An uncountable countable set
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From: Virgil on 28 Sep 2006 14:24 In article <451be86a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > What is the number of the ball which, when removed, > > makes the vase empty? > > > > There is no such number, since for each ball n removed, balls n+1 > through balls n*10 remain. I have maintained throughout that, despite > your "labeling schemes", 9/10 of the balls remain, if you add 10 and > remove 1 repeatedly. It is precisely like adding 10 gallons and removing > 1 per minute. The ocean will never empty. Think measure. So which balls are left after noon, TO?
From: stephen on 28 Sep 2006 15:27 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> Randy Poe <poespam-trap(a)yahoo.com> wrote: >>>> >>>> <snip> >>>> >>>>> What is the number of the ball which, when removed, >>>>> makes the vase empty? >>>>> I know the kind of nonsense you will spout in answer to >>>>> those questions, but the answers within our axiom system >>>>> are: (1) there is no t<noon which is the moment just >>>>> before noon. For any t<noon, there is t < t' < noon. >>>>> (2) There is no such ball. >>>>> Here are the Tony gobbledgook answers: >>>>> (1) noon - 1/oo >>>>> (2) Ball number omega >>>>> In TO-matics, one can confidently give an answer like >>>>> number 2 despite the fact that one can also agree >>>>> that no ball numbered omega is ever put into the >>>>> vase. >>>> In TO-matics, it is also possible to end up with >>>> an empty vase by simply adding balls. According to TO-matics >>>> >>>> ..1111111111 = 1 + 1 + 1 + 1 + ... >>>> >>>> and >>>> ..1111111111 + 1 = 0 >>>> >>>> So if you just keep on adding balls one at a time, >>>> at some point, the number of balls becomes zero. >>>> You have to add just the right number of balls. It is not >>>> clear what that number is, but it is clear that it >>>> exists in TO-matics. >>>> >>>>> But in mathematics and logic, we don't get to >>>>> keep a set of self-contradictory assumptions around, >>>>> only using the ones we want as needed. >>>>> - Randy >>>> Where's the fun in that? :) >>>> >>>> Stephen >> >>> You drew that from my suggestion of the number circle, and that ...11111 >>> could be considered equal to -1. Since then, I looked it up. I'm not the >>> first to think that. It's one of two perspectives on the number line. >>> It's either really straight, or circular with infinite radius, making it >>> infinitesimally straight. The latter describes the finite universe, and >>> the former, the limit. But, you knew that, and are just trying to have fun. >> >>> Tony >> >> I am just pointing out that according to your mathematics >> that if you keep adding balls to the vase, you can end up >> with an empty vase. The fact that other people may have >> considered a number circle does not change the fact that the >> number circle implies that if you keep on adding balls, eventually >> you will have zero balls. > That's a bastardization of the concept. There are two ways to look at > the number circle, and you are combining them in mutually contradictory > ways. How is it a bastardization of the concept? You claim that 1+1+1+1+ ... = ..11111111 and that ..11111111 + 1 = 0 Why does that not apply to balls in the vase? Each ball is a 1. If a add balls, I add 1's. Do I not eventually get 0? If it does not apply to balls, what does it apply to, and how do you determine when it applies? >> >> So why is it okay to end up with zero balls, when you never remove >> any at all, but it is not okay to end up with zero balls when >> each ball is clearly removed at a definite time? > Because the model of the number circle where all strings are positive is > incompatible with the number circle where any string with a > left-unending string of 1's (in binary) is negative. Duh. That seems to be a non-answer. The most I can glean from that is that the number circle is not relevant to the balls in the vase problem. Is the number circle relevant to anything? And how does one determine when it is relevant? If it is not relevant to anything, why did you bring it up in the first place? Like most of the other cranks, your "system" is only usable by yourself, as the only way to know when one of your "rules" applies is by asking you. So apparently sometimes 1+1+1+1+ ... = ...1111111 and sometimes ...1111111+1 = 0 but at other times they equal something different and there appears to be know way to know which is which. Stephen
From: stephen on 28 Sep 2006 15:34 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Virgil wrote: >> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>>Randy Poe wrote, about the Balls in a Vase problem: >>> >>>>It definitely empties, since every ball you put in is >>>>later taken out. >>> >>>And _that_ individual calls himself a physicist? >> >> Does Han claim that there is any ball put in that is not taken out? > Nonsense question. Noon doesn't exist in this problem. Yes it is a nonsense question, in the sense that it is non-physical. You cannot actually perform the "experiment". Just as choosing a number uniformly from the set of all naturals is a non-physical nonsense question. You cannot perform that experiment either. Stephen
