An uncountable countable set
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From: Virgil on 24 Oct 2006 17:02 In article <453e4c19(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David R Tribble wrote: > >>> Tony Orlow wrote: > >>>> You have agreed with everything so far. At every point before noon balls > >>>> remain. You claim nothing changes at noon. Is there something between > >>>> noon and "before noon", when those balls disappeared? If not, then they > >>>> must still be in there. > >>> Of course there is a "something" between "before noon" and "noon" where > >>> each ball disappears. At step n, time 2^-n min before noon, ball n is > >>> removed. This happens for every ball, since there is a step n for > >>> every ball. The balls are removed, one by one, one at each step, > >>> before noon. > >>> > >> 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? The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does TO answer?
From: Tony Orlow on 24 Oct 2006 17:14 Virgil wrote: > In article <453e4a85(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Your examples of the circle and rectangle are good. Neither has a >>>>>>>> height >>>>>>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, >>>>>>>> because the circle actually exists there. To ask about its height at >>>>>>>> x=9 >>>>>>>> is like asking how the air quality was on the 85th floor of the World >>>>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what >>>>>>>> happens at noon. There is no vase at noon. >>>>>>> Do you really mean to say that there is no vase at noon or do you mean >>>>>>> to say that the vase is not empty at noon? >>>>>> If noon exists at all, the vase is not empty. All finite naturals will >>>>>> have been removed, but an infinite number of infinitely-numbered balls >>>>>> will remain. >>>>> "If noon exists at all"? How do we decide? >>>>> >>>> We decide on the basis of whether 1/n=0. Is that possible for n in N? >>>> Hmmmm......nope. >>> So, noon doesn't exist. And, there is no vase at noon. I thought you >>> were saying the vase contains an infinite number of balls at noon. >>> >> If the vase exists at noon, then it has an uncountable number of balls >> labeled with infinite values. But, no infinite values are allowed i the >> experiment, so this cannot happen, and noon is excluded. > > So did the North Koreans nuke the vase before noon? > > The only relevant issue is whether according to the rules set up in the > problem, is each ball inserted before noon also removed before noon?" > > An affirmative confirms that the vase is empty at noon. > A negative directly violates the conditions of the problem. > > How does TO answer? You can repeat the same inane nonsense 25 more times, if you want. I already answered the question. It's not my problem that you can't understand it.
From: Randy Poe on 24 Oct 2006 18:08 Tony Orlow wrote: > Virgil wrote: > > In article <453e4a85(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> Your examples of the circle and rectangle are good. Neither has a > >>>>>>>> height > >>>>>>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, > >>>>>>>> because the circle actually exists there. To ask about its height at > >>>>>>>> x=9 > >>>>>>>> is like asking how the air quality was on the 85th floor of the World > >>>>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what > >>>>>>>> happens at noon. There is no vase at noon. > >>>>>>> Do you really mean to say that there is no vase at noon or do you mean > >>>>>>> to say that the vase is not empty at noon? > >>>>>> If noon exists at all, the vase is not empty. All finite naturals will > >>>>>> have been removed, but an infinite number of infinitely-numbered balls > >>>>>> will remain. > >>>>> "If noon exists at all"? How do we decide? > >>>>> > >>>> We decide on the basis of whether 1/n=0. Is that possible for n in N? > >>>> Hmmmm......nope. > >>> So, noon doesn't exist. And, there is no vase at noon. I thought you > >>> were saying the vase contains an infinite number of balls at noon. > >>> > >> If the vase exists at noon, then it has an uncountable number of balls > >> labeled with infinite values. But, no infinite values are allowed i the > >> experiment, so this cannot happen, and noon is excluded. > > > > So did the North Koreans nuke the vase before noon? > > > > The only relevant issue is whether according to the rules set up in the > > problem, is each ball inserted before noon also removed before noon?" > > > > An affirmative confirms that the vase is empty at noon. > > A negative directly violates the conditions of the problem. > > > > How does TO answer? > > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. Your response requires that the vase contains balls which were never, by the stated rules, put in. You keep saying things like "if the clock runs till noon there are balls with infinite numbers on them" even though the rules say there are no balls with infinite numbers on them. How do you reconcile that? If I put in balls 1, 2, 3 and stop, can the clock tick till noon without requiring a 4th ball? If I specify times for balls 1-1000 only, can the clock till noon without requiring a 1001-th ball? 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? - Randy
From: Lester Zick on 24 Oct 2006 18:23 On Tue, 24 Oct 2006 13:30:59 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Mon, 23 Oct 2006 15:00:58 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >> >> [. . .] >> >>>>> Unfortunately, transfinitology exists, despite the fact that it makes no >>>>> sense underneath the hood. When it comes to arithmetic on them, it's one >>>>> big kludge. But, there are forms of infinite numbers upon which one can >>>>> define arithmetic. They just have nothing whatsoever to do with omega or >>>>> the alephs. >>>> Not sure what you're talking about here, Tony. Lots of things exist in >>>> the sense of having been defined. That doesn't make them true and >>>> doesn't mean they form any basis for the truth of other things defined >>>> on them. There's no shortage of things other than infinity on which to >>>> define arithmetic. >>>> >>> What is log2(0)? >> >> -00? Not sure what this is in aid of, Tony. What is log0(0)? For that >> matter what is log0(00) or log-00(00) or x!