An uncountable countable set
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From: mueckenh on 2 Oct 2006 10:00 Virgil schrieb: > > By the only meaningful and consistent definition: A n eps |N : > > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > Do you challenge its truth? > > I challenge the "truth" of its being the ONLY meaningful and consistent > definition. > If we insist on unique results in mathematics, then this definition and the bijection exclude each other. > > > Let a be the number such that a = 4 and a = 5 > > > is erroneous? I think not. It is a proper definition, but there is just > > > no 'a' that satisfies the definition. > > > > It is erroneous, because you say let a *be* which is false, if a cannot > > *be*. > > There is no such thing as an "erroneous" definition, except possibly in > the sense of a grammatically incorrect one. A definition may lack any > instantiation, such as a 4 sided triangle, but as a definition is valid. A definition may be nonsense like that above. You may accept it nevertheless, because you got used to it. I call it erroneous or false. Regards, WM
From: mueckenh on 2 Oct 2006 10:13 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
From: Han de Bruijn on 2 Oct 2006 10:14 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
From: stephen on 2 Oct 2006 10:10 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
From: mueckenh on 2 Oct 2006 10:24
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. > 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 |