An uncountable countable set
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From: Virgil on 27 Oct 2006 02:38 In article <ehs4lg$6k1$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: > > stephen(a)nomail.com wrote: > >> Tony Orlow <tony(a)lightlink.com> wrote: > >>> stephen(a)nomail.com wrote: > >>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> stephen(a)nomail.com wrote: > >>>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>>>>>> imaginatorium(a)despammed.com wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> David Marcus wrote: > >>>>>>>>>>> Tony Orlow wrote: > >>>>>>>>>>>> As each ball n is removed, how many remain? > >>>>>>>>>>> 9n. > >>>>>>>>>>> > >>>>>>>>>>>> Can any be removed and leave an empty vase? > >>>>>>>>>>> Not sure what you are asking. > >>>>>>>>>> If, for all n e N, n>0, the number of balls remaining after n's > >>>>>>>>>> removal > >>>>>>>>>> is 9n, does there exist any n e N which, after its removal, leaves > >>>>>>>>>> 0? > >>>>>>>>> I don't know what you mean by "after its removal"? > >>>>>>>> Oh, I think this is clear, actually. Tony means: is there a ball > >>>>>>>> (call > >>>>>>>> it ball P) such that after the removal of ball P, zero balls remain. > >>>>>>>> > >>>>>>>> The answer is "No", obviously. If there were, it would be a > >>>>>>>> contradiction (following the stated rules of the experiment for the > >>>>>>>> moment) with the fact that ball P must have a pofnat p written on > >>>>>>>> it, > >>>>>>>> and the pofnat 10p (or similar) must be inserted at the moment ball > >>>>>>>> P > >>>>>>>> is removed. > >>>>>>> I agree. If Tony means is there a ball P, removed at time t_P, such > >>>>>>> that > >>>>>>> the number of balls at time t_P is zero, then the answer is no. After > >>>>>>> all, I just agreed that the number of balls at the time when ball n > >>>>>>> is > >>>>>>> removed is 9n, and this is not zero for any n. > >>>>>>>> Now to you and me, this is all obvious, and no "problem" whatsoever, > >>>>>>>> because if ball P existed it would have to be the "last natural > >>>>>>>> number", and there is no last natural number. > >>>>>>>> > >>>>>>>> Tony has a strange problem with this, causing him to write mangled > >>>>>>>> versions of Om mani padme hum, and protest that this is a "Greatest > >>>>>>>> natural objection". For some reason he seems to accept that there is > >>>>>>>> no > >>>>>>>> greatest natural number, yet feels that appealing to this fact in an > >>>>>>>> argument is somehow unfair. > >>>>>>> The vase problem violates Tony's mental picture of a vase filling > >>>>>>> with > >>>>>>> water. If we are steadily adding more water than is draining out, how > >>>>>>> can all the water go poof at noon? Mental pictures are very useful, > >>>>>>> but > >>>>>>> sometimes you have to modify your mental picture to match the > >>>>>>> mathematics. Of course, when doing physics, we modify our mathematics > >>>>>>> to > >>>>>>> match the experiment, but the vase problem originates in mathematics > >>>>>>> land, so you should modify your mental picture to match the > >>>>>>> mathematics. > >>>>>> As someone else has pointed out, the "balls" and "vase" > >>>>>> are just an attempt to make this sound like a physical problem, > >>>>>> which it clearly is not, because you cannot physically move > >>>>>> an infinite number of balls in a finite time. It is just > >>>>>> a distraction. As you say, the problem originates in mathematics. > >>>>>> Any attempt to impose physical constraints on inherently unphysical > >>>>>> problem is just silly. > >>>>>> > >>>>>> The problem could have been worded as follows: > >>>>>> > >>>>>> Let IN = { n | -1/(2^floor(n/10) < 0 } > >>>>>> Let OUT = { n | -1/(2^n) } > >>>>>> > >>>>>> What is | IN - OUT | ? > >>>>>> > >>>>>> But that would not cause any fuss at all. > >>>>>> > >>>>>> Stephen > >>>>>> > >>>>> It would still be inductively provable in my system that IN=OUT*10. > >>>> So you actually think that there exists an integer n such that > >>>> -1/(2^floor(n/10)) < 0 > >>>> but > >>>> -1/(2^n) >= 0 > >>>> ? > >>>> > >>>> What might that integer be? > >>>> > >>>> Stephen > >>>> > >> > >>> How do you glean that from what I said? Your "largest finite" arguments > >>> are very boring. > >> > >> How do I glean that? You claim that IN does not equal OUT. > >> IN contains all n such that > >> -1/(2^floor(n/10)) < 0 > >> and OUT contains all n such that > >> -1/(2^n) < 0 > >> > >> Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), > >> so presumably IN is bigger than OUT, and IN contains elements > >> that are not in OUT. The only way n can be an element of IN, > >> but not of out is if > >> -1/(2^floor(n/10)) < 0 > >> but > >> -1/(2^n) >= 0 > > > Incorrect. For every n, finite or infinite, when the nth ball is > > removed, 9n remain. Either you get to t=0, in which case all finite > > balls are indeed gone, but replaced by uncountably many infinite balls, > > or you don't get to noon. > > 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 } > > >> > >> So apparently you do not think such an n exists, yet you > >> think there are elements in IN that are not in OUT. > >> Those are contradictory positions. > > > No, it is the only conclusion consistent with the notion that a proper > > subset is alway smaller than the superset. It is contradictory to the > > notion that a simple bijection indicates equal size for infinite sets. > > The mapping formulas must be taken into account for proper comparison in > > that case. > > But OUT is not a proper subset of IN, unless you believe that > there exists an n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > > If you claim that OUT is a proper subset of IN, you must > be able to identify an element that is in IN that is not in OUT. > > Stephen That reminds me of the story about the twin baby skunks.
