An uncountable countable set
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From: Virgil on 16 Oct 2006 22:49 In article <453437c4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4533d0ff(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45319846(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>>> Functions can exist at points at which their limits do not. > >>>>> There are even functions with domain R which are discontinuous at every > >>>>> rational argument but continuous at every irrational one. > >>>> That sounds vaguely interesting. Can you give an example? > >>> It is standard fare for anyone who knows any analysis. > >>> > >>> Let f: R --> R, be such that for each irrational x f(x) = 0, > >>> and for each rational x whose expression in lowest terms is p/q, > >>> let f(x) = 1/q. Then that function is provably continuous at each > >>> irrational and provably discontinuous at each rational except 0. It is > >>> provably the case that for each real a, lim_{x --> a} f(x) = 0, which > >>> establishes the claim. > >> Can you give an example of two irrational numbers, between which there > >> are no rational numbers? > > > > Irrelevant to the example above. > > > > For the function,f, defined above one has > > at every irrationals value of x, lim_Py -> x} f(y) = 0 = f(x) > > but at every rational value, x = p/q in lowest terms, > > lim_Py -> x} f(y) = 0 but f(p/q) = 1/q != 0 > > Uh, I suppose that establishes it in basic terms. Of course, that > doesn't have anything to do with vase, but okay. Since TO cannot understand the vase problem either, okay.
From: Virgil on 16 Oct 2006 22:56 In article <45343843(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4533d18b(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45319b8c(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Given any finitely numbered ball, we can calculate its entry and exit > >>>> times. However, we can also say that when it exits, there are more balls > >>>> in the vase than when it entered. If you had any upper bound to your set > >>>> of naturals, you'd see your logic makes no sense, but there is none. > >>> > >>> When expressed as functions of time, rather than the number of > >>> operations, there is no problem with having an empty vase at noon. > >> When expressed in terms of iterations, the conclusion is quite the > >> opposite, so you cannot claim to be working with pure unquestionable > >> logic. You must choose which logical construction of the two is valid, > >> if either. > > > > As the problem is stated in terms of the times at which events occur, > > expressing things in terms of time is natural and indicated. > > > > When expressed in terms only of events, there is no longer any > > requirement that there even be an event of completing all the > > insertions-removals, or if there is, that it be close in time to any > > other event. You cannot dismiss time and still have it. > > You can say, "What if we do this an infinite number of times?", or, "If > we do this n times, how many balls are in the vase?" In the original problem each action is specifically linked to time and the question asked is also specifically linked to time. So that TO's , and other's, attempts to ignore time as an essential part of the problem are trying to obscure the problem. > > > > >>>> Before noon, there are balls. At noon, there are not. What happened? > >>> > >>> They were one by one removed. > >> "One by one" and all removed, but no last one. Vigilogic at its worst. > > > > It is apparently TOlogic, that if one moves from point A to point B one > > must first cover half the distance then half the remaining distance, and > > so on ad infinitum, so that one never reaches point B. > > It is Virgilogic that one can go from one such midpoint to the next "one > > by one" but still reach point B in finite time. It is something that I manage to do every time I move. It has never yet taken me infinitely long to get to any point I have been to any other point I have ever got to. However it does seem to be taking TO an infinitely long time to reach common sense. Perhaps it is just too far for him to reach.
