An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 10:25 Virgil wrote: > In article <4540d345(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <453fb693(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> David Marcus wrote: >>>>> Virgil wrote: >>>>>> In article <453e824b(a)news2.lightlink.com>, >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> Virgil wrote: >>>>>>>> In article <453e4a85(a)news2.lightlink.com>, >>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>> 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? >>>>>>> You can repeat the same inane nonsense 25 more times, if you want. I >>>>>>> already answered the question. It's not my problem that you can't >>>>>>> understand it. >>>>>> It is a good deal less inane and less nonsensical than trying to >>>>>> maintain, as TO and his ilk do, that a vase from which every ball has >>>>>> been removed before noon contains any balls at noon that have not been >>>>>> removed. >>>>> Ah, you are forgetting the balls labeled with "infinite values". Those >>>>> balls haven't been removed before noon. Although, I must say I'm not too >>>>> clear on when they were added. >>>>> >>>> At noon >>> Where in the original problem does it say anything like that? >> It doesn't. It specifically excludes noon as a time in the experiment by >> specifying that all balls are finitely numbered and all events are >> finitely before noon. Duh. > > How can there be a before without what comes after? > How can there be a before noon without a noon? > Does God cause the sun to stand still and stop time? Uh, God doesn't limit Godself to finite numbers.
From: Tony Orlow on 27 Oct 2006 10:27 Virgil wrote: > 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. That means either before noon, or at noon. No balls are removed at noon. Balls remain at every time before noon. You're busted.
From: Tony Orlow on 27 Oct 2006 10:31 David Marcus wrote: > Virgil wrote: >> In article <4540d217(a)news2.lightlink.com>,Tony Orlow <tony(a)lightlink.com> wrote: >> >>> Noon does not exist in the experiment, or else you have infinitely >>> numbered balls. >> It is specifically mentioned in the experiment as the base time from >> which all actions are determined, so that if it does not exist then none >> of the actions can occur. >> >> If there is no noon then there can be no one minute before noon at which >> the first ball is inserted, so the vase is frozen in a state of >> emptiness. > > That's a good point. Not particularly. Noon may be used as a time origin, but if all events happen such that n e N, and all events happen such that t=-1/n, then t<>0. Nothing can occur at noon, and the vase is not empty before noon, so it cannot be empty at noon. That's obvious. > >>> Like something occurring in time without at least a moment in which it >>> occurred. >> In the physical world, nothing happens instantaneously. In the >> mathematical world, pretty much everything does. >> >> In the mathematical world of the experiment, the balls move in and out >> of the vase instantaneously, and must be allowed to do so or the >> experiment cannot be performed at all. >> >> So either things can happen instantaneously or the experiment impossible. >> >> If TO allows a finite change of number of balls in the vase to occur >> instantaneously, what is so difficult about allowing an "infinite" >> change in the number of balls to occur instantaneously? > > That's another good point. > Yeah, sure, except that I never objected to an infinite number of events happening in one moment. I objected to them happening in NO moment. So, that's not a response.
