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From: Nam Nguyen on 28 Dec 2009 21:10 zuhair wrote: > On Dec 28, 8:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> zuhair wrote: >>> On Dec 28, 1:33 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> zuhair wrote: >>>>> On Dec 27, 5:02 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> zuhair wrote: >>>>>>> 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. >>>>>> I think it's a bad idea to assert that there's a unique "set" binary relation >>>>>> you've referred to as 'this relation "e"' that doesn't any intrinsic >>>>>> vagueness in it. >>>>>> So long as we talk about a kind of set formal systems that could be used >>>>>> to encode the naturals (which most common set theories - as well as your >>>>>> alluded set theories - are) then there's always vagueness in these >>>>>> set concepts. For instance, is there any clarity of the truth of >>>>>> "There are infinite counter examples of GC" in these set systems? >>>>>> One certainly could make the set concept more implicated (e.g. by introducing >>>>>> multiple binary epsilon relations) but I think one should abandon the idea >>>>>> that one would have "better" explanation (as you've tried in this thread) >>>>>> of what set concept is than the existing set formal systems, given the >>>>>> constraint/limitation mentioned above. >>>> I meant "more complicated". Sorry for the typo. >>>>> Well one can try! >>>> Why should we try something we shouldn't try? >>> Why we shouldn't? if we can have a more intuitive system, that can >>> give us better results, then we can try even if the system is a little >>> bit more complex, however if it doesn't give use any new results and >>> it appears more complex formally so that it is more difficult to work >>> with, then even if it is intuitively appealing we will discard it. >> I think you misunderstood my criticism here. It's not the "trying" per se >> I've discouraged; it's that your overall claim here seems to be misleading: >> given the *intrinsic* vagueness in *all* relevant (set) systems there can't >> be the one that's "better" than the rest of them - in term of vagueness. >> >> Put it differently, as long as your set theories are still with one epsilon >> symbol and are still able to encode the naturals, then *anyone* including >> you would still be able to point out some vagueness (in semantic and truth) >> in those theories. So why insist that they have clarity while the canonical >> ones don't? > > First, Who said that my set theories have ONE epsilon symbol? Didn't *you* say the the following at the beginning (above): >>>>>>> One of the facts of set theory is that the primitive relation >>>>>>> "e" is totally vague relation, >>>>>>> Here I will present a simple trial to understand this relation "e" >>>>>>> and to give it some informal intuitive background. > Second, I agree that there must be some vague area at the end. > That is inescapable. But my aim is to lessen this area. As I said above, the vagueness is "*intrinsic*" given the limitations. How could one "lessen" intrinsic - "inescapable" consequences - without changing the limitations? And if you do change the limitations, would your theories still be able to , say, encode the naturals? > > The approach of the two memberships here, do simplify our intuitive > understanding of what sets might be (although on a limited level, till > now I didn't solve the intuitive background of circular sets). > > Epsilon used in canonical set theories, is actually a totally > mysterious relation, we actually understand nothing of it other than > its name. That statement of mine might seem too harsh, but really I > didn't see any intuitive background explaining it. It seems that we'll > end up with what John Jones always say, a purely syntactical rule > following game, which in reality has no meaning outside rule > following. Anyhow, this is a complex matter. > > I am till now not satisfied even of the intuitive-formal account that > I made here to give an intuitive base for epsilon. It might be funny > to say that but Sets seems to be containers trying to mimic > collections, But WHY? > > Zuhair
From: zuhair on 28 Dec 2009 22:28 On Dec 28, 9:10 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > zuhair wrote: > > On Dec 28, 8:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> zuhair wrote: > >>> On Dec 28, 1:33 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> zuhair wrote: > >>>>> On Dec 27, 5:02 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> zuhair wrote: > >>>>>>> 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. > >>>>>> I think it's a bad idea to assert that there's a unique "set" binary relation > >>>>>> you've referred to as 'this relation "e"' that doesn't any intrinsic > >>>>>> vagueness in it. > >>>>>> So long as we talk about a kind of set formal systems that could be used > >>>>>> to encode the naturals (which most common set theories - as well as your > >>>>>> alluded set theories - are) then there's always vagueness in these > >>>>>> set concepts. For instance, is there any clarity of the truth of > >>>>>> "There are infinite counter examples of GC" in these set systems? > >>>>>> One certainly could make the set concept more implicated (e.g. by introducing > >>>>>> multiple binary epsilon relations) but I think one should abandon the idea > >>>>>> that one would have "better" explanation (as you've tried in this thread) > >>>>>> of what set concept is than the existing set formal systems, given the > >>>>>> constraint/limitation