Cantor Confusion
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From: mueckenh on 16 Nov 2006 05:06 William Hughes schrieb: > > You say so. But if so, then there must be a bijection such that a > > column is mapped on a line for every column and every line. All finite > > values are bijected without problem n <--> n. But there remains a taken > > supremum to be bijected with a not taken supremum. That is impossible. > > No. The taken supremum has nothing to do with the bijection. Make it easier for you to understand it by considering the initial segments of the first column. The initial segment 1 2 3 .... n stand in bijection with the line 1,2,3,...,n by the diagonal element d_nn. If we add 1 element to every initial segment, then we obtain from the complete column (which has not an omegath element but allegedly does exists having omega elements) a segment of order type omega + 1. And it there was a bijection in the original version, then there will be a bijection in the extended version. > > > > > > > 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}). > > > > of {n | n in N}). > > > > > 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}). > > > > If there was a bijection between columns and lines in A, then there is > > a bijection of columns and lines in B. > > No B has the same number of columns as A. B has one > more line than A. You cannot extend the bijection from A > to a bijection for B. I said: "If there was a bijection between columns and lines". That the extension is impossible shows that there was no bijection in A. > > Two different sets can have the same number of elements. > > As before let the set of lines be S, and the set of columns be U. > Then the elements of S are different than the elements of U > .However, the number of elements of S is the same as the > number of elements of U. Look above. The initial segments of the first column are exactly the same as the lines, except that the former are noted vertically. > Yes but the first column is the union of all of the initial > segments. No line is. Every initial segment is the union of all preceding segments. Every line is the union of all preceding lines. If there is a union of all initial segments but no union of all lines, then this union of all initial segments has no partner in the bijection > The set of lines cannot have an order type, it is not a sequence. The set of lines has the order type omega. It is the set of natural numbers. Regards, WM
From: mueckenh on 16 Nov 2006 05:16 Dik T. Winter schrieb: > > Take the set of natural numbers in form of a list or matrix: > > > > 1 > > 11 > > 111 > > ... > > > > This matrix has length omega and width omega. And its diagonal has > > length omega. No line has length omega. Therefore the width is larger > > than any line. And the diagonal is longer than any line. This is > > impossible. > > No, that is very possible. If you assert that there is no line longer than the diagonal, you have good reasons, which can be proved. If you assert that the diagonal can be longer than any line, then you have no reasons, because the diagonal consists of line elements and cannot be where no line is. So your second assertion is outside of logic and outside of any mathematics. Therefore I am not willing to discuss this topic further. The correct result: There must be at least one line which is exactly as long as the diagonal. There must be an infinite natural number. That is impossible. Therefore, there is no actually infinite number of natural numbers, Regards, WM
From: mueckenh on 16 Nov 2006 05:23 Dik T. Winter schrieb: > In article <1163604980.169629.197680(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > But you do not allow elliptic or hyperbolic > > > geometry? If not, why not? > > > > I do not forbid it. It is clear that already the simple geometry on a > > sphere does not yield two parallels. Euclid simply did not consider > > such kind of plane. In another system we have other axioms (or better > > fundamental truths). > > Well, the same in set theory. In one form it has AC as a fundamental > truth, in another version it is not a fundamental truth. What is the > essential difference between the cases? > That there is not an "underlying plane" which it is related to. > > > > > > I do not ask what you think. Reread my question. What is true about a > > > set of numbers in nature or reality? > > > > There are so many truths. Take order, 1 < 11, commutativity of addition > > and multiplication, n + m = m + n. These things do not become invalid > > or to be proved only because matrix multiplication or quaternions were > > invented. > > Again, no answer. If you cannot understand this answer, then we should stop here. > What is true about a set of numbers in nature or reality? > > > I think there is a great difference. It is not necessary to call > > negative solutions "false" solutions as even Descartes did, (because it > > was customary at his time. Although this custom was justified as long > > as only positive numbers were called numbers.) But it is necessary to > > distinguish between negative and positive numbers or real and complex > > numbers or Euclidean and non Euclidean spaces. > > And you were vehemently objecting at calling the irrational numbers numbers. The reason is that these "numbers" have no decimal representation, hence cannot be used to prove that they are uncountably many. > What is the essential difference between all those cases? I once asked > you for a definition of "number", and you did never supply a proper > definition. Now you say yourself that the definition has been changed in > the course of time. I may note that there is no trichonomy between the > complex numbers, it is not possible to consistently order them. You may call these entities numbers or not. My opposition stems from their use in Cantor's list and the lacking trichotomy. Regards, WM
