An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 17:21 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>>>> Han de Bruijn wrote: >>>>>>> Virgil wrote: >>>>>>> >>>>>>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>>>>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>>>>>> >>>>>>>>> Randy Poe wrote, about the Balls in a Vase problem: >>>>>>>>> >>>>>>>>>> It definitely empties, since every ball you put in is >>>>>>>>>> later taken out. >>>>>>>>> And _that_ individual calls himself a physicist? >>>>>>>> Does Han claim that there is any ball put in that is not taken out? >>>>>>> Nonsense question. Noon doesn't exist in this problem. >>>>>>> >>>>>>> Han de Bruijn >>>>>>> >>>>>> That's the question I am trying to pin down. If noon exists, that's when >>>>>> the vase supposedly empties, >>>>> Why does the existence of noon imply there is a time >>>>> which is the last time before noon? >>>>> >>>>> It doesn't. >>>>> >>>>> - Randy >>>>> >>>> I never said it did. When did I say that? >>> I was responding to Han, who said that "If noon exists, that's when >>> the vase empties". >>> >>> Noon exists. >>> >>> But in order for the vase to transition from not-empty >>> to empty, there would have to be a last non-empty >>> moment. That would be the last time before noon. > > But there is no "last moment before noon". So....you're correcting yourself? Okay...... > >> Yes, and at that last moment the last ball would have to be removed, > > There is no "last moment" and no "last ball" There is no spoon, and there is no empty vase, except initially. > >> and >> yet, at the moment before 10 balls would have to have been added. Can >> the vase contain -9 balls? :) > > There is no "moment before the last moment". There are successive iteration defined in the problem. > > Have you not yet figured out yet that given any two different > times, there are times in between them? That there's no > such thing as the "next moment"? There are successive iterations. In this gedanken, the next event after 11:59 is 30 seconds later, eh? > >>>> I will offer this simple >>>> logical argument. If the vase ever became empty, it would be because one >>>> ball was removed, >>> Hence my continued statement that the vase does not >>> "become empty". It is non-empty at certain times and >>> empty at others. >> How do you reconcile.... >> >> There is no transitional moment. >> ...with... >> >>> Noon is the first moment at which the vase is empty. >> Does the vase not go from non-empty to empty at noon? > > No. So, it stays non-empty? > >> You're making no >> sense. If you can't answer that simple question > > I answered it. The answer is "no". So, it doesn't go from non-empty to empty? > > Somebody is asking you to think about infinitely high strips, > and the situation is analogous. > > Think about the graph of tan(x), which you may or may not know > grows without bound as x approaches 90 degrees. For values of > x just above 90 degrees, tan(x) is large negative. Quite. Like 1/x around x=0. > > Here's the graph: > http://mathworld.wolfram.com/Tangent.html > > If you increase x from 0 to a point just above 90 degrees, > the following things are true: > (a) For every value of x below 90 degrees, y = tan(x) is positive. > (b) For every value of x above 90 degrees, y = tan(x) is negative. > (c) There is no point where y transitioned from positive to > negative. It did, at 90 degrees, where tan is both positive and negative, given that cos is 0, which is both positive and negative. This is part of the concept with the number circle which makes it intuitively satisfying. It gives continuity to some functions considered discontinuous. > > If you plot the number of balls in the vase vs. time, it behaves much > like one of those curves. You have a curve rising asymptotically > toward t = noon, and a flat line at t >= noon. But there is no > "transition" from the rising curve to the flat line, any more than > there is a "transition" from the curve of tan(x), x<90 deg to > tan(x), x>90 deg. There is no equivalence between oo and 0 (though I have heard people claim otherwise). Between oo and -oo there can sometimes be. > > The fact that this bothers you does not constitute my "getting > into trouble". > > - Randy > Oh, you're in trouble, Buster.
From: Tony Orlow on 29 Sep 2006 17:26 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <451bac34(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>>>> If the vase is empty at noon, but not before, how can that not be the >>>>>>>> moment that it becomes empty? >>>>>>> Saying that it is empty is quite different from saying anything about a >>>>>>> "last ball". andy does not deny that the vase becomes empty, he just >>>>>>> does not say anything about any "last ball out". >>>>>> Does that answer the question of **when** this occurs? Of course not. >>>>> It does answer the question of "whether" it occurs. "When" is of lesser >>>>> importance. >>>> So, you have no answer. >>> If something doesn't occur, the question "when does it occur" >>> does not have an answer. >> "[R]andy does not deny that the vase becomes empty". That sounds like it >> occurs. >> >>> If I ask you what date you took a trip to Mars last year, >>> would you have an answer? >> Does the vase become empty? > > Virgil and I differ on terminology here. As I have already said, > you are trying to pin down an identifiable pair of contiguous > moments where the vase is non-empty in one, and empty > in the next. As I have already said, a verb like "emptying" > conveys to me the existence of a PAIR of moments with > that property, of an identifiable "change moment". I would > not use the word "become" for the same reason. > > So I will continue to say what I have said. The vase is empty > at noon, because before noon every ball put in was taken > out. > > There is no moment when the vase "becomes empty". The > first time when the vase IS empty is noon. > > - Randy > Sorry, Randy, that's just daft. It's not, then it is, but it didn't become, because that would mean you'd have to think about that moment of becoming, and face the fact that you would need to have -9 balls two iterations beforehand to get an empty vase. Can you say "avoidance"? When something is at one moment, and wasn't before that moment, that's pretty much the definition of "become". Sorry. You're drifting into transfinitological mysticism here. Tony
From: Virgil on 29 Sep 2006 17:46 In article <451d6602(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I wouldn't put it that way. ...1111 can be interpreted as the largest > binary natural, if you claim all bit positions are finite. Calling all bit positions finite does not require that there only be a finitely many bit positions, and the binary string representation of every finite natural n requires <= ln(n+1)/ln(2) bit positions. > > > and that > > ..11111111 + 1 = 0 > > ...11111 can be interpreted indeed as -1, as is done every millions of > times per microsecond all over the world in computers. Which of the worlds computers can work with an infinitely long string of binary digits?
