Cantor Confusion
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From: mueckenh on 6 Nov 2006 10:50 MoeBlee schrieb: > David Marcus wrote: > > Obviously, I could be wrong, but I think WM means map it on a line of > > the list. He seems to think that because we construct the diagonal from > > the list, the diagonal must be one of the lines in the list. Why he > > thinks this, I have no clue. > > You mean map the diagonal (or the range of the diagonal, or whatever) > onto one of the finite sequences that is in the range of the infinite > sequence of those finite sequences? I.e., map the diagonal onto a > member of the range of S? A 1-1 map? If so, yes, I would share your > bafflement as to why we should think there is such a mapping or what > contradiction there is in there not being such a mapping. There is a mapping of the diagonal on a (each) column. There is no mapping of the diagonal on any line. The diagonal cannot have more elements than the width of the matrix is. The number of elements of the diagonal is assumed to be omega. That is wrong, because only the supremum is omega. Regards, WM Regards, WM
From: mueckenh on 6 Nov 2006 11:01 Virgil schrieb: > In article <1162563567.735020.246810(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > WM merely repeated his automatic error several more times here. > > > > > > WM claims that a list in which the nth listed element is a string of > > > length at least n characters cannot produce a diagonal of length > > > greater that any finite number of characters. > > > > > > > The diagonal needs an element from every line, the n-th element from > > the n-th line. Therefore it cannot be longer than every line. > > The "diagonal"must be longer than every finite "line". But it cant, because each of its elements stems from a line. > The only way it > will ever fail to be longer than some "line" is if that line is itself > infiitely long. Which is impossible, because each of its elements stems from a line. Regards, WM
From: mueckenh on 6 Nov 2006 11:08 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > No. I believe that the length of the diagonal is the supremum of > > > the length of the lines. This supremum does not have to be the > > > length of any line. Call it omega. Then omega is the *supremum* > > > of the set of lines (equally the supremum of the set of columns). > > > > The set of lines has infinitely many elements, no line has an infinite > > number of columns. That is a difference. > > So? Two different sets can have the same supremum. > > > > > If you consider omega to be not the maximum but only the supremum of > > the set of lines, then we agree that actual infinty does not exist. > > No, the supremum exists and the supremum is actual infinity. The number of elements of a column is omega. The number of elements of the diagonal is omega. Both are not suprema but maxima. The width of the matrix has not the maximum omega. > > > But > > we both then disagree with ZF which states *there exits* the set of > > lines. They all are there. And there does not exist a set of infinitely > > many columns (= an infinite natural number) > > No, there does exist a set of infinitely many columns. However, this > set is not a natural number. It is the supremum of a set of natural > numbers. A supremum of a set A does not have to be a member > of A. But it can be. The set of lines has a maximum, namely omega. Or does the complete set not exist at all? > So while this set is an infinite number it is not a natural > number. Why is the set of columns smaller than the set of lines? Or isn't it? Regards, WM
From: William Hughes on 6 Nov 2006 11:39 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > No. I believe that the length of the diagonal is the supremum of > > > > the length of the lines. This supremum does not have to be the > > > > length of any line. Call it omega. Then omega is the *supremum* > > > > of the set of lines (equally the supremum of the set of columns). > > > > > > The set of lines has infinitely many elements, no line has an infinite > > > number of columns. That is a difference. > > > > So? Two different sets can have the same supremum. > > > > > > > > If you consider omega to be not the maximum but only the supremum of > > > the set of lines, then we agree that actual infinty does not exist. > > > > No, the supremum exists and the supremum is actual infinity. > > The number of elements of a column is omega. The number of elements of > the diagonal is omega. Both are not suprema but maxima. In each case you are talking about a single element, so the supremum is equal to the maximum. Recall. If a set has a single element then this element is both the maximum and supremum. If a set has a maximum element, then this element is also the supremum and the supremum is an element of the set. If a set does not have a maximum element, then the supremum (if it exists) is not an element of the set. >The width of the matrix has not the maximum omega. The width of the matrix is the supremum of W, the widths of the lines of the matrix. Since there is no longest line, there is no maximum element of W. Therefore the supremum is not an element of W. The fact that every element of W is finite and the supremum is not finite is not a contradiction because the supremum does not have to have the same properties as the elements of W. > > > > > But > > > we both then disagree with ZF which states *there exits* the set of > > > lines. They all are there. And there does not exist a set of infinitely > > > many columns (= an infinite natural number) > > > > No, there does exist a set of infinitely many columns. However, this > > set is not a natural number. It is the supremum of a set of natural > > numbers. A supremum of a set A does not have to be a member > > of A. > > But it can be. The set of lines has a maximum, namely omega. Or does > the complete set not exist at all? The set of lines does not have a maximum. There is no line that is as long or longer than every other line. The complete set exists, however it does not have a maximum. > > > So while this set is an infinite number it is not a natural > > number. > > Why is the set of columns smaller than the set of lines? Or isn't it? > The set of lengths of columns, C, consists of the single element omega. The set of widths of lines, W, is the set of all natural numbers. W has more elements than C, but every element of W is smaller than every element of C. The supremum of W is omega. The width of the matrix is the supremum of the set of widths of the lines, i.e. omega. Thus the width of the matrix is equal to the height of the matrix. - William Hughes
From: David Marcus on 6 Nov 2006 13:22
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > No. I believe that the length of the diagonal is the supremum of > > > > the length of the lines. This supremum does not have to be the > > > > length of any line. Call it omega. Then omega is the *supremum* > > > > of the set of lines (equally the supremum of the set of columns). > > > > > > The set of lines has infinitely many elements, no line has an infinite > > > number of columns. That is a difference. > > > > Correct (in ZFC). > > Correct in any theory which claims "an infinite set of finite natural > numbers". > > > > > If you consider omega to be not the maximum but only the supremum of > > > the set of lines, then we agree that actual infinty does not exist. > > > > Omega is the sup of the set of *lengths* of lines. > > > > > But we both then disagree with ZF which states *there exits* the set of > > > lines. They all are there. > > > > Why would William disagree that there is a set of lines? I'm sure he > > doesn't. > > Please read carefully: Thanks, but I have been reading carefully. You are the one who has been sloppy. > If he considers omega to be not the maximum but only the supremum of > the set of lines, then we agree that actual infinty does not exist. Omega is the least ordinal greater than every natural number. In that sense, omega is the sup of the set of line numbers. How does this imply that the set of line numbers does not exist? Really, it is kind of a silly thing for you to say since how can you take the sup of something that doesn't exist? Does "actual infinity does not exist" mean the same as when you say you disagree that "there exists the set of lines"? -- David Marcus |