Cantor Confusion
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From: mueckenh on 15 Nov 2006 07:23 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > >> Let he original matrix be A. > [...] > > 1 > > 12 > > 123 > > ... > > Something is missing here: > > 1uuu... > 12uu... > 123u... > ... No. We discuss the question whether an infinite set of naturl numbers requires an infinite number as an element. The naural numbers are expressed by the digits of a line. We could also use 1 11 111 .... The "u" are not appropriate. > > Usually a matrix m(a,b) is a function of domain A x B (it is > rectangular not a triangle) and some co-domain. Here A equals B equals > omega by definition. A matrix with domain A x B is called (generalized) > square iff A = B. > > "u" stands for undefined (empty), not set. The "triangle" you see is not > the structure of the matrix but its occupancy. We discuss just this triangle as our marix. > > Prima facie "Adding x to each column" ("at the end") is not "possible", > since the columns have no end (no last element). If 1,2,3,... has the ordinal number omega and if it is possible to construct 2,3,4,...,1 and if it is meaningful to denote the ordinal number of 2,3,4,...,1 by omega + 1, then your argument fails. But just these antecedents are assumed. Regards, WM
From: William Hughes on 15 Nov 2006 07:31 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > 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. > > > > Let he original matrix be A. > > > 1 > 12 > 123 > ... > > > Let B be the matrix we get by adding one element to each > > of the columns of A. This matrix does not have a diagonal. > > Every matrix has a diagonal. The question is only whether all lines and > columns are crossed by the diagonal. Not every matrix has a bijection > between lines and columns. OK. We say that every matrix has a diagonal, but only a matrix with the same number of lines and columns has a diagonal which crosses every line and every column. The matrix A has the same number of lines (omega each column has omega elements) and columns (omega the supremum of {n | n in N}). The matrix B does not have the same number of lines (omega+1 each column has omega+1 elements) and columns (omega the supremum of {n+1 | n in N}). > > > > Let C be the matrix we get by adding one element to each of > > the lines of A. This matrix has a diagonal. > > The matrix C has the same number of lines (omega each column has omega elements) and columns (omega the supremum of {n+1 | n in N}). > > Let D be the matrix we get by adding one element to > > each of the lines of B. This matrix does not have a diagonal. > The matrix D does not have the same number of lines (omega+1 each column has omega+1 elements) and columns (omega the supremum of {n+1 | n in N}). > You want to say that A has a diagonal, but addition of one element to > each line, each column and the diagonal eliminates this property? Yes, because adding one element to each column changes the number of lines, but adding one element to each line does not change the number of columns. > Of > course you must say so because otherwise you had to confess that set > theory is self contradictory. On the other hand you could argue with a > split diagonal. Just a bit counter intuitive, but a hard-boiled set > theorist will swallow that without feeling nausea. > > No, William, if a matrix has a diagonal crossing every line and every > column, and if one element is added to every line and every column and > to the diagonal itself, then this diagonal again crosses every line and > every column. Briefly: > > If D(A_ij) then D(A_i+1,j+1) > > And if the property D(A_i+1,j+1) is false, then the property D(A_ij) is > false. > > Just that is what I assert and what has only been rejected by you with > the argument a diagonal could have more element than every line. Here > you see that you are forced to postulate a square matrix changes into a > non-square matrix when lines and rows are extended symmetrically. > The original matrix has the same number of lines and columns, however,the lines and columns are different. Doing the same thing to the lines and columns does not have the same result. > > > > Only for matrixes B and D is there a last line. > > > > So if there is a diagonal, there is no last line. > > How about adding the same elements not at the ends but at the > beginnings of the lines and the columns? Adding an element to the beginning of an infinite sequence does nothing. So this is the same as adding an element to the end of each line, but not adding an element at the end of each column. > > In that case you claim the existence of a diagonal. No problem, nobody > can check it (in particular because any infinity is a void claim). But > if we find a way to circumvent this spiritual existence and to fasten > it by adding probes at the end, then the diagonal mysteriously > disappers? > > The diagonal is nothing but the means to biject lines to columns. If > there is no diagonal crossing all lines and columns, this simply means > that there is no bijection between lines and columns. And that is fact > after increasing each one by one element as well as before. > No. If you add one line, but you do not add one column you do not have a bijection. > > > > >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 > > > > No. If there is a diagonal there is no last line. > > A new theorem? > Can it be applied generally, or only in this special > case? Only in this special case. > If we may apply it generally, then let's consider Cantor's list. > Cantor's list has a diagonal. If we add a last line, then the diagonal > argument can no longer be applied? > > Then let's construct some Cantor list, take any line, for instance the > first, and add it to the end of the list as a last line. This you cannot do. There is no end of the list. You can add it after the list, but then you have a transfinite sequence not a list. Cantor's argument applies to lists. - William Hughes