From: Virgil on 29 Sep 2006 17:49 In article <451d66c0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >> Virgil wrote: > > > >>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>> Randy Poe wrote, about the Balls in a Vase problem: > >>>> > >>>>> It definitely empties, since every ball you put in is > >>>>> later taken out. > >>>> And _that_ individual calls himself a physicist? > >>> Does Han claim that there is any ball put in that is not taken out? > > > >> Nonsense question. Noon doesn't exist in this problem. > > > > Yes it is a nonsense question, in the sense > > that it is non-physical. You cannot actually perform > > the "experiment". Just as choosing a number uniformly > > from the set of all naturals is a non-physical nonsense > > question. You cannot perform that experiment either. > > > > Stephen > > Yes, they both sound equally invalid, and it all goes back to omega, but > Han has a point about the density of the set in the naturals throughout > its range, and the overall statistical probability of selecting one of > that subset from the naturals, even if having probabilities of 1/omega > for each natural presents problems. > > Tony Do statistical probabilities have to satisfy the condition that their sum over all indivisible outcomes must equal 1?
From: Virgil on 29 Sep 2006 18:01
In article <451d68a5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451bec94(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <451b3296(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Randy Poe wrote: > >>>>>>> Tony Orlow wrote: > >>>>>> You must have been a strange 10 year old, like that kid > >>>>>> down the block that used to pull the legs off of roaches. > >>>>> Only those that looked like TO. > >>>>> > >>>>>>>>> So the reason I don't say it's full "an infinitesimal time > >>>>>>>>> before noon" or "some other time before noon" is that > >>>>>>>>> I don't say it's full. > >>>>>>>> But, you do say it's full or empty, right? > >>>>> One can easily say that it is empty at any time at which every ball > >>>>> that was put in has been taken out again. > >>>>> > >>>>> Does TO suggest that at any time after noon there is any ball that was > >>>>> put in that was not also taken out? > >>>> Yes, at any given time 9/10 of the balls inserted remain. > >>> Which ball does not have a definite time at which it > >>> is removed? > >>> > >> Any ball which does not have a definite time at which it is inserted. > > > > That excludes every ball, since each has a specific time of insertion > > and an equally specific time of removal. > > At any of those specific times are there balls in the vase? If there are > balls in the vase at every time before noon, but not at noon, then the > final ball is removed at noon. Which ball is that? Your left one, TO. Why should there be a "final ball" taken out when there is no final ball put in? If there were a last one out, it would have to be the last one in, or one of the last ones in if 10 are put in at a time. So when TO can say which 10 are the last ones in, I will take a guess at which of them is the last one out. > > > > No labelling scheme can be allowed to change any label once the ball > > with that label is inserted in the vase. > > Why not? Why can't I take out 1, 11, 21 etc, leave 9/10 of the balls in > the vase, then switch all the labels and make them disappear? That would > be neat. I am sure that's a theorem in transfinitology. It's got to be. > Please say it is. ;) It may be a theorem in TO-ology, but the original problem had fixed labels. > > >>> But in general if: > >>> (a) Every ball has a label n which is a finite natural number. > >>> (b) Every ball has a time t_n at which it is removed. > >>> (c) There exists a supremum T of the set {t_n, n in N} > >>> then for any time t >= T, the vase is empty. > >> What is this "supremum", in terms of iterations? > > > > It is the least upper bound of the times, and any set of times which is > > bounded above must have a supremum. > > > > But is that supremum WITHIN the set of times? Eh? Not necessarily. > > > > >> Here's one iteration: > >> (a) 10 balls added AND > >> (b) 1 ball removed IMPLIES > >> (c) net 9 balls added > >> > >> How many iterations? n? Fine. 9n balls remain. > > > > Which ones? > > Balls n/10+1 through n. Duh. After noon? > > > > > Unless one is postulating that the balls, like electrons, have no > > individual identities, but are totally interchangeable at all times, > > even when labelled to give them identities, the question of which ones > > is relevant. > > Then why is the answer so obvious before you introduce your labels? The > labeling scheme is just a parlor trick for entertaining girls. > > "See, ladies? Poof! The vase is empty. Tada!" TO wants to change the rules of the game because he doesn't like way the game comes out, which marks him either as a poor sportsman or a cheater. The rules of the vase game have been set, so either play by those rules, TO, or don't play at all. Trying to change the rules of a game in mid play is called cheating. |