Date: Jan 13, 2013 4:51 PM
Subject: Re: Matheology § 191
On 13 Jan., 22:45, Virgil <vir...@ligriv.com> wrote:
> In article
> 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!
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.
Above you see the enumeration of the set