An uncountable countable set
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From: Virgil on 5 Oct 2006 03:53 In article <541bc$4524b3c5$82a1e228$28433(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <b8b9$45236b93$82a1e228$13353(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote: > >> > >>>That the "series" diverges means that there is no such thing as a limit, > >>>so that method does not say anything about the result. > >> > >>That method says there is no result at noon. > > > > Not so. That method merely says that IT cannot tell what happens at noon. > > Don't know who IT is. That method merely says that that method cannot tell what happens at noon. > > > It definitely does NOT say that there is no other method that can tell > > what happens at noon. > > There is no other method. That method does not say that, and the whole point is that there are, in some instances, other methods. > > >>You can find this in any > >>first year calculus text book. > > > > I have looked in several calculus books, starting with Apostol's, and > > found no such thing in any of them. They are all careful to say that, > > absent convergence, limit definitions say nothing about what happens. > > There are no other definitions of the infinite than limit definitions. Perhaps not in the books HdB chooses to read, but there are in some of the books I choose to read.
From: Virgil on 5 Oct 2006 03:56 In article <22028$4524b4dc$82a1e228$28724(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <161ca$4523bb80$82a1e228$8996(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>imaginatorium(a)despammed.com wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>The question is: how many balls are there in the vase at noon. > >>>>This question is meaningless, because noon is never reached. > >>> > >>>Really? When's lunch, then? > >> > >>Time is _suggested_, but not present, in the Balls in a Vase problem. > > > > Without any time to put balls into the vase, the vase is empty. > > In real time there are no singularities. With the Balls in a Vase there > is a singularity at noon. Therefore the quantity called "time" resembles > real time sometimes, but not always. There is no part of the gedankenexperiment as stated that can be run in "real time", so the constraints allegedly imposed by properties of "real time" are irrelevant.
From: Han de Bruijn on 5 Oct 2006 04:01 Virgil wrote: > In article <22028$4524b4dc$82a1e228$28724(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <161ca$4523bb80$82a1e228$8996(a)news2.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>imaginatorium(a)despammed.com wrote: >>>> >>>>>Han de Bruijn wrote: >>>>> >>>>>>The question is: how many balls are there in the vase at noon. >>>>>>This question is meaningless, because noon is never reached. >>>>> >>>>>Really? When's lunch, then? >>>> >>>>Time is _suggested_, but not present, in the Balls in a Vase problem. >>> >>>Without any time to put balls into the vase, the vase is empty. >> >>In real time there are no singularities. With the Balls in a Vase there >>is a singularity at noon. Therefore the quantity called "time" resembles >>real time sometimes, but not always. > > There is no part of the gedankenexperiment as stated that can be run in > "real time", so the constraints allegedly imposed by properties of > "real time" are irrelevant. Precisely! Such as the constraint that something must happen at noon. Han de Bruijn
From: mueckenh on 5 Oct 2006 04:17 Tony Orlow schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Tony Orlow schrieb: > > > > > >>>> Why not? Each and every number of the list terminates. That one is a number > >>>> that does *not* terminate. > >>>> > >>>> > If you think that 0.111... is a number, but not in the list, > > > >>> It is me who insists that it is not a representation of a number. > >> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" > >> number. It's still some kind of quantitative expression, even if it's > >> unbounded. Would you agree that ...333>...111, given a digital number > >> system where 3>1? > > > > That is the similar to 0.333... > 0.111.... But all these > > representations exist only potentially, in my opinion. The difference > > is, that 0.333... can be shown to lie between two existing numbers, so > > we can calculate with it, while for ...333 this cannot be shown. > > I think it can be shown to lie between ...111 and ...555, given that > each digit is greater than the corresponding digit in the first, and > less than the corresponding digit in the second. Yes, but only if we define, for instance, A n eps |N : 111...1 < 333...3 where n digits are symbolized in both cases. This approach would be comparable with the "measure" which gives A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. I don't know whether these definitions are of any use, but I am sure that they are not less useful than Cantor's cardinality. Regards, WM ..
From: mueckenh on 5 Oct 2006 04:22
Tony Orlow schrieb: > Han de Bruijn wrote: > > stephen(a)nomail.com wrote: > > > >> Han.deBruijn(a)dto.tudelft.nl wrote: > >> > >>> Worse. I have fundamentally changed the mathematics. Such that it shall > >>> no longer claim to have the "right" answer to an ill posed question. > >> > >> Changed the mathematics? What does that mean? > >> The mathematics used in the balls and vase problem > >> is trivial. Each ball is put into the vase at a specific > >> time before noon, and each ball is removed from the vase at > >> a specific time before noon. Pick any arbitrary ball, > >> and we know exactly when it was added, and exactly when it > >> was removed, and every ball is removed. > >> Consider this rephrasing of the question: > >> > >> you have a set of n balls labelled 0...n-1. > >> > >> ball #m is added to the vase at time 1/2^(m/10) minutes > >> before noon. > >> > >> ball #m is removed from the vase at time 1/2^m minutes > >> before noon. > >> > >> how many balls are in the vase at noon? > >> > >> What does your "mathematics" say the answer to this > >> question is, in the "limit" as n approaches infinity? > > > > My mathematics says that it is an ill-posed question. And it doesn't > > give an answer to ill-posed questions. > > > > Han de Bruijn > > > > Actually, that question is not ill-posed, and has a clear answer. The > vase will be empty, if there is any limit on the number of balls, and > balls can be removed before more balls are added, but it is not the > original problem, which states clearly that ten balls are inserted, > before each one that is removed. That's the salient property of the > gedanken. Any other scheme, such as labeling the balls and applying > transfinitology, violates this basic sequential property, and so is a ruse. There are two equivalent truths: (1) (Each) ball number n will come out before noon. (2) When ball number n comes out, more than n balls remain in the vase. Both are absolutely correct. This shows that one can not consistently calculate with infinity. I published a similar but less lucid example as the 100 Euro question on my homepage: http://www.fh-augsburg.de/~mueckenh/Infinity/PreisfrageE%27.pdf For all sequences {2,4,6,...,2n} there exists z eps {2,4,6,...,2n} such that |{2,4,6,...,2n}| < z. But in the limit there is not such a z eps {2,4,6,...}. We have always |{2,4,6,...}| > z. Or briefly lim [n-->oo] 2n/n < 1. The first proof of this is worth 100 Euro. Regards, WM |