Cantor Confusion
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From: mueckenh on 12 Oct 2006 06:17 Virgil schrieb: > In article <1160561733.901224.261070(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1160397914.738238.238220(a)m7g2000cwm.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > > > > forever then no part of his biography would remain unwritten". That is > > > > wrong, because the major part remains unwritten. > > > > > > What part? > > > > That part accumulated to year t, i.e., 364*t. > > For any give t, there will be a time t_1 at which time the events of t > will have been written down. But there will never be a time at which we could say: there does not remain any part unwritten. > > Or does "Mueckenh" posit an only finitely remote end to time? > > > > If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > > What I think is that there is always some time t_1 enough larger than > any t so that the events of time t are written down by time t_1. But what you forget is that connected with this t_1 there is an even larger amount of unwritten days. (You would be a good secretary of finance.) Regards, WM
From: mueckenh on 12 Oct 2006 06:25 Virgil schrieb: > In article <1160577085.758246.228800(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > If discontinuous functions were easily allowed everywhere, why then do > > you think that > > lim{n-->oo} n < 10 > > or > > lim{n-->oo} 1/n > 10 > > would be wrong? > > Since N is not normally considered to be a topological space, continuity > of functions with a non-topological domain N is a contradiction in > terms. Apply your knowledge to the balls of the vase. > > On the other hand, limits of real sequences (functions from N to R) have > been quite adequately defined. One such definition is: > Give f:N --> R and L, then > lim_{n in N} f(n) = L (or lim_{n --> oo} f(n) = L > is defined to mean > For every real eps > 0, Card({n: Abs(f(n)-L) > eps}) is finite. For the vase problem with the number n(t) of balls in the vase after t transactions we can find always a positive eps such that for t > t_0: 1/n(t) < eps, hence n(t) larger than an arbitrary positive number. Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. Regards, WM
From: mueckenh on 12 Oct 2006 06:29 Virgil schrieb: > In article <1160578088.974689.303450(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > [...] > > > > > > > There is no largest natural! There is a finite set of > > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > > > > I can only conclude you have knocked youself out. > > > > Try the following gedanken-experiment to become accustomed with it: > > a) How many different natural numbers can you store using a maximum of > > 100 bits? > > b) What is the largest natural number you can store with a maximum of > > 100 bits? > > > > Regards, WM > > > > Answer to a) less than 100. > > I make the answer to A as 2^100, if each bit must be allowed to have its > own personal bit position. And it must have its own personal value 0 or 1. Therefore you can either represent one number with 100 bits or 100 numbers with one bit, but the latter would not yield different numbers. Therefore the answer is "less than 100". Regards, WM
From: mueckenh on 12 Oct 2006 06:37 David R Tribble schrieb: > Virgil schrieb: > >> Note, the question originally asked was very careful to > >> distinguish between the questions " Will the whole autobiography > >> be written?", and "Will certain pages of the autobiography > >> be written?, so my repharasing is accurate. > > > > mueckenh wrote: > >> Yes, but the assertion of Fraenkel and Levy was: "but if he lived > >> forever then no part of his biography would remain unwritten". That is > >> wrong, because the major part remains unwritten. > > > > David R Tribble wrote: > >> What part? > > > > mueckenh wrote: > > That part accumulated to year t, i.e., 364*t. > > It's stated that he lives forever, so what value of t you are using? > You can use any positive value of t and prove that the unwritten part n(t) for t > t_0 is larger than the unwritten part for t_0. You can even use the formal convergence criterion for the convergent function 1/n(t). There is no room for he assumption that the written part could ever surpass the unwritten part. > > > If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > > I don't think anyone has said that. I merely asked which pages (days) > in the "major part" of the book don't get written. Do you have a > certain t in mind? I merely answer that it is completely irrelevant to speak of certain t. The paradox is raised only by the asumption that the set of all t did exist. Regards, WM
From: mueckenh on 12 Oct 2006 06:42
Dik T. Winter schrieb: > In article <1160578088.974689.303450(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > [...] > > > > > > > There is no largest natural! There is a finite set of > > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > > > > I can only conclude you have knocked youself out. > > > > Try the following gedanken-experiment to become accustomed with it: > > a) How many different natural numbers can you store using a maximum of > > 100 bits? > > b) What is the largest natural number you can store with a maximum of > > 100 bits? > > What is the relevance? To inform the set theorist about the possible existence of sets with finite cardinality but without a largest number. Cantor assumed "the infinite set of finite numbers". That is impossile. Possible and truly existing are finite sets with unboundet number size. Regards, WM |