An uncountable countable set
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From: Tony Orlow on 1 Oct 2006 15:20 Virgil wrote: > 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. There are many riddles and joke which throw in extraneous information to distract one from the obvious answer. >>> 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. The something occurs at noon which requires explanation. Roll the tape backwards from noon, one iteration at a time, and you'll see what I mean. >>> 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. No, you haven't. Assume the labels AREN'T fixed. Can I do that now? > > If TO wants to play a different game, fine, but he cannot then pretend > that it is not a different game. >> Tony Assume it is. What then? How far does Virgilogic stretch?
From: Tony Orlow on 1 Oct 2006 15:22 Virgil wrote: > 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. It is nothing more or less than its contents. There is no sack, or Gucci bag in your case. Get it? Case? haha. ahem. >>>>>>>> ...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. But, no.
From: Tony Orlow on 1 Oct 2006 15:23 Virgil wrote: > In article <451df438(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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? > > All of them. How many in quantitative terms? All the people in the world doesn't describe the global population as a quantity.
From: Tony Orlow on 1 Oct 2006 15:31 Virgil wrote: > In article <451df5af(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > Violating one of the rules is not the same as omitting an irrelevant > detail. And statements which force those violations are fallacious. Saying the balls have labels is not a "rule". You are following rules about the labels which are not stated in the problem itself. > >> According to the gendanken, EITHER version, 10 balls are added, then one >> is removed, and you repeat the process. > > Insufficient for the actual gedanken which requires numbered balls and > specifies which numbers are moved in or out at each move. There were two presented in the original post, by a wonderful not-even-neophyte with a friend, one of which supposedly emptied, and the other of which did not, all depending on the labels. What happens if we don't know what ball we're removing? Can you answer than question? No, because much to Han's chagrin, you will not even consider statistical methods over this set. > >> 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. > > How does 10 - 1 come out -9? final removal: x-1=0 final addition: y+10=x What is y, at the second-to-last iteration? > TO's arithmetic is getting as sloppy as his thinking. > I saw you correct yourself twice the other day. Take a vitamin and have some eggs. Seriously. >>>>> 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? > > t is much more precise and detailed than that. Ho hum. Except it ignores the minor detail of the specified order of events. Sure. >>>>> 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. > > Which I never denied. The how claims you it be some convergence to 0? sum(x=1->oo: 9)=0? > >> You're playing "natural" labeling games based on the >> unboundedness of the finite realm. It's hocus pocus. > > > It follows the rules of the gedankenexperiment precisely and exactly. Except for order of operations, but what's that matter, right? > > That TO is incapable of grokking what mathematicians regularly grok does > not make it hocus pocus. It only points out TO's limitatains. I also don't grok what evangelistic creationist christians who preach love and wage war grok. I don't see that as a deficiency on my part.
From: Tony Orlow on 1 Oct 2006 15:49
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: >>>> imaginatorium(a)despammed.com wrote: >>>>> Tony Orlow wrote: >>>>>> imaginatorium(a)despammed.com wrote: >>>>>>> Consider a (notional, theoretical, mathematical, not physical) x-y >>>>>>> plane. That is, an area in which there is a point (0,0) in some >>>>>>> particular place, an x-axis, y-axis, and points are identified by >>>>>>> coordinates x and y, using (in normal maths) real values for these >>>>>>> coordinates. Consider (for convenience) that this plane is embedded in >>>>>>> a notional graphics application, with a "Fill" function. So if we draw >>>>>>> the circle x^2 + y^2 = 49 (centre origin, (constant! Zick, be quiet!) >>>>>>> radius 7), then click with the Fill function on the point (2,1), it >>>>>>> fills the circle, and no paint spills outside that radius 7. >>>>>>> >>>>>>> Now suppose we have the graphs of x=2 and x=5. Vertical lines, >>>>>>> extending up and down without limit. Suppose we click with the Fill >>>>>>> function on the point (3, 4), what would you say happens? Obviously >>>>>>> paint fills the vertical strip of width 3. Would you say that any paint >>>>>>> was able to "spill" around the (nonexistent!) "top" of either of the >>>>>>> graphs, and somehow fill more of the plane than this strip, or