An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 19:03 Virgil wrote: > 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. > Right. If I give you numbers, you can give them back. That's my point. If I say n balls went in, I can tell you the numbers which remain. If you say "all naturals", that has no defined boundary or value range. That's why you can't name the last ball in or out, or the numbers of balls that remain. It's spooooky, but nothing more significant than that. It's time to get past Zeno already. > >>> 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. And, given AFL, yes, ZFCFL predicts the vase will be empty. >>>>> 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. Yes. Go on. >>>> 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? Yup >>> 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. That's a laugh. I mean, I'm actually laughing. :D > > The rules of the vase game have been set, so either play by those rules, > TO, or don't play at all. Yeah, sure, Coach. > > Trying to change the rules of a game in mid play is called cheating. What, I can't call flying kings? I say the labels are now moveable after the fact. Just humor me. Can the vase become empty from the exchanging of labels?
From: Virgil on 29 Sep 2006 19:45 In article <451da3b6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. > > > > If all bit positions are finite, and the string up to any finite bit > position can only have a finite value, then there is no position in the > string where it achieves anything but a finite value. Precisely! > > Are you saying that aleph_0 naturals only require ln(aleph_0+1)/ln(2) > bit positions? Not at all. I am talking about indvidual natural numbers as members of N, ,not N itself, which is not a member of N. > > > > >>> 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? > > The fact works for an arbitrary number of bits, including in the 2-adics. Irrelevant. That does not tell me anything about which , if any, actual computers deal with infinitely long strings of binary digits. > > Besides, any Turing machine can process an infinite string given > infinite time. Go watch one, and don't come back till its finished.
From: Virgil on 29 Sep 2006 19:47 In article <451da475(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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? > > Yes, and that requires, for a uniform distribution, that we have a > precise count. If aleph_0 is some kind of precise number, then we can > say that a uniform probability distribution over a set of this size > yields a probability of 1/aleph_0, and if aleph_0 is infinite, this > individual probability is infinitesimal. Except that this does not happen within the standard reals of probability theory. > > Then you can add them all up and get the unit of universal inevitability. :)
From: Virgil on 29 Sep 2006 19:53 In article <451da650(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > Right. If I give you numbers, you can give them back. That's my point. > If I say n balls went in, I can tell you the numbers which remain. If > you say "all naturals", that has no defined boundary or value range. > That's why you can't name the last ball in or out, or the numbers of > balls that remain. It's spooooky, but nothing more significant than > that. It's time to get past Zeno already. It is TO who is having problems with it because he won't play by the rules. Those of us who follow the rules have no troubles. > > >> Balls n/10+1 through n. Duh. > > > > After noon? > > Yup Then TO is getting ball from somewhere else and putting them in the vase. > > >>> 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. > > That's a laugh. I mean, I'm actually laughing. :D > > > > > The rules of the vase game have been set, so either play by those rules, > > TO, or don't play at all. > > Yeah, sure, Coach. > > > > > Trying to change the rules of a game in mid play is called cheating. > > What, I can't call flying kings? I say the labels are now moveable after > the fact. Just humor me. Can the vase become empty from the exchanging > of labels? If TO wants to play a different game, fine, but he cannot require our game to come out according to another game's rules.
From: Tony Orlow on 29 Sep 2006 21:58
Virgil wrote: > In article <451da650(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >> Right. If I give you numbers, you can give them back. That's my point. >> If I say n balls went in, I can tell you the numbers which remain. If >> you say "all naturals", that has no defined boundary or value range. >> That's why you can't name the last ball in or out, or the numbers of >> balls that remain. It's spooooky, but nothing more significant than >> that. It's time to get past Zeno already. > > It is TO who is having problems with it because he won't play by the > rules. Those of us who follow the rules have no troubles. Hah! While you try to justify the contradiction between your nonsense and the formulation in terms of infinite series ((+10,-1)... diverges), by saying you can rearrange all the terms and postpone 9/10 of the +10's, making it all "balance out" to zero, that's specifically violating the sequence set forth in the premise. You changed horses and fell into the stream, on a rock. You add 10, then remove 1. Start with 0, an empty case, and try rolling the tape backwards. In two steps you have a negative set. Is that allowed? > >>>> Balls n/10+1 through n. Duh. >>> After noon? >> Yup > > Then TO is getting ball from somewhere else and putting them in the vase. No Ma'am. >>>>> 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. >> That's a laugh. I mean, I'm actually laughing. :D >> >>> The rules of the vase game have been set, so either play by those rules, >>> TO, or don't play at all. >> Yeah, sure, Coach. >> >>> Trying to change the rules of a game in mid play is called cheating. >> What, I can't call flying kings? I say the labels are now moveable after >> the fact. Just humor me. Can the vase become empty from the exchanging >> of labels? > > If TO wants to play a different game, fine, but he cannot require our > game to come out according to another game's rules. Similarly, if you want to explain the behavior of infinite sequences, you cannot violate the manner in which they are described sequentially. If, for every element removed, 10 have been previously added, then if the set becomes empty, it must have had -9 elements right before that. Seems fishy, to say the least. Not that I don't like fish. Just not that Shakespearean Danish type. :) Tony |