Cantor Confusion
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From: Dik T. Winter on 9 Oct 2006 10:48 In article <1160404669.240794.298920(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > 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? You can not use limits, because the > > limit does not exist when you use standard mathematics. > > But we can safely say that lim{n-->oo}n = 0 is false. Yes, it is false because that liit does not exis. > lim{n-->oo}n can be estimated by lim{n-->oo} 1/n = 0. You can not estimate something that does not exist. > So using standard > > definitions there is no answer. More precise, given the sequence of sets: > > {1, ..., 10) > > {2, ..., 20} > > {3, ..., 30} > > etc., is there a limit? Well, no, there is no defined limit unless you > > define what a limit of sets looks like. I have never seen a definition > > that tells me how the limit of a sequence is defined. The limit of the > > size of the sets also gives no answer, because that limit does not exist. > > Strange enough, when somebody goes on to define things, *you* question his > > definitions, rather than the result. > > The limit {1,...,n} for n-->oo is N, if N does exist. By what definitions? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 9 Oct 2006 10:52 Dik T. Winter schrieb: > 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. Then the infinite intersection of the cardinal numbers A(t) with t = 1, 2, 3, ... of the set in the vase after completing action t does also exist. It is 9. Regards, WM
From: mueckenh on 9 Oct 2006 10:58 William Hughes schrieb: > 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. That is pure intuition. Why should it exist at noon? Is there an axiom of ZFC? > 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. No. I have the estimation Lim{n-->}n > 0. If you call this "intuition", then some more fundamental laws of mathematics are invalid than can be recovered by set theory. Regards, WM
From: mueckenh on 9 Oct 2006 11:13 William Hughes schrieb: > > > In my view we have not gotten very far. We still have > > > the result that there is no list of all real numbers > > > > That is not astonishing, because there are only those few real numbers > > which can be constructed. > > Few? Few compared to what. Compared to the assumed set of uncountably many. > The real numbers that cannot > be constructed? According to you they don't exist. But even > these "few" real numbers cannot be listed! Nevertheless the diagonal proof shows only that there are elements of a countable set which have not yet been constructed. > > > > > > (we need to reinterpret our terms, real numbers are > > > computable real numbers, and a list is a computable > > > function from the natural numbers to the (computable) real > > > numbers). > > > > > > If it gives you a warm fuzzy to say that > > > "Every ball will be removed at some time before noon", > > > > No. To say that every ball will be removed, is wrong, because there is > > not every ball. > > > > If it gives you a warm fuzzy to say > > "For any natural N, the ball numbered N will be removed from > the vase before noon" There is not "any natural" but only those which we can define. There is a largest natural which ever will be defined. Hence mathematics in the universe and in eternity has to do with only a very small sequence of naturals. Writing 1,2,3,... is but cheating Regards, WM
From: mueckenh on 9 Oct 2006 11:18
William Hughes schrieb: > > 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 > > Straight quatifier dyslexia. The fact that "for every x there exists > a y such that" does not imply "there exists a y such that for every x" A nonsense argument. Your assertion is wrong in a linear set. Give an example where the linear set covers a number which is not covered by one member of the linear set. Regards, WM |