An uncountable countable set
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From: Ross A. Finlayson on 27 Oct 2006 20:52 Virgil wrote: ... > > 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. Virgil's among the few people here ever called a liar. His overgeneralizations are generally wrong. Is that like the one about wrestling with a pig? Virgil, are you considering attempting to converse directly with me? Last time that happened I showed you were wrong, no? You piped up with one of your little barbs and ate it. Virgil, I don't care for what you say and don't think you have anything else to tell me, so, don't, punk. Virgil, you're snotnosed and rude, fraud. Is it a bitter pill, Virgil? I've found mistakes in the CRC, Knuth, Mathworld, maybe HMF, etc. To paraphrase Fraenkel: reliance on transfinite cardinals is a mistake. Measure theory is about quantitative continua, there are no applications of transfinite cardinals, and the integral calculus is a nilpotent infinitesimal analysis. Re-Vitali-ize measure theory, polydimensionally. Hilbert requests a well-ordering of the reals. There is no universe in ZFC. The only theory with no axioms is A theory. The Finlayson numbers are all the numbers, the Finlayson reals are the reals. Select. Ross
From: Tony Orlow on 27 Oct 2006 20:52 cbrown(a)cbrownsystems.com wrote: > 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). > There is no contradiction between them, is there.... >>> 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)". > You claim no balls are added at noon, because nothing can be, but then, nothing can be removed at noon, either. Either it grows or stays un-zero. >>> 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. > It i larger than 1, and at an infinite rate of insertion, yes, an infinite number of balls are added at t=0. >>> 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).
From: Virgil on 27 Oct 2006 20:53 In article <4542a695(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > And the gedankenexperiment occurs in standard mathematics. > > t=-1/n ^ t=0 -> -1/n=0. T v F? "t=-1/n ^ t=0 -> 1 = 2" is just as true as "t=-1/n ^ t=0 -> -1/n = 0"
From: Tony Orlow on 27 Oct 2006 20:55 Lester Zick wrote: > On 27 Oct 2006 11:01:57 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >> Lester Zick wrote: >>> Ah, Brian, ever the amanuensis. >> Zick, ever the nuisance. > > Ah, Moe, truth is often a nuisance. > > ~v~~ Stop confusing me with facts, Lester. Not and/or the inverse of not the notness of it all. I just don't get it. :) TOny
From: Tony Orlow on 27 Oct 2006 20:55
Lester Zick wrote: > On 27 Oct 2006 11:38:10 -0700, imaginatorium(a)despammed.com wrote: > >> David Marcus 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. >> Gosh, you obviously haven't seen Lester when he's in full swing. (Have >> _you_ searched sci.math for "Zick transcendental"?) > > No but obviously you have, Brian. > > ~v~~ I think Brian may have "been in the swing" at the time. Dunno. 01oo |