An uncountable countable set
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From: mueckenh on 9 Jul 2006 17:48 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > 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. > > What exactly does "larger than all naturals together" mean? Uncountable. Nonsense. > > >> 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. Regards, WM
From: mueckenh on 9 Jul 2006 17:50 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. Regards, WM
From: Franziska Neugebauer on 9 Jul 2006 18:18 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? F. N. -- xyz
From: Franziska Neugebauer on 9 Jul 2006 18:34 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> After each of your countably many transpositions, there is still a >> >> countably infinite subset of the rationals left which are not in >> >> natural order. That is a necesssary consequence of the entire set >> >> being originally well ordered. >> > >> > After each of his countably many exchanges there is still a >> > countably infinite subset of the list numbers left which are not >> > yet exchanged. > > Excuse me. There is not a single exchange carried out. All aere > simultaneously defined. What exactly is defined? And how? > More is not required. Have you ever tried to > well-order the set of rationals for more than, say, 20 elements. > Nobody does carry out such exercise. The well order is given by the > principle. The order by magnitude is given by my principle too. I don't see a principle but merely an empty phrase. >> 1. There is no "exchance". >> 2. d_i def= f (a_ii) is done A i e N. There are no "list numbers >> left". > > 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, Are you writing about an infinite sequence of transpositions? > and all is instantaneously ordered by magnitude. What do you want to transpose and oder by magnitude? F. N. -- xyz
From: Franziska Neugebauer on 9 Jul 2006 18:41
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > 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. >> >> What exactly does "larger than all naturals together" mean? > > Uncountable. Nonsense. Ever heard of the set of all denumerable ordinals? >> >> 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"? F. N. -- xyz |