Cantor Confusion
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From: William Hughes on 11 Oct 2006 12:50 Tony Orlow wrote: > If you want to put this > in limit terms, with the corrections I suggest, you have the size of the > set being lim(n->oo: 9n). That's not 0 by any stretch of the imagination. > This only works if you make the totally unjustified assumption that the size of the limit is the limit of the sizes. -William Hughes
From: imaginatorium on 11 Oct 2006 13:21 William Hughes wrote: > Tony Orlow wrote: > > > If you want to put this > > in limit terms, with the corrections I suggest, you have the size of the > > set being lim(n->oo: 9n). That's not 0 by any stretch of the imagination. > > > > This only works if you make the totally unjustified assumption > that the size of the limit is the limit of the sizes. Yes, of course, but I believe this is an axiom in Orlovia. If the limit of the lengths of the staircases is 2, then the length of the limit of the staircases just *has* to be 2. (It's a sort of bendy straight line, seen from an infinitesimal viewpoint. Or something like that.) Brian Chandler http://imaginatorium.org
From: Virgil on 11 Oct 2006 14:28 In article <1160552537.366273.288860(a)m73g2000cwd.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > Virgil schrieb: > > > Unless there is a natural number upper bound on the number of digits > > for numbers in the list, the m,n,o,..., the number of digits in the > > constructed what-ever-it-is will have to be greater than any finite > > number, and thus the constructed whatever-it-is will NOT be a number at > > all. > > But by construction the number must be finite since at every step it is > build up by finite addition to a finite number. By that argument 0.333..., as a representation of 1/3, must be of finite "length" because every digit is in a finite position.
From: Virgil on 11 Oct 2006 14:30 In article <1160560772.307075.84370(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <b7f51$452a1029$82a1e228$25909(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Mathematicians have found another name for scientitic facts. They call > > > them "just an opinion". > > > > Mathematicians do not contest the alleged factualness of scientific > > "facts", but do contest their relevance in determining what > > mathematicians are to be allowed to think. > > > Who defined what they are allowed to think? "Mueckenh" has claimed the right to do so. "Mueckenh" claims mathematicians are not to be allowed to think of anything that cannot be directly derived from observation of the physical world.
From: cbrown on 11 Oct 2006 14:37
briggs(a)encompasserve.org wrote: > In article <1160546562.540946.205860(a)e3g2000cwe.googlegroups.com>, cbrown(a)cbrownsystems.com writes: > > Dik T. Winter wrote: > >> In article <virgil-372F10.17374709102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > >> > In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > >> > wrote: > >> > > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil > >> > > <virgil(a)comcast.net> writes: > >> > > > > Dik T. Winter wrote: > >> > > ... > >> > > > > > The balls in vase problem suffers because the problem is not > >> > > > > > well-defined. Most people in the discussion assume some implicit > >> > > > > > definitions, well that does not work as other people assume other > >> > > > > > definitions. How do you *define* the number of balls at noon? > >> > > ... > >> > > > How about the following model: > >> > > > >> > > And you also start with definitions, or a model. I did *not* state that > >> > > it was difficult to define, or to make a model. But without such a > >> > > definition or model we are in limbo. I think other (consistent) definitions > >> > > or models are possible, giving a different outcome. > >> > > >> > Can you suggest one? One that does not ignore the numbering on the balls > >> > as some others have tried to do. > >> > >> That does not matter, nor is that the problem. You gave a model where you > >> find 0 as answer. I only state that I think there are also models where > >> that is not the answer. Why is a limit of the number of balls over time > >> not an answer? > >> > >> Let's give a simpler problem. At step 1 you add ball 1. At step n you > >> remove ball n-1 and add ball n (simultaneously, I presume). > > > > When you say "at step n", do you have some particular time t associated > > with that step? > > That's somewhat irrelevant. What matters is not what numeric time t is > associated with each step. What matters is the [partial] ordering on > the steps. Associating a numeric time t with each step is a way to > ensure a total ordering. But that's more than we need. What time ensures in this problem is that the notion "after all steps have completed, the state of the vase is..." can be well-defined from the problem statement. > > In particular, arranging matters so that all the step times come before a > particular finite time is irrelevant -- it's a trick designed to fool > our intuitions into delusions of physicality and all the implicit > assumptions that come with physicality. > On the contrary, the red herring here is to assume that the problem is of the form: let V = {V_n} be a sequence of subsets of N; what subset of N which corresponds to the lim n->oo V_n? Now, there is an obvious notion of limit we can apply here (pointwise convergence); but as Dik asserted, in that case we need to /define/ that notion, independent of the given formulation of the problem. This opens the door to complaints such as "noon never arrives" (which is essentially the claim that there is no such thing defined in the problem as the limit of the sequence V); or that some other form of convergence should be used. For example, suppose the steps occured at t=1, t=2, etc. We still have the total ordering you describe below, but now one can argue that "when all steps are completed, the vase is empty" is a meaningless statement, because there is no such time "when" this state is achieved. What the element of time in the problem statement provides us with is the same definition of convergence you outline below. It allows us to conclude that if t > -1/n, then ball n is not in the vase at time t; and this holds for /all/ t > -1/n (not just for those t such that t = -1/m for some natural number m). In particular, it holds for t=0; which is equivalent to using pointwise convergence to find the limit of V (and that is why I disagreed with Dik regarding whether or not the problem is well-defined). Cheers - Chas |