An uncountable countable set
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From: Tony Orlow on 30 Sep 2006 00:36 Virgil wrote: > In article <451dd293(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451d83c4(a)news2.lightlink.com>, > > >>> So what balls remain in the vase at noon, oh waffler extraordinary? >> For n balls inserted, balls n/10+1 through n remain at the end of any >> iteration n. You specify the number of iterations, I'll give you the >> sum. What was it? Aleph_0? > > All iterations executed before noon. And that would be how many?
From: Tony Orlow on 30 Sep 2006 00:42 Virgil wrote: > In article <451dd5cf(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> What Randy is avoiding is TO's fallacious insistence that there must be >>> a last ball removed if one is ever to achieve a state where the balls >>> are all removed. >> >> When the gedanken specifically states that only one ball is removed at a >> time, what is fallacious about that statement? > > It omits that it is not just any ball which is removed. If one changes > which one is to be removed, one changes the game and the result. Omitting an irrelevant detail does not make a statement fallacious. According to the gendanken, EITHER version, 10 balls are added, then one is removed, and you repeat the process. Only one ball is removed at any time, immediately preceding which 10 balls have been added. You had to have -9 balls in your vase for that to have occurred. Is that possible? > > \ >>> In a physical world that might be the case, but in an ideological one it >>> need not be. >> It is stated as a condition. Insert 10, remove 1, repeat. > > It is much more precise and detailed than that. Is that condition not part of the specification of the sequence of events? > >>> TO tries to change, or break, those rules, which is a form of cheating. >> Oh, by rearranging a sequence and violating a specified order? No, that >> was your trick. > > I followed exactly the sequencing defined by the problem. > > It is TO who tries to change or ignore those rules. No, when I expressed it as an infinite sequence, you tried to rearrange terms to make it add up. When the two events, adding 10 and removing 1, are coupled as one iteration, then you have a net gain of 9 per iteration. You're playing "natural" labeling games based on the unboundedness of the finite realm. It's hocus pocus. Tony Orlow
From: imaginatorium on 30 Sep 2006 02:07 Tony Orlow wrote: > imaginatorium(a)despammed.com 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. > > > > No, but I think the problem is elsewhere, slightly. Is there a formal > > definition of what "transition" means? (Not in a nearby pocket "Dict. > > of maths." for example) > > > > Seems to me that if you had the graph y = (1 if x<0; 2 if x>=0), and > > and associated state transition diagram, then there would be a > > "transition" from 1 to 2 "at" x=0. But such terminology does not > > capture what the state is "at the point of" the transition, which may > > be why it isn't used much. But if a vase has balls in it for values (-1 > > <= t < 0), I can't see anything actually wrong with saying there is a > > transition from empty to non-empty "at" t=-1 and a transition from > > non-empty to empty "at" t=0. You need to be careful not to deduce > > anything about the _state_ at the two transition values. > > > > After all, consider the graph of y = -x. This is positive for x<0, and > > negative for x>0. It's undefined for x=0, but is there not a transition > > from positive to negative at x=0? > > That all makes very good sense, Brian. I can't see that there isn't a > point of transition, especially when that point is very explicitly > defined to be noon, and that it is at least then, and not until then, > that the vase goes from being non-empty to empty. Yes, but using this (slightly loose?) "transition" terminology, you have to be careful that this tells you _nothing_ about the state _at_ noon. Seems to me that all of the following graphs have a "transition" from positive to negative at x=0, but very different things are true _at_ 0 y=-1/x (at x=0 this is undefined) y = (1 if x<0; -1 if x>=0) : y(0) = -1 y = (1 if x<=0; -1 if x>0) : y(0) = 1 f = 1 if x<0; -1 if x>0; purple unicorn if x=0 : well, what can I say? > Is y=-x really "undefined" at x=0? I don't think so. It's 0, which > equals -0. Sorry, that's a (surely obvious?) typo - I meant y=-1/x. No ok, perhaps it isn't obvious. But another example: y=-x : y(0) = 0, which is perfectly well defined, but neither positive nor negative (au moins en anglais) (I'll carry on with the blue sliver when I have time, which may be in a day or so) Brian Chandler http://imaginatorium.org
From: Virgil on 30 Sep 2006 02:24 In article <451df1cb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451dcf42(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >>> 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? > > > > TO seems to have delusions of sanity. > > The operative word being "seems", indicating the subjectivity of the > statement. > > > The only question is whether the vase is empty after noon. > > At noon and possibly thereafter. > > > Since there is a specific time prior to noon at which any given numbered > > ball is removed, one must conclude that they have all been removed by > > noon. > > Numbers aside The whole point is that according to the definition of the problem, one is not allowed to put number aside. > > > > Suppose the instead of being put in in batches of 10, they are all put > > in when the first one is put in, but removed according to the original > > scheduled. In this case it is clear that every ball is removed. > > In that case addition is all performed first, then all subtraction. > Sure, that can work. But the measure of the addition is divorced from > that of the subtraction. If one measures the whole process in some kind > of common time frame, given the numbers for the additions and > subtractions, one can get a "rate of filling", which isn't going to > change in this case and cause the thing to suddenly empty. I beg to differ! There are in this game infinitely many balls in the vase at every instant after the balls are put in andbefore noon, and none left at noon. > > > > > So that TO is claiming that putting the balls in earlier, but taking > > them out the same way, leaves fewer balls than the original way. > > At least I have an explanation. You still haven't explained why I can't > relabel the balls afterwards and make them disappear from the vase. :) Yes I have. If TO wants to play a different game, fine, but he cannot then pretend that it is not a different game. > > Tony
From: Virgil on 30 Sep 2006 02:27
In article <451df41c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451dd1f2(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > > > >>>> 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 also for every set of contiguous naturals starting at 0 EXCEPT for > >> N. Why EXCEPT for N? > >> > > For the same reason that a paper sack holding oranges is not an orange. > > > > A set is a sack? It is nothing besides the elements it includes. A set is a container, and is not one of the objects that it contains. > > > > >>>>>> ...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. > >> Like, none, man, unless you droppa lotta 'cid, dude. > > > > Then it is irrelevant to infinite strings. > > > > But, no. But yes. |