An uncountable countable set
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From: Han.deBruijn on 8 Oct 2006 13:59 Virgil wrote: > In article <45286ce5$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > David R Tribble wrote: > > > > What about: > > > sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) > > > The left half specifies the number of balls added to the vase, and > > > the right half specifies those that are removed. > > > > Do you mean: > > sum{n=0 to oo} (10) - sum{n=0 to oo} (1)? > > That sounds like what you re describing, and termwise the difference is > > sum(n=0 to oo) (9). That's infinite, eh? > > But the sums are not given termwise in the question, but sumwise, so > cannot be calculated termwise in your answer, but must be done sumwise. > > And sumwise they are no different. Wrong. Go back to Calculus school. Sumwise they are both undefined. Han de Bruijn
From: Lester Zick on 8 Oct 2006 14:08 On Sat, 7 Oct 2006 00:16:20 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <1160127267.123550.306050(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: >... > > > No reason at all, but when you invent a new notation, it would be better > > > if you did define the notation before use. > > > > Ok, but as we have agreement now, we can return to he main question: > > Why do you think that 0.111... with the index sequences 1,2,3,... or > > k+1,k+2,k+3 or -k, -k+1, -k+2, ... represents exactly *one* number > > only, as you asserted? > >Why do you still maintain that I think it represents a number? How many >times do I need to state that, without proper definition, it only is >a sequence of symbols that I on occasion call a "number". Because I have >not yet seen a definition of "number", and you have stated that you are >not able to give one... But whatever. As a sequence of symbols, >igoring the "0.", it is in bijection with N. It also is in bijection >with {k+1,k+2,...} for every k. As {k+1,k+2,...} is in bijection with N. Why does any of this matter? I mean why does anyone care whether you can match sets in such a way? What is it we can do or not do if this is true assuming it is true? Can we still do arithmetic? All I see is sets match and are equal in size if the number of elements in each matches one to one. We might just as well count the number of commas. ~v~~
From: Tony Orlow on 8 Oct 2006 14:26 Virgil wrote: > In article <45251b3e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> 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. >> Within the standard reals, it's one number, if it's closer than any >> finite distance of a that number. > > In Standard reals,"infinitesimal", if it means anything, merely means > very small but not zero. > In The Robinson, or similar, non-standard models, infinitesimals are > different from standard numbers but still non-zero. > In both, they are bisectable, and between two distinct numbers, even > when only infinitesimally different, there is always another. Yes, but that infinitesimal difference doesn't count in the STANDARD reals, does it? There is no requirement that any two standard reals which are infinitesimally different have another between them, because there RAE no two stanard reals that are infinitesimally different.
From: Tony Orlow on 8 Oct 2006 14:28 Virgil wrote: > In article <45251bc5(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> 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. >>> >> It also says which balls remain when each is taken out, namely, when >> ball n is removed, balls n+1 through 10n remain. > > For a while. But the fact remains that for eacn n in N, there is a > specific time before noon at which ball n is removed. > >>> 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. >> Uh, no, the very conclusion that the vase empties, when at most one ball >> is removed at a time, implies that there is a last ball removed > > That is TO's assumption, contrary to the facts required by the > experiment. > > Infinite processes can end in finite time or else Zeno's 'paradoxes' > would prevent all action. Zeno's paradoxes involve a continuous motion at finite speed over finite time. The error is in considering each successively smaller time slice to be equal and add up to an infinite time. The vase problem is similar, but it's a paradox, not a fact. It's resolved with infinite series.
From: Han.deBruijn on 8 Oct 2006 14:30
David Marcus wrote: > Consider this situation: At time 5 one ball is added to a vase. At time > 6, the ball is removed. > > Is the following a valid translation into mathematics? > > Let the value 1 denote that that the ball is in the vase and the value 0 > that the ball is not in the vase. Let A(t) be the location of the ball > at time t. Let > > A(t) = { 1 if 5 < t < 6; 0 if t < 5 or t > 6 }. No. And IMHO you are introducing a different model here than the one we all agreed upon: http://groups.google.nl/group/sci.math/msg/d2573fcb63cbf1f0?hl=en& Let's give a different, but analogous example. Let p(V) be the pressure in a closed vessel having variable volume V. Let C > 0 and p(V) = { C/V if V > 0; 0 if V <= 0 } . You can "define" what you want, but this is _not_ the proper model of an ideal gas for V <= 0 . Likewise for the balls in a vase. The function A(t) may be a valid translation only for times t _before_ noon. The time domain is limited to t < 0. And therefore (t) is not really like physical "time", which would flow smoothly through t = 0. But ah, a picture says more than a thousand words: http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg Han de Bruijn |