An uncountable countable set
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From: Randy Poe on 2 Oct 2006 12:05 Tony Orlow wrote: > David R Tribble wrote: > > stephen wrote: > >>> In TO-matics, it is also possible to end up with > >>> an empty vase by simply adding balls. According to TO-matics > >>> ..1111111111 = 1 + 1 + 1 + 1 + ... > >>> and > >>> ..1111111111 + 1 = 0 > >>> > >>> So if you just keep on adding balls one at a time, > >>> at some point, the number of balls becomes zero. > >>> You have to add just the right number of balls. It is not > >>> clear what that number is, but it is clear that it > >>> exists in TO-matics. > > > > Tony Orlow wrote: > >> You drew that from my suggestion of the number circle, and that ...11111 > >> could be considered equal to -1. Since then, I looked it up. I'm not the > >> first to think that. It's one of two perspectives on the number line. > >> It's either really straight, or circular with infinite radius, making it > >> infinitesimally straight. The latter describes the finite universe, and > >> the former, the limit. But, you knew that, and are just trying to have fun. > > > > You are drawing geometric conclusions that are not warranted. > > The Projective Real Line is simply R U {oo}. Adding unsigned oo > > to the set allows certain arithmetic operations to be performed > > that are undefined in the regular real set. > > > > But simply adding the limit point oo to the set does not actually make > > it a "circle", because oo has no predecessor or successor, and > > certain operations like oo+1 and oo+oo are still meaningless within > > the set. > > You do realize that my statements involve a considerable amount of > personal reflection, don't you? There is more to the number circle than > "proven". In the binary number circle, "100...000" is both positive and > negative infinity. > > > > > See: > > http://en.wikipedia.org/wiki/Projective_line#Real_projective_line > > http://en.wikipedia.org/wiki/Division_by_zero#Real_projective_line > > http://en.wikipedia.org/wiki/Extended_real_number_line > > > > You obviously have something else in mind when you talk about > > the "number circle". Perhaps you could actually define it some time? > > > > There are a number of concepts in this area. Get acquainted with those > pages, think, and come back and talk. Translation: "No, I am no more able to define that than I am able to define anything else." - Randy
From: stephen on 2 Oct 2006 12:43 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> 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 That is a perfectly reasonable answer. But you do agree that for this problem, the vase is empty at noon for any finite n. So one wonders what criteria you used to determine that this infinity cannot be approached via limits. Stephen
From: Virgil on 2 Oct 2006 14:22 In article <d84b8$4520cdd3$82a1e228$26333(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > But that does not define a uniform distribution of a countably infinite > > set of naturals as that would require that enough 0's will add up to 1. > > Due to the mainstream mathematics doctrine that there is a JUMP between > "the" countably infinite set of naturals and all contiguous finite sets > of naturals {1,2,3,4,5 , ... , n}. Resulting in n.1/n becoming n.0 . > But nature does not jump (Leibniz: "Natura non facit saltus"). Nature and mathematics are not coextensive. So that mathematics can jump where nature cannot even reach.
From: MoeBlee on 2 Oct 2006 14:25 Han de Bruijn wrote: > Constructively valid proof. Intuition precedes axioms. A mathematician > is like an architect who builds his mathematics. Take a look at some > material concerning intuitionism and constructivism in the first place. What particular books or articles do you endorse? And what passages do you cite as claiming that constructivism is incompatible with axiomatic theory? MoeBlee
From: Virgil on 2 Oct 2006 14:26
In article <3d6a2$4520cf8f$82a1e228$26921(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <451d66c0(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>stephen(a)nomail.com wrote: > >> > >>>Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>> > >>>>Virgil wrote: > >>> > >>>>>In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > >>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>>>> > >>>>>>Randy Poe wrote, about the Balls in a Vase problem: > >>>>>> > >>>>>>>It definitely empties, since every ball you put in is > >>>>>>>later taken out. > >>>>>> > >>>>>>And _that_ individual calls himself a physicist? > >>>>> > >>>>>Does Han claim that there is any ball put in that is not taken out? > >>> > >>>>Nonsense question. Noon doesn't exist in this problem. > >>> > >>>Yes it is a nonsense question, in the sense > >>>that it is non-physical. You cannot actually perform > >>>the "experiment". Just as choosing a number uniformly > >>>from the set of all naturals is a non-physical nonsense > >>>question. You cannot perform that experiment either. > >> > >>Yes, they both sound equally invalid, and it all goes back to omega, but > >>Han has a point about the density of the set in the naturals throughout > >>its range, and the overall statistical probability of selecting one of > >>that subset from the naturals, even if having probabilities of 1/omega > >>for each natural presents problems. > > > > Do statistical probabilities have to satisfy the condition that their > > sum over all indivisible outcomes must equal 1? > > Precisely! Inasmuch as the sum over all indivisible outcomes must equal > 1 in the limit of the Riemann sum which represents the integral(0,1)dx. > > Han de Bruijn HdB is conflating probability over discrete spaces, like the naturals, with integration over continuous spaces, like the reals. The differences are sufficient to render his analogy fatally flawed. |