An uncountable countable set
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From: David Marcus on 24 Oct 2006 02:14 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Your examples of the circle and rectangle are good. Neither has a height > >>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, > >>>> because the circle actually exists there. To ask about its height at x=9 > >>>> is like asking how the air quality was on the 85th floor of the World > >>>> Trade Center yesterday. Similarly, it makes little sense to ask what > >>>> happens at noon. There is no vase at noon. > >>> Do you really mean to say that there is no vase at noon or do you mean > >>> to say that the vase is not empty at noon? > >> If noon exists at all, the vase is not empty. All finite naturals will > >> have been removed, but an infinite number of infinitely-numbered balls > >> will remain. > > > > "If noon exists at all"? How do we decide? > > > > We decide on the basis of whether 1/n=0. Is that possible for n in N? > Hmmmm......nope. So, noon doesn't exist. And, there is no vase at noon. I thought you were saying the vase contains an infinite number of balls at noon. -- David Marcus
From: David Marcus on 24 Oct 2006 02:17 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> Virgil wrote: > >>>>>>> In article <4533d315(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>>>> Then let us put all the balls in at once before the first is removed and > >>>>>>>>> then remove them according to the original time schedule. > >>>>>>>> Great! You changed the problem and got a different conclusion. How > >>>>>>>> very....like you. > >>>>>>> Does TO claim that putting balls in earlier but taking them out as in > >>>>>>> the original will result in fewer balls at the end? > >>>>>> If the two are separate events, sure. > >>>>> Not sure what you mean by "separate events". Suppose we put all the > >>>>> balls in at one minute before noon and take them out according to the > >>>>> original schedule. How many balls are in the vase at noon? > >>>> empty. > >>> Why? > >> Because of the infinite rate of removal without insertions at noon. > > > > OK. Just to recall, this vase has all the balls put in at one minute > > before noon, then taken out on the usual schedule. How many balls are in > > this vase at times before noon? > > Some supposedly infinite number, as only a finite number have been removed. But, for this vase, at all times before noon, there are an infinite number of balls in the vase. So, how does this vase become empty at noon? -- David Marcus
From: David Marcus on 24 Oct 2006 02:19 Tony Orlow wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >> You have agreed with everything so far. At every point before noon balls > >> remain. You claim nothing changes at noon. Is there something between > >> noon and "before noon", when those balls disappeared? If not, then they > >> must still be in there. > > > > Of course there is a "something" between "before noon" and "noon" where > > each ball disappears. At step n, time 2^-n min before noon, ball n is > > removed. This happens for every ball, since there is a step n for > > every ball. The balls are removed, one by one, one at each step, > > before noon. > > > > As each ball n is removed, how many remain? 9n. > Can any be removed and leave an empty vase? Not sure what you are asking. -- David Marcus
From: Tony Orlow on 24 Oct 2006 12:27 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: >> David Marcus wrote: >>> Your assumptions seem consistent with the following formulation of the >>> problem. >>> >>> For n = 1,2,..., define >>> >>> A_n = -1/floor((n+9)/10), >>> R_n = -1/n. >>> >>> For n = 1,2,..., define a function B_n by >>> >>> B_n(t) = 1 if A_n < t < R_n, >>> 0 if t < A_n or t > R_n, >>> undefined if t = A_n or t = R_n. >>> >>> Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? >> Yes; and in fact I chose (1)-(8) to be consistent with just about any >> sensible interpretation of the problem as given. >> >> But I'm currently trying with Tony to completely avoid numerical >> arguments such as the above, which rely on a complicated definition of >> "the number of balls at time t", in favor of the much simpler to agree >> with statement "either there is a ball in the vase at time t, or the >> number of balls in the vase at time t is 0; and not both". > > It is worth a try. Although, I think you'd do better to limit your > replies to a few connected paragraphs rather than reply to so many > individual paragraphs of Tony's post. While carrying on many threads of > a discussion in one post saves time (like I'm doing in this post), it > only seems to work if both people are basically on the same wavelength. > Otherwise, people pick and choose which comments they reply to. > >> I mean, given his confusion over simple logical arguments like "If >> (A->not A), then not A", I shudder to think what subtle >> misunderstandings exist in his version