An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 18:23 imaginatorium(a)despammed.com wrote: > Forgive me if I blunder in on Chas's carefully constructed argument, > but... > > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>> <snip> >>> >>>>> My question : what do you think is in the vase at noon? >>>>> >>>> A countable infinity of balls. >>>> >>>> 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. >>> No one disagrees with the above statements. >>> >>>> The vase can only become empty through the removal of balls, >>> Note that this is not identical to saying "the vase can only become >>> empty /at time t/, if there are balls removed /at time t/"; which is >>> what it seems you actually mean. >>> >>> This doesn't follow from (1)..(8), which lack any explicit mention of >>> what "becomes empty" means. However, we can easily make it an >>> assumption: >>> >>> (T1) If, for some time t1 < t0, it is the case that the number of balls >>> in the vase at any time t with t1 <= t < t0 is different than the >>> number of balls at time t0, then balls are removed at time t0, or balls >>> are added at time t0. >>> >> Well, you have (8), which is kind of circular, but related. >> >>>> 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. >>> Now your logical argument is complete, assuming we also accept >>> (1)..(8): If the number of balls at time t = 0, then by (7), (5) and >>> (6), the number of balls changes at time 0; and therefore by (T1), >>> balls are either placed or removed at time 0, implying by (5) and (6) >>> that there is a natural number n such that -1/n = 0; which is absurd. >>> Therefore, by reductio ad absurdum, the number of balls at time 0 >>> cannot be 0. >>> >>> However, it does not follow that the number of balls in the vase is >>> therefore any other natural number n, or even infinite, at time 0; >>> because that would /equally/ require that the number of balls changes >>> at time 0, and that in turn requires by (T1) that balls are either >>> added or removed at time 0; and again by (5) or (6) this implies that >>> there is a natural number n with -1/n = 0; which is absurd. So again, >>> we get that any statement of the form "the number of balls at time 0 is >>> (anything") must be false by reductio absurdum. >>> >>> So if we include (T1) as an assumption as well as (1)..(8), it follows >>> logically that the number of balls in the vase at time 0 is not >>> well-defined. >> That is correct. Noon is incompatible with the problem statement. >> >>> Of course, we also find that by (1)..(8) and (T1), it /still/ follows >>> logically that the number of balls in the vase at time t is 0; and this >>> is a problem: we can prove two different and incompatible statements >>> from the same set of assumptions >> Right. Your conclusion is at odds with the notion that only removals may >> empty the vase, which seems to be an obvious assumption, no other means >> of achieving emptiness having been mentioned. >> >>> So at least one of the assumptions (1)..(8) and (T1) must be discarded >>> if we are to resolve this. What do you suggest? Which of (1)..(8) do >>> you want discard to maintain (T1)? >> I don't believe any of those assumptions are the problem. (2) should >> state that t<t0, not t<=t0, at any event. But, that's irrelevant. The >> unspoken assumption on your part which causes the problem is that noon >> is part of the problem. Clearly, it cannot be, because anything that >> happened at t=0 would involve n s.t. 1/n=t. Essentially, the problem >> produces a paradox by asking a question which contradicts the situation. >> Nothing happens at noon. The process never completes the unending set. > > Here's something I don't understand. I believe, Tony, that you think > that if every one of these pofnat-labelled balls is inserted one minute > earlier (so *informally*, instead of a "sliver" tapering to zero width, > we have an endless boomerang shape, with the width tending to 1 as you > go ever up the y-direction), then at noon no balls are left. Presumably > because once all the balls are IN (at 11:59), there is only removal, > tick, tick, tick, ... and all are gone at noon. But why doesn't this > stuff about "noon being incompatible" apply here too? Is there a > *principled* way in which you determine which arguments apply at > particular points? (I'm sure it appears to most non-cranks here that > there isn't.) That's very simple, Brian. The limit of balls as n->noon is 0. That's not the case in the original problem. There, there is no limit. The sum diverges, as it does in this case until 11:59. Those points of infinitely quick iterations ultimately include an uncountable number of iterations, unless their countability is specified, in which