Cantor Confusion
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From: mueckenh on 4 Dec 2006 08:40 Virgil schrieb: > > Yes. But the mapping is not the usual one (one edge --> one path). The > > edges are subdivided in shares. > > Then it does not establish a one-to-one relationship. In one-to-one you > mast math whole with whole and not fractionate. Yes, so people think, franticly closing their eyes in front of a fractional relation. But in order to kill your prejudice: Map the edges on the paths one-to one by random choice. 0. /a \ 0 1 /b \c / \ 0 1 0 1 ............................. Map edge a on one of the paths beginning with 0.0 (for instance on 1/3). Map edge b on one of the paths beginning with 0.00, but not on that one which carries edge a. Map edge c on one of the paths beginning with 0.01, but not on that one which carries edge a. By this random mapping, we have a bijection from the set of edges onto the set of path. This is proved by the fact that always, when two paths become separated, one edge is available to be mapped on a path not yet carrying an edge. Regards, WM
From: mueckenh on 4 Dec 2006 08:47 Virgil schrieb: > > Therefore we can denote a set by X and we can say that the set X grows. > > When one calls X a variable, that does not mean that its value varies > but that its value can be any member of some set of allowable values. That is one of several interpretations. In this interpretation there is no place for potential infinity. That is a good reason to reject it. But more important, it is the loudmouth-claim of the availability of all members of many sets. That even forces clear thinkers to reject it. Regards, WM
From: mueckenh on 4 Dec 2006 08:47 Virgil schrieb: > > Therefore we can denote a set by X and we can say that the set X grows. > > When one calls X a variable, that does not mean that its value varies > but that its value can be any member of some set of allowable values. That is one of several interpretations. In this interpretation there is no place for potential infinity. That is a good reason to reject it. But more important, it is the loudmouth-claim of the availability of all members of many sets. That even forces clear thinkers to reject it. Regards, WM
From: William Hughes on 4 Dec 2006 08:53 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > We extend this to potentially infinite sets: > > > > A function from the set potentially infinite set A to the > > potentially infinite set B is a potentially infinite set of > > ordered pairs (a,b) such that a is an element of A and b is > > an element of B. > > > > We can now define bijections on potentially infinite sets > > and extend the bijection equivalence relation to include > > potentially infinite sets. Thus we can define > > equivalence classes under bijection of potentially infinite sets. > > Thus we can define "cardinal numbers" of potentially > > infinite sets. > > > There is only one "cardinal number". In order to apply any of Cantor's > proofs of higher cardinal numbers, a set of aleph_0 must be complete. > But it cannot be complete in potential infinity. You now agree that a potentially infinite set can have a cardinal number and that this cardinal is not a natural number. We have: there exists a bijection between sets or potentially infinite sets A and B iff the cardinal number of A is the same as the cardinal number of B. Now apply this. The natural numbers form a potentially infinite set. The diagonal contains the potentially infinite set of natural numbers. Therefore the cardinal of the set of elements of the diagonal is not a natural number. For any line, the cardinal of the elements of the line is a natural number. Therefore there is no bijection between the diagonal and any line. However, the line indexes contain the set of natural numbers and form a potentially infinite set. There can be a bijection from the diagonal to the line indexes. - William Hughes
From: stephen on 4 Dec 2006 08:59
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >>Do they define sets as allowed to grow? Not in the quote you supply. >> There they talk about set valued variables that can grow. > No. X and Y do not grow, they remain "X" and "Y". The set they denote > does grow. The number of EC states may be n. "n" does not grow. The > number denoted by "n" does grow. What do you mean by the 'the number denoted by "n" does grow'? Currently the number of EC states is 25. In a month it will be 27. Does that mean 25 is going to grow into 27? Will 25 no longer exist? Or will 25 now mean 27? What do you mean by 'the number denoted by "25" does grow'? The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. The idea that a set ever changes is equally ridiculous. Stephen |