Cantor Confusion
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From: Virgil on 17 Nov 2006 15:33 In article <1163782891.840480.212770(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > Make it easier for you to understand it by considering the initial > > > > > segments of the first column. The initial segment > > > > > > > > > > 1 > > > > > 2 > > > > > 3 > > > > > ... > > > > > n > > > > > > > > > > stand in bijection with the line 1,2,3,...,n by the diagonal element > > > > > d_nn. > > > > There is no bijection between the line indexes and > > the initial segments of the first column. There is > > almost a bijection, but not quite. > > Congratulations! After all you got it. This "not quite" is the > observation which is necessary to recognize that there are not actually > infinitely many numbers, i.e., that it is impossible to count the > number of natural numbers by a number omega, which can be increased by > 1. What Wm finds impossible, mathematicians find trivially easy. WM manages to swallow camels but strain at gnats. > > Perhaps you want to say that > > "The length of the diagonal cannot surpass > > the length of *every* line." This is false. The length of the > > diagonal can surpass the length of every line, if and only if the > > matrix does not have a longest line. > > If there is a longest line or not: The diagonal of the triangle is, by > definition, made of line indexes. Therefore it cannot be longer than > every line. The diagonal can and must be longer than every line for which there is a longer line. Does WM know of any line for which there is no longer line? If not WM is wrong again. > We see by your assertion of a diagonal longer than any line: There must > be more natural numbers than there are, in order to have infinitely > many. WM has wound himself into another Gordian knot, which only the logic of mathematics can cut through. And WM is no Iskander.
From: David R Tribble on 17 Nov 2006 15:46 Dik T. Winter wrote: >> If you want to find absolute truth you should not look at mathematics. > david petry schrieb: >> Perhaps we should replace "absolute truth" with "culturally neutral >> truth", or in other words, truth without any cultural, religious, or >> philosophical bias. [...] Thinking about this question >> leads most of us to believe that there is a core of mathematics which >> every such civilization will accept. > mueckenh wrote: > without axioms, yes. For instance: I + I = II (after translating "+" > and "="). Therefore I call this an absolute truth. Which axioms are you using to describe the "+" and "=" operators?
From: mueckenh on 17 Nov 2006 16:24 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > The natural numbers count themselves. Bijection of initial segments of > > column and lines > > > > 1 > > 2 > > 3 > > ... > > n <--> 1,2,3,...n > > > > If there is no infinite number then there are not infinitely many > > numbers. > > This is clearly the point of contention. > > Consider, N, the set of all natural numbers. > By definition N only contains natural numbers. > > Cases > i: There is a largest natural number,n_L, then N={1,2,3,...,n_L} > In this case there are n_L natural numbers. Deleted. > > ii: There is no largest natural number. We will > write this as N={1,2,3,...} (the ... represent only natural > numbers). Set N has infinitely many > elements. Please distinguish: iia: There is no number counting the elements of N. iib: There is a number omega counting the elements of N. > > iii: There are a finite number of natural numbers, but > there is no largest natural number. > Deleted. Athough this assertion is the only correct one, in my opinion, we can also delete it, because most mathematicians will not accept physical constraints. > > Case i is highly counterintuitive because of "What > about (n_L+1)?". > > I claim case 2 (existence of N and no largest number by the > axiom of infinity, infinitely > many numbers the fact that all the natural numbers have been used > to provide sizes of other sets). I claim case iia: There is a potentially infinite sequence N = 1,2,3,..., such that for any n there is n+1 but we cannot recognize or treat all of its elements. In particular we can never complete this set. We can never put it into a list. > > You want to show two things. > > a: case iii makes sense, i.e. there can be a finite number > of natural numbers but no largest natural number. Not here. > > b: case ii does not make sense. > Case iib does not make sense. > > To do a you need to define finite and show that this > definition does not lead the existence of a largest natural. > (note that an argument that there are only a finite number > of natural numbers because there are only a finite number > of available bits does not help. You still need to define finite). > > To do b you need to assume case ii and show this leads > to a contradiction. However, when you do this you cannot > use case iii. I don't use case iii. I use the fact that the diagonal (=bijection, d_nn) cannot be longer than every line, because it consists of what you have called the line indexes. For every element of the diagonal we must have a line index and, hence, a