case they do not reach those points of uncountability. Additionally, we have the fact that, if property p applies at all times before time t, and does not change state at time t, then it continues to apply at time t. In the case where we have a countably infinite number of balls at t=-1, no matter how they got there, and start removing them in Zeno fashion, we can conceptually empty the vase by time 0. Once time 0 is there, nothing else happens. So if all balls have been removed by then, that's the way it is at time 0. If all balls haven't been removed, due to a condition of the problem under consideration, then it's not empty. All balls have NOT been removed before noon in the gedanken. AT noon, no balls are removed. The vase can only be empty after having been non-empty if removals have occurred between those two times. Which of my statements above do you find objectionable, and why? That would be helpful to know. Thanks, Tony > > Note that in this scenario, at time noon- 1/n, there are, da-dah!, an > infinite number of balls in the vase. So the limit of the number of > balls in the vase at t approaches noon is infinity. Yet you (really?) > think that in this case the vase ends up empty? Do you have any sort of > *mathematical* argument for this
From: Tony Orlow on 27 Oct 2006 18:23 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> Tony Orlow wrote: >>> David Marcus wrote: >>>> Your question "Is there a smallest infinite number?" lacks context. You >>>> need to state what "numbers" you are considering. Lots of things can be >>>> constructed/defined that people refer to as "numbers". However, these >>>> "numbers" differ in many details. If you assume that all subjects that >>>> use the word "number" are talking about the same thing, then it is >>>> hardly surprising that you would become confused. >>> I don't consider transfinite "numbers" to be real numbers at all. I'm >>> not interested in that nonsense, to be honest. I see it as a dead end. >>> >>> If there is a definition for "number" in general, and for "infinite", >>> then there cannot both be a smallest infinite number and not be. > >> A moot point, since there is no definition for "'number' in general", as >> I just said. > >> -- >> David Marcus > > A very simple example is that there exists a smallest positive > non-zero integer, but there does not exist a smallest positive > non-zero real. If someone were to ask "does there exist a smallest > positive non-zero number?", the answer depends on what sort > of "numbers" you are talking about. > > Stephen Like, perhaps, the Finlayson Numbers? :)
From: Tony Orlow on 27 Oct 2006 18:24 MoeBlee wrote: > Tony Orlow wrote: >> I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. > > Surely fodder for a Jesse F. Hughes tagline. > > MoeBlee > Oh, surely.
From: Tony Orlow on 27 Oct 2006 18:27 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> 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. >>> So, noon doesn't exist in this case either? >> Nothing happens at noon, and as long as there is no claim that anything >> happens at noon, then there is no problem. Before noon there was an >> unboundedly large but finite number of balls. At noon, it is the same. > > So, noon does exist in this case? > Since the existence of noon does not require any further events, it's a moot point. As I think about it, no, noon does not exist in this problem either, as the time of any event, since nothing is removed at noon. It is also not required for any conclusion, except perhaps that there are uncountably many balls, rather than only countably many. But, there are only countably many balls, so, no, noon is not part of the problem here. As we approach noon, the limit is 0. We don't reach noon.
From: Tony Orlow on 27 Oct 2006 18:30
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> stephen(a)nomail.com wrote: >>>>> What are you talking about? I defined two sets. There are no >>>>> balls or vases. There are simply the two sets >>>>> >>>>> IN = { n | -1/(2^floor(n/10)) < 0 } >>>>> OUT = { n | -1/(2^n) < 0 } >>>> For each n e N, IN(n)=10*OUT(n). >>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT >>> (n)". So, you seem to be answering a question he didn't ask. Given >>> Stephen's definitions of IN and OUT, is IN = OUT? >> Yes, all elements are the same n, which are finite n. There is a simple >> bijection. But, as in all infinite bijections, the formulaic >> relationship between the sets is lost. > > Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > (The vertical lines denote "cardinality".) > Um, before I answer that question, I think you need to define what you mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do you define '-' on these "numbers"? |