Cantor Confusion
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From: David Marcus on 8 Nov 2006 23:58 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > 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. > > > > > > > > By "it cannot be longer than every line", do you mean its length can't > > > > be greater than the sup of the lengths of the lines? > > > > > > Its length can't be the length of a column, i.e. omega, if the width of > > > the matrix has only the supremum omega. > > > > All I asked was what "[the diagonal] cannot be longer than every line" > > meant. Was what you wrote supposed to be an answer to this question? > > Please follow the discussion with those who understand. Perhaps you > will understand later on too. Thanks for the advice, but I'd rather you answered my question. -- David Marcus
From: Virgil on 9 Nov 2006 00:02 In article <1162985563.583178.91070(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162828907.384237.5300(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > 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. > > > > Which line(s) is the diagonal NOT longer than when line n is at least of > > length n? > > > > > > > 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. > > > > When for all n in N, line n is of length n, which is quite possible, > > then for all n in N, the diagonal is of length >= n. > > > > That looks like an infinite diagonal to me. > > Actually infinitely long and potentially infinitely broad. If it is diagonal, it had better be as long as it is broad and as broad as it is long. That WM can imagine a diagonal longer then broad or broader than long is just more evidence of how far from reality he is. > > > > WM is assuming that since every natural is finite, there must be a > > largest natural, but in ZF and NBG, that is specifically false. > > Intermingling of actual infinity and potential infinity kept these > systems alive, until now. Those systems will continue to stay alive until there is much better evidence against them that WM has produced. WM has failed entirely to make his case. But refuses to see that truth.
From: David Marcus on 9 Nov 2006 00:02 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > Modern mathematics does not use the term "finished infinity". If you > > > > wish to use it, please define it. > > > > > > Modern mathematics assumes that there are sets which are larger than > > > infinite sets. That is the same as to say after one finished infinity > > > we consider the next infinity. > > > > I asked you for a definition of "finished infinity". Is what you wrote > > supposed to be the definition? Normally, a definition should start out > > something like "Finished infinity is ... ". > > It is tedious to explain things to you. How would you know, since you never explain anything you say? > Please consult a book. Hint: > "Some mathematicians object to the Axiom of Infinity on the grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity." (Karel Hrbacek and > Thomas Jech: "Introduction to set theory" Marcel Dekker Inc., New York, > 1984, 2nd edition.) > > Now substitute "entity" by the most significant property of the set N. So, "finished infinity" isn't even in the book you reference. Kind of silly for you to refer us to a book to look up what a phrase means when the book doesn't even use it. -- David Marcus
From: David Marcus on 9 Nov 2006 00:04 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > For the diagonal (d_kk) of the list > > > > > > > > > > 0 > > > > > 1 2 > > > > > 3 4 5 > > > > > 6 7 8 9 > > > > > ... > > > > > > > > > > we have the following mappings: > > > > > d_kk --> d_mk and d_kk --> d_km with k, m eps N. > > > > > > > > Normally, the word "mapping" means function. If you are trying to define > > > > functions, it isn't clear to me what you mean since m appears on the > > > > right, but not the left. > > > > > > > > Is d_mk the kth element in the mth row? > > > > > > Yes. That is the usual notation. > > > > > > > > It is curious that the set of terms of (d_km) has omega as a maximum > > > > > for every fixed m eps N while the set of terms of (d_mk) has omega not > > > > > as a maximum for every fixed m eps N. > > > > > > > > For m in N, you appear to be considering two sets: > > > > > > > > A = {d_km | k in N}, > > > > B = {d_mk | k in N}. > > > > > > > > If we take m = 0, then > > > > > > > > A = {d_00, d_10, d_20, ...} = {0,1,3,6,...}, > > > > > > This set has omega elements. > > > > > > > B = {d_00, d_01, d_12, ...} = {0, undefined, undefined, ...}. > > > > > > B = {0} > > > > > > Take m = 2, for instance: > > > > > > Then B' = {3,4,5} > > > > > > > Regardless, no set of natural numbers has omega as a maximum. I've no > > > > idea why you think your list shows that omega is the maximum of a set of > > > > natural numbers. > > > > > > What do you think is the difference between the first column A and any > > > line B of the above matrix, concerning the number of elements? Is there > > > a difference? > > > > If we let B_j be the set of numbers in line j, and we start labeling the > > lines at zero, then > > > > |A| = aleph_0, > > |B_j| = j + 1, for j in N. > > > > So, for any natural number j, |B_j| < |A|. Now what? > > No try to set up a bijection beween the lines and columns which is > necessary to prove that a matrix is a square matrix. Virgil just did that. His bijection maps line n to column n. Looks like a fine bijection to me. Now what? What do you say the length of the diagonal is? What do you say the length of the first column is? -- David Marcus
From: David Marcus on 9 Nov 2006 00:09
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > It is not true that a set cannot exist unless it has a maximum element. > > It is true, at least if all of its elements do exist. The set of natural numbers does not have a maximum element, so what you just wrote is clearly false. Want to try again? > But in order to force you to understand that simple truth, Calling something a "simple truth" does not make it more believable, especially when it is so easy to give an example to show that it is false. > consider the bijection between lines and columns. So, you define a bijection that maps line n to column n. This is a bijection of N to itself. > All lines do exist, all columns do not exist. It is quite silly for you to say "all columns do not exist". Which column do you say doesn't exist? -- David Marcus |