An uncountable countable set
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From: mueckenh on 11 Jul 2006 10:39 Franziska Neugebauer schrieb: > Mathematics is not dealing with believes but with proofs. > Why then do you *believe* in Cantor's beliefs of his "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? If infinity actually exists as an exhaustible set, then my ordering reaches its limit (that is the ordering by magnitude where no elements remain unordered). If infinity does not actually exists as an exhaustible set, then all of Cantor's arguments fail. > > > 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: > What is yours? The transposition (m, n) with m < n acts upon the well-ordered set which is currently indexed by 1,2,3,... (q_1, q_2, ..., q_m, ..., q_n, ...) and transforms it to the well-ordered set (q_1, q_2, ..., q_n, ..., q_m, ...) if q_m > q_n. Otherwise nothing is exchanged. Afterwards the current enumeration is actualized to 1, 2, 3, ..., m, ...., n, ... A finite example: The set (7, 4, 6) is ordered by size by my standard sequence of transpositions (1, 2) (2, 3) (1, 2) .... as follows given is the set with current indices (7_1, 4_2, 6_3) apply (1, 2), get (4_1, 7_2, 6_3) apply (2, 3), get (4_1, 6_2, 7_3) apply (1, 2), get (4_1, 6_2, 7_3). ready. Regards, WM
From: mueckenh on 11 Jul 2006 10:42 Franziska Neugebauer schrieb: > > Diese Methode leidet keinen Stillstand. > > Quote from Cantor? abbreviated. The original quote reads: "und es erfährt daher der aus unsrer Regel resultierende Zuordnungsprozeß keinen Stillstand." (Collected works, p. 239) > > > 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. "und es erfährt daher der aus unsrer Regel resultierende Zuordnungsprozeß keinen Stillstand." This wrote Cantor in the same context. But the defined limit is clearly stated: It is the order by magnitude, because then nothing happens further. > > 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) > ... This is not my case. (See my last answer to you.) > > > 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. > To prove the existence and uniqueness of (|Q_+, <)? You believe that it is proven by the axioms. That should be sufficient for you. Regards, WM
From: Franziska Neugebauer on 11 Jul 2006 10:54 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > 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? > > If infinity actually exists as an exhaustible set, then my ordering > reaches its limit (that is the ordering by magnitude where no elements > remain unordered). If infinity does not actually exists as an > exhaustible set, then all of Cantor's arguments fail. Where is the mathematical statement? Contemporary mathematics is not about: - /the/ infinity - /actual/ existence - /exhaustibility/ - orderings which /reach/ "its limits" - "currently existing" infinity - Cantors historical writings >> > 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: [Why did you delete my definition?] >> What is yours? > > The transposition (m, n) with m < n acts upon the well-ordered set > which is currently indexed by 1,2,3,... > (q_1, q_2, ..., q_m, ..., q_n, ...) > and transforms it to the well-ordered set > (q_1, q_2, ..., q_n, ..., q_m, ...) > if q_m > q_n. > Otherwise nothing is exchanged. > > Afterwards the current enumeration is actualized to 1, 2, 3, ..., m, > ..., n, ... > > A finite example: The set (7, 4, 6) is ordered by size by my standard > sequence of transpositions The set is obviously ordered by the sequence (2, 3, 1). > (1, 2) > (2, 3) > (1, 2) > ... > > as follows > > given is the set with current indices (7_1, 4_2, 6_3) > apply (1, 2), get (4_1, 7_2, 6_3) > apply (2, 3), get (4_1, 6_2, 7_3) > apply (1, 2), get (4_1, 6_2, 7_3). > ready. Now you have to define a meaning of an infinite sequence of your conditional transpostions. F. N. -- xyz
From: Franziska Neugebauer on 11 Jul 2006 11:01 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > 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. > > "und es erf?hrt daher der aus unsrer Regel resultierende > Zuordnungsproze? keinen Stillstand." This wrote Cantor in the same > context. But the defined limit is clearly stated: It is the order by > magnitude, because then nothing happens further. I don't want to discuss Cantor quote further. >> 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) >> ... > This is not my case. (See my last answer to you.) My example clearly demonstrates, that the notion of an limit of transpositions is not (yet) defined. >> > 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. >> > To prove the existence and uniqueness of (|Q_+, <)? I want to know what the application of all of your conditional transpositions to the ordered set *means*. I ask for a definition of the *result*. And it would be nice if you show if it exists and is unique. This is not (yet) proven "by the axioms". > You believe that it is proven by the axioms. That should be sufficient > for you. F. N. -- xyz
From: Virgil on 11 Jul 2006 12:50
In article <1152603589.521769.91480(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The *only* criterion about countable infinite sets is whether one can > > > determine *precisely* at which natural number something happens: After > > > how many steps in a > > > well-order of |Q the fraction 4711/235537 will appear, for instance, or > > > > Given an arbitrary bijection, f: N --> Q, "mueckenh" claims to be able > > to tell us the value of n such that f(n) = 4711/235537, even though that > > will differ from one such bijection to another. > > The initial well-order is arbitrary, but fixed. Are you really unable > to understand that much? > Regrads, WM As there are uncountably many well-orderings of a countably infinite set, and none of them can be squeezed into a well-ordered dense ordering by any means whatsoever, "mueckenh" is still wrong. |