An uncountable countable set
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From: Virgil on 26 Oct 2006 17:20 In article <4540d884(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > By "noon exists" do you mean there is a ball B_oo? There isn't. > > No kidding. That's why noon cannot be part of the experiment. Since everything is measured in terms of before noon, if there is no noon, nothing ever happens. > You are supposed to keep in mind the coupling of 10 additions with each > subtraction, and note that this sum of balls cannot converge to 0 no > matter how long you keep it up. But the sum is irrelevant when each and every ball is eventually removed.
From: Virgil on 26 Oct 2006 17:24 In article <4540db0f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> 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. > >>> I gave a Freshman Calculus interpretation/translation of the problem (no > >>> set theory required). Here is a suitable version: > >>> > >>> 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. > >>> > >>> Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? > >>> > >>> I suppose you either disagree with this interpretation/translation or > >>> you disagree that for this interpretatin V(0) = 0. Which is it? > >> t=0 is precluded by n e N and t(n) = -1/n. > > > > Sorry, I don't follow. Were you answering my question? I gave you a > > choice: > > > > 1. Disagree with the interpretation/translation > > 2. Agree with the interpretation/translation, but disagree that V(0) = 0 > > > > Are you picking #1 or #2? > > I'll choose #2 on the grounds that 0 does not exist in the experiment > and that V(0) is therefore without meaning. Since everything is keyed to a time that does not exist in TO's world. nothing happens in TO's world, no balls are ever inserted in TO's world, and the vase is always empty, at least in TO's world. > > Through some other mechanism than ball removal? Nope. > > > > > If you consider the vase becoming empty to be "something" rather than > > "nothing", then it is not true that nothing occurs at noon. If by > > "nothing occurs at noon", you mean no balls are added or removed, then > > it is true that nohting occurs at noon. > > And, if no balls are moved at noon, what causes the vase to become empty > at noon? Evaporation? A black hole? The fact that the rules require that every ball be removed before noon. > > The fact that there are balls at all times before noon and that no balls > are removed at noon imply that there are balls in the vase at noon, if > it exists in the experiment at all to begin with. Not in our experiment, whatever happens in TO's.
From: Virgil on 26 Oct 2006 17:26 In article <4540db63(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow 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? > >>>> > >>>> Events including insertions and removals only occur at times t of the > >>>> form t=-1/n, where n e N. Where noon means t=0, there is no t such that > >>>> -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? > >>>> > >>>> Balls can only leave the vase by removal, each of which must occur at > >>>> some t=-1/n. The vase can only become empty if balls leave. Therefore > >>>> the vase cannot become empty at noon. Agreed? > >>> Not so fast. What do "become empty" or "become empty at" mean? > >> "Not so fast"???? We've been laboring this point endlessly. The vase > >> goes from a state of balledness to a state of balllessness starting at > >> time 0. > > > > Agreed. > > > >> Balls have to have been removed for this transition to occur. > > > > Yes, but they don't have to have been removed at time 0. > > In order for emptiness to occur at that time, removals have to occur at > that time, if removals are what causes the emptiness. Was that too fast? In order to have emptiness at noon, all removals must take place no later than noon, which they are forced to do by the rules of the problem.
From: Virgil on 26 Oct 2006 17:29 In article <4540f9d0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > My question : what do you think is in the vase at noon? > > > > A countable infinity of balls. Which balls? Every ball inserted before noon must be removed before noon. > > This is very simple. Everything that occurs is either an addition of ten > balls or a removal of 1, and occurs a finite amount of time before noon. > At the time of each event, balls remain. At noon, no balls are inserted > or removed. The vase can only become empty through the removal of balls, > so if no balls are removed, the vase cannot become empty at noon. It was > not empty before noon, therefore it is not empty at noon. Nothing can > happen at noon, since that would involve a ball n such that 1/n=0. By noon every ball has already been removed, so nothing need to be removed at noon to make the vase empty.
From: Virgil on 26 Oct 2006 17:30
In article <4540fa1b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> t=0 is precluded by n e N and t(n) = -1/n. > > > > Really? > > > > I hope you will accept as true that noon occurred yesterday. > > > > Let's define noon yesterday as t=0. Now let's define a set of values > > t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE > > natural numbers n. > > > > Has my giving these names to those times somehow > > precluded noon yesterday from occurring? Retroactively? > > > > - Randy > > > > Do you live in the gedanken? Oy. Nothing happens at noon. Your desired > result does not happen before noon. Go back to yesterday and start over. If noon does not occur, then everything time that is measured from noon must also fail to occur and the vase is empty because no balls were ever inserted. |