An uncountable countable set
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From: mueckenh on 10 Jul 2006 11:27 Virgil schrieb: > > > Precisely where does the Cantor diagonal proof fail? > > > > In this and infinitely many infinite lists like this: > > 0.0 > > 0.1 > > 0.11 > > 0.111 > > ... > > > > In an infinitely long list the diagonal number 0.111... is not > > distingiushed from all the list numbers. > > If those numerals represent fractions then the thing represented by > 0.111... is also a fraction, but is not in the list. All positions which can be enumerated are in the list, by definition. Hence, the decimal representation of 1/9 does not exist. Regards, WM
From: mueckenh on 10 Jul 2006 11:30 Virgil schrieb: > In article <1152479719.404362.114580(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > Cantor's proof avoids this need to "reach the end" by giving a general > > > rule which works independently for each n in N without reference to any > > > other member of N. This allow the simultaneous creation of all the > > > digits. > > > > Cantor's proof allows to find the digit number n for every n. My proof > > allows to find the transposition number n for every n. > How can you tell what is going to happen after infinitely many > transpositions which have to be taken in a particular order to assure > that one knows what they are expected to do? If they would not do what they are expected to do they would not be the transpositions I employ. And all are enumerated. As long as the numbers are finite, the set of transpositions is finite. As there is no infinite natural number, the set remains finite. Regards, WM
From: mueckenh on 10 Jul 2006 11:32 Virgil schrieb: > > But in order to prove that the diagonal differs from every entry, there > > a limit is required but not available. > > Not by me. I can easily do it without any limit process at all. Because you are a blind believer. > > > The missing limit has not been remedied but has only been put aside. My > > objection remains: But that there is no satisfactory limit > > consideration becomes clear from the following: We know that 0.999... = > > 1.000... This leads to the result that a change of 1 in the limit where > > the digit number goes to oo does not have the effect which would be > > required in order to distinguish the diagonal number from the list > > numbers. > > Cantor's rules avoid that particular pitfall by never using either 0 or > 9 in diagonal constructions. Cantor did *not* consider this pitfall. But you would have believed him nevertheless. Regards, WM
From: Franziska Neugebauer on 10 Jul 2006 12:17 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > The set ordered by size and by natural indices is the limit of >> >> > my transpositions. >> >> >> >> 1. What is a limit of transpositions? >> >> 2. How do you prove its existence and uniqueness? >> > >> > How does Cantor prove the existence and uniqueness of the >> > well-ordered set of rationals or >> >> Once again: What is a limit of transpositions? How do *you* (not >> Cantor) prove its existence and uniqueness? > > Me, not Cantor? Sure. You are writing about a "limit of [your] transpositions". If you want to write meaningfully about it you should provide a proof. > If you agree that Cantor's ideas are wrong, I need not > prove anything else, because I prove just this. You may not assume that I agree or disagree with any of Cantor's ideas. Hence you have to prove any "new argument" regardless of whether it is Cantor's or yours. You may alternatively refer to *contemporary* mathematical literature . > If you believe that Cantor's ideas are correct, then I can utilize > them too. Mathematics is not dealing with believes but with proofs. > These ideas imply that a countable set is exhausted, if the > element to be mapped on n e |N is determined for every n. This is the > case for the set of my transpositions, which is countable and has > order type omega. What exactly do you want to posit? > The limit is given by the definition of the transpositions: If two > elements appear in order by magnitude, they are not exchanged, if not, > they are. My definition of a transposition is: Def: A *transposition* (m n) m, n e N is acting upon a sequence (b_i) i e N as follows: (a b) (..., b_m, ..., b_n, ...) = (..., b_n, ..., b_m, ...) (1) (2) (1) (2) (1) m - 1 Elements (2) n - m - 1 Elements What is yours? F. N. -- xyz
From: Franziska Neugebauer on 10 Jul 2006 13:10
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> Are you writing about an infinite sequence of transpositions? > > As you were not present, when I explained it first, I repeat it: > > Start with a well-ordered set of all positive rationals. (Q+, <*) > The initial indices remain with the rationals, but I use current > indices m, n in the following definition. > > A transposition (m,n) means: If the elements with current number m and > n are not in order by size, exchange these elements, if they are > already ordered by size, do nothing. This is a conditional transposition. > Now the following set of transpositions is applied: > > (1, 2) > (2, 3) > (1, 2) > (3, 4) > (2, 3) > (1, 2) > ... You mean an infinite sequence of conditional permutations (p_i) := ((1 2), (1 2)(2 3), (1 2)(2 3)(3 4), ...) i e N+ Please also memorize this sequence of (unconditional) transpositions: (q_i) := ((1 2), (1 2), (1 2), ...) i e N+ > The set of transpositions is countable and has order type omega. Sequence. A set is not ordered per se. > It is a sequence. As constructed. > And every term of this sequence is well defined, As explicitly constructed. > i.e., if > one wants to calculate how many transpositions it will take to have > the first j elements of he well-ordered set ordered by size too, she > can do that. The first j elements are then ordered by < and no longer by <*. The remaining infinitely many elements are still ordered by <*. > Diese Methode leidet keinen Stillstand. Quote from Cantor? > One never gets ready, but that is due to the fact that the well-order > has no last element. You need a defined limit in order to write about applying *all* of the conditional permutations. Please recall (q_i) now. As one easily sees for every j e N the permutation q_j q_j-1 ... q_1 can be applied to any sequence containing at least 2 members. Nonetheless the application of all q_i i e N is not defined: (1, 2) = (1, 2) q_1 (1, 2) = (2, 1) q_2 q_1 (1, 2) = (1, 2) q_3 q_2 q_2 (1, 2) = (2, 1) .... > All elements which do exist in the well-order will be in the > order by magnitude. Not yet. First you have to define the meaning of applying all conditional permutations and then prove the existence and uniqueness of the result. >> xyz > > What is the meaning of this magic sign? Is it to identify you articles > easier by the find function? Or is it a spell? xyz is the percentage of your posts containing objectionable reasoning. F. N. -- xyz |