An uncountable countable set
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From: mueckenh on 11 Jul 2006 13:42 Virgil schrieb: > > The first edge is mapped on the first path. > > Which of the infinitely many paths through that edge is the "first" one? > That at the root. > > > > If this splits in two, 1/2 of the first edge in addition to the new > > edge is mapped on the new paths. > > You are not allowed to split edges. Besides which, the first edge has > already been entirely used up. It is inherited by the childs of the first path. > > > If the new paths split, 1/4 of the first edge and 1/2 of the > > second and all of the third are mapped on each path. Do you know what a > > geometric series is? What do you object to this mapping? > > If one can split edges into infinitely many peices one can equally split > paths. it is only unsplit edges and unsplit paths that are to be matched > up in injections, surjection or bijections.. You dislike fractions in calculations? Several thousand years ago humans discovered that fraction can be useful. Why should they not be useful in set theory? Regards, WM
From: Virgil on 11 Jul 2006 13:44 In article <1152605227.458575.197240(a)35g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > And that is all you have to object against my proof? We have a fixed > > > scheme of transpositions. > > > > It is my objection to what you see as the similarity. > > But it is not justified. If a fixed rule is given which determines for > each rational when it will be included in the set ordered by magnitude > (and if this is determined for every rational), then the order by > magnitude is established. Remember: You believe in the existence of a > well order, although the smallest rational larger than 1/2 and the > largest rational smaller than 1/2 and so on do not appear. In any well ordering of all rationals, all appear, but in positions unrelated to their magnitude as rationals. "Mueckenh" claims to be able to well-order the rationals in such a way that there is no first rational in that ordering. That is no more possible than to have a natural number that is simultaneously even and odd. "Mueckenh" claims to do this from an arbitrary well ordering of the rationals using a sequence of "conditional" transpositions each of which transposes a successive pair of values if and only if they are not in their natural order. Since each is conditional on what has gone before, they must be executed in sequence to have the claimed effect. But such endless sequences never end. If it WERE possible to finish the process and have the rationals well ordered by magnitude. "mueckenh" should be able to produce the final result, a well ordering by magnitude, without the bother of any intermediate stages. Well, can yuh, punk?
From: Virgil on 11 Jul 2006 13:55 In article <1152628740.878598.96360(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Mathematics is not dealing with believes but with proofs. > > > Why then do you *believe* in Cantor's beliefs of his "proofs"? One doesn't. One believes in one's own reconstructions of a proof, if one can reconstruct it, not in the original author's faith. > > > > These ideas imply that a countable set is exhausted, if the > > > element to be mapped on n e |N is determined for every n. This is the > > > case for the set of my transpositions, which is countable and has > > > order type omega. > > > > What exactly do you want to posit? > > If infinity actually exists as an exhaustible set, then my ordering > reaches its limit (that is the ordering by magnitude where no elements > remain unordered). There are two ways to approach the "exhausting" of an infinite set. The one-member-at-a-time method always fails. The all-at-one-go method sometimes succeeds. "Mueckenh" is arguing a on-at-a-time approach. Cantor successfully argues an all-at-one-go approach in his second proof of the uncountability of the reals > If infinity does not actually exists as an > exhaustible set, then all of Cantor's arguments fail. That depends on which approach one uses. The all-at-one-go method can work whenever one can prove things for all of the given type of object. One can do that for naturals, e.g., for all naturals, odd plus odd is even.
From: Franziska Neugebauer on 11 Jul 2006 14:06 mueckenh(a)rz.fh-augsburg.de wrote: > 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. You have not yet been told what a proof is *in* the context of ZFC, haven't you? > Of course this is a contradiction. It *would* be a contradiction if you *had* proved that omega does not exist. But you have not (yet) proved that omega does not exist in the context of ZFC. Therefor there is not (yet) a contradiction. F. N. -- xyz
From: Virgil on 11 Jul 2006 14:08
In article <1152628953.131295.202010(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > You need a defined limit in order to write about applying *all* of the > > conditional permutations. > > "und es erf?hrt daher der aus unsrer Regel resultierende > Zuordnungsproze? keinen Stillstand." This wrote Cantor in the same > context. But the defined limit is clearly stated: It is the order by > magnitude, because then nothing happens further. But it is an unachieved limit, as limits tend to be. > > > > Please recall (q i) now. As one easily sees for every j e N the > > permutation q j q j-1 ... q 1 can be applied to any sequence > > containing at least 2 members. Nonetheless the application of all q i > > i e N is not defined: > > > > > All elements which do exist in the well-order will be in the > > > order by magnitude. > > > > Not yet. First you have to define the meaning of applying all > > conditional permutations and then prove the existence and uniqueness of > > the result. > > > To prove the existence and uniqueness of (|Q +, <)? NO! To prove the existence of an end to a process which does not end. After each conditional transposition on has a finite initial segment of rationals in numerical order followed by an infinite sequence of rationals in which there are infinitely many successive pairs in reverse order of size. This condition does not change for any finite number of transpositions which are applied. Thus if there is a "limit" it must be one with only a finite initial sequence ordered by magnitude and infinitely many size inversions. Otherwise "muecken" must claim that a finite number of transpositions suffices, which is nonsense. |