From: Tony Orlow on 27 Oct 2006 10:40 David Marcus wrote: > Tony Orlow wrote: >> Mike Kelly wrote: >>> Now correct me if I'm wrong, but I think you agreed that every >>> "specific" ball has been removed before noon. And indeed the problem >>> statement doesn't mention any "non-specific" balls, so it seems that >>> the vase must be empty. However, you believe that in order to "reach >>> noon" one must have iterations where "non specific" balls without >>> natural numbers are inserted into the vase and thus, if the problem >>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is >>> this a fair summary of your position? >>> >>> If so, I'd like to make clear that I have no idea in the world why you >>> hold such a notion. It seems utterly illogical to me and it baffles me >>> why you hold to it so doggedly. So, I'd like to try and understand why >>> you think that it is the case. If you can explain it cogently, maybe >>> I'll be convinced that you make sense. And maybe if you can't explain, >>> you'll admit that you might be wrong? >>> >>> Let's start simply so there is less room for mutual incomprehension. >>> Let's imagine a new experiment. In this experiment, we have the same >>> infinite vase and the same infinite set of balls with natural numbers >>> on them. Let's call the time one minute to noon -1 and noon 0. Note >>> that time is a real-valued variable that can have any real value. At >>> time -1/n we insert ball n into the vase. >>> >>> My question : what do you think is in the vase at noon? >> A countable infinity of balls. > > So, "noon exists" in this case, even though nothing happens at noon. Not really, but there is a big difference between this and the original experiment. If noon did exist here as the time of any event (insertion), then you would have an UNcountably infinite set of balls. Presumably, given only naturals, such that nothing is inserted at noon, by noon all naturals have been inserted, for the countable infinity. Then insertions stop, and the vase has what it has. The issue with the original problem is that, if it empties, it has to have done it before noon, because nothing happens at noon. You conclude there is a change of state when nothing happens. I conclude there is not. > >> 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. >
From: Tony Orlow on 27 Oct 2006 10:46
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> 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. >>>> >>>>>>> Given my interpretation/translation of the problem into Mathematics (see >>>>>>> above) and given that the "moment the vase becomes empty" means the >>>>>>> first time t >= -1 that V(t) is zero, then it follows that the "vase >>>>>>> becomes empty" at t = 0 (i.e., noon). >>>>>> Yes, now, when nothing occurs at noon, and no balls are removed, what >>>>>> else causes the vase to become empty? >>>>> No balls are added or removed at noon, but the vase becomes empty at >>>>> noon. >>>> Through some other mechanism than ball removal? >>>> >>>>> 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 cause of the vase becoming empty at noon is that all balls are >>>>> removed before noon, but at all times between one minute before noon and >>>>> noon, there are balls in the vase. >>>> 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. >>>> >>>>> Let me ask you the same question regarding the following problem. >>>>> >>>>> Problem: For n = 1,2,..., let >>>>> >>>>> 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 B_n(t). What is V(0)? Answer: V(0) = 0. >>>>> >>>>> Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) >>>>> is approaching infinity as t approaches zero from the left, what causes >>>>> V(0) to be zero? >>>> The fact that you have no upper bound to the naturals. This is the same >>>> technique, essentially, which equates the naturals with, say, the evens, >>>> or squares of naturals, even though those are proper subsets of the >>>> naturals. You can draw a 1-1 correspondence between the balls in and >>>> out, sure. There's a bijection there. Infinite bijections do not given >>>> any notion of measure unless they are parameterized. Here, you can look >>>> at number of balls in the vase as a function of n or of t. In either >>>> case, the sum diverges. It is only in trying to consider the unbounded >>>> set as completed that you come to this silly conclusion. >>> Let's see if I understand what you are saying. Consider this math: >>> >>> -------------------------- >>> For n = 1,2,..., let >>> >>> A_n = -1/floor((n+9)/10), >>> R_n = -1/n. >>> >>> For n = 1,2,..., define a function B_n: R -> R 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 B_n(t). >>> -------------------------- >>> >>> Are you saying that V(0) is not equal to zero? >> (sigh) I already answered this. Are you just trying to test for >> consistency in my statements? That gets a little tiresome. > > Sorry, but I think you answered a different question than I asked. > >> I am saying 0 doesn't happen in this experiment. All events are before >> 0. Any events occurring at 0, by the constraints of the experiment, must >> have an index n in the sequence such that 1/n=0, but this n cannot exist >> in the sequence of finite natural numbers. Therefore, nothing happens at >> t=0. If it did, all finite balls would be gone, but they would be >> replaced by an uncountable number of infinitely-numbered balls. >> >> At the time of each and every event before 0, without exception, more >> balls are left in the vase than were there before the event, and during >> (-1,0), the vase is never empty. >> >> Combine these two facts, and you get that the vase did not become empty, >> since nothing happens at noon to the balls to change the state of the >> vase, and the desired state does not occur before noon. It still has >> balls. And including noon doesn't change that, but only pushes the >> potential, countable infinity to the actual, uncountable point. > > You are mentioning balls and time and a vase. But, what I'm asking is > completely separate from that. I'm just asking about a math problem. > Please just consider the following mathematical definitions and > completely ignore that they may or may not be relevant/related/similar > to the vase and balls problem: > > -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R 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 B_n(t). > -------------------------- > > Just looking at these definitions of sequences and functions from R (the > real numbers) to R, and assuming that the sum is defined as it would be > in a Freshman Calculus class, are you saying that V(0) is n |