holˇ capek (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 ∈ 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 �∈ U . M. Holˇ capek (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 ∈ 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 �∈ U . M. Holˇ capek (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 ∈ 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 �∈ U . M. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 10 / 54
Basic properties Theorem Let x , y ∈ U and y i ∈ U for any i ∈ x . Then we have ∅ and { x } belong to U , 1 x × y , x ⊔ y , x ∩ y and y x belong to U , 2 if z ∈ U ∪ { L } and f : x → z , then f and R ( f ) belong to U , 3 if z ⊆ U and | z | ≤ | x | , then z belongs to U , 4 � i ∈ x y i , � i ∈ x y i and � i ∈ x y i belong to U . 5 M. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 11 / 54
Basic properties Theorem Let x , y ∈ U and y i ∈ U for any i ∈ x . Then we have ∅ and { x } belong to U , 1 x × y , x ⊔ y , x ∩ y and y x belong to U , 2 if z ∈ U ∪ { L } and f : x → z , then f and R ( f ) belong to U , 3 if z ⊆ U and | z | ≤ | x | , then z belongs to U , 4 � i ∈ x y i , � i ∈ x y i and � i ∈ x y i belong to U . 5 Theorem (Extensibility of sets in U ) Let x ∈ U . Then there exists y ∈ U such that | x | ≤ | y \ x | . M. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 11 / 54
Outline Motivation 1 Universes of sets over L 2 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 Graded equipollence of fuzzy sets in F ( U ) 4 Graded Cantor’s equipollence Elementary cardinal theory based on graded Cantor’s equipollence Conclusion 5 M. Holˇ capek (IRAFM) Fuzzy Sets and Fuzzy Classes in Universes of Sets Non–Classical Mathematics 12 / 54
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