Date: Jan 13, 2013 4:51 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 191
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.

Above you see the enumeration of the set

0.

0.0

0.1

0.00

0.01

0.10

0.11

...

Regards, WM