Another AC anomaly?
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From: Virgil on 9 Dec 2009 22:38 In article <QP-dnV0EIYPt2b3WnZ2dnUVZ_h2dnZ2d(a)giganews.com>, "K_h" <KHolmes(a)SX729.com> wrote: > "Virgil" <Virgil(a)home.esc> wrote in message > news:Virgil-1E8B09.00355309122009(a)newsfarm.iad.highwinds-media.com... > > In article > > <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d(a)giganews.com>, > > "K_h" <KHolmes(a)SX729.com> wrote: > > > >> It should be pointed out that N is a limit set even if N > >> is > >> initially given by a definition that doesn't involve the > >> notion of a limit. > > > > > > The issue between Dik and WM is whether the limit of a > > sequence of sets > > according to Dik's definition of such limits is > > necessarily the same as > > the limit of the sequence of cardinalities for those sets. > > > > And Dik quire successfully gave an example in which the > > limits differ. > > I suspect those definitions are not valid. Irrelevant to whether that definition "is valid" or corresponds to anyone else's. That is the definition under which WM claimed the two limit processes coincide, but they do not.
From: WM on 10 Dec 2009 01:53 On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > When using an intermediate reservoir, as shown in my > > lesson > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > 22 > > it becomes clear that N cannot be generated by adding > > number after > > number. > > Why not? Say we have an infinitely large sheet of paper and > we print each natural number, n, on the paper at time > t=1-1/(n+1). Certainly at time t=1 we have all the naturals > printed on the page. It seems so. But it is wrong. You see it if you consider the alternative process using an intermediate reservoir as "realized" in my lesson above. naturals - reservoir - paper N - { } - { } N/{1} - {1} - { } N/{1,2} - {2} - {1} N/{1,2,3} - {3} - {1,2} .... N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} .... The set in the middle contains a number at every time after t = 0. Hence this number cannot yet have been printed on the paper (because it will be printed only after its follower will have entered the reservoir). Regards, WM
From: Virgil on 10 Dec 2009 02:19 In article <b963b9aa-c345-43cf-bf78-e9e27401f539(a)c34g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > > > When using an intermediate reservoir, as shown in my > > > lesson > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > > 22 > > > it becomes clear that N cannot be generated by adding > > > number after > > > number. > > > > Why not? �Say we have an infinitely large sheet of paper and > > we print each natural number, n, on the paper at time > > t=1-1/(n+1). �Certainly at time t=1 we have all the naturals > > printed on the page. > > It seems so. But it is wrong. You see it if you consider the > alternative process using an intermediate reservoir as "realized" in > my lesson above. > > naturals - reservoir - paper > N - { } - { } > N/{1} - {1} - { } > N/{1,2} - {2} - {1} > N/{1,2,3} - {3} - {1,2} > ... > N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} > ... > > The set in the middle contains a number at every time after t = 0. > Hence this number cannot yet have been printed on the paper (because > it will be printed only after its follower will have entered the > reservoir). And when all the numbers have passed through your "reservoir", both into and out ofd it, as will have happened by t = 1, which numbers does WM claim will still be unprinted. When going through the terms of an infinite sequence, as in the above, EITHER the process hangs up on a particular term of that sequence OR it goes through every term of the sequence, TERTIUM NON DATUR.
From: WM on 10 Dec 2009 08:19 On 10 Dez., 08:19, Virgil <Vir...(a)home.esc> wrote: > In article > <b963b9aa-c345-43cf-bf78-e9e27401f...(a)c34g2000yqn.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > > > > When using an intermediate reservoir, as shown in my > > > > lesson > > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > > > 22 > > > > it becomes clear that N cannot be generated by adding > > > > number after > > > > number. > > > > Why not? Say we have an infinitely large sheet of paper and > > > we print each natural number, n, on the paper at time > > > t=1-1/(n+1). Certainly at time t=1 we have all the naturals > > > printed on the page. > > > It seems so. But it is wrong. You see it if you consider the > > alternative process using an intermediate reservoir as "realized" in > > my lesson above. > > > naturals - reservoir - paper > > N - { } - { } > > N/{1} - {1} - { } > > N/{1,2} - {2} - {1} > > N/{1,2,3} - {3} - {1,2} > > ... > > N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} > > ... > > > The set in the middle contains a number at every time after t = 0. > > Hence this number cannot yet have been printed on the paper (because > > it will be printed only after its follower will have entered the > > reservoir). > > And when all the numbers have passed through your "reservoir", both into > and out ofd it, as will have happened by t = 1, which numbers does WM > claim will still be unprinted. Then a last one would have had to leave the intermediate reservoir. > > When going through the terms of an infinite sequence, as in the above, > EITHER the process hangs up on a particular term of that sequence OR it > goes through every term of the sequence, TERTIUM NON DATUR.- Or it is proved that the assumption of finished infinity is nonsense. Secundum non datur. Regards, WM
From: Dik T. Winter on 10 Dec 2009 08:54
In article <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuAGqH.FrI(a)cwi.nl... .... > > Not at all. When you define N as an infinite union there > > is no limit > > involved, there is even no sequence involved. N follows > > immediately > > from the axioms. > > I disagree. Please note that I am not endorsing many of > WM's claims. There are many equivalent ways of defining N. > I have seen the definition that Rucker uses, in his infinity > and mind book, in a number of books on mathematics and set > theory: On page 240 of his book he defines: > > a_(n+1) = a_n Union {a_n} > > and then: > > a = limit a_n. But here an infinite union is *not* involved, that is the crucial difference. As stated, you may define N as a limit or not, and when it is defined as an infinite union as in: N = union {1, 2, ..., n} a limit is not involved. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |