Cantor Confusion
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From: Dik T. Winter on 8 Oct 2006 21:28 In article <MPG.1f92ed4f7727112398969c(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > mueckenh(a)rz.fh-augsburg.de wrote: .... > > My conclusion is: > > Either > > (S is covered up to every position <==> S is completely covered by at > > least one element of the infinite set of finite unary numbers <==> S is > > an unary natural) ==> Contradiction, because S can be shown to be not a > > unary natural. > > Are you saying that standard mathematics contains a contradiction or > that you think mathematics should be done differently? It is worse. Mueckenheim is stating, without proof that: S is covered up to every position <==> S is completely covered by at least one element of the infinite set of finite unary numbers. That implication is false. It is true if the set of natural numbers is finite. And when you change '<==>' to '<==' it also becomes true. You can talk at length with Mueckenheim, he will never give a proof, using standard mathematical proof technology. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 8 Oct 2006 21:21 In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > You got it several times already. According to the axiom of infinity > > > the set of all natural numbers does exist. And with it the following > > > statements are true: > > > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > empty. > > > 2) Before and at noon there are more balls in the vase than have come > > > out. > > > > How do you translate the words of the problem into mathematics? > > 0) There is a bijection between the set of balls entering the vase and > |N. > 1) There is a bijection between the set of escaped balls and |N. > 2) There is a bijection between (the cardinal numbers of the sets of > balls remaining in the vase after an escape)/9 and |N. How do you *define* division between cardinal numbers? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 8 Oct 2006 21:36 In article <virgil-F7008E.15353808102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: .... > If one chooses to work within ZFC or NBG, the vase is empty at noon. I doubt this. The problem is not defined with enough precision to state that. It has not been defined by most what is meant with "the number of balls in the vase at noon". Of course, you can use that the infinite intersection of sets does exist (and that is what you are using), and so get at the result. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 8 Oct 2006 21:44 In article <virgil-148331.15522708102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: .... > How is this: > > Let A_n(t) be equal to > 0 at all times, t, when the nth ball is out of the vase, > 1 at all times, t, when the nth ball is in the vase, and > undefined at all times, t, when the nth ball is in transition. > > Note that noon is not a time of transition for any ball, though it is a > cluster point of such times. > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > at any non-transition time t. > > B(t) is clearly defined and finite at every non-transition point, as > being, essentially, a finite sum at every such non-transition point. > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 9 Oct 2006 00:39
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > We know, not by > > > intuition, but by logic, that the vase at any time contains more balls > > > than have escaped. > > > > Absolutely false. > > Bold words, but unfortunately simply wrong. The contents of the vase > increases any time by 9 balls. > > > Before any balls are put in the vase, it is empty, and > > after al balls have been removed from the vase it is equally empty. > > It is only between these time that there are any balls in the vase at > > all. > > > We have a contradiction n ZFC > 1) Every ball will have left the vase at noon. > 2) At noon there are more balls in the vase than at any time before. > Nope. No one disagrees with 1. 2 does not follow. It is true that at any time t before noon, there will be more balls in the vase than at any time s before t. But it is not true that: anything that is true at any time t before noon must be true at noon (At any time t before noon only a finite number steps have been taken. This does not imply that at noon only a finite number of steps have been taken). Consider a set of propositions A, with the property that if p is in A and p is true at any time before noon, then p is true at noon. Some propositions (e.g. "the vase exists") are in A. Some propositions (e.g. "only a finite number of steps have been taken") are not in A. You have only intuition to tell you that "there are more balls in the vase than at any time previous" is in A. Intuition does not equal contradiction. - William Hughes |