Cantor Confusion
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From: Virgil on 14 Oct 2006 17:29 In article <1160834188.179162.299120(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > Of course, Tony, you are right! > > Regards, WM WM + TO = chaos.
From: William Hughes on 14 Oct 2006 17:35 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > Han de Bruijn wrote: > > > > > Set Theory is simply not very useful. The main problem being that finite > > > > > sets in your axiom system are STATIC. They can not grow. > > > > > > > > Set theory provides for capturing the notion of mathematical growth. > > > > Sets don't grow, but growth is expressible in set theory. If there is a > > > > mathematical notion that set theory cannot express, then please say > > > > what it is. > > > > > > Obviously the notion of "rational relation" as used in the binary tree > > > cannot be expressed by mathematical notion: > > > Consider the binary tree which has (no finite paths but only) infinite > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > edges is countable, because we can enumerate them > > > > > > 0. > > > /a \ > > > 0 1 > > > /b \c / \ > > > 0 1 0 1 > > > ............. > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > a is inherited by all paths which begin with 0.00, the other half of > > > edge a is inherited by all paths which begin with 0.01. > > > > So each finite path of length N is related to > > > > 1 + 1/2 +1/4 + ... + 1/2^N > > > > edges > > > > > Continuing in this manner in infinity, > > > > > > we get a limit which may or may not be related to anything. > > This geometric sum is defined if any infinity is defined. > > > > > we see that every single infinite path is > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > > other path. > > > > No, the statement that what holds for finite paths also > > holds for infinite paths needs proof. > > Your provide none. > > Who provides, if not I? What is needed is a proof that what holds for finite paths holds for infinite paths. > Operating with the infinite can only be > justified by the finite" (Hilbert). Correct > That I do. No. The fact that everything that is true about the infinite must be justified in the finite, does not mean that everything that can be justified in the finite must be true about the infinite. You prove that something is true in the finite case. You do not justify your transfer to the infinite case. > The axiom of infinity > applies to the paths. They are nothing but representations of real > numbers. These exist according to set theory, therefore the paths > exist too. > Yes, but the question is not "do the paths exist?". - William Hughes
From: Virgil on 14 Oct 2006 17:37 In article <1160834284.109153.282390(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > mueckenh wrote: > > > I merely answer that it is completely irrelevant to speak of certain t. > > > > Then why did you say "use any positive value of t"? > > > > > The paradox is raised only by the asumption that the set of all t did > > > exist. > > > > What paradox? > > The result Lim{n-->oo} 9n = 0 Since the original problem was stated in terms of time, or "t",, why would anyone say any t does not exist? And in terms of t, Lim_{t -> 0- } number_of_balls(t), need not be equal to number_of_balls(0), as the number_of_balls function is rife with discontinuities already. Those who try to require continuity a t = 0 are imposing a false model that is nowhere implied in the original problem.
From: Virgil on 14 Oct 2006 17:41 In article <1160834418.764253.292380(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > Han de Bruijn wrote: > > > Set Theory is simply not very useful. The main problem being that finite > > > sets in your axiom system are STATIC. They can not grow. > > > > Set theory provides for capturing the notion of mathematical growth. > > Sets don't grow, but growth is expressible in set theory. If there is a > > mathematical notion that set theory cannot express, then please say > > what it is. > > Obviously the notion of "rational relation" as used in the binary tree > cannot be expressed by mathematical notion: > Consider the binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1. The edges (like a, > b, and c below) connect the nodes, i.e., the binary digits. The set of > edges is countable, because we can enumerate them > > 0. > /a \ > 0 1 > /b \c / \ > 0 1 0 1 > ............. > > Now we set up a relation between paths and edges. The relation as described by "Mueckenh" does not establish the bijection he calms. Relate edge a to all > paths which begin with 0.0. Relate edge b to all paths which begin with > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > a is inherited by all paths which begin with 0.00, the other half of > edge a is inherited by all paths which begin with 0.01. Continuing in > this manner in infinity, we see that every single infinite path is > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, How does one take only a fraction of an edge in a path? No path can take anything less that an entire edge if it is ever to get to the next edge.
From: Virgil on 14 Oct 2006 17:49
In article <1160834804.812032.315980(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The rules to cover up with infinity are objectionable. > In decimal representations of irrational numbers the infinite string of > digits leads to an undefined result unless the factors 10^(-n) are > applied. In Cantor's diagonal proof each of the elements of the > infinite string is required with equal weight. Not in any view of the Cantor "diagonal" I have ever seen. It is arranged in that proof the diagonal D and the nth member of a given list L_n differ by at least 1/10^n, which , for every natural number n, is a strictly positive difference. According to standard mathematics any difference between two values, however small, is sufficient to establish that they are not equaL That in some people's view physics may not work that way does not mean that mathematics will not continue to work that way. |