An uncountable countable set
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From: David R Tribble on 1 Oct 2006 13:30 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. 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?
From: Han.deBruijn on 1 Oct 2006 13:51 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
From: Virgil on 1 Oct 2006 14:15 In article <1159698201.187574.90660(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1159648032.835876.237760(a)c28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Tony Orlow schrieb: > > > > > > >Do I "misunderstand" that if you remove balls 1, then 11, then > > > > 21, etc, that the vase will NOT be empty? > > > > > > 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. > > > > Then by all means. less us do way with the balls and keep only the > > numbers. > > So we will keep only the numbers. The vase will be empty at noon if it > emits 1,2,3,... but not empty if it emits 1, 11, 21, ...? Even for pure > numbers that point of split-brains view is lot too silly. Silliness, like beauty, is strictly in the mind of the beholder.
From: Virgil on 1 Oct 2006 14:17 In article <1159698322.721090.272040(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1159648683.273921.24350(a)e3g2000cwe.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > > > > > Dik T. Winter schrieb: > > > > > > > > > > > > The successor function *is* counting (+1). > > > > > > > > > > > > Wrong. > > > > > > > > > > After a while you will have run out of the predefined successor, > > > > > unavoidably. > > > > > > > > succ(x) = {x}. > > > > > > That is nothing else but a veiled form of +1. (This form of addition > > > of 1 is due to Zermelo's, a little bit different from that of von > > > Neumann's.) > > > > "Mueckenh" has it backwards, "+1" is merely a veiled form of "another", > > which predates counting by millennia. And both Zermelo's and von > > Nuemann's successor echo "another" faithfully without any requirement > > for "+1". > > Another one. Another of whatever there are some of. > Yes, that is +1. Only to those who are so compulsive about it that they cannot think. > > Regards, WM
From: Han.deBruijn on 1 Oct 2006 14:20
stephen(a)nomail.com schreef: > I suppose I should clarify this. You can approach the infinite > using the the limit concept, but you always have to be careful > when using limits, and you have to be precise about what you > mean by the limit. Okay. But the point is whether there exist infinities that can _not_ be approached using the limit concept. Obviously they exist, because how can we approach e.g. the Continuum Hypothesis by employing limts? Han de Bruijn |