From: RLG on 27 Oct 2006 03:21 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:4540f9d0(a)news2.lightlink.com... > > 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. Tony, I think your confusion results from imagining the balls without any labels. In this case at 1 minute before noon 10 balls are inserted into the vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 minute before noon, 9 more balls are inserted into the vase and, in general, at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you are saying that the number of vase balls at noon is: 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. Or, since one ball is removed each time ten more are added, we should write: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. Now, this divergent series is conditionally convergent. That means we can make the sum equal any value we like depending on how the terms are arranged. So if we choose 0 for the sum that is perfectly valid: 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. In this case there are no balls in the vase at noon. Without labels on the balls there is no criterion by which to select what the sum should be and the end state of the supertask is undefined. As I noted in an earlier post, if some of the balls are labeled with numbers that are not naturals, for example transfinite ordinal numbers, we can choose "Infinite" for the sum if the circumstances require it. Consider the following problem: Tony has a two gallon bucket and his job is to ensure that the amount of water in the bucket during the nth day is 1+sin(n) gallons. Since Tony's job never ends he will always be making daily changes in the bucket's water content and we have a full mathematical description of Tony's job. There is no problem with this. But if we changed Tony's job so that it had an end, say at noon, and the bucket had to contain 1+sin(n) gallons at (1/2)^n minutes before noon then we do not have a full description of Tony's activities. It is a mistake to assume the bucket's water content at noon is a function of its pre-noon state. At noon Tony puts whatever amount of water he wants into the bucket. -R
From: imaginatorium on 27 Oct 2006 04:11 David Marcus wrote: > imaginatorium(a)despammed.com wrote: > > > > Virgil wrote: > > > In article <45417528$1(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > <snip> > > > > > > For what it's worth, and I know this doesn't add a lot of credibility to > > > > Ross in your eyes, coming from me, but I think Ross has a genuine > > > > intuition that isn't far off with respect to what's controversial in > > > > modern math. Sure, he gets repetitive and I don't agree with everything > > > > he says, but his cryptic "Well order the reals", which I actually > > > > haven't seen too much of lately, is a direct reference to his EF > > > > (Equivalence Function, yes?) between the naturals and the reals in > > > > [0,1). The reals viewed as discrete infinitesimals map to the > > > > hypernaturals, anyway, and his EF is a special case of my IFR. So, to > > > > answer your question, I think Ross makes some sense. But, of course, > > > > coming from me, that probably doesn't mean much. :) > > > > > > Coming from TO it damns Ross. > > > > Even by your standards, Virgil, this is egregiously silly. TO skips the > > basic exposition in Robinson's book, but finds a sentence he likes. So > > this "damns" Robinson's non-standard analysis, does it? > > Virgil said "Ross", not "Robinson", I believe. Yes, of course. But Virgil's implication is that "TO says person P is right about something" implies P is wrong. This may, contingently, be true about Ross, but the argument could equally be applied to Robinson, in which case the conclusion is obviously not true. Brian Chandler http://imaginatorium.org
From: Tony Orlow on 27 Oct 2006 10:10 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. > > Cheers - Chas > Cheers!
