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From: zuhair on 24 Dec 2009 22:41 Hi all, One of the facts of set theory is that the primitive relation "e" is totally vague relation, there is no way of understanding this relation at all, even at informal level! Which is actually a great drawback, since in set theory we are building hierarchy of objects using this relation, a relation that nobody understand what it stands for? Here I will present a simple trial to understand this relation "e" and to give it some informal intuitive background. First I shall speak about the concept of set as used in ordinary language. When we say we have a set of objects x,y,z , then we usually mean that there is an entity that represent the *Whole* of x,y,z, so although x,y,z are distinct objects, but yet the *whole* that is constituted from these objects is considered the set of these objects, or the *collection* of these objects. However this *Whole* of x,y,z is in a sense these objects themselves. So a set is a form of describing multiplicity by unity, that unity is taken to be the totality of the Whole of these multiple objects. However if we collect ONE object, then it is pretty intuitive to say, that the set of all objects collected IS that object. Also it is pretty intuitive to say that if there are no objects collected, then there is NO set of objects collected, in other words there do not exist an empty collection. Of course the traditional membership relation "e" doesn't capture all these pretty intuitive concepts of the customary use of the word membership and of the word "set" or the word "collection". To try to put all of the above informal account into rigorous mathematical context, we can define a theory in FOL with identity and the primitive binary relation symbol e' , which stands for what I call "intuitive membership", or in reality "membership", since epsilon doesn't capture the intuitive concept of membership. So we can have the following axioms that characterize this binary relation e'. (1) Extensionality: For all z ( z e' x <-> z e' y ) -> x=y Define(singleton): x is singleton <-> Exist z for all y ( y e' x <-> y=z ) (2) Membership: x e' y -> x is singleton (3) Singletons: x is singleton -> x e' x theorem: x is singleton <-> x e' x Proof: we have x e' x -> x is singleton (2) and we have x is singleton -> x e' x (3) thus x is singleton <-> x e' x QED (4) Non Emptiness: For all x Exist y ( y e' x) (5) Comprehension: if Phi(y) is a formula in which at least y is free, and in which x is not free, then all closures of Exist y ( y is singleton Phi(y) ) -> Exist x for all y ( y e' x <-> ( y is singleton Phi(y) ) ) are axioms. Define[y|Phi]: Exist y ( y is singleton Phi(y) ) -> x=[y|Phi] <-> for all y ( y e' x <-> ( y is singleton Phi(y) ) ) Now we come to explain what "sets" in set theory mean: The approach here is that objects in a set theory like Zermelo's set theory, are actually nothing but "containers", that is their reality. However these containers in Z, have the property of being identical if they have the same contents, which is not a property that is inherent in the concept of containers, since containers of equal size can have exactly the same contents in them but yet can be different from each other, however this is not allowed in Z, so how come I say that these objects are containers. To understand that issue, lets assume that sets are containers, and lets assume the relation epsilon in the language of Z refers informally to "contained in" so x e y means informally x is contained in y. Of course "e" here is a primitive concept also, but in this approach it has an informal background as I illustrated above, it refers to "containment". Now lets have the following axiom: (6) Containment: x e y -> ( x is singleton & y is singleton ) were singleton is defined above ( note: the definition of singleton uses the relation e' so it is not the same as singletons present in customary set theories which uses the relation e). e can be read as " is a content of" so x e y is to be read as: x is a content of y. Now we define any collection of containers in which no two distinct containers have the same contents as a "discriminable collection" Now the union of two discriminable collections is not necessarily a discriminable collection. So we need to define discriminable union as the union of any two discriminable collections that is itself discriminable. Now lets define V as the largest discriminable union of discriminable collections (this notion needs further treatment as to what "largest" would mean, but for now it means that every discriminable union of discriminable collections would be strictly subnumerous to V) Now a 'set' might be defined as a member of V. So V here would stand as the model of that set theory. Then other axioms of Z, or ZF can be added ( were the variables in these axioms are members of V ). This way of understanding sets, although it is not a complete workup yet, but it does provide us with an insight to what are sets, what are those objects in ZFC set theory. They are Containers discriminable by their contents, nothing more, nothing less. The relation e' can be used here to build proper classes also. So it is a theory of this theory that: For all x,z,u,w ((u=[y|yex] & w=[y|yez] & u=w) ->x=z) Zuhair
