An uncountable countable set
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From: Virgil on 9 Jul 2006 14:41 In article <1152449931.980150.257340(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Relevance??? > 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. > > > 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. 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 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. But that does not prove that there is any number in that list for which Cantor's rule does not apply, and the finding of such a number is required to invalidate Cantor. On the other hand, "mueckenh" must show that his rearrangement can be completed, even though it can't be, in order to double order the rationals. > > > 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. 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. "Muecken's" transpositions must be applied sequentially, so cannot be applied simultaneously.
From: Virgil on 9 Jul 2006 14:53 In article <1152450138.710643.68660(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. 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. > 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. Except that Cantor's comparisons can all be done simultaneously whereas your transpositions must be done sequentially, so it takes infinitely longer to do all infinitely many of yours. And you have shown no evidence that even then you have completed your alleged rearrangement. > > > 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. And your notation sucks.
From: Virgil on 9 Jul 2006 14:59 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? 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. That does not require any "carry over to the limit" assumption.
From: Virgil on 9 Jul 2006 15:13 In article <1152450294.687475.39310(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But it cannot be "exhausted" by sequential operation. 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 "mueckenh"'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? What is one's measure of closeness to natural ordering? What are one's "delta"s and "epsilon"s ? Absent these, "muecken" has no case.
From: Virgil on 9 Jul 2006 15:15
In article <1152450354.585870.85210(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Except that with Cantor's method, one need not proceed through the list sequencially, but can deal with them all simultaneously, which with "mueckenh"'s transpositions is impossible. |