Date: Jan 13, 2013 5:44 PM
Author: Virgil
Subject: Re: Matheology � 191
In article

<0fa84faa-2103-4a2a-9c16-b6d498f23dd4@4g2000yqv.googlegroups.com>,

WM <mueckenh@rz.fh-augsburg.de> wrote:

> > 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

Since every path of a Complete Infinite Binary Tree must have infintely

many nodes (or, equivalently, infinitely many branchings), your list of

finite objects contains none of them

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