From: William Elliot on 25 Dec 2009 04:01 On Thu, 24 Dec 2009, zuhair wrote: > One of the facts of set theory is that the primitive relation > "e" is totally vague relation, there is no way of understanding this > relation at all, even at informal level! Which is actually a great > drawback, since in set theory we are building hierarchy of objects > using this relation, a relation that nobody understand what it stands > for? > > Here I will present a simple trial to understand this relation "e" > and to give it some informal intuitive background. > > First I shall speak about the concept of set as used in ordinary > language. > > When we say we have a set of objects x,y,z , then we usually mean that > there is an entity that represent the *Whole* of > x,y,z, so although x,y,z are distinct objects, but yet > the *whole* that is constituted from these objects is considered the > set of these objects, or the *collection* of these objects. > In ZF, by extensionality, there is no object but the empty set. Sets are containers and the containers can contain nothing but other containers. Thus regularity, as a container can contain itself as well as a snake can swallow itself. However, again by extensionality, an empty can is the same as an empty box. Thus a set is a generic container for other containers. Set theory is the infinite xmas present. Open it up and you'll find nothing but more to open. Riddle of the day. How many times, when unwrapping an infinite xmas present, will you open a container with nothing in it? > However this *Whole* of x,y,z is in a sense these objects themselves. > > So a set is a form of describing multiplicity by unity, that unity is > taken to be the totality of the Whole of these multiple objects. > > However if we collect ONE object, then it is pretty intuitive to say, > that the set of all objects collected IS that object. > > Also it is pretty intuitive to say that if there are no objects > collected, then there is NO set of objects collected, in other words > there do not exist an empty collection. > > Of course the traditional membership relation "e" doesn't capture all > these pretty intuitive concepts of the customary use of the word > membership and of the word "set" or the word "collection". > > To try to put all of the above informal account into rigorous > mathematical context, we can define a theory in FOL with identity > and the primitive binary relation symbol e' , which stands for > what I call "intuitive membership", or in reality "membership", since > epsilon doesn't capture the intuitive concept of membership. > > So we can have the following axioms that characterize this binary > relation e'. > > (1) Extensionality: > For all z ( z e' x <-> z e' y ) -> x=y > > Define(singleton): > x is singleton <-> Exist z for all y ( y e' x <-> y=z ) > > (2) Membership: x e' y -> x is singleton > > (3) Singletons: x is singleton -> x e' x > > theorem: x is singleton <-> x e' x > Proof: we have x e' x -> x is singleton (2) > and we have x is singleton -> x e' x (3) > thus x is singleton <-> x e' x > QED > > (4) Non Emptiness: For all x Exist y ( y e' x) > > (5) Comprehension: if Phi(y) is a formula in which at least y is free, > and in which x is not free, then all closures of > > Exist y ( y is singleton Phi(y) ) -> > Exist x for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > are axioms. > > Define[y|Phi]: > Exist y ( y is singleton Phi(y) ) -> > x=[y|Phi] <-> for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > Now we come to explain what "sets" in set theory mean: > > The approach here is that objects in a set theory like Zermelo's set > theory, are actually nothing but "containers", that is their reality. > > However these containers in Z, have the property of being identical if > they have the same contents, which is not a property that is inherent > in the concept of containers, since containers of equal size can have > exactly the same contents in them but yet can be different from each > other, however this is not allowed in Z, so how come I say that these > objects are containers. > > To understand that issue, lets assume that sets are containers, and > lets assume the relation epsilon in the language of Z refers > informally to "contained in" so x e y means informally > x is contained in y. > > Of course "e" here is a primitive concept also, but in this approach > it has an informal background as I illustrated above, it refers to > "containment". > > Now lets have the following axiom: > > (6) Containment: x e y -> ( x is singleton & y is singleton ) > > were singleton is defined above ( note: the definition of singleton > uses the relation