would >>>>>>> you say we just get a (vertically) unbounded strip of blue? (Goddabe >>>>>>> blue!) >>>>>> I'd have to agree that it would fill the strip only. Proceed, but it >>>>>> would be nice to know the context of the question. >>>>> Ok, well just as a diversion: suppose you were on >>>>> sci.comp.graphics.crank, and one of the residents produced a long, >>>>> rambling argument, including mention of Planck's constant, twin-slit >>>>> experiments and more, at the end of which was a claim that outside the >>>>> strip would also be a very pale (ok "infinitesimally pale"!?) blue. How >>>>> would you try to justify your claim that the blue fills the vertically >>>>> unbounded strip only? >>>> I'd have to see their fill algorithm to see what their malfunction is. >>> Remember this is not a practical example. It can't be, because no >>> actual physical computer can simulate a boundless x-y plane. >>> >>>>> Note that when discussing the behaviour of a real-world graphics >>>>> program, within a bounded window, it's possible to discuss the >>>>> paint-filling as a terminating procedure. With an unbounded strip, it >>>>> obviously isn't. So I would say something like the following: for the >>>>> paint to spill outside the vertical lines bounding the strip, there >>>>> must be a path from a point inside to a point outside. But since the >>>>> x-coordinate of the points on the path must go from (say) 4 to 6, at >>>>> some point it must be 5; and that point must be a point on the >>>>> boundary, so it would have crossed the boundary, and it's not allowed >>>>> to cross the boundary, so this can't have happened. >>>> Any general fill algorithm would probably leave some section of that >>>> infinite strip un-blued. >>> Really? We are talking about the (non-practical, not actually >>> implementable, [what mathematicians call 'infinite']) notional plane, >>> and a strip of width 3 that extends indefinitely up and down. An >>> algorithm by definition terminates, and this strip doesn't, so >>> obviously no algorithm could fill it. But there doesn't seem to be >>> anything difficult in painting a strip that goes on forever, other than >>> that the job can't be completed. Are you suggesting that somehow an >>> algorithm that purported to paint the whole strip would just stop at >>> some point? When the strip was a certain height? Even though it painted >>> without end, it would reach the end? (Oh dear, this all seems rather >>> familiar...) >> Yes, your imaginations don't seem to vary much. > > Meaning? You _can_ imagine that given an endless task, something could > segue to "the infinite realm" and get to the end of the endless task? > Hmm... Yes, that given an unending sequence with a limit that any infinite number of iterations will fall within any finite distance of that limit. That's pretty obvious to me. > >> I am just suggesting that a general-purpose fill algorithm which could >> fill any finite shape might not work on this shape. If it found a left >> most point, filled that vertical line up and down until it hit the >> boundaries of the shape, then went right to the next column, it would >> never get to that next column, because it would never finish with the >> first vertical column. > > In any discrete simulation of the area-filling task, this is true (for > some shapes). But we cannot be talking about a discrete simulation, > since we agree that the (to-be-painted-blue) sliver goes up without > end, getting narrower and narrower without a minimum width; any > discrete simulation would stop when the width was less than 1 pixel. I > don't want to get bogged down too much in this "fill algorithm" stuff - > we are not talking about real-world computation, but about a > mathematical idealisation (which is what maths _is_ f'goodness sake). The discrete isn't math? Why the hell did I take those discrete math classes, then? I didn't create this example. Get to the point. > >> If it started by finding a topmost or bottommost >> point, it would never even get to filling one pixel. > > But yes, this is certainly true. It could go around in discrete circles, and fill an arbitrary space, keeping track of where it's been. It would never fill this crescent. Is that what you're getting to, eventually? > >> .... Of course you could >> create an algorithm that worked its way back and forth a row at a time, >> perhaps going both up and down, and continue filling the shape forever. > > Good. That's OK then. Grrrrreat! (Sugar Frosted Flakes suck) > >>>>> ---- back to the point ---- >>>>> >>>>> Now consider some other graphs: >>>>> >>>>> y=1/x, fill from the point (0, 0) - get blue lower left and upper right >>>>> quadrants, plus filling out to the white lobes that almost fill the >>>>> upper left and lower right quadrants. OK? (Graph is a hyperbola) >>>>> >>>>> Now consider the following two hyperbola-halves: >>>>> >>>>> y1 = -1/x (for negative x) >>>>> y2 = -2/x (for negative x) >>>>> >>>>> Each of these is a lobe in the uppe |