The standard Set/Member relation is based on a xor connective. For example: Any given x is (a member) xor (not a member) of set A and there is no intermediate state. Fuzzy logic expands standard membership (0 xor 1) by using x, which defines the degree of a membership between 0 and 1 (0 or 1 are included too). If 0<x<1, and [0,x] belongs to set A then (x,1] does not belong to set A ( [0,x] xor (x,1] ) So, in both cases a xor connective is used as the logical basis of the Set/Member relations. 0, 1, [0,x] or (x,1] (where 0<x<1) are all local mathematical objects because we can clearly define their locality (they are "in" xor "out" of some mathematical object). An object that is not a set but can be a member of a set, is called an urelemnt (http://72.14.221.104/search?q=cache...g/wiki/Urelement urelement&hl=en&ct=clnk&cd=1 ). A sub-object is a part of an object. Since an urelement is not a set, it does include any sub-object as a part of it, or in other words, it is an atomic singleton. x is a urelement. If x is a member and not a member of A then x is a non-local mathematical object. The best way to notate this is: _{_} , where __ is a urelement. __ can be both a local ( {_} xor __{ } ) or a non-local ( _{_} ) urelement. If we wish to find the best way to notate a local urelement (a local atomic singleton) than we use . (a point). An example: {.} xor .{ } In the standard Set/Member relations each member is a sub-object xor not a sub-object of a set, and both the empty and any a non-empty set are defined by the ways that these sub-objects belong xor don't belong to them. Since . and __ are atomic singletons, they are not defined by each other (they are mutually independent exactly like two axioms) and we can expand the membership concept beyond the Set/Member dependency. ------------------------------------------------------------------------- 1. Introduction The Membership concept needs logical foundations in order to be defined rigorously. Definition 1: A is a common property of x xor y. An example: If A = thing, x = some , y = no then A is the common property of xA(something) and yA(nothing). By definition 1 the Memberships' logical basis is not less than A and (x xor y). If we ignore A then the Memberships' logical basis between x,y is: a) x xor y (preventing each other). b) x and y (contradicting each other). By not ignoring A we define a mathematical universe that is based on at least two levels of logical connectives, which enable to expand the Membership concept beyond (a) (b). Furthermore, if A and (x xor y) then we may fulfill Hilbert's organic paradigm of the mathematical language. Quoting Hilbert’s famous Paris 1900 lecture: “…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.” 2. A context and an axiom Definition 2: An Axiom is a statement that is considered "self-proved", that is, a statement that is not dependent on the conditions of any other statement. Definition 3: A Product is any mathematical object that can formally be deduced from one or more given axioms. Definition 4: A Context is a "frame" through which one defines a product which is consistent with at least two axioms. Let A and B be axioms of context Z. Definition 5: If A and B are not consistent with context Z then they cannot be consistent with each other. If each given object is disjoint from any other object but itself, then the exact way to define it logicaly is by a xor connective, for example: Let A,B,C,... be a non-finite set of variables, where each variable is disjoint from the rest the given variables. For example, if C is examined, then since C is not disjoint from itself, it get 1 (a true value), where each variable that is not C, gets 0 (a false value) because it is disjoint from C. Now we wish to define logicaly, what binary relation is true when C is disjoint from rest of the list of given variables, so we get: C(true) xor C(true) --> F C(true) xor A(false) --> T C(true) xor B(false) --> T ... Let us return to my argument. Z is a place holder of the Context concept. A, B, C, ... are place holders of independent (disjoint) and self-consistent (each axiom is consistent, when compared to itself) mathematical objects (axioms). Z without A, B, C, ... (a context, without objects) is not an interesting mathematical framework . If Z does not exist, then A, B, C, ... have nothing in common (they are disjoint) and no axiomatic system of more than a single axiom can be found. In that case, an interesting mathematical framework is at least Z and (A xor B xor C, ...) . Furthermore, A and B are mutually independent, if and only if they are consistent with context Z. Without context Z, A and B independency is too strong (they cannot interact with each other) and no interesting mathematical framework can be found. In that case we can conclude that context Z is a property that enables independent mathematical objects (axioms) to be related to each other without a mutual contradiction. We can generalize this notion in order to define a deeper level of consistency between consistent axiomatic systems, etc., etc. ad infinitum ... In other words, in order to define more general mathematical frameworks, there has to be some relation between independent objects and a context that enables to transform independency to mutual consistency. This transformation can be done if a context dose not depend on the independency that can be found among axioms. In other words, a context has the property to reduce axioms' independency up to the level that enables constructive interaction among them. From this point of view, the mathematical language is not less than a one organic fractal-like structure with infinitely many levels of consistency, and this organic structure is not less then separated, independent and local axioms, that are related to each other through a non-local context. In other words, the local and the non-local are mutually independent first-order properties of an organic mathematical universe. In order to research locality and non-locality, we first have to define their logical basis. It can be done by reexamining the Membership concept. 3. Locality and non-locality, basic terms and definitions The set is a fundamental concept used in many branches of mathematics. Although not rigorously defined, a set can be thought of as a collection of distinct members whose order is not important. If we expand the membership concept beyond the set/member relation, then new mathematical frameworks emerge. For example: or is a logical OR connective. xor is a logical EXCLUSIVE OR connective. and is a logical AND connective. = is "Equal to …". ≠ is "Not equal to …". Urelement is not a set, but can be a member of a set. Element is either a set or an urelement. Sub-element is an element that defines another element. € is a member of an element, but not necessarily a sub-element of an element. ₡ is the negation of €. x and A are placeholders of an element. Definition 6: If x € A xor x ₡ A, then x is local. Definition 7: If x € A and x ₡ A, then x is non-local. Axiom x (represented by "•" ) is a local urelement of context A (represented by "__" ), that is, x € A xor x ₡ A. Also Axiom x is not a sub-element of context A, because context A is an urelement (an example of locality: _•_ xor __ • ). Context x (represented by "__" ) is a non-local urelement of axiom A (represented by "•" ), that is, x € A and x ₡ A. Also Context x is not a sub-element of axiom A, because axiom A is an urelement (an example of non-locality: _•_ and _•_ ). The Membership concept is expanded beyond the set/member relation (In a set/member relation each x member is local and a sub-element of set A) and we logically define the relations between mutually independent urelements. For further reading please look at http://www.geocities.com/complementarytheory/TOUM.pdf Please reply your comments, thank you. Doron
