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From: Marshall on 28 Dec 2009 11:12 On Dec 28, 1:32 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > Actually, I disagree with this. We may, in fact, simply come in and > > put rules on matters at our own whim. The only *objective* objection > > that might arise is inconsistency; if we avoid that we are entirely > > justified. Whether we have produced something useful is of course > > another matter. > > What exactly do you mean by "inconsistency" here?[1] In the context of the axiomatic method (which I believe is the usual for zuhair,) the ability to prove A ^ ~A for some A. Since we're not talking about making up a new logic, we don't have to consider soundness, for example. Yes? > > Happy New Year! > > Happy ditto to you and all and sundry! > > Footnotes: > [1] I know you sometimes find my style a tad exasperating, but just > bear with me -- there's a point to this question, honest. (It is possible that you overestimate my exasperation; it is generally fleeting. In fact you are among the posters for whom I feel the most fondness, if that makes any sense to say of a person who I only know as a series of NNTP posts.) Marshall
From: Nam Nguyen on 28 Dec 2009 13:33 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?
From: zuhair on 28 Dec 2009 14:55 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. Zuhair
From: Nam Nguyen on 28 Dec 2009 20:00 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?
From: zuhair on 28 Dec 2009 20:43
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? Second, I agree that there must be some vague area at the end. That is inescapable. But my aim is to lessen this area. 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 |