In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 29 Okt., 01:56, William Hughes <wpihug...@gmail.com> wrote: > > On Oct 28, 5:31 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > If so, you have shown that b_oo does not exist, because it cannot be > > > distinguished from all the other elements whereas every other element > > > can be distinguished from all others. > > > > Trivially: forall n in N [ b_oo =/= b_n ] > > It is not meant how you distinguish b_oo from every b_n, but from all.
For all b_n there is a successor, b_(n+1), but for b_oo, there is no successor. > > > > So you must mean something else by "[b_oo] cannot be distinguished > > from all the other elements". > > Remember the Binary Trees. How do you distinguish the complete > infinite Binary Tree containing all infinite paths and the incomplete > infinite Binary Tree containing all paths except the infinite paths.
The former can be shown to exist and have uncountably many paths, one for each subset of |N, at least in ZFC and the like, but the latter cannot exist at all , since if it contains any finite path as a path it cannot contain any extension of that path, since paths are necessarily maximal, thus cannot contain ALL finite paths, and thus cannot exist at all. --