mentioned above. > >>>> I meant "more complicated". Sorry for the typo. > >>>>> Well one can try! > >>>> Why should we try something we shouldn't try? > >>> Why we shouldn't? if we can have a more intuitive system, that can > >>> give us better results, then we can try even if the system is a little > >>> bit more complex, however if it doesn't give use any new results and > >>> it appears more complex formally so that it is more difficult to work > >>> with, then even if it is intuitively appealing we will discard it. > >> I think you misunderstood my criticism here. It's not the "trying" per se > >> I've discouraged; it's that your overall claim here seems to be misleading: > >> given the *intrinsic* vagueness in *all* relevant (set) systems there can't > >> be the one that's "better" than the rest of them - in term of vagueness. > > >> Put it differently, as long as your set theories are still with one epsilon > >> symbol and are still able to encode the naturals, then *anyone* including > >> you would still be able to point out some vagueness (in semantic and truth) > >> in those theories. So why insist that they have clarity while the canonical > >> ones don't? > > > First, Who said that my set theories have ONE epsilon symbol? > > Didn't *you* say the the following at the beginning (above): > > >>>>>>> One of the facts of set theory is that the primitive relation > >>>>>>> "e" is totally vague relation, > > >>>>>>> Here I will present a simple trial to understand this relation "e" > >>>>>>> and to give it some informal intuitive background. > > > Second, I agree that there must be some vague area at the end. > > That is inescapable. But my aim is to lessen this area. > > As I said above, the vagueness is "*intrinsic*" given the limitations. > How could one "lessen" intrinsic - "inescapable" consequences - without > changing the limitations? And if you do change the limitations, would > your theories still be able to , say, encode the naturals? I don't think I have a full answer to this Nam. But yes for instance the theory would encode the naturals of course, it will have the limitations, but yet it would be less vague on *intuitive* level, what I meant by inescapable is there must be some degree of vagueness at the end, but that doesn't mean it cannot be lessened even given the same limitations. Anyhow that is a complex subject as it seems, and I really don't have answers, all what I am seeking now is a simple non circular intuitive explanation of membership weather by ONE relation symbol, or more than one primitive it doesn't really matter, what matter to me now is to give some intuitive light to set theory, so that one can understand what he doing, since it *MIGHT* be easier to work with, even if it turns to be formally more complex. Anyhow what I said here is pretty much in its infancy really. Zuhair > > > > > The approach of the two memberships here, do simplify our intuitive > > understanding of what sets might be (although on a limited level, till > > now I didn't solve the intuitive background of circular sets). > > > Epsilon used in canonical set theories, is actually a totally > > mysterious relation, we actually understand nothing of it other than > > its name. That statement of mine might seem too harsh, but really I > > didn't see any intuitive background explaining it. It seems that we'll > > end up with what John Jones always say, a purely syntactical rule > > following game, which in reality has no meaning outside rule > > following. Anyhow, this is a complex matter. > > > I am till now not satisfied even of the intuitive-formal account that > > I made here to give an intuitive base for epsilon. It might be funny > > to say that but Sets seems to be containers trying to mimic > > collections, But WHY? > > > Zuhair
From: John Jones on 9 Jan 2010 22:15 Virgil wrote: > In article <hh6rvu$urn$2(a)news.eternal-september.org>, > John Jones <jonescardiff(a)btinternet.com> wrote: > >> Virgil wrote: >>> In article <hh3m28$8ii$1(a)news.eternal-september.org>, >>> John Jones <jonescardiff(a)btinternet.com> wrote: >>> >>>> zuhair wrote: >>>>> 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 far so good, but that last statement is incorrect. There's no >>>> relationship between parts and their whole. >>>> >>>>> 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. >>>> The set "the set of all objects collected" doesn't look like a set to >>>> me. It begs the question of what a set is, and if its a collection then >>>> it looks tautologous. It just says "set"? >>>> >>>>> 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'. >>>> I don't think you will find the resources you need from what is >>>> currently available to you. I would look at the fringes. >>> A necessary and sufficient realization of the idea of a "set" is the >>> ability to distinguish between the objects one wishes to include in the >>> set from tho objects one wishes to exclude. >>> >>> I.e., if one can deal successfully with the issue of membership, then >>> one has grasped the idea of "set". >> But neither membership nor distinguishment identify a set. > > But they separate what "belongs" to the set from what doesn't. No, the set does that. A set isn't quantified, and certainly not by its membership. > > You might think of a set as a boundary, like a simple closed curve in > the plane, having an inside and an outside, but unlike such curves, not > having things as a part of that boundary. That's inadequate as a definition of a set because it appeals to a pictogram.
