An uncountable countable set
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From: Tony Orlow on 14 Oct 2006 11:42 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Randy Poe <poespam-trap(a)yahoo.com> wrote: >>> >>>> Tony Orlow wrote: >>>>> Mike Kelly wrote: >>>>>> Tony Orlow wrote: >>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>> Tony Orlow wrote: >>>>>>>>> Virgil wrote: >>>>>>>>>> In article <452d11ca(a)news2.lightlink.com>, >>>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>>> >>>>>>>>>>>> I'm sorry, but I can't separate your statement of the problem from your >>>>>>>>>>>> conclusions. Please give just the statement. >>>>>>>>>>>> >>>>>>>>>>> The sequence of events consists of adding 10 and removing 1, an infinite >>>>>>>>>>> number of times. In other words, it's an infinite series of (+10-1). >>>>>>>>>> That deliberately and specifically omits the requirement of identifying >>>>>>>>>> and tracking each ball individually as required in the originally stated >>>>>>>>>> problem, in which each ball is uniquely identified and tracked. >>>>>>>>> The original statement contrasted two situations which both matched this >>>>>>>>> scenario. The difference between them was the label on the ball removed >>>>>>>>> at each iteration, and yet, that's not relevant to how many balls are in >>>>>>>>> the vase at, or before, noon. >>>>>>>> Do you think that the numbering of the balls is not relevant to >>>>>>>> determining the answer to the question "Is there a ball labelled 15 in >>>>>>>> the vase at 1/20 second before midnight?" >>>>>>>> >>>>>>>> Cheers - Chas >>>>>>>> >>>>>>> If it's a question specifically about the labels, as that is, then it's >>>>>>> relevant. It's not relevant to the number of balls in the vase at any >>>>>>> time, as long as the sequence of inserting 10 and removing 1 is the same. >>>>>>> >>>>>>> Tony >>>>>> Ah, but noon is not a part of the sequence of iterations. No more than >>>>>> 0 is an element of the sequence 1, 1/2, 1/4, 1/8, .... >>>>>> >>>>>> The question asks how many balls are in the vase at noon. Not at some >>>>>> iteration. >>>>>> >>>>> Ah, but if noon is not part of the sequence, then nothing from the >>>>> sequence has anything whatsoever to do with how many balls are in the >>>>> vase at noon. >>>> No, there's one of your leaps again. >>>> That's a particularly weird one. >>>> "If the value at noon doesn't have THIS to do with the >>>> sequence, then it must not have ANYTHING to do with >>>> the sequence". >>>> There's no reason to make such a leap. >>>> - Randy >>> Actually I think Tony is right on this one. The >>> sequence Tony is talking about is >>> 1, 9, 18, 27, ... > >> Uh, starts with 0, but do go on... > >>> This sequence represents the number of balls at times before >>> noon. The sequence has nothing to do with the number of >>> balls at noon, as the value for noon does not appear in >>> the sequence. This is why nobody who argues that the >>> vase is empty at noon ever mentions such a sequence, and >>> instead point out the simple fact that each ball added >>> before noon is removed before noon. >>> >>> Stephen >>> > >> So, the infinite sequence of finite iterations where we can actually >> tell exactly how many balls are in the vase has nothing to do with the >> vase's state at noon, which is supposed to be the limit of this >> sequence? > > Who ever said it was the limit of this sequence? > >> Why even mention the gedanken at all then? > > I am not the one who brought it up. I am not even sure > why people think it has anything to do with set theory. It doesn't. It's a distinct SEQUENCE of events, not a set without order. Set theory doesn't apply. It's just another example of set theorists trying to claim that everything falls under set theory. This experimant obviously does not. Set theory is incapable of handling the concept of sequence in a well-defined way over such a set. > The whole argument is simply that if -(1/2)^floor(n/10) is > less than zero (the minutes before noon that the ball is added), > then -(1/2)^n is less than zero (the minutes before noon the > ball is removed). This really does not rely on set theory. No, set theory confuses the issue with its concentration on omega. There is no such distinct size of the finite naturals. The infinite iterations are all compressed to a point in this experiment, and since those operations are a combination of additions and subtractions, set theorists feel entitled to rearrange the events any way that gets them their magical results. It's pitiful. > >> I suppose every >> vase is empty at noon, or just whatever you feel like declaring. You're >> playing silly magic tricks. I'm ashamed for the planet. > > The only argument I am making is that each ball that is added > before noon is removed before noon. I don't disagree with that. After the removal of every such ball before noon, nine times as many balls remain as have been removed. That is true for every moment before noon. The conclusion as to what happens AT noon either does, or doesn't, have to do with this fact. Of course by supposing that > an infinite number of actions can be performed we are playing > silly magic tricks. This is not a physical problem. Insisting > on a physical answer to an unphysical problem is pointless. > > Stephen > If we start with a vase full of any number of balls, and remove one ball at each of these -1/2^n times, then it becomes empty at noon, or before if the number of balls is finite. There is no argument about that. However, in this case, no balls is removed without ten more being inserted, so the vase cannot become empty, despite set theoretical shenanigans.
