An uncountable countable set
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From: Tony Orlow on 2 Oct 2006 14:28 Han de Bruijn wrote: > Tony Orlow wrote: > >> They are saying that the vase empties because every ball inserted is >> removed. They agree that this does not occur before noon, when there >> are always balls in the vase, but by noon the vase is empty. But they >> cannot say that, even though there are balls before noon, and none at >> noon, that the vase "became empty" at noon, because they are claiming >> "the limit doesn't exist". So, don't ask me what they mean. I can't >> figure it out. > > So far for Tony Orlow. But neither can Han de Bruijn, neither can David > Petry, neither can Wolfgang Mueckenheim, neither can Jeroen Boschma and > neither can 'snapdragon', neither can 'rennie nelson' - to mention only > a few others - nobody of them can figure out what they mean. Therefore > I wonder if this mainstream mathematics "solution" still has a majority > of yes-voters behind it. > > Han de Bruijn > And, I thought it was just me. ;)
From: Tony Orlow on 2 Oct 2006 14:29 stephen(a)nomail.com wrote: > Han.deBruijn(a)dto.tudelft.nl wrote: >> stephen(a)nomail.com wrote: > >>> 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: >>>>> >>>>>> 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. >>>> But you _can_ do it at any time _before_ noon. There is no limit >>>> of the number of balls before noon which converges at noon. >>>> But you _can_ do it with any finite contiguous set of naturals. >>>> Then you find floor(n/a)/n and with limit(n -> oo) find 1/a . >>> But in neither case are you performing the actual "experiment". >>> In the balls in the vase "experiment", for every ball there >>> is a definite time at which it is removed. Your finite approximation >>> throws out that fact, so it is not surprising that it gets >>> the wrong answer. You have fundamentally changed the "experiment". > >> Worse. I have fundamentally changed the mathematics. Such that it shall >> no longer claim to have the "right" answer to an ill posed question. > >> Han de Bruijn > > 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? > > Stephen > Any true mathematics reserves decision until some number of iterations, at least as a variable, has been declared.
From: Tony Orlow on 2 Oct 2006 14:32 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. > >> Cardinality is a weak measure of size for infinite sets, the operative >> word here being "measure". > > It is a wrong measure. Cp. the vase. > > Regards, WM > It is a non-measure, being only a rough and ill-founded classification.
From: Tony Orlow on 2 Oct 2006 14:34 mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: >>> Another one. Yes, that is +1. >>> >>> Regards, WM >>> >> I dunno Wolfgang. I think you're going out on a limb to say that one is >> the same as '1', and that "+" means "in addition to what's already >> there". ;) > > Tony, I think that "1" is a short symbol for "one" and that "+1" is a > short symbol for "put another one" to the present set. But I don't > think that this is very important. > > Regards, WM > Oh Wolfgang, I was being completely sarcastic about you going out on a limb with that statement. T'was a joke. Your statement is self-evident. You seem to be standing on quite a strong branch, from where I hold the ladder. :) Tony
From: Virgil on 2 Oct 2006 14:37
In article <d7589$4520d3d9$82a1e228$28138(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1159727459.165196.109230(a)b28g2000cwb.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Nature does not jump, Leibnitz said. > > > > Who said anything about this being "nature"? > > "Nature" does not have an endless supply of balls in the first place, > > Definitely. And "Nature" does also not have an endless supply of dx's in > integral(0,1)dx = 1 . Yet this integral is of significance to physics. Is HdB referring to the integral from 0 to 1 of the constant function f(x) = 1 ? If so, it is defined as a limit process in which there are never more that a finite number of delta_x's needed to determine the limit. Does HdB insist that every part of mathematics that is not currently known to be significant to physics be round-filed? If so, he would stifle physics as well as mathematics, at least if history is any indication. History is rife with mathematical developments that, at the time of their development, had no uses in physics or any other science, but which later turned out to be essential to some science's further development. > > >>>Noon is the first moment at which the vase is empty. > >> > >>>But noon is not the transitional moment. There's no > >>>time just before noon where the transition happened. > >> > >>Wow ! And _that_ calls himself a physicist ... > > > > Leave it to a physicalist to insist that a nonphysical problem is > > physics. > > According to Virgil's criteria, most of theoretical physics would be > unphysical. Not at all. I keep my hands off declaring of what is or is not physics, and suggest that HdB keep his hands off declaring what is or is not mathematics. |