From: Ross A. Finlayson on 16 Oct 2006 23:05 Virgil wrote: > In article <45343843(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > In article <4533d18b(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Virgil wrote: > > >>> In article <45319b8c(a)news2.lightlink.com>, > > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > >>> > > >>>> Given any finitely numbered ball, we can calculate its entry and exit > > >>>> times. However, we can also say that when it exits, there are more balls > > >>>> in the vase than when it entered. If you had any upper bound to your set > > >>>> of naturals, you'd see your logic makes no sense, but there is none. > > >>> > > >>> When expressed as functions of time, rather than the number of > > >>> operations, there is no problem with having an empty vase at noon. > > >> When expressed in terms of iterations, the conclusion is quite the > > >> opposite, so you cannot claim to be working with pure unquestionable > > >> logic. You must choose which logical construction of the two is valid, > > >> if either. > > > > > > As the problem is stated in terms of the times at which events occur, > > > expressing things in terms of time is natural and indicated. > > > > > > When expressed in terms only of events, there is no longer any > > > requirement that there even be an event of completing all the > > > insertions-removals, or if there is, that it be close in time to any > > > other event. You cannot dismiss time and still have it. > > > > You can say, "What if we do this an infinite number of times?", or, "If > > we do this n times, how many balls are in the vase?" > > In the original problem each action is specifically linked to time and > the question asked is also specifically linked to time. > > So that TO's , and other's, attempts to ignore time as an essential part > of the problem are trying to obscure the problem. > > > > > > > >>>> Before noon, there are balls. At noon, there are not. What happened? > > >>> > > >>> They were one by one removed. > > >> "One by one" and all removed, but no last one. Vigilogic at its worst. > > > > > > It is apparently TOlogic, that if one moves from point A to point B one > > > must first cover half the distance then half the remaining distance, and > > > so on ad infinitum, so that one never reaches point B. > > > It is Virgilogic that one can go from one such midpoint to the next "one > > > by one" but still reach point B in finite time. > > It is something that I manage to do every time I move. It has never yet > taken me infinitely long to get to any point I have been to any other > point I have ever got to. > > However it does seem to be taking TO an infinitely long time to reach > common sense. Perhaps it is just too far for him to reach. Try taking nine steps backwards for each forwards. Virgil, people put up with you. Don't worry, I know I'm unliked. Do you know anything about physics? I just wonder if you ever heard the story of why one dimension of time was sufficient. Ross
From: Ross A. Finlayson on 16 Oct 2006 23:18 Tony Orlow wrote: > Virgil wrote: > > In article <4533d18b(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45319b8c(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Given any finitely numbered ball, we can calculate its entry and exit > >>>> times. However, we can also say that when it exits, there are more balls > >>>> in the vase than when it entered. If you had any upper bound to your set > >>>> of naturals, you'd see your logic makes no sense, but there is none. > >>> > >>> When expressed as functions of time, rather than the number of > >>> operations, there is no problem with having an empty vase at noon. > >> When expressed in terms of iterations, the conclusion is quite the > >> opposite, so you cannot claim to be working with pure unquestionable > >> logic. You must choose which logical construction of the two is valid, > >> if either. > > > > As the problem is stated in terms of the times at which events occur, > > expressing things in terms of time is natural and indicated. > > > > When expressed in terms only of events, there is no longer any > > requirement that there even be an event of completing all the > > insertions-removals, or if there is, that it be close in time to any > > other event. You cannot dismiss time and still have it. > > You can say, "What if we do this an infinite number of times?", or, "If > we do this n times, how many balls are in the vase?" > > > > >>>> Before noon, there are balls. At noon, there are not. What happened? > >>> > >>> They were one by one removed. > >> "One by one" and all removed, but no last one. Vigilogic at its worst. > > > > It is apparently TOlogic, that if one moves from point A to point B one > > must first cover half the distance then half the remaining distance, and > > so on ad infinitum, so that one never reaches point B. > > It is Virgilogic that one can go from one such midpoint to the next "one > > by one" but still reach point B in finite time. > > Zeno's paradox is long since explained. This one is too. Tony, you got something wrong: it's not "Vigilogic", it's "VBN." I think Toni understands I'm a post-Cantorian. Tony, I think for Virgil to call you a phony, ignore it. There are plenty of people who'll talk to you, as we all read these discussions until we get bored. Zeno's paradox appeases. Look it up in the dictionary. Ross
From: Virgil on 17 Oct 2006 01:17
