An uncountable countable set
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From: Virgil on 5 Oct 2006 15:21 In article <41462$45251982$82a1e228$28904(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Randy Poe wrote: > > > Han de Bruijn wrote: > > > >>Virgil wrote: > >> > >>>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. > > > > That's a property of the vase, not the clock. The clock > > ticks on independently of the imaginary process > > we're timing with it. > > No. The clock ticks because there is no Balls in a Vase in the universe. HdB's clock doesn't seem to be ticking at all. > > Han de Bruijn
From: Virgil on 5 Oct 2006 15:24 In article <3513a$452519d6$82a1e228$28904(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Randy Poe wrote: > > > Do you think anyone reading this thread, beside you, > > understands what you mean by that phrase? > > Sure. > > Han de Bruijn However "self-explanatory" HdB thinks it, he has presented no convincing evidence to support it.
From: Virgil on 5 Oct 2006 15:29 In article <45251b3e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4523c954$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David R Tribble wrote: > >>> Tony Orlow wrote: > >>>>> On the other hand > >>>>> I don't know why I said "neither can the reals". In any case, the only > >>>>> way the ordinals manage to be "well ordered" is because they're defined > >>>>> with predecessor discontinuities at the limit ordinals, including 0. > >>>>> That doesn't seem "real" > >>> Virgil wrote: > >>>>> In what sense of "real". There are subsets of the reals which are order > >>>>> isomorphic to every countable ordinal, including those with limit > >>>>> ordinals, so until one posits uncountable ordinals there are no > >>>>> problems. > >>> Tony Orlow wrote: > >>>> The real line is a line, with > >>>> each point touching two others. > >>> That's a neat trick, considering that between any two points there is > >>> always another point. An infinite number of points between any two, > >>> in fact. So how do you choose two points in the real number line > >>> that "touch"? > >>> > >> They have to be infinitely close, so actually, they have an > >> infinitesimal segment between them. :) > > > > But any "infinitesimal segment" within the reals is bisectable. > > Within the standard reals, it's one number, if it's closer than any > finite distance of a that number. In Standard reals,"infinitesimal", if it means anything, merely means very small but not zero. In The Robinson, or similar, non-standard models, infinitesimals are different from standard numbers but still non-zero. In both, they are bisectable, and between two distinct numbers, even when only infinitesimally different, there is always another.
From: Virgil on 5 Oct 2006 15:34 In article <45251bc5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4523cb30(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Mike Kelly wrote: > >>> 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? > >> Only if you change the order of events, or refuse to say when the vase > >> empties or how. Any "|N" aside, the problem clearly states that ten > >> balls are added and then one removed, per iteration > > > > It also says precisely which numbered balls are added at which times and > > which numbered balls are removed at which times. Absent that > > information, one has a different puzzle which has an indeterminant > > result. > > > > It also says which balls remain when each is taken out, namely, when > ball n is removed, balls n+1 through 10n remain. For a while. But the fact remains that for eacn n in N, there is a specific time before noon at which ball n is removed. > > > > That assumes that there would have to be a "last ball", which equally > > assumes that there would have to be a "last natural number", which > > destroys TO's analysis. > > Uh, no, the very conclusion that the vase empties, when at most one ball > is removed at a time, implies that there is a last ball removed That is TO's assumption, contrary to the facts required by the experiment. Infinite processes can end in finite time or else Zeno's 'paradoxes' would prevent all action.
From: Virgil on 5 Oct 2006 15:38
In article <1160066342.476227.262000(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > 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. > > This variant shows for another time that the result is completely > undefined. (I assume you made a Freudian typo but wanted to say: "but > now everyone will see that the vase is empty at noon". And you are > right, at least for those who think that all numbers exist.) "Mueckenh" still does not see the inevitable, so what I said was right. > > > > 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, for they know that to talk about all the balls yields nonsense. The nonsense is talked by those who claim others talk nonsense. |