line. This must hold in the extended version too. This is my point of departure. If you say that the diagonal can be longer than every line, then you say that there are more natural numbers (elements of the bijection d_nn) than natural numbers (indexes n). Then there is no reason to continue this discussion. Otherwise we have a contradiction. Together with this very systematic account of yours describing our arguing, we arrive at the present, and probably, final state. > In particular, we know that asumming case ii > means that we can find a bijection between a matrix with > an infinite number of lines, but for which every line is finite. This is exatly the point where you have to admit that there are more, namely omega, natural numbers (d_nn), than natural numbers n < omega. > (There is a line for every natural number, but no other > line. Therefore by ii there are an infinite number of lines. > Line n will be {1,2,3,...,n} and have length n, which is > finite) You have to explain the difference appearing after extending every line and every initial segment of the column. You cannot do it by claiming different things, because for all n, the initial segments and the lines are identical (see above). Regards, WM
From: mueckenh on 17 Nov 2006 16:27 Tony Orlow schrieb: > > > >>> The natural numbers count themselves. Bijection of initial segments of > >>> column and lines > >>> > >>> 1 > >>> 2 > >>> 3 > >>> ... > >>> n <--> 1,2,3,...n IET: 1 12 123 .... > Not, "more than any finite number"? I'd leave omega out of the > discussion, if I were me. :) "More than any finite number" is already the number of elements of the first column. But it is said that a set of finite numbers suffices to build this infinite set. This is wrong but it is not easy to show the error. Therefore I add one element in order to have a set which has not only infinitely many elements but which provably must have also a transfinite element w+1. Now I show that there is no image of w+1 among the set of finite elements, although these elements were also increased by 1. And that is the desired result. In order to explain it, one must assume that the diagonal of the equilateral infinite triangle (IET) must be longer than every line. But the diagonal consists of ends of lines and nothing else. Therefore this assertion is a self-contradiction: The set of ends of lines (d_nn) must contain more elements than the set of lines n. Regards, WM
From: mueckenh on 17 Nov 2006 16:37
Franziska Neugebauer schrieb: > > Therefore a set of ordinal number omega must exist (it is the first > > column) > > omega "is" not "the first column". What you may write is, that in your > sketch the first column _represents_ omega. That is a matter of taste. I is not uncommon to say N = omega. On the other hand omega represents what before Cantor was commonly abbreviated by oo. Omega is the first transfinite number. You need not interpret n as a set (though you can do it). This all shows that omega is a number. It is said to be the number of natural numbers. And omega > n for every n e N. > > Of course not. Two sets X and Y are "bijected" by "pairing" the > *elements* of the sets not by "pairing" the sets itself. If there should be infinitely many lines, then their number should be omega. If you say that the diagonal can or even must be longer than every line, then you say that there are more natural numbers (elements of the bijection d_nn) than natural numbers (indexes n). Then there is no reason to continue this discussion. Otherwise we have a contradiction. > > > But an explanation: The set of all sets is an impossible set. > > This mathematically reads: There is no set of all sets (in some > axiomatic systems). Shall I conclude from your "definition by example", > that you define > > possible := exists > impossible := does not exist There is no other interpretation possible, I think. > > >> Sets (including bijections) are neither possible nor impossible. They > >> exist or do not exist. > > > > If they exist, then they are possible. If not, then they are > > impossible. > > Shall I interpret this as "possible := exists, impossible := does not > exist"? So why don't you simply use the established terms "exist" and > "not exist"? Maliciousness? Knowledge of literature. Cantor. We should pay a bit more respect to the creator of set theory: "An mehreren Stellen meiner Arbeit werden Sie die Ansicht ausgesprochen finden, daß dies unmögliche, d. h. in sich widersprechende Gedankendinge sind, ..." > > Now the part which you have not answered: > I would plead to stop this discussion. We have arrived at a clear result. You claim that the diagonal of a matrix can be longer than every line. As the diagonal is defined to consist of the ends of terms of lines, this claim is easy to conradict. In order to defend your claim that there are infinitely many finite numbers, you could simply say: "There are more natural numbers d_nn than natural numbers n." This sentence is as true as the sentence: The diagonal (d_nn) of a matrix can be longer than every line n. But it shows that there is no point in further arguing on a logical basis with sound arguments. Regards, WM |