An uncountable countable set
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From: imaginatorium on 27 Oct 2006 12:34 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.) 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 (as opposed to intuition and hand-waving?) Brian Chandler http://imaginatorium.org
From: stephen on 27 Oct 2006 12:30 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
From: Tony Orlow on 27 Oct 2006 12:48 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.
From: Tony Orlow on 27 Oct 2006 12:50 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.
From: Tony Orlow on 27 Oct 2006 12:51
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> 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 not equal to >>> 0? >> On the surface, you math appears correct, but that doesn't mend the >> obvious contradiction in having an event occur in a time continuum >> without occupying at least one moment. It doesn't explain how a >> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >> that all finite naturals are removed by noon. I never disagreed with >> that. However, to actually reach noon requires infinite naturals. Sure, >> if V is defined as the sum of all finite balls, V(0)=0. But, I've >> already said that, several times, haven't I? Isn't that an answer to >> your question? > > I think it is an answer. Just to be sure, please confirm that you agree > that, with the definitions above, V(0) = 0. Is that correct? > Sure, all finite balls are gone at noon. |