An uncountable countable set
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From: Virgil on 11 Jul 2006 13:06 In article <1152603911.430496.144400(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > You said Cantor "proves" the existence of a well-ordering of the > rationals by proving the existence of a bijection from N to them. > That should be sufficient for me and my result too. "Mueckenh" claims to be able to arrange the rationals in such a way that the same order relation on them is simultaneously dense and a well ordering. > > > > Have you ever tried to well-order the set of rationals for more > > > than, say, 20 elements. > > > > One can do it for all of Q by establishing a rule for what rational > > follows any given rational. This works simultaneoulsy for all of Q, > > and has often been done. > > And I establish a rule which determines what well-order follows any > given well-order. Density under an order relation and well-ordering under an order relation are mutually exclusive. If you have either in a particular order relation, you cannot have the other in that same order relation. What "mueckenh" claims is no more possible that to have a natural number which is simultaneously even and odd. Why should anyone accept that "mueckenh" can produce the impossible when he so often has failed to produce the possible? > > > > > You are right. I responded only to Virgil's very naive belief > > > that there something really should be carried out. Of course it > > > is sufficient to give the principle by a short expression like > > > this > > > > > > (1,2) > > > (2,3) > > > (1,2) ... > > > > > > to intelligent people, and all is instantaneously ordered by > > > magnitude. > > > > The effect of transpositions (1 2)(2 3)(1 2) applied successively > > to the first 3 elements of any list of 3 or more is simply to > > reverse the order of those first 3. > > Transpositions are carried out only if the elements are not yet in > order by magnitude. In that case you have no more that a bubble sort of some finite initial segment. If "mueckenh"'s scheme were ever to be able to succeed, there would have to be some transposition after which your transposed "sequence" no longer would have a first element, as the rationals in natural order do not have a first element. Since if is clear that NO transposition can achieve this, "mueckenh"'s alleged final result never results.
From: Virgil on 11 Jul 2006 13:15 In article <1152604493.362026.129000(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1152450704.938408.69690(a)h48g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Virgil schrieb: > > > > What we have said, and what is quite true, is that for every natural > > > > there is a larger natural. > > > > > > Of course. And therefore it is impossible to exhaust all of them or to > > > find a set which is larger than all naturals together. > > > > But nowhere an attempt is made to exhaust them. There is only an axiom > > from which it can be derived that there is a set that contains them all. > > With that axiom, Cantor's argument is a proof. Without that axiom, > > Cantor's argument is meaningless. > > When Cantor's proof was published, there was not such an axiom. There was the equivalent, at least for Cantor, a general assumption that there one could speak of something being true "for all naturals", just as today we would agree that for ->all<- naturals the sum of an odd natural with another odd natural results in an even natural. > > > You are not arguing against Cantor's > > argument, you are arguing against that axiom. > > If an axiom states the existence of 100 natural numbers below 20, it > has to be abolished in order to save mathematics. The axiom of > infinity, interpreted as you do, is such an axiom. But it was built to > model Cantor's worldview. WE prefer to save mathematics from its self-promoting "savior" and retain the axiom of infinity, at least in ZF, ZFC and NBG.
From: mueckenh on 11 Jul 2006 13:23 Virgil schrieb: > > > It can only be proved > > after having counted the digits from 1 to n without leaving out a > > single one. > > Nonsense. The Cantor rule generates a digit to go in the nth decimal > place of the number being created without any reference to any other > decimal place. > You cannot identify any place without counting from 1 to that place. Regards, WM
From: mueckenh on 11 Jul 2006 13:24 Franziska Neugebauer schrieb: > >> So the elements of omega do exist "statically"? > > > > Neither nor. But if one claims their existence, I reserve the right to > > let them exist at my convenience. > > This is acceptable when doing proofs by contradiction or when you argue > within a special context. If you claim "The elements of a set do exist" > and "Elements of omega do neither exist nor exist statically" in the > *same* context and simultaneously then you violate the law of > noncontradiction. In ZFC the elements of a set do exist and omega does exist too. I did not *claim* that omega does not exist, but I *proved* that as a fact. Of course this is a contradiction. Regards, WM
From: mueckenh on 11 Jul 2006 13:28
Franziska Neugebauer schrieb: > Every "list number" (= sequence member) is a sequence of digits indexed > by every natural number. Not as an unary representation. The list number 0.1111 is indexed by the list number 4 = 0.1111 but not by the list number 3 = 0.111, because 3 cannot index the fourth 1 of 0.1111. > > > I say: What can be indexed by different list numbers can also be > > indexed by one alone. > > Meaning? Proof? The first 1 of 0.1111 can be indexed by 0.1, the second 1 of 0.1111 can be indexed by 2 = 0.11 etc. But all 1's of 0.1111 can be indexed by 4 = 0.1111 simultaneously. > > > If you do not believe that, > > I do not know what you mean. > > > then you should be able to show an example where more > > than one list number is required. Of course it must be a finite > > example, because there are only finite list numbers. And note: all > > list numbers are unary representations of natural numbers. > > But not all unary sequences are representations of natural numbers. > 0,111... is a sequence which does not represent a natural. And it cannot be completely indexed by natural numbers. Regards, WM |