Cantor Confusion
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From: mueckenh on 6 Nov 2006 10:11 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: 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. > > > And there does not exist a set of infinitely > > many columns (= an infinite natural number). > > What do you mean a "set of infinitely many columns"? We can consider the > set of column numbers. Is that what you mean? The number of columns is finite for every line considered. The numbers of lines is infinite for every column considered. Regards, WM
From: Sebastian Holzmann on 6 Nov 2006 10:16 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> Still, how do you *define* 1/omega? You just state 1/omega = 0? Oh, well. > > I state omega > 2 which implies1/omega < 2. Then I take 3 instead of 2. > And so on. To me, in the field of rational numbers, "1/n" is the element, that, when multiplied by n, yields 1 (the neutral element of multiplication in the field). This is a valid definition, since it allows to identify this element (there can be only one with that property). So, "1/omega" should be something, that, multiplied by omega, yields 1. What is "multiplied" here? Why is there such a thing in your theory? Can it be described in some other way? (The last question may be optional.)
From: mueckenh on 6 Nov 2006 10:21 David Marcus schrieb: > > > > "Whole number" is Cantor's name for his creation. > > > > > > > > My question is : Do you maintain omega > n for all n e N? > > > > > > Before I can answer the question, I need to know what you mean by the > > > words/terms. So, please define "omega", "N", and ">". Also, by > > > "maintain" do you mean that ZFC proves it? > > > > Do you keep on thinking that omega > n for all N, applying all > > words/terms as Kunen would understand them. > > Using Kunen's definitions, then ZFC proves that if n is a natural > number, then n < omega. Do you disagree that ZFC proves this? No. Therefore omega is larger than any finite natural number. Every natural number is finite. omega is not finite. Omega is the number of digits of the diagonal of 1 12 123 .... The number of digits of the first and any other column is omega. The number of digits of any line is less than omega. The length of the matrix is omega, as a maximum taken. The width of the matrix is omega, as a supremum not taken. This small problem contradicts ZFC. Regards, WM
From: mueckenh on 6 Nov 2006 10:30 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? Regards, WM
From: mueckenh on 6 Nov 2006 10:42
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. There is a difference between maximum and supremum which cannot be bridged. Regards, WM |