of "define a sequence of >> functions indexed by n in N". > > Perhaps. Although, I think he might have taken Calculus at some point. > Many people who take Calculus learn something about functions despite > being unable to reason logically. > >>> It is rather amazing. >> It's also sort of fascinating - how can one /not/ understand the >> argument, and yet give the impression of understanding /some/ sort of >> logic? It's like some sort of mental blind spot. > > Do you really think he understands any logic? I believe that 98% or more > of people don't think conceptually/logically. Instead they rely on the > brain's amazing ability to do pattern matching. Pattern matching is > extrememly useful, but it dosn't do logic. > >>> The logic seems to be that the limit of the number >>> of balls in the vase as we approach noon is infinity, so the number of >>> balls in the vase at noon must be infinity, but all numbered balls have >>> been removed, therefore the infinity of balls in the vase at noon aren't >>> numbered. It does have a sort of surreal appeal. >> If we assume at the start that the number of balls at t=0 is /anything >> but/ 0 (as TO apparantly does, although he has yet to realize it), then >> pretty much anything goes. Let your imagination roam! There are a prime >> number of cubical balls in the vase at noon! ZFC is inconsistent! >> Cantor is alive and living in Brooklyn New York! I am the current King >> of France! > > I don't agree that he is assuming that. I think he isn't reasoning > logically at all. The number of balls approaches infinity as time > approaches noon. If you imagine a vase filling up with an infinite > number of balls, it is rather hard to imagine them suddenly all > disappearing. Of course, mathematics isn't constrained by our > imagination. It relies on precise definitions and logic. And, functions > do not have to be continuous. > So, David, you think the fact that balls leave the vase only by being removed one at a time, and the fact that at all times before noon there are balls in the vase, and the fact that at noon there are no balls in the vase, is consistent with the fact that no balls are removed at noon? How can you not see the logical inconsistency of an infinitude of balls disappearing, not just in a moment, but at no possible moment? Are you so steeped in set theory that you cannot see that an unending sequence of +10-1 amounts to an unending series of +9's which diverges? What is illogical about that? In your set-theoretic interpretation of the experiment there is a problem which makes your conclusion incompatible with conclusions drawn from infinite series, and other basic logical approaches. It is not that I don't understand how your logic works. It's that I see clearly that it doesn't, and I'm trying to precisely pin down exactly where the error is. It's not an easy task, since this transfinite theory is rather well crafted and tweaked over the years. However, there are clear reasons, once the matter is fully investigated, why the logic fails. The conclusion produces clear contradictions in terms of a time of emptying and the requirement at some point of a negative number of balls in the vase in order for it to empty at all, and it all derives from using the Zeno schedule to complete a sequence which has no end, hiding this fact in a time singularity at t=0. Very basic logic would hold that, if the vase is not empty at any time t such that -1<=t<0, and the vase is empty at t=0, then balls were removed at t=0, since that's the only way the vase can become empty. However, t=0 corresponds, according to the stated schedule, to infinite index n in the sequence, and an infinite label on a ball, which is not allowed, as per the experiment. Therefore, no ball can be removed at t=0, and the vase cannot become empty at that point, or at any point before. I asked you when you thought the vase became empty. You avoided the question, saying it was interesting, and then going on with your same tired formulation of the problem, as if I haven't followed the logic and pointed out the flaw in the approach. So, answer the question. When does this miracle of emptiness occur? Tony
From: Tony Orlow on 24 Oct 2006 12:28
David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: >> stephen(a)nomail.com wrote: >>> With the added surreal twist that the limit of the number >>> of unnumbered balls in the vase as we approach noon is 0, >>> but the number of unnumbered balls in the vase at noon is >>> infinite. :) >> I think his response, when I pointed this out to him, was either "Oh, >> shut up!" or "Whatever." > > That is consistent with my suggestion that Tony is reasoning by > imagining a vase filling up. If you visualize the vase filling up in > your mind, you don't see the unnumbered balls in the picture. > If you have an infinite ocean wit 10 liter/sec flowing in, and 1 liter/sec flowing out (and no evaporation), will it ever empty? No. Same difference. |