From: Franziska Neugebauer on 16 Nov 2006 05:40 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > I propose to use "infinite triangle" in order to be clear >> > and to show that your commentary below fails to show anything. >> > Instead of "square" we should speak of "equilateral". So we have an >> > Equilateral Infinite Triangle: EIT. >> >> Misnomer. Your triangle lacks two vertices. If ever call it >> "monangle" or just "angle". > > After adding one element .... you have a different entity. > to every row (= every colun and every line), Equivocation as explained elsewhere. > then the triangle has at least two corners. After having proved a contradiction in ZFC you will be named "Fields Medal Winner". By now you are not. > And if there exist infinitely many natual numbers, i.e., if there was > a bijection between columns and lines, then it has even three corners. Non sequitur. >> > The following matrix is unsuitable to express natural numbers in >> > unary representation. >> >> Untenable assertion. > > The following matrix has omega elements in every line. By definition. > In my approach, the lines contain unary representations of natural > numbers. What the lines contain (occupancy) does not effect the number of sequence menbers. > No natural number has omega elements. I did not claim that. > Now clear enoug? > >> >> | 1xuu... >> >> | 12xu... >> >> | 123x... >> >> | ... > >> Equivocation: "Adding one element" names two different things >> (changing the occupancy vs. changing the domain). If you chose the >> matrix-view consequently you would have recognized your error. > > Your matrix cannot represent natural numbers. Proof? > I discuss natural numbers. Not mathematically and not set theoretically. > The question is whether there is a bijection between the > initial segments of the first column and the lines like > > 1 > 2 > 3 > ... > n > and 1,2,3,...n. ISOTFC := { {1}, {1, 2}, {1, 2, 3}, ... } L := { 1, 2, 3, ... } B := { <{1}, 1>, <{1, 2}, 2>, <{1, 2, 3}, 3>, ... } The bijection between the initial segments of the first column (ISOTFC) and the lines (L) is explicitly contructed. I have named it "B". I cannot see any reason to debate about the existence of B. > If this is possible, and if the ordinal of the first column is omega, What does "possible" mean? Sets are or are not. There is no set theoretical notion of "possibility of sets". > then adding one element to every initial segment of a column and to > every line maintains the bijection. Adding one element (named "x") to every initial segment gives us: ISOFTC' := { {1, x}, {1, 2, x}, {1, 2, 3, x}, ... } I have no clue what precisely you mean by adding one element to every line. Do you mean the set L' := { 1, 2, 3, ..., x } ? > The fact that it is not maintained proves that your asserted bijection > does ot exist. There exists a bijection between ISOTFC' and L' : B' := { <{1, x}, x>, <{1, 2, x}, 1>, <{1, 2, 3, x}, 2>, ... } proving |ISOTFC'| = |L '|. >> [...] >> >> As explained: In the matrix-view there is no change of ordinals at >> all when you "add one to each line". You simply change the occupancy >> of a sequence member which was not occupied before. > > I know. Just this fact is used to show that there is no set of > ordinality omega. Never shown now proven. >> Since there is no "last line" you cannot "add one to each column" >> without extending the matrix structure (domain from omega to omega + >> 1). > > I know. But it is asserted that there is a set of ordinality omega. "But" is inappropriate here. > If you increase this set by 1 element (which is possible - otherwise > not set of ordinality omega + 1 could exist) then you get omega + 1. omega + 1 = omega + 1. What is your point? > My arguing is only a little trick to show the impossibility Sets (including bijections) are neither possible nor impossible. They exist or do not exist. > to have a bijection in this EIT In what? F. N. -- xyz
From: Franziska Neugebauer on 16 Nov 2006 05:57
mueckenh(a)rz.fh-augsburg.de wrote: >> > If there are infinitely many finite numbers, >> >> Does your religious confession require you to eulogize "infinity"? Or >> do you simply want to render a disclaimer against "infinity"? >> >> > then there is a bijection between lines and columns and then line n >> > has exactly the same properties as the initial segment of the first >> > column: 1,2,3,...,n <--> 1,2,3,...,n. >> >> Whichs properties are you writing about? > > The natural numbers count themselves. Bijection of initial segments of > column and lines > > 1 > 2 > 3 > ... > n <--> 1,2,3,...n A n e omega. > If there is no infinite number then there are not infinitely many > numbers. 1. If *where* is no infinite number? 2. Define "infinitely many". > And, by definition, there is no infinite number. By definition there is no infinite *natural* number. F. N. -- xyz |