From: Virgil on 9 Jan 2010 23:42 In article <hibgno$g46$3(a)news.eternal-september.org>, John Jones <jonescardiff(a)btinternet.com> wrote: > Virgil wrote: > > In article <hh6rvu$urn$2(a)news.eternal-september.org>, > > John Jones <jonescardiff(a)btinternet.com> wrote: > > > >> Virgil wrote: > >>> In article <hh3m28$8ii$1(a)news.eternal-september.org>, > >>> John Jones <jonescardiff(a)btinternet.com> wrote: > >>> > >>>> zuhair wrote: > >>>>> 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 far so good, but that last statement is incorrect. There's no > >>>> relationship between parts and their whole. > >>>> > >>>>> 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. > >>>> The set "the set of all objects collected" doesn't look like a set to > >>>> me. It begs the question of what a set is, and if its a collection then > >>>> it looks tautologous. It just says "set"? > >>>> > >>>>> 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'. > >>>> I don't think you will find the resources you need from what is > >>>> currently available to you. I would look at the fringes. > >>> A necessary and sufficient realization of the idea of a "set" is the > >>> ability to distinguish between the objects one wishes to include in the > >>> set from tho objects one wishes to exclude. > >>> > >>> I.e., if one can deal successfully with the issue of membership, then > >>> one has grasped the idea of "set". > >> But neither membership nor distinguishment identify a set. > > > > But they separate what "belongs" to the set from what doesn't. > > No, the set does that. A set isn't quantified, and certainly not by its > membership. How can a set exist unless there is a distinction between what is a member of it from what is not a member of it? And how can one distinguish between sets other than by noting differences in membership? > > > > > You might think of a set as a boundary, like a simple closed curve in > > the plane, having an inside and an outside, but unlike such curves, not > > having things as a part of that boundary. > > That's inadequate as a definition of a set because it appeals to a > pictogram. It was clearly not intended as a definition, but merely as an analogy.
From: John Jones on 20 Jan 2010 20:58
Virgil wrote: > In article <hibgno$g46$3(a)news.eternal-september.org>, > John Jones <jonescardiff(a)btinternet.com> wrote: > >> Virgil wrote: >>> In article <hh6rvu$urn$2(a)news.eternal-september.org>, >>> John Jones <jonescardiff(a)btinternet.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <hh3m28$8ii$1(a)news.eternal-september.org>, >>>>> John Jones <jonescardiff(a)btinternet.com> wrote: >>>>> >>>>>> zuhair wrote: >>>>>>> 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 far so good, but that last statement is incorrect. There's no >>>>>> relationship between parts and their whole. >>>>>> >>>>>>> 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. >>>>>> The set "the set of all objects collected" doesn't look like a set to >>>>>> me. It begs the question of what a set is, and if its a collection then >>>>>> it looks tautologous. It just says "set"? >>>>>> >>>>>>> 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'. >>>>>> I don't think you will find the resources you need from what is >>>>>> currently available to you. I would look at the fringes. >>>>> A necessary and sufficient realization of the idea of a "set" is the >>>>> ability to distinguish between the objects one wishes to include in the >>>>> set from tho objects one wishes to exclude. >>>>> >>>>> I.e., if one can deal successfully with the issue of membership, then >>>>> one has grasped the idea of "set". >>>> But neither membership nor distinguishment identify a set. >>> But they separate what "belongs" to the set from what doesn't. >> No, the set does that. A set isn't quantified, and certainly not by its >> membership. > > How can a set exist unless there is a distinction between what is a > member of it from what is not a member of it? A set is distinguished by its name alone. It is an inadequate definition, but the fullest one that mathematics traditionally allows. > And how can one distinguish between sets other than by noting > differences in membership? >>> You might think of a set as a boundary, like a simple closed curve in >>> the plane, having an inside and an outside, but unlike such curves, not >>> having things as a part of that boundary. >> That's inadequate as a definition of a set because it appeals to a >> pictogram. > > It was clearly not intended as a definition, but merely as an analogy. The analogy uses a coralle or a boundary line. The analogy is thus one of a bag or container. But is a set a bag or a container? What mathematical decsription do we give to such a container? |