In article <email@example.com>, WM <firstname.lastname@example.org> 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!
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.