An uncountable countable set
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From: Virgil on 12 Jul 2006 13:29 In article <1152707786.981875.231380(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > When Cantor's proof was published, there was not such an axiom. > > > > Indeed, at that time it was not yet an axiom, but the thought was present > > that infinite sets did exist. > > There was not such an assumption in general. Infinite sets did not > exist (as they do not exist yet) That they do not exist in "mueckenh"'s mind does not bar them from existing in other minds. > > > > > 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. > > > > The first of these two axioms would lead to an inconsistent system. The > > axiom of infinity does not. It leads only to an inconsistency with *your* > > view. > > It leads to the inconsistency that you must demand 0.111... *- (0.1 + > 0.11 + 0.111 + ...) = 0.000... but 0.111... not being in the sum. I do not have to demand that any arithmetic involves things which are not numbers. If "mueckenh" feels compelled to do so, that is his personal problem, and does not affect anyone else.. > It > leads to a binary tree having infinitely many path mapped upon the very > same edge though no path can split without two new edges. And you must > refuse to calculate with fractions in mappings. That is inconsistency > enough. I do not choose to fractionate the indivisible. In a tree, there is no such thing as a fraction of an edge, since anything less than a link between two nodes has no meaning in a tree. Those who insist on unscrewing the inscrutable, will themselves be screwed in the end.
From: Virgil on 12 Jul 2006 13:50 In article <1152708585.848976.307310(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > The limit is the stable state where no further transpositions are > > > applied, because there are no two elements remaining, which were not > > > ordered by magnitude. > > > > That makes no sense as a definition, as you will never reach that stable > > state. > > That is not my problem but the problem of non-existing infinite sets. It is "mueckenh"'s problem when he claims what he cannot deliver, as he is doing here. > > Is it for rationals *without repetition*? How does it read? > > > What do you mean with a sentence like "in which line of Cantor's list a > > certain > > digit of the diagonal number will be placed"? > > How can you find out whether a certain real of Cantors list has the > number 5782712398413208741? It is never Cantor's list, it is always one provided to him for testing. Your question is somewhat ambiguous. Do you mean how can you tell if a number in the list given to Cantor is the 5782712398413208741th number in that list? If that is what you mean, all you have to do is see which n in N it corresponds to and compare that n with 5782712398413208741. > > The mapping from the naturals to the list gives the n-th element. It is > > in the *definition* of the list. > > Do you want to define a proof or to give a proof? Proofs are not "defined". > > > > > > A mapping that well-orders > > > > the rationals is *not* a sequential process. The determination of the > > > > n-th digit of the diagonal from a given list is *not* a sequential > > > > process > > > > > > Wrong. You cannot know line number n without knowing line number n-1. > > That depends entirely on how the list is given. If a formula is given allowing calculation of f(n) directly from n, one can do each f(n) individually. For example, f(n) = 1/sqrt(n+1) defines a list of reals in which it is not necessary to access f(n-1) in order to access f(n). > > Therefore you resist to count lines. Why do unneccessary work? But in case of proof (not > definition of proof) you are forced to count all of them. Not at all. One no more needs to do that than to prove (x+1)^2 = x^2 + 2*x + 1 by proving it separately for each and every value that x can take. There are general proofs that cover all members of a set simultaneously without any for a case by case approach. Those who cannot realize this are severely handicapped. > Why don't you apply this knowledge in case of Canor's list? In the > limit n --> oo it may well be that 4 cannot be distinguished from 5. Try getting stronger glasses. > No, he did not. But when extending his proof to functions, he applied > numbers 0 and 1, which obviously can also serve as m and w in the > orignal version. No they can't, since for some numbers either will serve.
From: Virgil on 12 Jul 2006 14:02 In article <1152708937.861956.22190(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > 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". > > Many things cannot be proven by ZFC+FOPL. ZFC is not all, as even > set-theorists increasingly recognize. > > (|Q_+, <) is the result, because if there was any q_i not yet ordered > by magnitude then it would become ordered by magnitude by my definition > (not by yours). Then your ordering "by magnitude" does not agree with the standard ordering by magnitude of the rationals. In the standard rationals, every rational has a (non-unique) representation as j/n, where j is a signed integer and n is a positive natural. Then the relation between j1/n1 and j2/n2 is "<" or "=" or ">" if and only if the relation between n2*j1 and n1*j2 is "<" or "=" or ">" respectively. Unless "mueckenh" can prove that his well ordering of the rationals satisfies this order rule, he has to made his case.
From: Virgil on 12 Jul 2006 14:10 In article <1152709074.647932.291630(a)35g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > What "mueckenh" claims is no more possible that to have a natural number > > which is simultaneously even and odd. > > Or to have an infinite set of finite numbers. There is an infinite set of finite rationals between every two rationals. > > > > Since it is clear that NO transposition can achieve this, "mueckenh"'s > > alleged final result never results. > > Just that is the essence of my proof. It shows that infinity can never > by reached and does not exist. It does not show that to me, nor to anyone else who chooses to work in ZF or ZFC or NBG. What it does show is "mueckenh"'s mental limitations.
From: Virgil on 12 Jul 2006 14:20
In article <1152718356.159898.303420(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > What he hasn't proved is his claim that, since each ordering in the > > sequence is order-isomorphic to the naturals, so is the limit. When > > challenged on this, he refuses to acknowledge the problem, just claiming > > that it is "clearly" true. Unless he attempts to prove this there is > > little point in arguing with him; you can't point out the errors in a > > non-existent proof. If you do continue, at least attack the part of his > > argument which fails, not the part that he has - just about - got right. > > I really can't see your problem. Every rational q_n_0 has an index n_0 > due to the initial well-ordering. This initial index will never get > lost (contrary to the current indices) because this initial index is > all that identifies this rational. If a rational numbers's value does not also identify it , then comparison of rationals by size is no longer possible, and so rearranging them into order by size is impossible. Normally, a transposition, or any other permutation, is regarded as the re-indexing of a set of objects being transposed, or permuted, after which the original indexing is regarded as being no longer relevant. "Mueckenh" seems to want to carry both the value and original index of real rational along as he transposes positions. Does he also want to carry along each of the indices that a rational is ever labeled with in this endless process? |