From: Tony Orlow on 14 Oct 2006 11:43 Virgil wrote: > In article <452fbf62(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>>> which is supposed to be the limit of this >>>> sequence? >>> Why is it the limit of any sequence? >>> And since the set of balls removed by noon includes every ball, how >>> does TO come up with any balls still waiting to be removed at noon? >> You tell me how many were removed, and I'll tell you how many remain. > > Card(N) were removed including the first numbered ball and each > successively numbered ball. That is not a number.
From: Tony Orlow on 14 Oct 2006 11:44 Virgil wrote: > In article <452fbe21(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <452ef411(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> The whole point of the Zeno machine is to conceive of completing this >>>> infinite series of events, and yet, it compresses the vast majority of >>>> events into a single moment at noon, making it impossible to distinguish >>>> them. >>> Actually, in either version of the original problem, NONE of the >>> transactions take place AT noon. Each of them precedes noon. >> And, after each of those transactions, before noon, there is an >> increased finite number of balls in the vase. So, it's nothing but >> finite and growing before noon. Then, at noon.....what? The linear >> growth implodes? It's true hogwash at its worst, Virgil, and you know it. > > Then how is it that in your analysis, by putting the balls in earlier, > but taking them out at the original times, one ends with fewer in the > vase? Now THAT is prime hogwash. Because, during the period that the balls are being removed none are being inserted.
From: Tony Orlow on 14 Oct 2006 11:47 Virgil wrote: > In article <452fbf0e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > > >>> So lets put them all in one minute earlier so they are all in before any >>> have to be removed and each ball will be in for a longer time, and then >>> remove them one at a time according to the original schedule. >>> >>> According to TO, putting them in earlier and taking them out as before >>> leaves FEWER in the vase at noon, even though there is no change in >>> removals. >> If you decouple the series of insertions with the series of removals, >> each series having its own point of condensation (say, fill up to noon >> and empty up to 12:01), then you have a different problem. > > So let us leave them coupled but merely change the coupling so that the > nth ball is inserted, say , 1/2^n minutes before it is removed. Both the > insertions and the removals are still all completed before noon, and it > is obvious that the vase is empty at noon. > > Then you are inserting balls one at a time, and removing them as you insert the next. What does that have to do with the original problem? > > >> If the >> series, which is a sequence, specifies that only one is removed for >> every ten added, in alternation, then that creates a relation between >> the insertions and removals that's so obvious, it really is weird that >> it even merits discussion. > > When infinitely many are inserted and all of them removed, what is > obvious to TO is false to logic. Your take on logic is very, shall we say, provincial.
From: Tony Orlow on 14 Oct 2006 11:50
David R Tribble wrote: > Tony Orlow wrote: >>> No, the inductive proof of an equality applies to all n, finite or >>> infinite. But "is finite" is an inequality, equivalent to "<oo". >>> lim(n->oo: n)=oo, not <oo. You can only increment a finite value a >>> finite number of times before you get infinite values out of it. > > David R Tribble wrote: >>> How many times? > > Tony Orlow wrote: >> Less than any infinite number of times. > > So now you're saying that a finite value can be incremented a > finite number of times (any number less than an infinite number > of times) and you'll get an infinite value? BEFORE you get an infinite value. Learn to read. Before you said that > an infinite value results when you increment a finite value an > infinite number of times. That is correct. That's one way to get an infinite value. > > And here I was thinking all this time that any finite value plus > another finite value always resulted in another finite value. > Where, oh where, did I go wrong? > It was in remedial reading class, when you fell asleep. |