Cantor Confusion
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From: Tony Orlow on 17 Nov 2006 11:26 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > >>> The natural numbers count themselves. Bijection of initial segments of >>> column and lines >>> >>> 1 >>> 2 >>> 3 >>> ... >>> n <--> 1,2,3,...n >> A n e omega. >> >>> If there is no infinite number then there are not infinitely many >>> numbers. >> 1. If *where* is no infinite number? > > There is no infinite number among the lines 1,2,3,...,n. There can be > no bijection with the completed first column. > >> 2. Define "infinitely many" [, please.] > > Here I mean with "infinitely many" a number which, like the column, by > adding 1 element yields a transfinite number omega + 1. > > Regards, WM > Not, "more than any finite number"? I'd leave omega out of the discussion, if I were me. :)
From: mueckenh on 17 Nov 2006 11:53 Virgil schrieb: > > The length of the diagonal cannot surpass the length of any line. (This > > is easy to show for every matrix.) > > it has never been shown because it is false for a list in which the nth > member is of length n. > > In such a list, for every n, the diagonal has "length" >= n, and in > fact, has "length">= n+1 (as here is always a n+1'st line) so: > > For every n in N, length(diagonal) > n. Yes. But for every digit of the diagonal there is a line supplying it. Therefore the diagonal cannot be longer than every natural number. ============= > Then, according to WM, one can have a set containing NO infinite > element, such as the rationals, and a proper subset of it which MUST > contain an infinite element, such as the integral rationals. > I find such self-contradictory systems quite unsatisfactory. > As do, I suspect, all those who are mathematically at least minimally competent. Please check your own competence. As usual you misunderstood and drew the wrong conclusion. In principle it was possible that the set 1/n had infinitely many elements. ================= > > Perhaps it can change its size. > A sequence, being a function, a sort of set, is of fixed "size", in the > sense that its membership does not vary with time or with any other > "variable". O, yeah, you are the guy who has imagined all the natural numbers and all the terms of all sequences. Already reached the last one? Or are you still working on it? Regards, WM
From: mueckenh on 17 Nov 2006 11:55 david petry schrieb: > Dik T. Winter wrote: > > > If you want to find absolute truth you should not look at mathematics. > > Perhaps we should replace "absolute truth" with "culturally neutral > truth", or in other words, truth without any cultural, religious, or > philosophical bias. We can reason about this concept by asking the > question: what will the mathematics of advanced alien civilizations > (i.e. from other planets) look like? Thinking about this question > leads most of us to believe that there is a core of mathematics which > every such civilization will accept. without axioms, yes. For instance: I + I = II (after translating "+" and "="). Therefore I call this an absolute truth. Regards, WM
From: mueckenh on 17 Nov 2006 11:57 Franziska Neugebauer schrieb: > > http://mathworld.wolfram.com/InitialSegment.html > > What you call "complete initial segment" is not an *initial* segment > but the whole set of lines. What is a name? What *counts* is this: It is asserted that the number of natural numbers is omega (the elements of the first column = number of lines). By adding 1 element to the first column we see that, if this assertion is correct, the number increases to omega + 1. But are there enough lines? No. That means: The finite natural numbers are not sufficient to stand in bijection with all omega elements of the column, but only with the finite ones. Therefore the unavoidable conclusion is: There are not omega finite numbers. Regards, WM
From: mueckenh on 17 Nov 2006 12:01
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. > Each line index is associated with > an initial segment of the form {1,2,3,...,n}, that is an > initial segment that ends. Each initial > segment that ends is associated to a line index. > However, there is one initial segment, {1,2,3,...} that does not end. > This initial segment is not associated with any > line index. So the fact that there is no bijection > between the initial segments of the first column > and the lines, does not mean that there is > no bijection between the line indexes and > the column indexes. Of course not. It only mans that there is not a number omega of lines. Equivalently, there is no actually infinite set of finite natural numbers. ======== > 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. We are discussing whether there can be infinitely many natural numbers. They all are finite. Therefore the triangle is appropriate. 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. Regards WM |