Cantor Confusion
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From: mueckenh on 18 Oct 2006 03:55 Virgil schrieb: > In article <1161027429.194776.277830(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > > > > > But the end time of the problem (noon) does not correspond to > > > > > an integer (neither in standard mathematics, nor in your > > > > > system, whether or not you interpret the problem as dealing > > > > > with infinite integers as well as finite integers). So the function > > > > > 9n does not have a value at noon. There is no way > > > > > it can be continuous at noon. And since there is no > > > > > value of n that corresponds to noon, 9n cannot be used > > > > > to determine the number of balls in the vase at noon. > > > > > > > > But the function n can be used to determine the number of balls removed > > > > from the vase at noon? > > > > > > > > > > Nope. [There are no balls removed from the vase at noon] > > > > Arbitrary misunderstanding? > > > > > The function 9n has nothing to do with the number of > > > balls in the vase at noon. > > > > But the function n can be used to determine the number of balls having > > been removed > > from the vase at noon? > > Not even that. How then do you know that all the balls have been removed? Regards, WM
From: mueckenh on 18 Oct 2006 03:56 Virgil schrieb: > In article <1161027538.870754.34000(a)k70g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > The connection between finite paths and partial sums of edges > > leads to > > (1-(1/2)^n+1)/(1 - 1/2) edges per path. > > Which, as written, is negative for all positive naturals n. Indeed? In your world? Then I can guess why your vase is sometimes empty. Regards, WM
From: mueckenh on 18 Oct 2006 03:58 Virgil schrieb: > Unless someone can dispute it, how can anyone claim that there are any > balls in the vase at noon? It is obviously nonsense to see only the one side as set theorists do. Infinity has two sides. The other side says the opposite. Further the result o the gedankenexperiment must not depend on switching numbers. Removing balls 1, 11, 21, ... does not change the quantities in fact, but according to set theory it does. Therefore set theory has been contradicted. Regards, WM
From: mueckenh on 18 Oct 2006 03:59 MoeBlee schrieb: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > David Marcus schreef: > > > > > Han de Bruijn wrote: > > > > > > > > How can we know, heh? Can things in the real world be true AND false > > > > (: definition of inconsistency) at the same time? > > > > > > That is not the definition of "inconsistency" in Mathematics. On the > > > other hand, I don't know of any statements in Mathematics that are both > > > true and false. If you have one, please state it. > > > > What then is the precise definition of "inconsistency" in Mathematics? > > How many times does it have to be posted? > > G is inconsistent <-> G is a set of formulas such that there exists a > formula P such that P and its negation are both members of G. Like: The vase is empty a noon and the vase is not empty at noon. Regards, WM
From: mueckenh on 18 Oct 2006 04:02
David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > However, you wish to do more. You want to show > > > > > that claiming "N does not have an upper bound and > > > > > N exists as a complete set" leads to a contradiction.] > > > > > > > > That is true too. And it is easy to see: If we define Lim [n-->oo] > > > > {1,2,3,...,n} = N, then we can see it easily: > > > > > > > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > > > > than |{2,4,6,...,2n}| = n. > > > > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > > > > Contradiction, because there are only natural numbers in {2,4,6,...}. > > > > > > You appear to have written the following: > > > > > > Let N be the set of natural numbers. For all n in N, > > > > > > 2n > |{2,4,6,...,2n}| = n, > > > > No. I have written the following: For all n e N we have {2,4,6,...,2n} > > contains larger natural numbers than |{2,4,6,...,2n}| = n. I did not > > explicitly mention 2n. > > > > > > n < |{2,4,6,...}| = alpheh_0, > > > > > > {2,4,6,...,2n} is a subset of N. > > > > > > I follow this. But, you have the word "contradiction" in your last > > > sentence. Are you saying there is a contradiction in standard > > > Mathematics? If so, what is it? I don't see it. > > > > By induction we find the larger the set the larger the number of > > numbers contained in the set and surpassing its cardinality. The > > assumption that the infinite set would not contain such numbers > > neglects the question of what kind of numbers can increase the > > cardinality without increasing the number sizes. > > The problem is the same, in principle, as the vase and its balls. > > You didn't mention 2n, but you said "larger natural numbers than n". 2n > is one of those numbers that is in the set and is larger than n, so I > wrote "2n". Is that not what you meant? It is not what I meant, because the main observation is that the number of numbers larger than n grows with n. It is just the vase-ball situation. In addition there is no largest natural number as could be suggested by the last number 2n of the finite sequences. > > As for your final paragraph, I'm sorry, but I can't follow it. Please > start with the following and add whatever is missing to give the full > proof. > > Let N be the set of natural numbers. For all n in N, > > |{2,4,6,...,2n}| = n, > there exists m in {2,4,6,...,2n} such that m > n, and the number of these numbers m grows with n. There is no finite natural number n, such that the number of m's in the set {2,4,6,...,2n} is larger than the number of m's in the set {2,4,6,...,2n, 2(n+1)}. or briefly: The cardinality of the set {n,n+1,n+2,...,2n} is strictly increasing. Therefore, in the limit {2,4,6,...} there are infinitely many finite natural numbers m > |{2,4,6,...}| The assumption |{2,4,6,...}| = alpheh_0 > n forall n e N is false. The only alternative: We cannot at all conclude anything about infinite sets from considering limits. But then infinite sets are a tool of pure arbitrariness and completely useless. Regards, WM |