From: Virgil on 28 Sep 2006 16:36 In article <451bec94(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <451b3296(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > >>>> You must have been a strange 10 year old, like that kid > >>>> down the block that used to pull the legs off of roaches. > >>> Only those that looked like TO. > >>> > >>>>>>> So the reason I don't say it's full "an infinitesimal time > >>>>>>> before noon" or "some other time before noon" is that > >>>>>>> I don't say it's full. > >>>>>> But, you do say it's full or empty, right? > >>> One can easily say that it is empty at any time at which every ball > >>> that was put in has been taken out again. > >>> > >>> Does TO suggest that at any time after noon there is any ball that was > >>> put in that was not also taken out? > >> Yes, at any given time 9/10 of the balls inserted remain. > > > > Which ball does not have a definite time at which it > > is removed? > > > > Any ball which does not have a definite time at which it is inserted. That excludes every ball, since each has a specific time of insertion and an equally specific time of removal. > > >>>>> So your conclusion from my statement that I would never > >>>>> say it's full is that sometimes I would say it's full? > >>>> Uh, you would say it contains an infinite number of balls in some > >>>> circumstances, as I understand it. > >>> Then you misunderstand it. > >> No, your labels misconstrue the problem with your silly fixation on > >> omega. Do I "misunderstand" that if you remove balls 1, then 11, then > >> 21, etc, that the vase will NOT be empty? > > > > We have different variants of the problem setup. Before > > discussing too many details, we need to agree on > > what EXACTLY are the starting assumptions. > > The subject is whether that makes any difference or not. It doesn't. Then TO should not object to any specific starting assumptions, as he claims they make no difference. > Your dual gedankens imply that changing the labeling scheme after noon > makes the balls all disappear. That's ridiculous. No labelling scheme can be allowed to change any label once the ball with that label is inserted in the vase. > > > > > But in general if: > > (a) Every ball has a label n which is a finite natural number. > > (b) Every ball has a time t_n at which it is removed. > > (c) There exists a supremum T of the set {t_n, n in N} > > then for any time t >= T, the vase is empty. > > What is this "supremum", in terms of iterations? It is the least upper bound of the times, and any set of times which is bounded above must have a supremum. > Here's one iteration: > (a) 10 balls added AND > (b) 1 ball removed IMPLIES > (c) net 9 balls added > > How many iterations? n? Fine. 9n balls remain. Which ones? Unless one is postulating that the balls, like electrons, have no individual identities, but are totally interchangeable at all times, even when labelled to give them identities, the question of which ones is relevant.
From: cbrown on 28 Sep 2006 21:06
Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: <snip> > >> Like I said, there were > >> terms in my infinitesimal sections of moving staircase which differed by > >> a sub-infinitesimal from those in the original staircase. So, they could > >> be considered to be two infinitesimally different objects in the limit. > > > > Here's a thing that confuses me about your use of the term "limit". > > > > In the usual sense of the term, every subsequence of a sequence that > > has as its limit say, X, /also/ has a limit of X. > > > > For example, the sequence (1, 1/2, 1/2, 1/3, ..., 1/n, ...) usually is > > considered to have a limit of 0. And the subsequence (1/2, 1/4, 1/6, > > ..., 1/(2*n), ...) which is a subsequence of the former sequence has > > the same limit, 0. > > > > But the way you seem to evaluate a limit, the sequence of staircases > > with step lengths (1, 1/2, 1/3, ..., 1/n, ...) is a staircase with > > steps size 1/B, where B is unit infinity; but the sequence of > > staircases with step lengths (1/2, 1/4, 1/6, ..., 1/(2*n), ...), which > > is a subsequence of the first sequence, would seem to have as its limit > > a staircase with steps of size 1/(2*B). > > > > Unless steps of size 1/B are the same as steps of size 1/(2*B), I don't > > see how that can be possible. > > > > Cheers - Chas > > > > It's possible because no distinction is currently made between countable > infinities, even to the point where a set dense in the reals like the > rationals is considered equal to a set sparse in the reals like the > naturals. Where there is no parametric understanding of infinity, > infinity is just infinity, and 0 is just 0. Uh, OK. I assume that you somehow resolve this lack of "parametric understanding" in /your/ interpretation of T-numbers. > Where there is a formulaic > comparison of infinite sets as n->oo, the distinction can be made. The > fact that you have steps of size 1/n as opposed to steps of size 1/(2*n) > is a reflection of the fact that the first set has twice the density on > the real line as the first. As a proper superset, it SHOULD be larger. > So, it's quite possible to make sense of my position, with a modicum of > effort. Well, let me ask you this: Suppose we have the original sequence of staircases, with step lengths (1, 1/2, 1/3, ..., 1/n, ...). Let S be the T-limit staircase; you claim that it has step sizes 1/B, where B is unit infinity. Now, suppose we just forget about the very first staircase, but otherwise continue normally. Now we have step lengths (1/2, 1/3, 1/4, ...., 1/(n+1), ...). Do you claim that the T-limit staircase has step size 1/(B+1)? Do you propose that 1/B is or is not equal to 1/(B+1)? I mean, suppose you actually started constructing staircases, starting with side length 1; I then enter the room when you start to construct the second staircase with length 1/2. I copy everything that you do from thereon, just as you do it: you make a staircase of length 1/2, and at the same time, so do I. You make a staircase with side length 1/3, and at the same time, so do I. And so on. But somehow, we end up with different results "in the infinite case"? Cheers - Chas |