Cantor Confusion
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From: Virgil on 8 Nov 2006 18:04 In article <1162981705.150031.164160(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > 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? > > If the maximum of the set of enumerated lines does exist as aleph_0 That would require a line numbered omega, which does not exist if each line number is merely one larger than its predecessor. If there were a line numbered omega the there would have to be a line with a finite number x such that x + 1 = omega, as that is how the lines are numbered. And until WM can tell us the actual value of that x, we will contintue to believe that he is wrong, as the logic tells us. > > Does "actual infinity does not exist" mean the same as when you say you > > disagree that "there exists the set of lines"? > > It means the same as to say "finished infinity is nonsense" for he set > of lines as well as for the set of numbers in a line. Then WM is arguing against himself.
From: Virgil on 8 Nov 2006 18:06 In article <1162981932.768984.319700(a)k70g2000cwa.googlegroups.com>, 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. For each n in N there is a line n, for each n in N there is a column n, so line(n) <--> column(n) does it nicely.
From: Virgil on 8 Nov 2006 18:13 In article <1162982061.323607.278430(a)m73g2000cwd.googlegroups.com>, 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. If it takes a value from the nth column in the nth row, then it will have a value for every column n in N. If every row is finite but at least of length equal to its row number, then WM is dead wrong. > > 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. Every part of the discussion is crystal clear except for WM's maunderings which seem more and more to be deliberately obfuscating. If every line is finite, say, for example, the nth line is of length 2*n, and the diagonal is not finite , then the diagonal MUST be longer that every line.
From: Virgil on 8 Nov 2006 18:17 In article <1162982318.830699.287740(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > In my tree argument, the edges are not physically divided. The > following edge > 1 > / > 0 > remains the connection between this 1 and this 0. It is only the > "ownership" which is shared equally by all the paths going through this > edge. As infinitely many paths pass through each edge no path can "own" more than an infinitesimal share of that edge according to that "equal sharing" rule. And unless we are in something like Robinson's non-standard analysis, less than infinitesimal means zero.
From: Virgil on 8 Nov 2006 23:22
In article <1162983169.303386.11260(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > > Here is my translation: It is allowed to understand the new number > > > > > omega as limit to which the (natural) numbers n grow, if by that we > > > > > understand nothing else than: omega shall be the first whole number > > > > > which follows upon all numbers n, i.e., which is to be called larger > > > > > than each of the numbers n. > > > > > > > > That's not a definition. It is just a remark. > > > > > > It is a definition. You should not try to judge about topics which you > > > must reject because you do not understand them. Please learn: A > > > definition is an explanation in other words. Nothing more can be done. > > > > That's a wrong definition of the word "definition", at least in > > Mathematics. > > As you believe? As mathematicians believe. A definition, at least in the mathematical sense, is merely a form of abbreviation. It allows one to say more briefly what could be said without it but at greater length. > > > > Would you understand: "The new number omega is the limit to which the > > > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > > > because you don't know the meanings of "number", "limit", and "grow"? Which natural number(s) does WM suggest grows? If Wm means that the set of natural numbers capped by some natural n is larger for larger values of n and is only limited by the set of all naturals, he is saying it very badly. > > > > Sure, I understand it as a remark. But, it still isn't a definition or > > theorem unless the other words are defined. > > The problem is that words cannot be defined unless other words are > known. As it seems you don't know words like "number", "limit", and > "grow". Therefore it is impossible to define something so that you can > understand it. > > Please note: In mathematics definitions are not necessary at all. All > expressions can be given using some primitives. (Therefore your belief > expressed above is wrong.) But without those primitive words no > discurse is possible. What are the primitives of WM's version of set theory? |