An uncountable countable set
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From: mueckenh on 9 Jul 2006 08:58 Virgil schrieb: > > No one has shown that any sequence of transpositions, nor any sequence > of "finite" permutations (which leave all but finitely many elements > fixed), can convert a well-ordering into a non-well-ordering. No one has shown that any sequence of exchanges can convert a number with finitely many digits into a number with infinitely many digits. After each of his countably many exchanges Cantor has still a countably infinite subset of the list numbers left which are not yet exchanged. > While a transposition can increase by one the *finite* initial segment > of the list of rationals which are also ordered by size, it can never > decrease the infiniteness of the terminal segment which is not so > ordered. While an exchange of the diagonal digit can increase by one the *finite* initial segment of the list which have been treated already, it can never decrease the infiniteness of the terminal segment which has not yet been treated. > So that each new transposition faces exactly the same problem as its > predecessor, an infinite terminal list of rationals not in natural order. So that each new exchange faces exactly the same problem as its predecessor, an infinite list of nubers not yet exchanged. > If there could be any permutation in the list of permutations which did > not face the initial problem all over again, there would have to be a > first one, but that is impossible. 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. Thanks, very good arguments! Regards, WM
From: mueckenh on 9 Jul 2006 09:02 Virgil schrieb: > > That does not make their sizes infinite. It is completely irrelevant > > here. > > What IS irrelevant is the notion that in order to have infinitely many > naturals (more than any finite number of them) one must somehow have one > of them being infinitely large. Then consider a set without an infinitely large element: {1,2,3,...,n}. It is finite. Remember: Sets are fixed entities. They do not grow. > > 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. 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, . . .), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eï? = (aï?,1, aï?,2, . . .,aï?,ï?®, . . .), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regards, WM
From: mueckenh on 9 Jul 2006 09:03 David Hartley schrieb: > In message <vmhjr2-E08407.11521207072006(a)news.usenetmonster.com>, Virgil > <vmhjr2(a)comcast.net> writes > >In article <1152282125.757193.28320(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> There is no limit other than in Canrtor's diagonal. EVERY set initially > >> indexed by natural numbers will remain indexed by natural numbers. And > >> it will unavoidably become ordered by size too. > > > >At best you have proved that a countable subset of the rationals have > >ben restored to their usual ordering, but you have nowhere shown that > >your process ever re-orders ALL of them. > > > >After each of your countably many transpositions, there is still a > >countably infinite subset of the rationals left which are not in natural > >order. That is a necesssary consequence of the entire set being > >originally well ordered. > > Suppose x, y are any two rationals. Suppose they are indexed by m and n > in the first enumeration, and let k = max(m,n). From the k-th stage on, > x and y are in their natural order. Under any reasonable definition of > the limit of a sequence of orderings, here it will be the natural > ordering. > > (WM's fallacy is his claim that the (order-preserving) indexation by the > naturals carries over to the limit.) Cantor's fallacy is the claim the indexation of list numbers and diagonal digits carries over to the limit. Regards, WM
From: mueckenh on 9 Jul 2006 09:04 Virgil schrieb: > > What axiom is not obeyed by my transpositions? > > What system of axioms are you claiming your transpositions are operating > in? The same as Cantor's diagonal proof is operating in. > > And it may transpire that it is some theorem deduced from the axioms of > that system as a whole that you are violating, which theorem requires a > complex of axioms, not merely a single axiom. > > But as far s I can see, in any appropriate axiom system, say ZF or ZFC > or even NBG, you merely show that you can natually order some finite > initial segment of your list, not the entire list. 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. Regrads, WM
From: mueckenh on 9 Jul 2006 09:05
Virgil schrieb: > > > But it reaches past every finite value!!! > > > > There is always a finite value past every finite value. So we are and > > remain sufficiently save within the domain of finit values. > > So one never reaches an end. That is what "infinite" for a sequence like > the naturals means, "without end". Correct. And that is valid for Cantor's list too. Regards, WM |