Cantor Confusion
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From: mueckenh on 18 Oct 2006 10:41 Dik T. Winter schrieb: > > It is impossible to show a "valid" mathematical proof against set > > theory. > > Ah, so you agree that you can not prove an inconsistency using mathematical > terms of proof. I agree that set theorists will ever accept any proof of inconsistency as valid. > > > We have discussed the vase and I would not have believed in > > advance that anybody could maintain arguments here like Virgil and > > William and others. > > No. You never would believe that anybody would use mathematical proofs > against your intuition. It is not *intuition* to find a proof of lim{n-->oo} n = 0 is wrong. > In actual mathematics you can state that > sqrt(2) squared is exactly equal to 2 And in actual life you can state that all problems are exactly solved. I wonder whether our politicans should not get a better mathematical education. > Redo your calculations. With 100 bits there are 2^100 possibilities, so the > cardinality for the set of numberss represented is <= 2^100. Here is a simplified problem: With one bit here are two numbers possible, namely 0 and 1, but the set of numbers realized with one bit has cardinality 1, namely either the number 0 or he number 1 but not both can be realized. > In the case of N it is known how the numbers are built up. It is known how a light years high scyscraper can be built up, but there are not enough bricks. > > Nothing, as soon as we withdraw to call set theory mathematics. > Oh. In that case, please do not call it mathematics. The copyright of this name is protected for a science that is one of the eldest on earth. It has been occupied illegally by a gang of gamblers but will be reinstalled within the next years. Regards, WM
From: mueckenh on 18 Oct 2006 10:42 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Virgil schrieb: > > > > > > > > > > According to the ZFC system: The vase is empty at noon, because all > > > > > > natural numbers left it before noon. > > > > > > By means of the ZFC system we can formulate sequences and their limits > > > > > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > > > > > And from this it follows that the vase is not empty at noon. > > > > > > > > > > By what axiom do you conclude that the limit as t increases towards noon > > > > > of any function and the value of that function at noon must be the same? > > > > > > > > By that or those axiom(s) which lead(s) to the result lim {t-->oo} 1/t > > > > = 0. > > > > > > Here is your theorem: Let f(x) be any function f:R->R. Then > > > lim(x->0-) f(x) = f(0). That is, the limit of f(x) as x approaches > > > 0 from the left is f(0). > > > > > > Can you show me how the axiom(s) you describe prove > > > that theorem? > > > > > > Can you then show me how the theorem applies to this > > > function? f(x) = 1 if x<0, f(x) = -1 if x>=0. > > > > If there is no stepwise continuity in f(t) = n, can you show me why the > > set of balls/numbers removed from the vase is containing all natural > > numbers at noon after the number of transactions t --> oo? > > Simple. Because n being a natural number => there is a removal time > t_n < noon. Therefore every natural is a member of the set of > balls removed before noon. The cardinal numbers of the sets of balls residing in the vase are also natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can such a function take on the value zero? > > What I don't understand is how anyone can think there are balls > with removal times which are still in the vase. > What you don't understand is that it s impossible to catch the whole set N. if you think you have done so, then there are many other infinite sets of natural numbers not yet caught. It is simply Hilberts hotel. It s simply an inconsistency of the idea that N could be completed consistently. Believe Hilbert's words: "Summing up: The infinite is nowhere realized. It is neither present in nature nor is it admissible as the foundation of our rational thinking - a remarkable harmony between being and thinking." Das Gesamtergebnis ist dann: das Unendliche findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in unserem verstandesmäßigen Denken zulässig - eine bemerkenswerte Harmonie zwischen Sein und Denken" [D. HILBERT: Über das Unendliche, Math. Ann. 95 (1925) p. 190]. Regards, WM
From: mueckenh on 18 Oct 2006 10:47 jpalecek(a)web.de schrieb: > > > > > > 0. > > > > > > /a \ > > > > > > 0 1 > > > > > > /b \c / \ > > > > > > 0 1 0 1 > > > > > > ..................... > > > > > > > > An edge is related to a set of path. If the paths, belonging to this > > set, split in two different subsets, then the edge related to the > > complete set is divided and half of that edge is related to each of the > > two subsets. If it were important, which parts of the edges were > > related, then we could denote this by "edge a splits into a_1 and a_2". > > But because it is completely irrelevant which part of an edge is > > related to which subset, we need not denote the fractions of the edges. > > 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. 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.) > This is > impossible for infinite paths. 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. > 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? The distance between any two edges of one path is infinite? Regards, WM
From: Dave Seaman on 18 Oct 2006 10:49 On Wed, 18 Oct 2006 14:14:33 GMT, Dik T. Winter wrote: > In article <J7B4p3.ItG(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > In article <eh2fe1$j3e$1(a)mailhub227.itcs.purdue.edu> Dave Seaman <dseaman(a)no.such.host> writes: > > > On Tue, 17 Oct 2006 02:48:55 GMT, Dik T. Winter wrote: > > ... > > > > One of the most serious errors can he found in the statement that > > > > "to count sets of first cardinality you need ordinals of the second > > > > class" > > > > > > Are you sure you are quoting him correctly? Cantor did say (in fact, > > > this is a section heading in boldface): > > > > It was in an article explaining transfinite "counting". And I am quite sure > > I did quote him reasonably correct. But the book is on my desk at work, so > > I will try to find it tomorrow. > Gesammelte Abhandlungen, Hildesheim, 1962, p. 213: > ... der Unterschied ist nur der, da?, w?hrend die Mengen erster > M?chtigkeit nur durch (mit Hilfe von) Zahlen der zweiten Zahlenklasse > abgez?hlt werden k?nnen, die Abz?hlung bei Mengen zweiter M?chtigkeit > nur durch Zahlen der dritten Zahlenklasse, bei Mengen dritter > M?chtigkeit nur durch Zahlen der vierten Zahlenklasse u. s. w. > erfolgen kann. > or translated: > ... the difference is only that, while sets of the first cardinality > can be counted only through (with the aid off) numbers of the second > class, the counting of sets of the second cardinality only through > numbers of the third class, with sets of the third cardinality only > through numbers of the fourth class, etc. > From: > ?ber unendlichen lineare Punktmannigfaltigkeiten, Nr. 6, sec. 15. I may be wrong, but I interpret the quoted passage to mean: 1. The cardinality of the set of all finite ordinals is aleph_0. 2. The cardinality of the set of all ordinals having cardinality aleph_0 is aleph_1. 3. The cardinality of the set of all ordinals having cardinality aleph_1 is aleph_2. and so on. This fits with the quotation I provided yesterday. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: mueckenh on 18 Oct 2006 10:52
jpalecek(a)web.de schrieb: > > > Every list of real numbers supplies a diagonal number which > > > is not contained in the list. > > > > > > so any way you cut it there is no complete list of real numbers. > > > > And there is no complete list of computable numbers. But they are > > countable. Hence the diagonal argument does not prove anything. > > This is nonsense. You say "my proof is flawed, therefore your proof is > flawed". You'd better think more about your proof. In order to avoid computable lists and to have a common base think of constructible numbers. A constructible number is a number of which every digit can be obtained by a given formula. Every diagonal number of a given list is a constructible number. The set of constructible numbers is obviously countable. Every list of constructible numbers yields a constructible diagonal number. Every list of real numbers yields a real diagonal number. There is absolutely no difference. Regards, WM |