Cantor Confusion
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From: mueckenh on 14 Nov 2006 07:35 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > Well, since I already agree there is no infinite column this does > > > not change anything. > > > > If you add only one element to each column, you get the order type > > omega + 1 for the length of he matrix. > > If you add only one element to each line, the order type of the width > > of the matrix remains omega. > > > > Correct. > > No column has an infiite term. > Each column has as infinite number of terms. Therefore: If we add one term to a column, then it has the order type omega + 1, hasn't it? > No line has an infinite term > No line has an infinite number of term > The supremum of the number of line indexes is > the same as the supremum of the number of column indexes. After having added one term to each colunmn, the ordinal number of each column and of the length of the matrix is omega + 1, isn't it? > > However, the fact that the supremums are the same does > not mean that the sets are the same. The suprema are not the same. After havin added one term to each line, each line has the ordinal number n+1 < omega. And the width of the matrix has the ordinal number omega. > If two sets A and B > are not the same, adding one element to A may not > give the same result as adding one element to B. May be. But he diagonal bijects the columns to the lines. The element d_nn maps the n-th column to he n-th line. It should do so, at least. Regards, WM
From: William Hughes on 14 Nov 2006 07:58 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > Well, since I already agree there is no infinite column this does > > > > not change anything. > > > > > > If you add only one element to each column, you get the order type > > > omega + 1 for the length of he matrix. > > > If you add only one element to each line, the order type of the width > > > of the matrix remains omega. > > > > > > > Correct. > > > > No column has an infiite term. > > Each column has as infinite number of terms. > > Therefore: If we add one term to a column, then it has the order type > omega + 1, hasn't it? > Yes. > > No line has an infinite term > > No line has an infinite number of term > > The supremum of the number of line indexes is > > the same as the supremum of the number of column indexes. > > After having added one term to each colunmn, the ordinal number of > each column and of the length of the matrix is omega + 1, isn't it? Yes. > > > > However, the fact that the supremums are the same does > > not mean that the sets are the same. > > The suprema are not the same. After havin added one term to each line, > each line has the ordinal number n+1 < omega. And the width of the > matrix has the ordinal number omega. So? Adding one element may or may not change the supremum. The supremum of the lengths of the initial segments of the columns is omega. The supremum of lengths of the lines is omega. The supremums are the same. However, we can make the supremums different by adding elements. If we add one element to each of the columns then the supremum of the lengths of the initial segments of the columns changes to omega +1. If we add one elelment to each of the lines the supremum of the lengths of the lines does not change, it remains omega. > > > If two sets A and B > > are not the same, adding one element to A may not > > give the same result as adding one element to B. > > May be. But he diagonal bijects the columns to the lines. The element > d_nn maps the n-th column to he n-th line. It should do so, at least. > And it does. There is no line with infinite index. There is no column with infinite index There is no element of the diagonal with infinite index - William Hughes
From: Franziska Neugebauer on 14 Nov 2006 07:59 mueckenh(a)rz.fh-augsburg.de wrote: > {1,2,3} is the collection of, and a convenient expression to write > {that we are talking about, the numbers 1 ,2, and 3. What are we talking about, when we write { } ? F. N. -- xyz
From: mueckenh on 14 Nov 2006 08:13 William Hughes schrieb: > If we add one element to each of the columns then > the supremum of the lengths of the initial segments of the columns > changes to omega +1. If we add one elelment to each of the lines > the supremum of the lengths of the lines does not change, it > remains omega. Fine. This new matrix has a diagonal, if the old matrix had one. It bijects the columns to the lines. The element d_nn maps the n-th column to the n-th line. It should do so, at least. > > And it does. Could you construct this bijection? It is easy to begin: Column 1 <--> Line 1 by the diagonal element d_11 Column 2 <--> Line 2 by the diagonal element d_22 Column 3 <--> Line 3 by the diagonal element d_33 .... Column ? <--> Line omega+1 by the diagonal element d_?? But t is difficult to end, although there is a last element in the set of lines. > > There is no line with infinite index. But there is a line enumerated by the number omega+1, because each column has an element omega+1. > There is no column with infinite index Just that seems to be the problem. Therefore, no column is enumerated by the number omega+1. > There is no element of the diagonal with infinite index But the downwards part o the diagonal has an element omega+1. Is it too much intuition to require that a diagonal does not split in two parts "in the infinite"? Regards, WM
From: mueckenh on 14 Nov 2006 08:16
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > {1,2,3} is the collection of, and a convenient expression to write > > {that we are talking about, the numbers 1 ,2, and 3. > > What are we talking about, when we write > > { } > > ? The same as when we write Regards, WM |