In article <45343a40(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4533d315(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45319e2d(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <453108b5(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>>> So how many balls are left in the vase at 1:00pm? > >>>>>>> > >>>>>> If you paid attention to the various subthreads, you'd know I just > >>>>>> answered that. Where the insertions and removals are so decoupled, > >>>>>> there > >>>>>> is no problem. Where the removal of a ball is immediately preceded and > >>>>>> succeeded by insertions of 10, the vase never empties. > >>>>> It may that it never "empties", but at noon, and thereafter, it has > >>>>> "become empty". > >>>>> > >>>> And when did that happen? > >>> At the invisible transition from forenoon to noon. > >> Oh, you mean the moment between all moments before noon and the moment > >> of noon? > > > > > > No! The set of all moments before noon is, in the real number model of > > times, an open set whose least upper bound, noon, is not a member of > > that set but such that there are no times between that set and its LUB, > > noon. > > So that as sets of times, the forenoon and what comes after, form a > > partition of times into disjoint sets with every time being in one or the > > other but none in both. > > > > So, something happens at this partition, which is not actually a point > in time, but a conceptual separation between this point and everything > that came before it? One can separate the reals into everything before 0 as one set and 0 and everything after it as the other. Does TO claim time is less seperable? > Is this your quantum math again? Do we step out of > time for less than a moment to make the balls disappear? Any partition of time into before and after must have a point of separation, like the noon of this case. Everything happens before that point. > > > > >>>>> The only necessary constraint on insertions of balls into the vase and > >>>>> removals of balls from the vase is that each ball that is to be removed > >>>>> must be inserted before it can be removed, and, subject only to that > >>>>> constraint, the set of balls remaining in the vase at the end of all > >>>>> removals is independent of both the times of insertion and of the times > >>>>> of removal. > >>>>> > >>>>> To argue otherwise is to misrepresent the problem. > >>>> You already said that.....WRONG!!!! > >>> What is wrong about it? > >> This: > >> > >>>> There is the additional constraint that, before removing any ball, ten > >>>> have been inserted. > > > >>> > >>> Then let us put all the balls in at once before the first is removed and > >>> then remove them according to the original time schedule. > >> Great! You changed the problem and got a different conclusion. How > >> very....like you. > > > > Does TO claim that putting balls in earlier but taking them out as in > > the original will result in fewer balls at the end? > > If the two are separate events, sure. Now that is really illogical: TO claims that having the balls in for longer times will leave fewer of them at the end. > > > If so, by what logic, and if not, what difference does it make? > > That the removal of balls in the second phase is unrelated to the > addition of balls in the first stage. In the original problem, no ball > is removed with the immediately preceding addition of ten. > >>> Does TO claim that by putting balls in earlier there can be at ANY time > >>> fewer balls in the vase that when putting them in later? > >> Yes, I've already explained that. Not to the satisfaction of anyone but yourself. > > > > Not to anyone else's satisfaction. > > I'm sure to that of some. You're never satisfied except with yourself. I am not satisfied that having more balls in for longer times gives less balls in at any time. One certainly starts with more balls. At what time do more balls become less balls? And why? > > >>> But when one puts them all in early enough, it becomes obvious that the > >>> vase must be empty at noon. > >> Yes, if all insertions occur before all removals. > > > > How does that change things? Is there any time at which putting balls in > > earlier forces fewer balls to be in the vase? If so, at what time does > > the number of balls from the earlier insertions become less that the > > number of balls from the later insertions. > > > > As far as I can see, putting balls in earlier can only increase the > > number of balls in the vase at some times, but not decrease the number > > at any time. > > > > > > Why, did you put more balls in, when you added them earlier? When some balls are to be added earlier then at all times there will be at least as many balls as there would be when entering them later. Anything else is idiotic. >You really > don't understand the implications of the Zeno machine, do you? I do not understand how having more balls in the vase for longer times can produce less balls in the urn at any time. > > >>> So TO must argue that having more balls in the vase at all times before > >>> noon results in less balls in the vase at noon. > >>> > >>> Now that is ...REALLY... WRONG!!! > >>> > >>> > >> It is correct to say that decoupling the insertions from the removals > >> such that each sequence is distinct time-wise and with its own point of > >> condensation creates a different situation. > > > > As long as each ball is inserted at least as early as originally and no > > ball is removed earlier that originally, if there is to be any set of > > times at which there are fewer balls that originally then it must be > > after the first ball is inserted in the new schedule, so the times are > > bounded below and must have a GLB (greatest lower bound). > > What is that GLB? > > > > Unless you can provide one, your claim fails. > > No it doesn't. It does with anyone of sense. |