An uncountable countable set
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From: Virgil on 27 Oct 2006 17:15 In article <45422534(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > >> > >> Tach me more about mathematical logic and consistency. > > > > We've been trying, but you don't seem to want to learn. > > I guess you missed the sarcastic tone there. TO missed the spell checker there, too. >I have a hard time taking > lessons in logic from people who think things can happen in some kind of > time without being anywhere in time. We have a hard time taking seriously someone who thinks he can stop time because he does not like what happens. > > > > > 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. Then why does TO deliberately conflate them with real numbers? > > If there is a definition for "number" in general, and for "infinite", > then there cannot both be a smallest infinite number and not be. As there is no definition for 'number' in general, but a variety of definitions in particular, one must particularize 'number' to a sort which allows infinites before specifying "infinite'. TO does not do this. So he is then speaking nonsense. > Thanks for the logic lesson, but if transfinitologists are going to > claim to have a "correct" answer, and that any other interpretation is, > as Virgil would say, WRONG!!!, well then, they should feel obligated to > explain what's wrong with non-standard analysis, infinitesimals, > infinite series, limits, infinite-case induction, and other approaches. On the contrary. The standard stuff is already well established in texts and other works. If someone like TO wants to do anything non-standard, it is HE who must justify his case. It is the position of cranks that their claims stand without proofs unless they are refuted. It is the positions of non-cranks, that what has already been established and accepted shall persist until refuted. By those standards, TO is definitely cranky.
From: MoeBlee on 27 Oct 2006 17:17 Lester Zick wrote: > According to MoeBlee's recent lectures on the subject of exhaustive > mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled versions as if they are what I said. (And your claim that that is "a perfectly acceptable forensic modality" is typical Zickian rot.) Or, you'll just keep doing it until, like a child demanding to play "peek-a-boo" and forever tugging on the coats of adults, you tire of your own silly game. MoeBlee
From: MoeBlee on 27 Oct 2006 17:30 Lester Zick wrote: > According to MoeBlee's recent lectures on the subject of exhaustive > mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled versions as if they are what I said. MoeBlee
From: MoeBlee on 27 Oct 2006 17:31 Lester Zick wrote: > According to MoeBlee's recent lectures on the subject of exhaustive > mathematical definitions one cannot simply define IN and OUT, one must > use a placeholder such as IN(x) and OUT(x) to establish the domain of > discourse. Please stop mangling what I've said and then representing your mangled interpretations as if they are what I said. MoeBlee
From: cbrown on 27 Oct 2006 17:36
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 |