e' so it is not the same as singletons present in > customary set theories which uses the relation e). > > e can be read as " is a content of" > > so x e y is to be read as: x is a content of y. > > Now we define any collection of containers in which no two distinct > containers have the same contents as a "discriminable collection" > > Now the union of two discriminable collections is not necessarily > a discriminable collection. > > So we need to define discriminable union as the union of any two > discriminable collections that is itself discriminable. > > Now lets define V as the largest discriminable union of > discriminable collections (this notion needs further treatment as to > what "largest" would mean, but for now it means that every > discriminable union of discriminable collections would be strictly > subnumerous to V) > > Now a 'set' might be defined as a member of V. > > So V here would stand as the model of that set theory. > > Then other axioms of Z, or ZF can be added ( were the variables in > these axioms are members of V ). > > This way of understanding sets, although it is not a complete workup > yet, but it does provide us with an insight to what are sets, what are > those objects in ZFC set theory. > > They are Containers discriminable by their contents, nothing more, > nothing less. > > The relation e' can be used here to build proper classes also. > > So it is a theory of this theory that: > > For all x,z,u,w ((u=[y|yex] & w=[y|yez] & u=w) ->x=z) > > Zuhair > > > > > > > > > > > >
From: zuhair on 25 Dec 2009 08:08 On Dec 25, 4:01 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Thu, 24 Dec 2009, zuhair wrote: > > One of the facts of set theory is that the primitive relation > > "e" is totally vague relation, there is no way of understanding this > > relation at all, even at informal level! Which is actually a great > > drawback, since in set theory we are building hierarchy of objects > > using this relation, a relation that nobody understand what it stands > > for? > > > Here I will present a simple trial to understand this relation "e" > > and to give it some informal intuitive background. > > > First I shall speak about the concept of set as used in ordinary > > language. > > > When we say we have a set of objects x,y,z , then we usually mean that > > there is an entity that represent the *Whole* of > > x,y,z, so although x,y,z are distinct objects, but yet > > the *whole* that is constituted from these objects is considered the > > set of these objects, or the *collection* of these objects. > > In ZF, by extensionality, there is no object but the empty set. Though I don't know what you mean by that, but Regularity would ensure that the empty set is a member of the transitive closure of every set, so ZF is a whole buildup of cans of cans of cans....finitely many time of canning of an empty can at the end, that is correct. However in ZF minus Regularity, that is not necessarily the case, for example the "Recursive Cardinals" that I've defined do not have this property. > > Sets are containers and the containers can contain nothing but other > containers. Yes, that is in ZF and related system that has no Ur-elements, that is correct. Actually this statement of yours can be generalized, to say that Ur- elements are also cans but they are indiscriminable cans, so in a universe of discourse V of any set theory if there exist at least two cans that have the same contents but they are yet distinct from each other (i.e. not identical), then we say that V is containing a Ur-element, so your statement can be made to be true. Thus regularity, as a container can contain itself as well as > a snake can swallow itself. But Regularity is not a container, Regularity is an axiom that sets one aspect of the behavior of these containers. IF you actually violate Regularity then yes you can have the snake example container you were taking about. However, again by extensionality, an empty > can is the same as an empty box. Thus a set is a generic container for > other containers. do you mean that the empty set is the abstraction of all empty boxes. hmmm.... Set theory is the infinite xmas present. Open it > up and you'll find nothing but more to open. Yes, I agree. > > Riddle of the day. How many times, when unwrapping an infinite xmas > present, will you open a container with nothing in it? Well, with Regularity, it must be infinite number of times! Without Regularity, if this xmas present had the structure of a recursive singleton (or any of the x-recursive cardinals) then you will never open a container with nothing in it. Happy Xmas William. Zuhair > > > However this *Whole* of x,y,z is in a sense these objects themselves. > > > So a set is a form of describing multiplicity by unity, that unity is > > taken to be the totality of the Whole of these multiple objects. > > > However if we collect ONE object, then it is pretty intuitive to say, > > that the set of all objects collected IS that