From: Tony Orlow on 27 Oct 2006 10:23
Virgil wrote: > In article <4540d217(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <453fb285(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Mike Kelly wrote: >>>>> Randy Poe wrote: >>>>>> How is it, in your world, that when I specify times for all natural >>>>>> numbered >>>>>> balls, I am required to put in balls that don't have natural numbers? >>>>> The problem is that Tony thinks time is a function of the number of >>>>> insertions you've gone through. In order to "get to" any particular >>>>> time you have to perform the insertions "up to" that point. He then >>>>> thinks that if you want to "get to" noon, you have to have performed >>>>> some "infinite" (whatever that means) iterations, where balls without >>>>> natural numbers are inserted. That this is obviously not what the >>>>> problem statement says doesn't seem to bother him. Nor that it's >>>>> absolutely nothing like an intuitive picture of what time is. >>>> Time is ultimately irrelevant in this gedanken, but if it is to be >>>> considered, the constraints regarding time cannot be ignored. Events >>>> occurring in time must occupy at least one moment. >>> >>> How is time irrelevant when every action is specified by the time at >>> which it is to occur? >> Please specify the moment when the vase becomes empty. > > It IS empty at noon, but not before. But I do not know what TO means by > "becomes". Become: To assume a state not previously assumed. This can happen over a period, or in an instant, but it must happen sometime. If noon is the first moment when the vase is empty, then it emptied at noon, but nothing happens to the balls at noon. Contradiction. >>> The only relevant question is "According to the rules set up in the >>> problem, is each ball inserted at a time before noon also removed at a >>> time before noon?" >>> >>> An affirmative answer confirms that the vase is empty at noon. >> Not if noon is proscribed the the problem itself, which it is. > > How so? I see nothing in the statement of the problem which "proscribes" > noon. Nothing can occur at noon because that implies 1/n=0, false for all natural numbers. >>> A negative answer directly violates the conditions of the problem. >>> >>> How does TO answer this question? >>> >>> As usual, he avoids such relevant questions in his dogged pursuit of the >>> irrelevant. >>> >> 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. No, time begins at -1, such that t(n)=-1/n. n never becomes infinite, so t never becomes 0. > > 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. At t=-1=-1/n, n=1. Are you saying 1 is not a natural number? I thought the labels were the most important aspect of this for you. Now you want to ignore them? Huh! >>>>> Obviously, time is an independent variable in this experiment and the >>>>> insertion or removal or location of balls is a function of time. That's >>>>> what the problem statement says: we have this thing called "time" which >>>>> is a real number and it "goes from" before noon to after noon and, at >>>>> certain specified times, things happen. There are only >>>>> naturally-numbered balls inserted and removed, always before noon. >>>>> Every ball is removed before noon. Therefore, the vase is empty. > >>>> No, you have the concept of the independent variable bent. The number of >>>> balls is related to the time by a formula which works in both directions. > > Where does the problem say that the numbers on balls being moved > determines the time? Of each event? Where it says that ball n is inserted at time -1/n and removed at time -1/10n. That was a dumb question. >>> As time is a continuum and the numbers of balls in the vase is not, >>> there is no way of inverting the realtionship in the way that TO claims. >> Your times are as discontinuous as the number of balls, if no events can >> happen at any other moments than those specified. > > That hardly means that there are no other times in between. > > Time is a continuum. Or does TO claim that time is quantized? Where real time is continuous, there is always something happening. That's not the case here. The moments during events are a countable subset of the uncountable interval. >>>> So, when does the vase become empty? Nothing can occur at noon, as far >>>> as ball removals. AT every time before noon, balls are in the vase. So, >>>> when does the vase become empty, and how? >>> The vase is empty when every ball has been removed, and that occurs at >>> noon. >> So, that occurs AT noon? The vase becomes empty, when no balls are being >> removed? Remember, every ball was removed BEFORE noon, and upon the >> removal of each and every ball, more balls resided still in the vase. >> So, how does the vase empty, when no balls are removed? > > So how is the vase be not empty after every ball is removed? There is no "after". You are hiding your largest finite in a moment of infinite processing, but it's leaving a hole in your logic. If every natural is removed, then t=0, and infinite balls are added. That's how. >>>>> If you follow the sequence of insertions and removals you never "get >>>>> to" noon but this doesn't imply that noon is never reached, or that >>>>> iterations involving non-naturally numbered balls occur. It just >>>>> implies that all insertion and removal is performed before noon. >>>>> >>>>> Tony won't let himself understand this. He is delusional. His problem. >>>>> >>>> I won't let myself accept self-contradictory conclusions. >>> >>> At least not unless they are TO's own personal self-contradictory >>> conclusions. Like the existence of balls in a vase from which all balls >>> have been removed. >>> >>> >> 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. Uh, yeah, at specific times. But there is no non-self-c |