An uncountable countable set
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From: imaginatorium on 4 Oct 2006 15:09 Lester Zick wrote: > On 4 Oct 2006 11:36:27 -0700, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: > >> On 3 Oct 2006 23:15:04 -0700, imaginatorium(a)despammed.com wrote: > ><snip> > ><snop> > >Very disappointed with your answer, Lester. You didn't use the word > >"technically" even once. > > "Technically" is reserved for answers, Brian, not questions. And > needless to say you've gotten so snippy with questions lately it's a > little difficult to tell if any answers qualify for a "technically". You can be irksome sometimes Herr-Professor Zick. OK, here's what you said: > Well perhaps not quite so easy as you might imagine, Brian. Perhaps > you'd care to spell out exactly how you choose a first point that's so > easy? Now please don't try to beg off from the problem under the > pretext that actually choosing points is easy since it's an excercise > in physics where you don't excel but not in the zen of mathematics > where you don't seem to excel either. Just tell us how the trick is > done that's so easy that's anything other than you're saying it's so. Is the first sentence of this not a statement? An answer? Is it not an answer that might be raised to something more resembling your usual intellectual level by saying - for example - "Well technically, perhaps not quite so easy as you might imagine, Brian." Incidentally, you think "choosing a point" is something in physics? Like "choosing a refrigerator" is something in domestic science? Hmm, well, I confess: I really hadn't thought of that. If I merely chose pi+2 as a point on the real line what would that be then? (Don't be too zen about it if possible. Of course, since in the end we all learn by observing a master at work, if there's something unsatisfactory about my choice, I'm sure you'll demonstrate your own techniques for doing it.) Brian Chandler http://imaginatorium.org
From: Virgil on 4 Oct 2006 15:11 In article <4523c954$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.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: > >> 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"? > > > > They have to be infinitely close, so actually, they have an > infinitesimal segment between them. :) But any "infinitesimal segment" within the reals is bisectable.
From: Virgil on 4 Oct 2006 15:19 In article <4523cb30(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> Han de Bruijn schrieb: > >> > >>> 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. > >> You are right, but the illness does not begin with the vase, it beginns > >> already with the assumption that meaningful results could be obtained > >> under the premise that infinie sets like |N did actually exist. > > > > The meaningful result is that if you allow "|N exists" then the vase > > empties at noon. Even if you don't allow that in your mathematics, you > > can surely accept the logical conclusion that IF you allow that THEN > > the vase is empty at noon. No? > > Only if you change the order of events, or refuse to say when the vase > empties or how. Any "|N" aside, the problem clearly states that ten > balls are added and then one removed, per iteration It also says precisely which numbered balls are added at which times and which numbered balls are removed at which times. Absent that information, one has a different puzzle which has an indeterminant result. > so if the vase > emptied, it could only be with the removal of that 1 ball > not with the > addition of the ten balls, since that would require that there had been > -10 balls in the vase. But, for there to be 1 ball left, which when > removed left an empty vase, ten would have been inserted right > beforehand, meaning there had to have been -9 balls in the vase. Neither > negative count is possible, therefore the vase could not have emptied. That assumes that there would have to be a "last ball", which equally assumes that there would have to be a "last natural number", which destroys TO's analysis.
From: Lester Zick on 4 Oct 2006 15:29 On 4 Oct 2006 10:21:09 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >Lester Zick wrote: >> On 4 Oct 2006 05:19:38 -0700, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: >> >> >Tony Orlow wrote: >> >> [. . .] >> >> >Note : I agree with those who say it makes no sense in physical terms >> >to have an infinite number of balls. But mathematics is an idealisation >> >so it can make sense to talk about the infinite, even if it is >> >physically impossible. >> >> So physics has to make sense but math doesn't? > >Math has to be logical. It doesn't have to be physically >realizable. Did anyone say it does? And in response to your contention that math has to be logical I would just paraphrase my objection above: So math has to be logical but physics doesn't? >> I think you'd find >> plenty of quantum theorists, hyperdimensionalists and relativists >> who'd disagree. > >I think you underestimate their sophistication. Quite possibly so. Indeed I usually tend to suspect a higher degree of sophistry than sophistication when people appeal to counter intuitive arguments to support their intuitions. Perhaps not so high a degree as in math because empiricism is pretty much all guesswork to begin with whereas modern mathemagicians simply appeal to a higher logical counter intuition to support their intuitions. > Physicists learn at >a young age that things can exist in mathematics that are not >physically realizable and yet are very useful. Such as quantum effects, hyperdimensionality, relativity, and counter intuition among which perhaps the most useful is counter intuition in support of their intuitions. > Point masses >and continuous mass distributions for instance. And they also >realize that things can exist in mathematics that aren't even >approximations of a physical realizable. That aren't physically >sensible in other words. Nor logical apparently. Especially when the counter intuition of "continuously distributed point masses" come into play. >You are conflating "physically sensible" with "logical", the same >way you mistake "agrees with my untrained common sense" with >"sensible". I never conflated your "untrained common sense" with "sensible". I do however conflate your counter intuitive rationalizations for your intuitions as nonsensical whether in physics or math. ~v~~
From: Virgil on 4 Oct 2006 15:33
In article <4523cd45(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> If distinguishing balls gives a less exact answer, > > > > Less exact how? > > > >> and a nonsensical one to boot > > > > It makes sense to me that if you put a ball into a vase and later > > remove it then it isn't there. It also makes sense to me that if you > > put a ball in a vase and don't remove it then it is still there. What > > *doesn't* make sense to me is that if you put some number of balls in a > > vase and remove them all then there are still some left. That seems to > > be what you are claiming. > > > > Note : I agree with those who say it makes no sense in physical terms > > to have an infinite number of balls. But mathematics is an idealisation > > so it can make sense to talk about the infinite, even if it is > > physically impossible. > > It makes no sense that adding ten balls and removing one will ever do > anything but increase the number of balls in the vase. It makes no sense > to choose a unfounded theory over basic logic which states that if you > have 0 balls at any iteration, you had -9 balls in the previous. And TO has just assumed a LAST iteration in an endless sequence of iterations. So TO is calling on HIS last natural number again. > It > makes no sense to choose labels without end over infinite series. The 10 and 9 and 1 as numbers of balls are equally labels with a first a second and a third, etc., ball. If we can't use numbers then TO can't either. > This > theory is at odds with everything around it. TO's theories always are at odds with everything around them, and with themselves, as well. > > But the number of balls in the vase at noon *isn't* the limit of that > > sum, Tony. Nobody disagrees that that sum diverges (of course, we might > > disagree that it diverges to a "specific actually infinite value", but > > I digress...), people disagree that the limit of that sum is the same > > thing as the number of balls in the vase at noon. > > Add 10, remove 1, repeat. > (+10-1)+(+10-1)+.... > 9+9+.... > > How many times were we doing this? You can name the balls after Pokemon > for all I care, this sum doesn't not approach 0. No one has said that if does "approach" 0. Consider the function defined on the "extended" real interval, [0,oo] by f(x) = x*sin(x) for x in [0,oo) and f(oo) = 0. According to TO's theories, the value of that function at oo cannot be 0. > > It seems a little barmy to spend a year arguing something that nobody > > disagrees with - that the sum 9, 18, 27... diverges. You should instead > > try to argue that the limit of that sum is equal to the number of balls > > in the vase at noon - that is what people are disagreeing with! Just > > saying "clearly" doesn't quite cut it. > > Limits are > math. Limit ordinals are not. That there is a great deal that is math to mathematicians but is not math to TO has long been obvious. |