object. > > > Also it is pretty intuitive to say that if there are no objects > > collected, then there is NO set of objects collected, in other words > > there do not exist an empty collection. > > > Of course the traditional membership relation "e" doesn't capture all > > these pretty intuitive concepts of the customary use of the word > > membership and of the word "set" or the word "collection". > > > To try to put all of the above informal account into rigorous > > mathematical context, we can define a theory in FOL with identity > > and the primitive binary relation symbol e' , which stands for > > what I call "intuitive membership", or in reality "membership", since > > epsilon doesn't capture the intuitive concept of membership. > > > So we can have the following axioms that characterize this binary > > relation e'. > > > (1) Extensionality: > > For all z ( z e' x <-> z e' y ) -> x=y > > > Define(singleton): > > x is singleton <-> Exist z for all y ( y e' x <-> y=z ) > > > (2) Membership: x e' y -> x is singleton > > > (3) Singletons: x is singleton -> x e' x > > > theorem: x is singleton <-> x e' x > > Proof: we have x e' x -> x is singleton (2) > > and we have x is singleton -> x e' x (3) > > thus x is singleton <-> x e' x > > QED > > > (4) Non Emptiness: For all x Exist y ( y e' x) > > > (5) Comprehension: if Phi(y) is a formula in which at least y is free, > > and in which x is not free, then all closures of > > > Exist y ( y is singleton Phi(y) ) -> > > Exist x for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > > are axioms. > > > Define[y|Phi]: > > Exist y ( y is singleton Phi(y) ) -> > > x=[y|Phi] <-> for all y ( y e' x <-> ( y is singleton Phi(y) ) ) > > > Now we come to explain what "sets" in set theory mean: > > > The approach here is that objects in a set theory like Zermelo's set > > theory, are actually nothing but "containers", that is their reality. > > > However these containers in Z, have the property of being identical if > > they have the same contents, which is not a property that is inherent > > in the concept of containers, since containers of equal size can have > > exactly the same contents in them but yet can be different from each > > other, however this is not allowed in Z, so how come I say that these > > objects are containers. > > > To understand that issue, lets assume that sets are containers, and > > lets assume the relation epsilon in the language of Z refers > > informally to "contained in" so x e y means informally > > x is contained in y. > > > Of course "e" here is a primitive concept also, but in this approach > > it has an informal background as I illustrated above, it refers to > > "containment". > > > Now lets have the following axiom: > > > (6) Containment: x e y -> ( x is singleton & y is singleton ) > > > were singleton is defined above ( note: the definition of singleton > > uses the relation e' so it is not the same as singletons present in > > customary set theories which uses the relation e). > > > e can be read as " is a content of" > > > so x e y is to be read as: x is a content of y. > > > Now we define any collection of containers in which no two distinct > > containers have the same contents as a "discriminable collection" > > > Now the union of two discriminable collections is not necessarily > > a discriminable collection. > > > So we need to define discriminable union as the union of any two > > discriminable collections that is itself discriminable. > > > Now lets define V as the largest discriminable union of > > discriminable collections (this notion needs further treatment as to > > what "largest" would mean, but for now it means that every > > discriminable union of discriminable collections would be strictly > > subnumerous to V) > > > Now a 'set' might be defined as a member of V. > > > So V here would stand as the model of that set theory. > > > Then other axioms of Z, or ZF can be added ( were the variables in > > these axioms are members of V ). > > > This way of understanding sets, although it is not a complete workup > > yet, but it does provide us with an insight to what are sets, what are > > those objects in ZFC set theory. > > > They are Containers discriminable by their contents, nothing more, > > nothing less. > > > The relation e' can be used here to build proper classes also. > > > So it is a theory of this theory that: > > > For all x,z,u,w ((u=[y|yex] & w=[y|yez] & u=w) ->x=z) > > > Zuhair
