In article <firstname.lastname@example.org>, WM <email@example.com> 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.
The decisive point is that only the existence of a complete listing, for which there could not be any anti-diagonal, could prove the set of decimals, or binaries, or whatever, to be countable So an anti-diagoal, proving ANY such list is necessarilyan incomplete listing destroys any claim of the countability of any such a set of sequences.
By definition, a set is countable ONLY IF there is a complete listing of its members. Cantors antidiagonal argument shows that for listings of ,say, binary sequences, no such complete listings of all of them can exist.
Thus, according to that STANDARD definition of countability, the set of ALL infinite binary sequences is NOT a countable set. --