An uncountable countable set
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From: Lester Zick on 27 Oct 2006 14:48 On Fri, 27 Oct 2006 00:07:16 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >MoeBlee wrote: >> Tony Orlow wrote: >>> I share your and Godel's concerns about point set theory >> >> Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says >> such things, "I share Godel's concerns about point set theory." Too bad >> Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's >> concerns, or mathematical logic, or set theory, or point set topology, >> or topology. >> >> MoeBlee >> > >Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) Of course a little levity, like motions to adjourn, is always in order, Tony. Mathematikers take all this drollery way too seriously. They get all huffy and self righteous when forced to actually explain things they're used to outsourcing to books. Thanks for the comment. ~v~~
From: stephen on 27 Oct 2006 14:47 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Lester Zick wrote: >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: >> >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. >> >> So non zero integers are not real? > That's a pretty impressive leap of illogic. Using Lester IllLogic it is easy to prove that 2 is not prime. 2 is the largest even prime integer. There is no largest prime integer. Therefore 2 cannot be prime. Stephen
From: cbrown on 27 Oct 2006 15:27 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > 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. > >>> > >>>> 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. > >>> > >>> 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 > >>> > >>> 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)? > >>> > >>> Cheers - Chas > >>> > >> This is a very good question, Chas. Thanks. I'll have to think about it, > >> and I'm rather tired right now, but at first glance it seems like it > >> could be a sound analysis. I've cut and pasted for perusal when I'm > >> sharper tomorrow. > >> > > > > Here's some of my thoughts: > > > > When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > > time t, we mean a real number t" > > That doesn't mean t has to be able to assume ALL real numbers. The times > in [-1,0) are all real numbers. And I would say that assuming that by a time t, we mean a real number in [-1,0) is a different assumption than (1). > > > > > When you say "if we always add more balls than we remove, the number of > > balls in the vase at time 0 is not 0", I think "he doesn't accept (8): > > if the numbers of balls in the vase is not 0, then there is a ball in > > the vase." > > No, I accept that. There is no time after t=-1 where there is no ball in > the vase. > I.e., there is a ball in the vase. But then by the argument I previously gave, there is then a ball in the vase which is not in the vase. Your reference to "unspecified" balls in the vase at noon I interpret to be a way of saying that (8) should instead state something like "if the number of balls in the vase is not 0, then there may be no /specific/ ball in the vase (because there is instead an /unspecific/ ball in the vase)". > > > > When you say "an infinite number of balls are removed at time 0", I > > think "he does not agree with (6) if balls are removed at some time t, > > they are removed in accordance with the problem statement: i.e. there > > exists some natural number n s.t. n = -1/t and (some other stuff)". > > I didn't say that exactly. If 0 occurs, then all finite balls are gone, > but infinite balls have been inserted such that 1/n=0 for those balls. > So, at noon the vase is not empty, even if it occurs in the problem, > which it doesn't. > If infinite balls are inserted at some time t = -1/n = 0, then by (5) each of them are inserted at time t; and at that time exactly 10 balls are inserted. 10 is not infinite. > > > > All these assertions follow a simgle theme: "If I require that my > > statemnents be /logically/ consistent, does the given problem make > > sense; and if so, what is a reasonable resonse?". > > > > Cheers - Chas > > > > That there is a contradiction in your conclusion if you assume that all > events must occur at some time... The "occurence" of these events (ball insertions and removals at particular times) is described by (1), (5), (6), and (7). > ... and that becoming empty is the result of > events that happen in the vase. There is no "becoming" empty described in (1)..(8). There is only "being" empty; which is described by (1), (2), (3), and (4), and (8). > It cannot become empty until noon, when > nothing happens to cause it. And that is your premise that I call (T1). It is incompatible with (1)..(8); so either we must reject something in (1)..(8), or we must reject (T1) in favor of my previously described (*). Cheers - Chas
From: Virgil on 27 Oct 2006 15:57 In article <PcydnR9ql7biLtzYnZ2dnUVZ_vGdnZ2d(a)comcast.com>, "RLG" <Junk(a)Goldolfo.com> wrote: > "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. Although one might make a reasonable argument that the value should b e between 0 and 2 gallons, inclusive.
From: Virgil on 27 Oct 2006 16:03
In article <1161936661.828422.190110(a)e3g2000cwe.googlegroups.com>, imaginatorium(a)despammed.com wrote: > 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. Since Ross has, by his own posts, shown himself to be as far out of touch with reality as TO, TO's approval is only piling Pelion on Ossa. If TO were to support someone reasonably in touch with mathematical reality, I should not have regarded it as a "last straw" situation. |