Cantor Confusion
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From: Virgil on 18 Oct 2006 17:12 In article <1161182838.521344.142200(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > Sorry, but your "proof" doesn't work. Imagine an infinite path in the > > tree. Which is the edge it inherits as a whole? Whenever you give me > > that edge, I can tell you're lying because if a path inherits an edge > > as a whole, it means that the path terminates by that edge. > > How should I be able to name the last term of a sum which has no last > term? But while we cannot name any individual edge we can prove: No > path splits into two paths without the supply of two new edges, one > edge for each path. This implies there cannot be less edges than paths. And from each edge paths bifurcate endlessly, showing that there are infinitely many paths through each edge. > > > Or the other way round: Assume there were more paths than edges, then > at least two paths could no be distinguished. (A path can be > distinguished from every other path by at least one edge.) If each path is identified by and endless sequence of branchings each being a left or right branching from the previous node, then any two paths will differ at some node from which one branches left and the other right. > > > This is > > impossible for infinite paths. This is inevitable for distinct paths, finite or infinite, having a common root node. > > Of course that is impossible. Therefore the sum 1 + 1/2 + 1/4 + ... is > a infinite sum. But nevertheless your argument covers only half of the > story. Whenever you give me two infinite paths, I can name an edge > which belongs to only one of them. And whenever you give me an edge, I can name uncountably many paths containing that edge. > > > The same argument applies to other terms > > in the sum. (That edge is inherited by an infinite path by 1/1024! > > Ok, but that means that the path terminates 10 levels lower). This > > means that infinite path inherit zero edges in your proof. > > Then the series 1 + 1/2 + 1/4 + ... contains zeros? In "Mueckenh"'s arithmetic perhaps, but as it is irrelevant in path analysis, who else cares? > The distance between any two edges of one path is infinite? By what metric? In any one path there are only finitely many edges between any two given edges, just as for any two natural numbers, there are only finitely many others between them. "Mueckenh" is off in his dream world again. Way off. > > Regards, WM
From: Virgil on 18 Oct 2006 17:57 In article <1161183237.727249.154740(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? Induction. > > > (2) 2) is not because of continuity > > Why then? Induction. > > > (3) no continuity reasoning can lead to the result that the balls > > remaining at noon is the set N. > > But more than 1. Induction says none. > > Regards, WM
From: Virgil on 18 Oct 2006 17:58 In article <1161183394.556713.197770(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 If each ball inserted is still removed before noon, no.
From: Virgil on 18 Oct 2006 18:01 In article <1161183696.325721.158140(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. And it is a shame that "Mueckenh" still doesn't understand binary trees.
From: Virgil on 18 Oct 2006 18:05
In article <1161183817.516177.215940(a)e3g2000cwe.googlegroups.com>, 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: > > > > > > > > > > > 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. Induction makes it possible to say that all the balls were inserted before noon and that all the balls were removed before noon, and since there is not a smallest numbered ball left in the vase at noon, there are no balls left in the vase at noon. |