An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: mueckenh on 10 Jul 2006 11:07 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > The set ordered by size and by natural indices is the limit of my > >> > transpositions. > >> > >> 1. What is a limit of transpositions? > >> 2. How do you prove its existence and uniqueness? > > > > How does Cantor prove the existence and uniqueness of the well-ordered > > set of rationals or > > Once again: What is a limit of transpositions? How do *you* (not Cantor) > prove its existence and uniqueness? Me, not Cantor? If you agree that Cantor's ideas are wrong, I need not prove anything else, because I prove just this. If you believe that Cantor's ideas are correct, then I can utilize them too. 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. The limit is given by the definition of the transpositions: If two elements appear in order by magnitude, they are not exchanged, if not, they are. Regards, WM
From: mueckenh on 10 Jul 2006 11:09 Franziska Neugebauer schrieb: > Are you writing about an infinite sequence of transpositions? As you were not present, when I explained it first, I repeat it: Start with a well-ordered set of all positive rationals. The initial indices remain with the rationals, but I use current indices m, n in the following definition. A transposition (m,n) means: If the elements with current number m and n are not in order by size, exchange these elements, if they are already ordered by size, do nothing. Now the following set of transpositions is applied: (1, 2) (2, 3) (1, 2) (3, 4) (2, 3) (1, 2) .... The set of transpositions is countable and has order type omega. It is a sequence. And every term of this sequence is well defined, i.e., if one wants to calculate how many transpositions it will take to have the first j elements of he well-ordered set ordered by size too, she can do that. Diese Methode leidet keinen Stillstand. One never gets ready, but that is due to the fact that the well-order has no last element. All elements which do exist in the well-order will be in the order by magnitude. > xyz What is the meaning of this magic sign? Is it to identify you articles easier by the find function? Or is it a spell? Regards, WM
From: mueckenh on 10 Jul 2006 11:11 Franziska Neugebauer schrieb: > >> >> Right! There are more than any finite number of finite naturals, > >> >> indeed, an endless supply of them which collectively form an > >> >> infinite set. > >> > > >> > There is no "supply". The elements of a set do exist, > >> > instantaneously and immediately. Sets are static. > >> > >> Sudden insight? > > > > Clever application of different standpoints. > > 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. Regards, WM
From: mueckenh on 10 Jul 2006 11:21 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > >> > Either K is in the list (which would contradict analysis in the > >> > same way as my original example), then there is an n with 10^(-n) = > >> > 0. Or K is not in the list. > >> > >> K is not in the list. > >> > >> > Then there must be a position which cannot be enumerated by natural > >> > numbers > >> > >> non sequitur. All positions are indexed by definition of decimal > >> representation. > > > > All positions are indexed <==> K is in the list. It is a logical > > equivalence for linear sets like my list. > > What kind of "logic" is employed here? That one which deserves its name, and which you unfortunately seem to be not familiar with. The logic of a linear sequence like this sequence of list numbers 0.1 0.11 0.111 .... If "all positions are indexed by list numbers" is not equivalent to "K is in the list", then you seem to imply that more than one list number is required to index the digits of K. I say: What can be indexed by different list numbers can also be indexed by one alone. If you do not believe that, 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. And note: It is unimportant whether there is a definition which says this or that. Regards, WM
From: mueckenh on 10 Jul 2006 11:25
Virgil schrieb: > That alleged ontradiction is only in the mind of the beholder who > insists that a property of finite trees must hold for infinite trees. If you are an admirer of the mysteries of the infinite, why then do you believe that the bijection n <--> 2n or the formula SUM{1 to n} 1/2^n = 1 - 2^(-n) remains unaltered in infinity? But if so, why do you believe that the element | o /\ of the binary tree is altered? There is no point at all to maintain or to drop only one of these three laws. No, it is only the wilful restriction of one's ratio in order to keep a beloved but useless toy. > > > Infinite sets do not behave like finite sets. It is "mueckenh"'s eyes > that are remaining "wide shut". > That would only work if there were only one path through any edge. > > There are, in, fact, infinitely many paths thru each edge. The first edge is mapped on the first path. If this splits in two, 1/2 of the first edge in addition to the new edge is mapped on the new paths. 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? > > Do you believe that the infinite paths exist there without consisting > > of edges? Do you believe that above diagram becomes invalid somewhere? > > No finite diagram represents the infinite case. But a finite brain can think of the infinite. But a finite formula can represent the infinite. What is the diagram else than the representation of a formula? Regards, WM |