An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 3 Oct 2006 03:29 In article <452204aa(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45216233(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David R Tribble wrote: > >>> Virgil wrote: > >>>>> Except for the first 10 balls, each insertion follow a removal and with > >>>>> no exceptions each removal follows an insertion. > >>> Tony Orlow wrote: > >>>> Which is why you have to have -9 balls at some point, so you can add 10, > >>>> remove 1, and have an empty vase. > >>> "At some point". Is that at the last moment before noon, when the > >>> last 10 balls are added to the vase? > >>> > >> Yes, at the end of the previous iteration. If the vase is to become > >> empty, it must be according to the rules of the gedanken. > > > > But the "rules of the gedanken" specifically forbid any "last 10 balls", > > by specifying an ENDLESS sequence of 10 ball additions. > > No, the gedanken starts with adding ten balls, then removing one, then > repeating forever. That is not how mine starts, or continues. Mine follows the rules of the original problem, TO's does not.
From: Tony Orlow on 3 Oct 2006 03:30 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>> Tony Orlow wrote: >>>>> <snip> >>>>> >>>>>>>> Like I said, there were >>>>>>>> terms in my infinitesimal sections of moving staircase which differed by >>>>>>>> a sub-infinitesimal from those in the original staircase. So, they could >>>>>>>> be considered to be two infinitesimally different objects in the limit. >>>>>>> Here's a thing that confuses me about your use of the term "limit". >>>>>>> >>>>>>> In the usual sense of the term, every subsequence of a sequence that >>>>>>> has as its limit say, X, /also/ has a limit of X. >>>>>>> >>>>>>> For example, the sequence (1, 1/2, 1/2, 1/3, ..., 1/n, ...) usually is >>>>>>> considered to have a limit of 0. And the subsequence (1/2, 1/4, 1/6, >>>>>>> ..., 1/(2*n), ...) which is a subsequence of the former sequence has >>>>>>> the same limit, 0. >>>>>>> >>>>>>> But the way you seem to evaluate a limit, the sequence of staircases >>>>>>> with step lengths (1, 1/2, 1/3, ..., 1/n, ...) is a staircase with >>>>>>> steps size 1/B, where B is unit infinity; but the sequence of >>>>>>> staircases with step lengths (1/2, 1/4, 1/6, ..., 1/(2*n), ...), which >>>>>>> is a subsequence of the first sequence, would seem to have as its limit >>>>>>> a staircase with steps of size 1/(2*B). >>>>>>> >>>>>>> Unless steps of size 1/B are the same as steps of size 1/(2*B), I don't >>>>>>> see how that can be possible. >>>>>>> >>>>>>> Cheers - Chas >>>>>>> >>>>>> It's possible because no distinction is currently made between countable >>>>>> infinities, even to the point where a set dense in the reals like the >>>>>> rationals is considered equal to a set sparse in the reals like the >>>>>> naturals. Where there is no parametric understanding of infinity, >>>>>> infinity is just infinity, and 0 is just 0. >>>>> Uh, OK. I assume that you somehow resolve this lack of "parametric >>>>> understanding" in /your/ interpretation of T-numbers. >>>> Well, yes. That's the whole point, but I'm not sure which particular >>>> numbers you mean. Maybe the T-riffics, which are center-indefinite >>>> digital numbers? >>>> >>> I mean, whatever type of number the /particular/ number "x" is; when >>> you state "/the/ limit of the staircases has steplength 1/x". >>> >> If you say you have some particular infinite number of steps x... > > I /don't/ say that; you do. What do you mean by it? I mean that it can be specified in a number system by a unique string. Good? > >> , then you >> can consider 1/x to be the specific infinitesimal size of each. As long >> as 1/x is considered nonzero > > Well, do you, or do you not, consider it non-zero? Yes, as long as x is specified with a unique string, it's inverse can be as well. > >> , there is a ratio between the x and y >> offsets of any given segment which gives a direction at that point. It >> is clear that when defined this way the directions of the segments in >> the staircase do not at all approach the directions of the corresponding >> segments in the diagonal, even the locations of the endpoints of the >> segments do become arbitrarily close. > > But they don't become arbitrarily close. For example, since 1/x^2 < 1/x > in your system, and yet your points do not get closer than 1/x, there > is still "more room" for them to get closer - they /don't/ get > arbitrarily close.. > On the finite level, they do. One requires more bit positions on the right of the digital point than any finite number to represent this infinitesimal difference of 1/x. It's infinitely worse with 1/x^2, and negligible. >>>>>> Where there is a formulaic >>>>>> comparison of infinite sets as n->oo, the distinction can be made. The >>>>>> fact that you have steps of size 1/n as opposed to steps of size 1/(2*n) >>>>>> is a reflection of the fact that the first set has twice the density on >>>>>> the real line as the first. As a proper superset, it SHOULD be larger. >>>>>> So, it's quite possible to make sense of my position, with a modicum of >>>>>> effort. >>>>> Well, let me ask you this: >>>>> >>>>> Suppose we have the original sequence of staircases, with step lengths >>>>> (1, 1/2, 1/3, ..., 1/n, ...). Let S be the T-limit staircase; you claim >>>>> that it has step sizes 1/B, where B is unit infinity. >>>> Well, where B is some infinite number of iterations for both staircase >>>> and diagonal. >>> "Some" number? Do you mean that there is no "the limit" of the >>> staircases, but instead many possible "a limit" to the staircases? >> No, the limit is the limit. Once you have applied an infinite number of >> steps you have become infinitesimally close to the limit, and have >> essentially reached it. > > "Essentially" reached it? How is that different from "reaching" it? It's only different on some sub-infinitesimal, negligible way. > >> Any infinite value will do. In the limit, as a >> segment sequence, the infinite staircase continues to have a length of >> 2/x for each step, and x steps, for a length of 2. >> > > But if x = B, then the steps have length 1/B. If x = 1/B^2, then the > steps have a /different length/; so these two "limits" can't be the > same. > That wasn't even stated correctly. Limits are different from specific infinite or infinitesimal values. Get used to it. >>>> We can call it the unit infinity if you want, especially >>>> since it covers a space of 1 unit. :) >>>> >>> One might as well start somewhere. >>> >>> Starting from 0 and 1, we can construct the naturals. Add the concept >>> of "subtraction", and (arguably) we get the intgeres. Add the concept >>> of multiplication and division, and we get the rationlas. >> Throw in exponentiation, logs and roots, and we get the reals. > > If you include roots of polynoimals, you get the algebraic closure of Q > (normally called the algebraic numbers or A), which does not include > transcendental numbers such as pi, e, and so on. In order to get the > transcendentals, including "exponentiation and logs", you need a > concept of limit; and your limit concept is currently too weak to allow > this. Um, perhaps. It sounds like you may have a point to make. I'm willing to hear it. > >> I suppose >> I should get
From: Han de Bruijn on 3 Oct 2006 03:31 Quoted from a poster by Tony Orlow: > mueckenh(a)rz.fh-augsburg.de wrote: > >> Han de Bruijn schrieb: >> >>> My mathematics says that it is an ill-posed question. And it doesn't >>> give an answer to ill-posed questions. >> >> You are right, but the illness does not begin with the vase, it beginns >> already with the assumption that meaningful results could be obtained >> under the premise that infinie sets like |N did actually exist. Yes. That's why I launched the "Naturals Construction Set", a while ago: http://groups.google.nl/group/sci.math/msg/13795822737a77ca?hl=en& Han de Bruijn
From: Virgil on 3 Oct 2006 03:32 In article <45220791(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45216360(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <1159784963.257471.99490(a)i42g2000cwa.googlegroups.com>, > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>> Virgil wrote: > >>>> > >>>>> The problem as I recall it was this: > >>>>> > >>>>> Given an infinite set of balls numbered with the infinite set of > >>>>> naturals and an "infinitely large" initially empty vase, and a positive > >>>>> time interval in seconds, t, and a small positive time interval in > >>>>> seconds, epsilon ( much smaller than t/2). > >>>>> (1) At time t before noon balls 1 through 10 are put into the vase and > >>>>> at time t - epsilon before noon ball 1 is removed. > >>>>> (2) At time t/2 before noon balls 11 through 20 are put into the vase > >>>>> and at time (t - epsilon)/2 before noon ball 2 is removed. > >>>>> ... > >>>>> (n) At time t/2^(n-1) before noon balls 10*(n-1)+1 through 10*n are put > >>>>> in the vase and at time (t-epsilon)/2^(n-1) before noon, ball n is > >>>>> removed. > >>>>> ... > >>>>> > >>>>> The question is what will be the contents of the vase at or after noon. > >>>> There is no noon in this problem. > >>> There is in mathematics. The problem does not exist in physics at any > >>> time. > >> So, the limit "exists". > > > > There is a function, f, from the reals, R, representing time in suitable > > units to the set of von Neumann naturals,N = {0,1,2,3,...}, such that > > f(t) = the number of balls in the vase at time t, for all t. > > Yes, f(t)=9*floor(-log2(noon-t)), in minutes. That may work for TO's game but it does not for the actual game. > > > This function has an isolated jump discontinuity at each time at which > > any balli is inserted or removed from the vase and and at noon there is > > a condensation point of such discontinuities and the function is > > unbounded to the left in every neighborhood of noon, but is still > > right-continuous at noon. > > > > There is no jump. That's a leap of faith. Functions may have jumps but do not leap. > > > > > > > Is that what you're saying? There's "the > >> infinite case"? > > > > There is the function case, as described above. > > "I can neither confirm nor deny." Thanks. Meaning TO sees he is wrong but won't admit it.
From: Tony Orlow on 3 Oct 2006 03:32
Virgil wrote: > In article <4521fc40$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1159797618.679513.221400(a)i3g2000cwc.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>> Virgil schrieb: >>>> >>>> >>>>>> By the only meaningful and consistent definition: A n eps |N : >>>>>> |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>>>>> Do you challenge its truth? >>>>> I challenge the "truth" of its being the ONLY meaningful and consistent >>>>> definition. >>>>> >>>> If we insist on unique results in mathematics, then this definition and >>>> the bijection exclude each other. >>> Uniqueness of results depends on what one is doing. >>> If one is considering finding a possible order relation on a set of two >>> or more elements, would "Mueckenh" insist that there is a unique result? >>> >>> Does "Mueckenh" claim that there is only one function mapping >>> {1,2,3,...} to {2,4,6,...} or vice versa? >> There is only one natural order-isomorphic relation, defined by y=2x. >> The inverse clearly indicates the second set is half the first, whether >> subset or no. > > That presumes, among other things, that a proper subset must be in > every sense smaller than its superset, but that is not true, as the > bijection y = 2x proves.. How presumptuous of me! |