Another AC anomaly?
in [Logic]
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From: K_h on 9 Dec 2009 19:44 "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. The definition I used is the one on wikipedia and is generally `standard' -- as I've seen it in numerous places, including books and websites. That definition is below. Using this definition it is possible to prove that N is a limit and I presented a proof of this in my previous post. Let /\ = Intersection Let \/ = union Define infimum and supremum as follows: liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] (n-->oo) limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] (n-->oo) > Here is another way to see that N is a >> limit even if you consider it bad taste to define it in >> those terms: >> >> Let \/ = union >> >> Let /\ = Intersection >> >> Define infimum and supremum as follows: >> >> liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] >> (n-->oo) >> >> limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] >> (n-->oo) >> >> If these two are the same then the limit exists and is >> both >> of them. > > The issue is not whether the naturals are such a limit but > whether for > every so defined limit the cardinality of the limit equals > the limit > cardinality of their cardinalities, which is different > sort of limit. It doesn't sound like a valid limit. With the definitions I provided it is possible to prove, in your words, that "the cardinality of the limit equals the cardinality of their cardinalities". It is actually intuitively obvious: the set of finite cardinals 0,1,2,3,... and the set of finite ordinals 0,1,2,3,... both have cardinality ALEPH_0. k
From: K_h on 9 Dec 2009 19:58 "WM" <mueckenh(a)rz.fh-augsburg.de> wrote in message news:b6fe23bb-f753-4cfc-a9d6-79711f3a1e5d(a)r40g2000yqn.googlegroups.com... > On 9 Dez., 08:35, Virgil <Vir...(a)home.esc> wrote: >> In article >> <c8idnc6JG6hv34LWnZ2dnUVZ_q-dn...(a)giganews.com>, > > >> However, in the discussion between Dik and WM, Dik gave >> SPECIFIC >> definition of what HE meant by the limit of a seqeunce of >> sets which >> differed from that in your citation. > > Dik: > Given a sequence of sets S_n then: > lim sup{n -> oo} S_n contains those elements that occur > in > infinitely > many S_n I don't think this is a good definition for the limit of a sequence of sets S_n. For instance, consider the alternating sets: S_0 = {0, 2, 4, 6, 8,...} S_1 = {-1, -3, -5, ...} S_2 = {0, 2, 4, 6, 8,...} S_3 = {-1, -3, -5, ...} S_4 = {0, 2, 4, 6, 8,...} S_5 = {-1, -3, -5, ...} .... To me, I don't see the S_n converging to anything, certainly not the set of negative odd numbers union the set of even numbers. Like Thompson's lamp alternating between on and off it reaches no limit in any intuitive sense of what a limit is. > lim inf{n -> oo} S_n contains those elements that occur > in all S_n > from > a certain S_n (which can be different > for each > element). > lim{n -> oo} S_n exists whenever lim sup and lim inf are > equal. >> >> > Define infimum and supremum as follows: >> >> > liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] >> > (n-->oo) >> >> > limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] >> > (n-->oo) >> >> > If these two are the same then the limit exists and is >> > both >> > of them. >> >> The issue is not whether the naturals are such a limit >> but whether for >> every so defined limit the cardinality of the limit >> equals the limit >> cardinality of their cardinalities, which is different >> sort of limit > > If the limit set emerges continuously (i.e. growing > one-by-one) from > the sets of the sequence, then the cardinality will be > bound to this > process and will also continuously emerge from the > cardinalities of > the sets of the sequence. > > If there is a gap in one sequence and no gap in the other, > then this > indicates that at least one of the limits does not exist, > probably > both do not exist. > > This has been substanciated by the following > thought-experiment: > If N is generated as a limit, then it is generated by > adding number > after number. > 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 natruals printed on the page. > Therefore neither N nor Card(N) are meaningful concepts - > at least not > in mathematics. k
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. |