Date: Jan 13, 2013 4:45 PM
Subject: Re: Matheology � 191
WM <email@example.com> 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.
At least not outside of WMytheology!