An uncountable countable set
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From: mueckenh on 10 Jul 2006 05:26 Virgil schrieb: > > No one has shown that any sequence of exchanges can convert a number > > with finitely many digits into a number with infinitely many digits. > > Relevance??? Cantor's diagonal, of course. If you replace a_n by b_n, this cannot be done and cannot be proved without having counted carefully from 1 to n. But it is impossible to count over all natural numbers. Therefore the prescription replace a_n by b_n does not cover all natural numbers, it is void. > > After each of his countably many exchanges Cantor has still a countably > > infinite subset of the list numbers left which are not yet exchanged. > > Cantor performs no exchanges at all, he merely establishes a rule by > which for an arbitrary list of reals one may construct a number not in > that list. > That is the same I do, you see? I merely established a rule by which for an arbitrary well-ordering of all rationals one (for instance you) may construct an order by size without destroying the initial map of indices, i.e. the initial well-order. In fact, you can calculate the complete order by size with the same pocket calculator as one can calculate the complete diagonal. > > Cantor only states a rule of construction. Those who dispute that the > rule succeeds must show that there exists some member of the list for > which it fails. So do I. It is really more comfortable to publish only the rule and let others do the calculation if they do not believe in the rule. So start up and don't forget: You can calculate the diagonal to any place n you want (if you have carefully counted the lines up to there, to be sure to find the line number n in the given list) and you can order by size any set of rationals. There is no rational inaccessible (in principle) as no digit of the diagonal numbers is inaccessible (in principle). > > > > On the other hand, Mueckenheim must show that his rearrangement can be > completed, even though it can't be, in order to double order the > rationals. Your arguing does not prove that there is any number in that list for which my rule does not apply, and the finding of such a number is required to invalidate me. > > If there could be any exchange in the list which did > > not face the initial problem all over again, there would have to be a > > first one, but that is impossible. > > What "initial" problem is that? Cantor's rule can apply to any member of > the list of reals independently of any other member, so applies to all > simultaneously. So there is no "initial" problem at all. > In order to apply the rule to a given list, you must count its lines. But how far you may count, the remaining part is infinite and does not decrease. > Mueckenheim's transpositions must be applied sequentially, so cannot be > applied simultaneously. They must not be applied at all. They are given. No one need to do any calculation if he is able to do abstract thinking. The order-by-size is established with the simple rule: Order all rationals by size which can be reached within the well-order. Or somewhat formalized: (1,2) (2,3) (1,2) .... Regards, WM
From: mueckenh on 10 Jul 2006 05:30 Virgil schrieb: > > > You have posited an infinite sequence of infinite sequences of > > > transpositions to perform your alleged miracle. As you can never even > > > finish the first subsequence in finite time, you must reorder your > > > transpositions into a single sequence to even consider its effect. > > > > I did so. > > When? Since reordering disrupts the effect of sequences of overlapping > transpositions (non-commutativity), you have not done so to anyone's > satisfaction but your own. (1,2) (2,3) (1,2) (3,4) (2,3) (1,2) .... > > > No I have only one infinite sequence of transpositions. They > > are as well defined and as fast to be executed as Cantors sequence of > > replacements of digits. > > > Cantor does not have a list. Only those who challenge him need to have > > > lists. What Cantor does is to refute each list presented to him... > > > > Cantor does have a list, it was the first one, constructed by himself: > > > > E1 = (a1,1, a1,2, . . .,a1,nu, . . .), > > E2 = (a2,1, a2,2, . . .,a2,nu, . . .), > > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > > E??? = (a???,1, a???,2, . . .,a???,???, . . .), > > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > > That is not a list so much as an N by N matrix over the decimal digits > into which any list can be fitted. It is an abstract list. You should learn to think in abstract terms. > > And your notation sucks. There were Greek letters, mu and nu, but completely irrelevant for recognizing the list as a list (if one is able to think in abstract notions - and of no further help, if one is not). Regards, WM
From: mueckenh on 10 Jul 2006 05:34 Virgil schrieb: > In article <1152450210.343596.317970(a)m79g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Cantor's fallacy is the claim the indexation of list numbers and > > diagonal digits carries over to the limit. > > What limit is that? n --> oo > > All Cantor assumes is that if two reals differ in some place in > restricted certain ways (to avoid the double representation problem of > certain reals) then they are not equal as reals. But this double representation problem shows that differing by digits is not sufficient a criterion to differ by size in the limit n --> oo. > > That does not require any "carry over to the limit" assumption. All I assume is that if two rationals differ in the well-order then they are not equal by size and can be ordered by size. Regards, WM
From: Franziska Neugebauer on 10 Jul 2006 05:39 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > They must not be applied at all. They are given. There is no defined *result* of your transpostions yet. Please consider the infinite sequence of the same transposition (1 2) (1 2), (1 2), ... acting upon the sequence (1, 2). The result is undefined though the sequence of transpositions is defined. > No one need to do any calculation if he is able to do abstract > thinking. You should prefer doing calculations. F. N. -- xyz
From: mueckenh on 10 Jul 2006 05:40
Virgil schrieb: > > According to ZFC there is an infinite set of natural numbers which can > > be exhausted. This set is the set of indices of my transpositions. > > But it cannot be "exhausted" by sequential operation. Then the set of the algebraic numbers cannot be exhausted, i.e., they cannot be well-ordered? Poor numbers. Now they have lost a high-rank position and degenerate to an amorphous, hence uncountable, mass. The countability proof of Dedekind estimates but does not biject. It is the same approach as with the paths in my binary tree. > > There may be "limit"definable, but that requires in some definition of > what the limit is to be like and how one is to measure at each finite > stage the "distance" between that finite stage and that limit. > > Since in Mueckenheim's process, one has, after any finite number of > transpositions, only a finite initial segment ordered by size, and an > infinite terminal segment not yet ordered by size, how is one supposed > to define such a limiting process? Give me a rational number of the well-order which cannot be put in the order by magnitude with its predecessors. If you can, then you are right. I you cannot, then you are wrong, as usual. Regards, WM |