=0? There are all kinds of >> restrictions around 0 and 00 precisely because 0 is not a natural >> number. >> > >It was just a little example of where an infinity arises naturally. But the problem is that infinity doesn't arise in connection with the naturals. >>>>>> Yet I've also been considering what it looks like you're trying to do >>>>>> with trans finite arithmetic.In particular it occurs to me that if one >>>>>> takes +00 to be larger than any positive finite -00 correspondingly >>>>>> must be smaller than any negative finite such that your concept of >>>>>> circularity among arithmetic numbers might be combined in the >>>>>> following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only >>>>>> difference would be that whereas +00 represents the number of >>>>>> infinitesimals, -00 would represent the size of infinitesimals. Thus >>>>>> we'd have a positive axis with the number of infinitesimals and a >>>>>> negative axis with the size of infinitesimals. At least that's the >>>>>> best I can make of the situation. >>> Technically, the number of reals in the unit interval (0,1] is Big'un. >> >> But the point is that they're within the interval. There is no >> infinite set 1, 2, 3 . . . 00 outside of some interval. > >Sure there is, in an infinite interval. Of course, that requires the >existence of infinite numeric values, but that well within our capabilities. No this is wrong, Tony. There is no infinite interval because the interval itself is defined by the naturals. Don't forget the term "infinity" means "undefined". And without "finite" or "defined" boundaries there is nothing which can be defined in arithmetic terms. If you stop considering zero and conversely infinity naturals the problem goes away. Otherwise it becomes intractable as the various rules governing arithmetic processes with zero and infinity attest. I don't say zero and infinity aren't useful only that rules governing their operations are different from the naturals. >Ross's EF is a special case of my IFR, where the mapping function is >f(x)=1/Big'un. That maps the set of Big'un hypernaturals over the real >line to the set of Big'un infinitesimals, natural multiples of Lil'un, >within the first unit interval. >> >>> That's also the infinite length of the real number line, in unit >>> intervals. The unit infinitesimal is Lil'un, or 1/Big'un. Now they're >>> all specific and related to spatial measure and quantity. :) >>> >>>> On the other hand if you want to do transfinite arithmetic you might >>>> ask yourself what the results of 00-00 or 00/00 are. The latter can be >>>> addressed through application of L'Hospital's rule but I don't know >>>> any way to address the former. >>>> >>>> ~v~~ >>> The formulas that lend themselves to L'Hospital's Rule usually cannot be >>> simplified any further to resolve that problem. Subtracting one simple >>> formula from another is just a matter of combining like terms and >>> finding the most significant to see if you get a finite result through >>> mutual cancellations. >> >> But L'Hospital's rule applies to ratios, Tony. It only gives the >> finite ratio between infinities. If you subtract 1/0 from 2/0 what do >> you get? They both already have common denominators so the answer >> would seem to be 1/0 which still remains infinite. Kluge is the right >> word for transfinite arithmetic. >> >> ~v~~ > >You had oo-oo. I assume that's something like, say, the derivative of >1/x - 3/x^2 at x=0. Clearly, the second term dominates, and this tends >to -oo. I don't see the difficulty there, but it's probably not important. But I suspect it's vitally important when considering the actual location of infinities whether inside or outside any defined interval. As far as I can tell the set of naturals doesn't include 0 or 00. If you add a finite to a finite such as 1 to 1 to generate the set of naturals you only wind up with a finite set regardless of how long the process is thought to go on. Thus any set of naturals is finite. Infinity can only be defined where infinitesimal subdivision occurs because the "definition" for the interval is supplied by pairs of finites and the infinitesimal subdivision can only occur between those limits. ~v~~
From: Virgil on 24 Oct 2006 18:35
In article <453e824b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453e4a85(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> Your examples of the circle and rectangle are good. Neither has a > >>>>>>>> height > >>>>>>>> outside of its x range. The height of the circle is 0 at x=-1 and > >>>>>>>> x=1, > >>>>>>>> because the circle actually exists there. To ask about its height at > >>>>>>>> x=9 > >>>>>>>> is like asking how the air quality was on the 85th floor of the > >>>>>>>> World > >>>>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what > >>>>>>>> happens at noon. There is no vase at noon. > >>>>>>> Do you really mean to say that there is no vase at noon or do you > >>>>>>> mean > >>>>>>> to say that the vase is not empty at noon? > >>>>>> If noon exists at all, the vase is not empty. All finite naturals will > >>>>>> have been removed, but an infinite number of infinitely-numbered balls > >>>>>> will remain. > >>>>> "If noon exists at all"? How do we decide? > >>>>> > >>>> We decide on the basis of whether 1/n=0. Is that possible for n in N? > >>>> Hmmmm......nope. > >>> So, noon doesn't exist. And, there is no vase at noon. I thought you > >>> were saying the vase contains an infinite number of balls at noon. > >>> > >> If the vase exists at noon, then it has an uncountable number of balls > >> labeled with infinite values. But, no infinite values are allowed i the > >> experiment, so this cannot happen, and noon is excluded. > > > > So did the North Koreans nuke the vase before noon? > > > > The only relevant issue is whether according to the rules set up in the > > problem, is each ball inserted before noon also removed before noon?" > > > > An affirmative confirms that the vase is empty at noon. > > A negative directly violates the conditions of the problem. > > > > How does TO answer? > > You can repeat the same inane nonsense 25 more times, if you want. I > already answered the question. It's not my problem that you can't > understand it. It is a good deal less inane and less nonsensical than trying to maintain, as TO and his ilk do, that a vase from which every ball has been removed before noon contains any balls at noon that have not been removed. TO only objects to my repeating it because he has no effective answer to its pelucid logic. |