Cantor Confusion
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From: David Marcus on 11 Oct 2006 18:37 Dik T. Winter wrote: > In article <MPG.1f93b0b45c6e7119896a5(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > > Dik T. Winter wrote: > ... > > > Hm. I humbly submit that the probability for a particular rational number > > > in the range [0,1) the probability to get it when doing a random choice > > > is 0. > > > Nevertheless, the sum of all the probabilities is 1. The sum of countably > > > many 0's is not always 0. > > > > I don't follow. Usually, probability measures are countably additive. > > I just gave a counterexample. Sorry. As I said, I didn't follow your counterexample. If we take Lebesgue measure on [0,1) as our probability measure P, then P({q}) = 0 for any rational q, but P({q|q rational and q in [0,1)}) = 0, not 1. -- David Marcus
From: David R Tribble on 11 Oct 2006 19:17 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? > 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?
From: MoeBlee on 11 Oct 2006 20:10 Albrecht wrote: > What's wrong with Russell's [Easterly] argument but right with Cantor's? That has been answered in ponderous detail too many times already. My guess is that anyone who doesn't understand this matter after more than two explanations will never understand it, as a function of actually not wanting to understand it. MoeBlee
From: MoeBlee on 11 Oct 2006 20:13 Albrecht wrote: > Which subject? That the axiomatic method dominate modern math cause > Cantor "invented" the axiom of infinity Who said Cantor invented the axiom of infinity? MoeBlee
From: Dik T. Winter on 11 Oct 2006 21:38
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? Graham's number (which is so large that special notation had to be deviced to even indicate the number) has been used in a proof. You think that number is not a natural number? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |