An uncountable countable set
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From: David R Tribble on 4 Oct 2006 01:22 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: > In the sense that the real world is continuous, and you don't just have > these beginnings with nothing before them. 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"?
From: imaginatorium on 4 Oct 2006 02:15 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: > > In the sense that the real world is continuous, and you don't just have > > these beginnings with nothing before them. 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"? Me! Me!! I can do this one!! Er, well, here's how. You choose a first point (P). Easy, yes? Now choose another point (Q) fairly close to P. Of course, you're right, there's going to be another point P1 between P and Q - so first, calculate the time to lunch, halve it, and when that time comes choose a new point P1, midway between P and Q. Repeat this procedure, so when half the remaining time to lunch has elapsed you choose P2, midway between P1 and Q. And so on. Well, at the instant lunchtime comes, the last point (Pz) you chose must be next to Q. Proof: Suppose R is a point between Pz and Q, then we would have chosen it before lunchtime, but now it's time for lunch. Contradiction. We have shown two adjacent points Pz and Q. You can't argue about how far apart they are (obviously an infinitesimal, since infinitesimals have the desired properties in all context), because I've gone to lunch. Brian Chandler http://imaginatorium.org
From: Virgil on 4 Oct 2006 02:21 In article <1159942504.753886.300900(a)h48g2000cwc.googlegroups.com>, imaginatorium(a)despammed.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: > > > In the sense that the real world is continuous, and you don't just have > > > these beginnings with nothing before them. 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"? > > Me! Me!! I can do this one!! > > Er, well, here's how. You choose a first point (P). Easy, yes? Now > choose another point (Q) fairly close to P. Of course, you're right, > there's going to be another point P1 between P and Q - so first, > calculate the time to lunch, halve it, and when that time comes choose > a new point P1, midway between P and Q. Repeat this procedure, so when > half the remaining time to lunch has elapsed you choose P2, midway > between P1 and Q. And so on. Well, at the instant lunchtime comes, the > last point (Pz) you chose must be next to Q. Proof: Suppose R is a > point between Pz and Q, then we would have chosen it before lunchtime, > but now it's time for lunch. Contradiction. > > We have shown two adjacent points Pz and Q. You can't argue about how > far apart they are (obviously an infinitesimal, since infinitesimals > have the desired properties in all context), because I've gone to > lunch. > > Brian Chandler > http://imaginatorium.org Bon appetit!
From: Han de Bruijn on 4 Oct 2006 03:35 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>We can say that the number of balls Bk at step k = 1,2,3,4, ... is: >>Bk = 9 + 9.ln(-1/tk)/ln(2) where tk = - 1/2^(k-1) for all k in N . >>And that's ALL we can say. The version of the problem used here is >>the first experiment in: > >>http://groups.google.nl/group/sci.math/msg/d2573fcb63cbf1f0?hl=en& > > Why can't we say that every ball that is added is also > removed? > > ball #m is added to the vase at time 1/2^(floor(m/10)) minutes > before noon. > > ball #m is removed from the vase at time 1/2^m minutes > > Every ball is removed before noon, no matter how many > balls there are. The question is: how many balls are there in the vase at noon. This question is meaningless, because noon is never reached. Han de Bruijn
From: Han de Bruijn on 4 Oct 2006 03:55
Dik T. Winter wrote: > In article <40ef7$452210c2$82a1e228$23007(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > ... > > We can say that the number of balls Bk at step k = 1,2,3,4, ... is: > > Bk = 9 + 9.ln(-1/tk)/ln(2) where tk = - 1/2^(k-1) for all k in N . > > And that's ALL we can say. > > Why the obfuscation? Why not simply Bk = 9 + 9.(k - 1) = 9.k? Because of the time stamps. > > And that's ALL we can say. The version of the problem used here is > > the first experiment in: > > Not only of the first, but also of the second experiment. Good. > But strange as it may appear, the two experiments give different results. > In the first experiment there is no ball that escapes from being taken > out. In the second experiment there are quite a few balls that are never > taken out. The whole point is that you can not use limits to determine > what is the valid answer. If you can not use limits to determine what is the valid answer with infinities, then there _is no_ valid answer. Approaching the infinite without limits (or rather taking it as a finished thing) demonstrates how mathematics has become an un-discipline instead of a discipline. There is nothing "strange as it may appear", or "accepting the counter intuitive" as with e.g the Theory of Relativity or Quantum Mechanics. Instead, there's only banality in Cantorian infinitary mathematics. Han de Bruijn |