Cantor Confusion
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From: mueckenh on 18 Oct 2006 10:53 Dik T. Winter schrieb: > In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > The inconsistency is that > > 1) For the balls inserted until noon, you can find the result: It is > > the set N. > > 2) For the balls removed until noon, you can find the result: It is the > > set N. > > 3) For the balls remaining at noon, the same arguments of continuity > > which lead to (1) and (2) cannot apply. > > There are quite a few obvious reasons. > (1) 1) is not because of continuity Why then? > (2) 2) is not because of continuity Why then? > (3) no continuity reasoning can lead to the result that the balls > remaining at noon is the set N. But more than 1. Regards, WM
From: mueckenh on 18 Oct 2006 10:56 imaginatorium(a)despammed.com schrieb: > > But the function of balls/numbers removed from the vase is a > > continuously (stepwise) increasing one, containing all natural numbers > > at noon? > > Uh, yes, unless I mysteriously misunderstand you... If takenout() is a > function from time to the power set of the integers (i.e. it maps to a > set of integers) then each natural number m is included in the set that > takenout() maps to from time = -1/m. So by time zero, all natural > numbers are included. > > Was there a question with that? And this result would change, if the numbers of the balls were exchanged, for instance multiplied by 10 after having been inserted? Regards, WM
From: mueckenh on 18 Oct 2006 10:59 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > A good, if no the best source to learn about the different meanings of > > > > infinity would be Cantor's collected works. > > > > > > Set theory has advanced since Cantor. The best source to learn about > > > current set theoretic definitions of 'infinite' is not Cantor. > > > > There is no definition what infinity is and there is no definition what > > a set is. > > There is a definitition in Z set theory of 'is infinite'. And we may > also have a definition in Z set theory of 'is a set' (though that > particular definition may be considered trivial). Could you please provide me with the definition of a set? > > > Current set theory has forgotten about theoretic definitions > > of the infinite at all and uses the notion "infinity" just as seems > > necessary to avoid too obvious contradictions. > > You just don't know what you're talking about. Why don't you just read > a textbook in set theory? I read more than one. There I found different means to test for infinity, but very rarely the important distinction between potential and actual infinity, and I did not find what a set is. Regards, WM
From: mueckenh on 18 Oct 2006 11:01 MoeBlee schrieb: > > First you say the notion of 'rational relation' (whatever that means) > "cannot be expressed by mathematical notion". Then you challenge me to > say what part of your proof is in conflict with set theory. What is the > notion of 'rational relation' that "cannot be expressed by mathematical > notion"? Are defining a certain relation in set theory or are you > definining a relation you claim not to exist in set theory? Meanwhile there are many who understand the binary tree. Perhaps you will follow the discussion, then you may understand it too. Regards, WM
From: mueckenh on 18 Oct 2006 11:03
William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > 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? > > > > > > > > > No. There are no balls removed from the vase at noon. > > > > > > Note, that there is no time "just before noon". At any time > > > before noon there remain an infinite number of steps. > > > > > > So no value of n is close to the end. > > > > > > The balls are removed during an infinite number of > > > steps. > > > > Please read carefully: But the function n can be used to determine the > > number of balls *having been* removed from the vase at noon? (That > > means up to noon.) > > > > > No. The function can be used to determine the number of > balls having been removed from the vase at any time before noon. Correct. > The function cannot be use to determine the number of balls > having been removed from the vase at noon, because the function does > not have a value at noon. Correct. It is not possible to have the value "all natural numbers". > We can take the limit of the function as time > approaches noon, but we cannot say that this limit is the number > of balls having been removed from the vase at noon without further > analysis. Correct. Therefore the assertion that all natural numbers were outside the vase at noon is unjustified like any quantitative assertion about all natural numbers. Regard, WM |