From: William Hughes on 15 Nov 2006 07:59 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > 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. > > > > > > If the supremum is a maximum, it will be changed. If not, it will not > > > be changed. > > > > > > > > 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. > > > > > > This shows that they were not quite the same before. > > > > > > > > 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. > > > > > > Correct. And if we add one element to the diagonal, then its supremum > > > changes form omega to omega + 1 > > > > Correct, but to add one element to the diagonal we > > have to add one element to the lines and one > > element to the columns. > > Not necessarily. We add just one element to the diagonal which, > according to your assertion is existing ad has ordinal omega. Now it > has ordinal omega +1. > We can of course add one element to the diagonal sequence. However, after we do this it is no longer a diagonal of a matrix with one more line but the same number of columns. > > If we just add one element > > to the lines we no longer have a diagonal. > > Therefore I recommended to add one element to every line, to every > column, and to the diagonal. No we need to add a column. Adding one element to every line does not add a column. > > . > > > There is an element of every column with transfinite index. > > There is no element of a line with transfinite index. > > There is no diagonal. > > That means, there is no bijection between lines and columns. Correct > Therefore, > there was no such bijection before. Wrong. If there is a bijection f between A and B, and we add one elment to A to form A' and no elements to B to form B', we cannot extend f to a bijection between A' and B' - William Hughes
From: Franziska Neugebauer on 15 Nov 2006 08:11 William Hughes wrote: >> You want to say that A has a diagonal, but addition of one element to >> each line, each column and the diagonal eliminates this property? > > Yes, because adding one element to each column changes the number > of lines, but adding one element to each line does not > change the number of columns. What did I tell you!? [...] > The original matrix has the same number of lines and columns, > however,the lines and columns are different. Doing the same thing to > the lines and columns does not have the same result. You are *not* doing the same thing! F. N. -- xyz
From: Franziska Neugebauer on 15 Nov 2006 08:13
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > William Hughes schrieb: >> >> Let he original matrix be A. >> [...] >> > 1 >> > 12 >> > 123 >> > ... >> >> Something is missing here: >> >> 1uuu... >> 12uu... >> 123u... >> ... > > No. Wrong. You have been discussing matrices. At least William Hughes did. Are you both writing at cross purposes? > We discuss the question whether an infinite set of naturl numbers > requires an infinite number as an element. There is nothing to debate as this is consistently *not* required in current set theory. > The naural numbers are expressed by the digits of a line. Perhaps your "expression by the digits of a line" is unapt to properly represent the naturals and sets thereof? > We could also use > > 1 > 11 > 111 > ... If you in advance define what exactly this idea sketch stands for. > The "u" are not appropriate. The "u"s are necessary if you are writing about matrices. They denote the remaining positions which are not occupied. >> Usually a matrix m(a,b) is a function of domain A x B (it is >> rectangular not a triangle) and some co-domain. Here A equals B >> equals omega by definition. A matrix with domain A x B is called >> (generalized) square iff A = B. >> >> "u" stands for undefined (empty), not set. The "triangle" you see is >> not the structure of the matrix but its occupancy. > > We discuss just this triangle as our marix. Do you want to redefine "matrix"? Twee! >> Prima facie "Adding x to each column" ("at the end") is not >> "possible", since the columns have no end (no last element). > > If 1,2,3,... has the ordinal number omega and if it is possible to > construct 2,3,4,...,1 and if it is meaningful to denote the ordinal > number of 2,3,4,...,1 by omega + 1, then your argument fails. I don't know what this means. But I see that you have not commented on the remaining part which nicely explains where you are in error: ,----[ <455aeccb$0$97262$892e7fe2(a)authen.yellow.readfreenews.net> ] | 1uuu... | 12uu... | 123u... | ... | xxxx.... | | This operation changes the domain A into A + 1 (omgea + 1). The order | type of domain A has changed. Since the domain has changed this new | matrix has no longer the original structure. And since omega + 1 is | not equal to omega, this matrix is no longer a square one. | | It is not possible to "add x to each column" and to retain the domain. | | "Adding x to each row" means simply changing the occupancy of the | matrix without changing its domain: | | 1xuu... | 12xu... | 123x... | ... | | There is no change in domain "necessary" to "perform" this operation. | [...] |> No, William, if a matrix has a diagonal crossing every line and every |> column, and if one element is added to every line and every column |> and to the diagonal itself, then this diagonal again crosses every |> line and every column. Briefly: | | As explained above: "Adding x to each row" is entirely different from | "Adding x to each column". You are comparing apples and oranges. `---- F. N. -- xyz |