Cantor Confusion
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From: Virgil on 16 Oct 2006 15:33 In article <1161003336.658410.101590(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > By the way: Every means to draw conclusions and to calculate results is > > > mathematics. There is no need to prefer a certain language (unless > > > there is someone who cannot speak another one). Would you assert > > > Archimedes did not do mathematics, because he used only the Greek > > > language and had not yet special symbols but Greek letters to denote > > > numbers? > > > > The language of Mathematics has evolved over time. > > And it is going on to evolve, like mathematics itself. > > > > If you assert that the ball and vase problem shows that modern > > Mathematics contains a contradiction, then please state the problem > > using the language of modern Mathematics. "Balls" and "vases" are not > > part of Mathematics, although people may use such language to talk > > informally about Mathematics. > > I did mention already that "balls" is an abbreviation for "natural > numbers" but that I prefer to use balls in order not to intermingle > these numbers with the natural numbers used for the transactions t. > > > So, please state the ball and vase problem > > using the language of Mathematics (e.g., "sets", "functions", > > "integers", "reals") so that we can see what mathematical problem you > > are talking about. > > > > What you wrote above (i.e., X(t), Y(t), Z(t)) uses the language of > > Mathematics, but (as I pointed out above) is not the statement of a > > problem since it doesn't end with a question. > > It is not the statement of a problem but the proof of a contradiction. There is no such proof within the mathematics without invoking assumptions not required by the statement of the problem.
From: mueckenh on 16 Oct 2006 15:34 William Hughes schrieb: > > One cannot compute a list of all computable numbers. By this > > definition, > > (1) the computable numbers are uncountable. > > (2) There is no question, that the computable numbers form a countable > > set. > > This is a contradiction. It is not necessary to come up with a list of > > all computable numbers. > > Nope. You are mixing two approaches.. > > A set X is countable if there exists a surjective function,f, > from the natural numbers to X > > There are two possibilities No. > > A: you allow arbitrary functions f > > B: you allow only computable functions f What is a function which is not computable? > > In case A, the computable reals are countable [ but you > also have arbitrary reals, and the set of reals (computable > and arbitrary) is uncountable.] > > In case B. the computable reals are not countable > (i.e. there is no list of computable reals) > > You cannot arrive at a contradiction by taking one result from > case B (the computable reals are uncountable) > and one result from case A > (the computable reals are countable). I take not at all results from your mysterious cases. I see: Every list of computable numbers supplies a diagonal number which is computable but not contained in the list. And I see: Every list of real numbers supplies a diagonal number which is not contained in the list. There is absolutely no difference. Regards, WM
From: mueckenh on 16 Oct 2006 15:37 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? Regards, WM
From: Virgil on 16 Oct 2006 15:37 In article <1161004418.858411.313520(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > In standard terminology, a "relation between paths and edges" means a > > set of ordered pairs where the first element of a pair is a path and the > > second is an edge. Is this what you meant? > > Yes, but this notion is developed to include fractions of edges. Then one must be allowed to fractionalize paths as well , so that one matches any fraction of an edge to no more than the corresponding fraction of a path. If one deals only with whole edges and whole paths, it transpires that there are countably many edges and uncountably many paths and no bijection between whole edges and whole paths.
From: mueckenh on 16 Oct 2006 15:38
William Hughes schrieb: > mueck...(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > 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. > > > > Who has ever justified such a proof? In fact that is impossible because > > there is no infinity. Therefore all such "proofs" are false. But if we > > assume the existence of the infinite, then the sum of the geometric > > series is the most reliable entity at all. (Niels Abel: With the > > exception of the geometric series no series has ever been calculated > > precisely.) > > If you wish to assme that infinity does not exist, knock yourself > out. However, if you are trying to show that the assumption > that infinity does exists leads to a contradiction you need > to justify the proofs that you make using that assumption. > > > > > > > > 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?". > > > > There are two questions: Do the infinite paths exist and does the > > geometric series wit q = 1/2 have a limit? I don't need any further > > infinities. > > > > No, there is a third question: "What is the connection between the > infinite paths and the limit of the series?" You have only shown a > connection between finite paths and partial sums. Wrong. The connection between finite paths and partial sums of edges leads to (1-(1/2)^n+1)/(1 - 1/2) edges per path. Regards, WM |