From: William Elliot on 25 Dec 2009 10:07 >>> One of the facts of set theory is that the primitive relation >>> "e" is totally vague relation, there is no way of understanding this >>> relation at all, even at informal level! Which is actually a great >>> drawback, since in set theory we are building hierarchy of objects >>> using this relation, a relation that nobody understand what it stands >>> for? >> >>> Here I will present a simple trial to understand this relation "e" >>> and to give it some informal intuitive background. >> >>> First I shall speak about the concept of set as used in ordinary >>> language. >> >>> When we say we have a set of objects x,y,z , then we usually mean that >>> there is an entity that represent the *Whole* of >>> x,y,z, so although x,y,z are distinct objects, but yet >>> the *whole* that is constituted from these objects is considered the >>> set of these objects, or the *collection* of these objects. >> >> In ZF, by extensionality, there is no object but the empty set. > > Though I don't know what you mean by that, but Regularity would ensure > that the empty set is a member of the transitive closure of every set, > so ZF is a whole buildup of cans of cans of cans....finitely many time > of canning of an empty can at the end, that is correct. > Objects do not have elements for they are not sets. By extensionality if ob is an object, since .. . for all x, (x in ob iff x in nulset) ob = nulset. The empty set is OTOH, an object (the only object in ZF) by the criterian of not having elements > However in ZF minus Regularity, that is not necessarily the case, for > example the "Recursive Cardinals" that I've defined do not have this > property. >> Sets are containers and the containers can contain nothing but other >> containers. � > > Yes, that is in ZF and related system that has no Ur-elements, that is > correct. > > Actually this statement of yours can be generalized, to say that Ur- > elements are also cans but they are indiscriminable cans, so in a > universe of discourse V of any set theory if there exist at least two > cans that have the same contents but they are yet distinct from each > other (i.e. not identical), then we say that V is containing > a Ur-element, so your statement can be made to be true. > > Thus regularity, as a container can contain itself as well as > a snake can swallow itself. > > But Regularity is not a container, Regularity is an axiom that sets > one aspect of the behavior of these containers. IF you actually > violate Regularity then yes > you can have the snake example container you were taking about. > By regularity, a container can contain itself as well as a snake can swallow itself. > �However, again by extensionality, an empty >> can is the same as an empty box. �Thus a set is a generic container for >> other containers. � > > do you mean that the empty set is the abstraction of all empty boxes. > hmmm.... > No. When looking at the elements/objects: a b c d e f one can visualize several sets. A set is the mental construct, a gestalt, of seeing some of the elements bound, connected or related as parts of an imagined thing. Within a b c d e f one can visualze upto 2^6 different bundlings of those letters or elements. > Set theory is the infinite xmas present. �Open it >> up and you'll find nothing but more to open. > > Yes, I agree. >> >> Riddle of the day. �How many times, when unwrapping an infinite xmas >> present, will you open a container with nothing in it? > > Well, with Regularity, it must be infinite number of times! > Without Regularity, if this xmas present had the structure of a > recursive singleton (or any of the x-recursive cardinals) then you > will never open a container with nothing in it. > Occam's axiom: regularity, GCH, no unreachables. Occam's strong axiom: V = L, no unreachables.
From: zuhair on 25 Dec 2009 12:23
On Dec 25, 10:07 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >>> One of the facts of set theory is that the primitive relation > >>> "e" is totally vague relation, there is no way of understanding this > >>> relation at all, even at informal level! Which is actually a great > >>> drawback, since in set theory we are building hierarchy of objects > >>> using this relation, a relation that nobody understand what it stands > >>> for? > > >>> Here I will present a simple trial to understand this relation "e" > >>> and to give it some informal intuitive background. > > >>> First I shall speak about the concept of set as used in ordinary > >>> language. > > >>> When we say we have a set of objects x,y,z , then we usually mean that > >>> there is an entity that represent the *Whole* of > >>> x,y,z, so although x,y,z are distinct objects, but yet > >>> the *whole* that is constituted from these objects is considered the > >>> set of these objects, or the *collection* of these objects. > > >> In ZF, by extensionality, there is no object but the empty set. > > > Though I don't know what you mean by that, but Regularity would ensure > > that the empty set is a member of the transitive closure of every set, > > so ZF is a whole buildup of cans of cans of cans....finitely many time > > of canning of an empty can at the end, that is correct. > > Objects do not have elements for they are not sets. that is *your* terminology of *object* this is not the standard use of the word "object". in the standard use of the word object all classes weather they are sets or proper classes are "objects", also Ur-elements are objects, in the standard terminology