In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> Am Donnerstag, 5. Dezember 2013 20:35:08 UTC+1 schrieb Zeit Geist: > > On Thursday, December 5, 2013 12:59:55 AM UTC-7, WM wrote: > > > > > Am Donnerstag, 5. Dezember 2013 08:49:37 UTC+1 schrieb fom: > > > > > > > > > > > > The property of differing or not depedns on the d_n only. Note that > > > > > Cantor's diagonal argument only uses the d_n. > > > > > > > > > > > No. It also uses the fact that the listing may be > > > > > > arbitrarily given, > > > > > > > > > > Why say no? Of course the listing may be arbitrarily given. Nevertheless > > > the decisive point is that d_n =/= a_nn. > > > > > > > > > > And it forgets two crucial aspects: 1) A list could contain all rational > > > numbers. 2) No d_n is out of all rational numbers. > > > > > > > > > > > > > But, d might Not be rational, and if your given List contains only > > Rationals then the anti-diagonal is not. > > If you insist that the irrational antidiagonal differs from every entry of > the list, you must use a digit d_oo, because up to every d_n it does not > differ. It differs from d)1 at position 1 and d_2 at position 2, from each d_n at position n. I was not aware that it had to differ from any entry in more than one place to be differnt from it.
> If you don't say so, you must agree that an irrational number cannot be > defined by its digits but only by a formula supplying every digit.
That is irrelevant to Cantor's diagonal argument.
At least outside of WM's wild weird world of WMytheology, the DEFINITION of a set being countable is that there must exist a surjection from |N to that set, in other words a list of all that set's members. so unless WM his a different definition of countability that he uses only in his wild weird world of WMytheology, so either all reals can be put into a list or the set of reals is uncountable. Cantor proved that not all reals can be put into a list.
And WM hs not proved tha all eals CAN be listed. And until he has, the set of reals will remain NOT countable. --