Cantor Confusion
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From: mueckenh on 16 Nov 2006 04:13 Virgil schrieb: > WM cannot have it both ways. Either he must accept everything Cantor > says if he is to use Cantor as authority, or nothing Cantor says, unless > otherwise supported, if he does not use Cantor as authority. That may be true for someone who cannot judge. I pick the correct ideas and drop the wrong ones. Regards, WM
From: mueckenh on 16 Nov 2006 04:36 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 to every row (= every colun and every line), then the triangle has at least two corners. 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. > > The following matrix is unsuitable to express natural numbers in unary > > representation. > > Untenable assertion. The following matrix has omega elements in every line. In my approach, the lines contain unary representations of natural numbers. No natural number has omega elements. 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. I discuss natural numbers. 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. If this is possible, and if the ordinal of the first column is omega, then adding one element to every initial segment of a column and to every line maintains the bijection. The fact that it is not maintained proves that your asserted bijection does ot exist. > > [...] > > > 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. > > 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. 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. My arguing is only a little trick to show the impossibility to have a bijection in this EIT, which proves that the Gequatsche of an actually infinite set of finite natural numbers is inappropriate. Regards, WM
From: Virgil on 16 Nov 2006 04:43 In article <1163668423.690913.280250(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > WM cannot have it both ways. Either he must accept everything Cantor > > says if he is to use Cantor as authority, or nothing Cantor says, unless > > otherwise supported, if he does not use Cantor as authority. > > > That may be true for someone who cannot judge. I pick the correct ideas > and drop the wrong ones. Then you must present what you consider correct as being correct in your own judgment, not citing those others from whom you pick and choose as authoritative only when you concur. To do otherwise is deliberately misleading.
From: mueckenh on 16 Nov 2006 04:44 > > 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 If there is no infinite number then there are not infinitely many numbers. And, by definition, there is no infinite number. It is very simple and independent of any religion (as far as I am concerned.) Regards, WM
From: mueckenh on 16 Nov 2006 04:48
Dik T. Winter schrieb: > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > In > > > principle no axiom is necessary. But you need a few to have some start > > > to work with. > > > > That's the question. By means of axioms you can produce conditional > > truth at most. I am interested in absolute truth. Axioms will not help > > us to find it. I don't think we need any axioms. > > If you want to find absolute truth you should not look at mathematics. > -- Not at that what today is called mathematics, I agree. I + I = II is very fine and reliable mathematics. Absolutely true. And this approach can be put forward --- very far. Regards, WM |