An uncountable countable set
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From: Virgil on 24 Oct 2006 16:49 In article <453e41c5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > Then what makes you think it's > >> empty? I know, I know. You have your logic. But, it amounts to > >> artificially creating an upper bound to a boundless set, > > > > The only "upper bound" is on the times at which balls are to be moved, > > and that has least upper bound of noon. > > > > No, you are using that to pretend the set is completed, that you have > counted the last countable finite natural. That's bull. All your > arguments end up being some "largest finite" argument of one sort or > another. Omega is a phantom. Learn Non-Standard Analysis. Standard analysis is quite sufficient. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > > > > >> The fact remains that it doesn't become empty before noon, and > >> nothing happens at noon, so it doesn't empty. > > > > It is emptying, in a sense, at each t = -1/n, in such a way that it is > > empty at noon. > > It's emptying with a net gain of 9 balls, accelerating exponentially? > Curious. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? > > > > > Let those who disagree name the natural number on any ball remaining in > > the vase at noon. > > I don't claim any natural number remains at noon. No one does. So, go > knock down your straw man while you try to say when the vase empties. So according to TO one has a vase containing some naturally numbered balls (as these are the only balls available and the vase is not allowed to be empty) but simultaneously does not contain any naturally numbered balls. Neatest trick of the century.
From: Virgil on 24 Oct 2006 16:53 In article <453e4568(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > < endless reiterations of the following > > > > > The only question is "According to the rules set up in the problem, is > > each ball which is inserted into the vase before noon also removed from > > the vase before noon?" > > > > An affirmative answer confirms that the vase is empty at noon. > > A negative answer violates the conditions of the problem. > > > > Which answer does TO choose? > > God, are you a broken record, or what? Let's take this very slowly. Ready? > > Each ball inserted before noon is removed before noon, but at each time > before noon when a ball is removed, 10 balls have been added, and 9/10 > of the balls inserted remain. Therefore, at no time before noon is the > vase empty. Agreed? Since TO agrees that each ball inserted is also removed before noon, whatever else goes on he will not be able to find any ball in the vase at or after noon. > > It is not empty, and it does not become empty, then it is still not > empty. Agreed? Since TO has already agreed above that it must be empty, which should I agree to anything so silly? > > When you bring t=0 into the experiment, if anything DOES occur at that > moment, then the index n of any ball removed at that point must satisfy > t=-1/n=0, which means that n must be infinite. So, if noon comes, you > will have balls, but not finitely numbered balls. In this experiment, > however, t=0 is excluded by the fact that n e N, so noon is implicitly > impossible to begin with. The only relevant question is "According to the rules set up in the problem, is each ball which is inserted into the vase before noon also removed from the vase before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. TO has answered in the affirmative above.
From: Virgil on 24 Oct 2006 16:57 In article <453e4a3f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> Also, supposing for the sake of argument that there are "infinitely > >>>>> number balls", if a ball is added at time -1/(2^floor(n/10)), and > >>>>> removed > >>>>> at time -1/(2^n)), then the balls added at time t=0, are those > >>>>> where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > >>>>> then -1/(2^n) = 0 (making some reasonable assumptions about how > >>>>> arithmetic > >>>>> on these infinite numbers works), so those balls are also removed at > >>>>> noon and > >>>>> never spend any time in the vase. > >>>> Yes, the insertion/removal schedule instantly becomes infinitely fast in > >>>> a truly uncountable way. The only way to get a handle on it is to > >>>> explicitly state the level of infinity the iterations are allowed to > >>>> achieve at noon. When the iterations are restricted to finite values, > >>>> noon is never reached, but approached as a limit. > >>> Suppose we only do an insertion or removal at t = 1/n for n a natural > >>> number. What do you mean by "noon is never reached"? > >> 1/n>0 > > > > Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. > > > > But, I don't follow. Translating "-1/n < 0" back into words, I get "all > > insertions and removals are before noon". However, I asked you what > > "noon is never reached" means. Are you saying that "noon is never > > reached" means that "all insertions and removals are before noon"? > > > > Yes, David. What else happens in this experiment besides insertions and > removals of naturals at finite times before noon? If the infinite > sequence of events is actually allowed to continue until t=0, then you > are talking about events not indexed with natural numbers, so you're not > talking about the same experiment. If noon is not allowed, and all times > in the experiment are finitely before noon, well, at none of those times > does the vase empty, as we all agree. This is why I am asking when this > occurs. It can't, given the constraints of the problem. TO, as usual, assumes what is not in evidence, that Zeno was right and no infinite process can occur in finite time. But that includes not being able to move from point A to point B in finite time, and all the other Zeno paradoxes that argue the impossibilities of everyday events. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer?
From: Virgil on 24 Oct 2006 16:59 In article <453e4a85(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > 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. > > > > If the vase exists at noon, then it has an uncountable number of balls > labeled with infinite values. But, no infinite values are allowed i the > experiment, so this cannot happen, and noon is excluded. So did the North Koreans nuke the vase before noon? The only relevant issue is whether according to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative confirms that the vase is empty at noon. A negative directly violates the conditions of the problem. How does TO answer?
From: Virgil on 24 Oct 2006 17:01
In article <453e4aec$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > 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? > > > > Using the time singularity of the Zeno machine, where there is a > condensation point in the sequence that allows an infinite number of > iterations to occur in a moment. Luckily for the vase, no one is > inserting extra balls on the same schedule. TO is trying to reincarnate the Zeno paradoxes. But if he does, all action becomes impossible. The only relevant question is "According to the rules set up in the problem, is each ball which is inserted into the vase before noon also removed from the vase before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer? |