An uncountable countable set
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From: Han de Bruijn on 4 Oct 2006 04:03 Tony Orlow wrote: > Dik T. Winter wrote: > >> In article <40ef7$452210c2$82a1e228$23007(a)news1.tudelft.nl> Han de >> Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: >> ... >> > We can say that the number of balls Bk at step k = 1,2,3,4, ... is: >> > Bk = 9 + 9.ln(-1/tk)/ln(2) where tk = - 1/2^(k-1) for all k in N . >> > And that's ALL we can say. >> >> Why the obfuscation? Why not simply Bk = 9 + 9.(k - 1) = 9.k? >> >> > And that's ALL we can say. The version of the problem used here is >> > the first experiment in: >> >> Not only of the first, but also of the second experiment. >> >> But strange as it may appear, the two experiments give different results. >> In the first experiment there is no ball that escapes from being taken >> out. In the second experiment there are quite a few balls that are never >> taken out. The whole point is that you can not use limits to determine >> what is the valid answer. > > Hi Dik. I think the point that WM, Han, myself and others are trying to > make is that limits gives a more reasonable answer than transfinite set > theory. Why is it more credible to have the balls disappear due to > labeling, than to apply the infinite series and see that it diverges? Precisely! Every mathematician living before Cantor would have taken the mere fact that the infinite sequence diverges as decisive evidence that searching for an answer "at noon" is quite meaningless. But not so with the current un-discipline which still dares to call itself mathematics. Han de Bruijn
From: Han de Bruijn on 4 Oct 2006 04:06 Virgil wrote: > In article <45231542$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>Dik T. Winter wrote: >> >>>In article <40ef7$452210c2$82a1e228$23007(a)news1.tudelft.nl> Han de Bruijn >>><Han.deBruijn(a)DTO.TUDelft.NL> writes: >>>... >>> > We can say that the number of balls Bk at step k = 1,2,3,4, ... is: >>> > Bk = 9 + 9.ln(-1/tk)/ln(2) where tk = - 1/2^(k-1) for all k in N . >>> > And that's ALL we can say. >>> >>>Why the obfuscation? Why not simply Bk = 9 + 9.(k - 1) = 9.k? >>> >>> > And that's ALL we can say. The version of the problem used here is >>> > the first experiment in: >>> >>>Not only of the first, but also of the second experiment. >>> >>>But strange as it may appear, the two experiments give different results. >>>In the first experiment there is no ball that escapes from being taken >>>out. In the second experiment there are quite a few balls that are never >>>taken out. The whole point is that you can not use limits to determine >>>what is the valid answer. >> >>Hi Dik. I think the point that WM, Han, myself and others are trying to >>make is that limits gives a more reasonable answer than transfinite set >>theory. Why is it more credible to have the balls disappear due to >>labeling, than to apply the infinite series and see that it diverges? > > 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. You can find this in any first year calculus text book. Han de Bruijn
From: Mike Kelly on 4 Oct 2006 06:31 mueckenh(a)rz.fh-augsburg.de wrote: > Han de Bruijn schrieb: > > > 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. > > You are right, but the illness does not begin with the vase, it beginns > already with the assumption that meaningful results could be obtained > under the premise that infinie sets like |N did actually exist. The meaningful result is that if you allow "|N exists" then the vase empties at noon. Even if you don't allow that in your mathematics, you can surely accept the logical conclusion that IF you allow that THEN the vase is empty at noon. No? There is no logical contradiction in the existence of |N, by the way. You're free to reject it in your versions of mathematics (which I have no doubt are of great use and value) but it's absurd to claim that mathematics that does accept it is meaningless. If you are arguing that the existence of |N is contradictory and anything derived from it is meaningless then what is the point of arguing about the vase problem? What is the point of arguing about the random natural problem? You don't accept a premise that those problems mean nothing without. Restrict yourself to attempting to show that the axiom of infinity (or existence of |N, if you like) is contradictory. -- mike.
From: Mike Kelly on 4 Oct 2006 06:49 Tony Orlow wrote: > Virgil wrote: > > In article <1159798407.217254.275710(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Tony Orlow schrieb: > >> > >> > >>>> This is an extremely good example that shows that set theory is at > >>>> least for physics and, more generally, for any science, completely > >>>> meaningless. Because the numbers on the balls cannot play any role > >>>> except for set-theory-believers. > >>> Yes. I was flabbergasted by this example of "logic". The amazing thing > >>> is, set theory is supposed to apply where we know nothing except for the > >>> membership status of each element in a set, and yet, here is applied > >>> this property of labels that set theorists claim is crucial to answering > >>> the question. Set theory in the finite sense is a fine thing, but when > >>> it comes to the infinite case, set theorists don't even know anymore > >>> what they're TRYING to do. > >> One should think that set theory, if useful at all, should be capable > >> of treating problems like this. But here we see it fail with > >> gracefulness and mastery. > >> > >> Set theorists always see only the one ball escaping the vase but not > >> the 9 remaining there. So they can accept that there are as many > >> natural numbers as rational numbers. I cannot understand how this > >> theory could invade mathematics and how I could believe it over many > >> years without a shade of doubt. > >> > >> Regards, WM > > > > If we alter the problem to start with all the balls in the vase and > > remove them according to the original schedule then every ball spends at > > least as much time in the vase as before, but not everyone will see that > > the vase is empty at noon. > > > > Those who argue that having the balls spend less time in the vase leaves > > more of them in the vase as noon, have some explaining to do. > > No, if you start with all the balls in the vase, it clearly becomes > empty, but when you are adding more balls per iteration than you are > removing, the eventual emptying of the vase is impossible. If I rub the > vase, will a genie come out? Even though exactly the same balls are added and removed in each case? If that's part of your mathematics then I really do find it counter-intuitive! Back to the drawing board? -- mike.
From: imaginatorium on 4 Oct 2006 08:11
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? Brian Chandler http://imaginatorium.org |