An uncountable countable set
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From: Virgil on 4 Oct 2006 13:55 In article <47d15$45236440$82a1e228$12308(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>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. The version of the problem used here is > >>the first experiment in: > > > >>http://groups.google.nl/group/sci.math/msg/d2573fcb63cbf1f0?hl=en& > > > > Why can't we say that every ball that is added is also > > removed? > > > > ball #m is added to the vase at time 1/2^(floor(m/10)) minutes > > before noon. > > > > ball #m is removed from the vase at time 1/2^m minutes > > > > Every ball is removed before noon, no matter how many > > balls there are. > > The question is: how many balls are there in the vase at noon. > This question is meaningless, because noon is never reached. It was yesterday. At least here.
From: Virgil on 4 Oct 2006 13:57 In article <c107e$45236903$82a1e228$12673(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > there's only banality in Cantorian infinitary mathematics. Not as much as amount of the banality in opposing it.
From: Virgil on 4 Oct 2006 14:00 In article <858a$45236ad5$82a1e228$13144(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >! 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. And how does HdB know this? Has he contacted each and every one of them on his Ouija board?
From: Virgil on 4 Oct 2006 14:06 In article <b8b9$45236b93$82a1e228$13353(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <45231542$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >>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. Not so. That method merely says that IT cannot tell what happens at noon. It definitely does NOT say that there is no other method that can tell what happens at noon. > 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.
From: Virgil on 4 Oct 2006 14:09
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. |