of FOLs the term objects refers to something that is not a relation. > By extensionality if ob is an object, since > . . for all x, (x in ob iff x in nulset) > ob = nulset. I should confess, that I am not following you here. what do you want to say exactly. > > The empty set is OTOH, an object (the only object in ZF) by > the criterian of not having elements OK. > > > > > However in ZF minus Regularity, that is not necessarily the case, for > > example the "Recursive Cardinals" that I've defined do not have this > > property. > >> Sets are containers and the containers can contain nothing but other > >> containers. > > > Yes, that is in ZF and related system that has no Ur-elements, that is > > correct. > > > Actually this statement of yours can be generalized, to say that Ur- > > elements are also cans but they are indiscriminable cans, so in a > > universe of discourse V of any set theory if there exist at least two > > cans that have the same contents but they are yet distinct from each > > other (i.e. not identical), then we say that V is containing > > a Ur-element, so your statement can be made to be true. > > > Thus regularity, as a container can contain itself as well as > > a snake can swallow itself. > > > But Regularity is not a container, Regularity is an axiom that sets > > one aspect of the behavior of these containers. IF you actually > > violate Regularity then yes > > you can have the snake example container you were taking about. > > By regularity, a container can contain itself > as well as a snake can swallow itself. No that is incorrect, you can only have that if you violate Regularity. The axiom of Regularity doesn't allow you to have containers containing themselves, you must violate Regularity to have that, that is crystal clear. > > > However, again by extensionality, an empty > >> can is the same as an empty box. Thus a set is a generic container for > >> other containers. > > > do you mean that the empty set is the abstraction of all empty boxes. > > hmmm.... > > No. > When looking at the elements/objects: a b c d e f > one can visualize several sets. A set is the mental > construct, a gestalt, of seeing some of the elements > bound, connected or related as parts of an imagined thing. Yes, these are the sets that I am speaking of in the beginning of this post here, but it is NOT the objects the set theories call as sets, you are confusing the two concepts. However I prefer to rephrase it to the following: A set is the mental construct, a gestalt, of seeing some of the elements bound, connected or related as *members* of an imagined thing. (members here refers to the trivial membership e' and not to the customary e set theorists use, so don't confuse them) I refuse Mereology as a background for set theory in this post, you should have anticipated that from my first five axioms, and from me naming a primitive binary relation of membership e', instead of using the binary relation "part" which is used in Mereology. > > Within a b c d e f one can visualze upto 2^6 different > bundlings of those letters or elements. Yea, these buildings are what I call "true collections" Notice, that if you think of this matter along the same lines, you'll see that you cannot have a singleton that is not a member(e' ) of itself, and notice that you cannot have an object empty of objects having the relation e' to it. This matter needs some contemplation though, it is not easy. You should differential between *true collections* (which capture the informal notion you were speaking about above) and between *sets* as used in set theories which are actually captured by the notion of "containers" rather than the informal background you were speaking of. > > > Set theory is the infinite xmas present. Open it > >> up and you'll find nothing but more to open. > > > Yes, I agree. > > >> Riddle of the day. How many times, when unwrapping an infinite xmas > >> present, will you open a container with nothing in it? > > > Well, with Regularity, it must be infinite number of times! > > Without Regularity, if this xmas present had the structure of a > > recursive singleton (or any of the x-recursive cardinals) then you > > will never open a container with nothing in it. > > Occam's axiom: regularity, GCH, no unreachables. > Occam's strong axiom: V = L, no unreachables. Occam's axioms are not common grounds for all set theories so that you put much weight to them.Indeed there are many non well founded set theories that do not care the hell about these axioms at all. The classical answer to your question, is that we'll have infinitely many empty boxes if we presume Regularity, if we don't presume Regularity, then we can have an infinite descending membership chains, and thus we do not necessarily come to have an infinite number of empty boxes. However physically speaking, in this finite world of ours, there is no infinite Xmas present, so the question is not applicable to the real physical world we live in. Zuhair |