On 13 Jan., 22:45, Virgil <vir...@ligriv.com> wrote: > In article > <bf5b8afa-bd4d-4f8e-9414-ff26ba2b7...@w8g2000yqm.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > There are not uncountably many finite (initial segments of) paths. And > > also any anti-diagonal can only differ from other paths in its (and > > their) finite initial segments. Unless your silly idea of nodes at > > level aleph_0 was correct (it is not) there is no chance to differ at > > other places than finite (initial segments of) paths. But that is > > impossible if all of them are already there. And the latter is > > possible, because they form a countable set. > > A set which WM cannot count!
Even you can count it!
0 1 2 3 4 5 6 ... > > The definition of a set being countable is that there is a surjection > from |N to that set. > > Thus in order to PROVE a set is countable one must show a surjection > from |N to that set, which is just a listing, possibly with repetitions, > of that sets members. > > But any listing of the paths of a Complete Infinite Binary Tree (as > infinite binary sequences) proves itself incomplete. > > Thus the set of paths cannot be made to fit the "countable" definition.