Date: Dec 26, 2012 10:49 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Distinguishability of paths of the Infinite Binary tree???
On 26 Dez., 13:24, gus gassmann <g...@nospam.com> wrote:

> On 26/12/2012 7:29 AM, Zuhair wrote:

>

>

>

> >> Depends on the level of distinguishability at issue.

>

> >> For any finite set of such strings, finite initial segments suffice to

> >> distinguish all of them from each oterhbut for at least some infinite

> >> set, no finite set of finite initial segments suffices.

>

> > Yes but a countable set of them suffices! no?

>

> Of course. And how many such countable sets are there? Cantor showed

> that there are uncountably many.

Cantor showed that by digits or nodes.

And I showed that they cannot be distinguished by nodes.

>

> There are at least two counter-intuitive notions when dealing with

> infinities: There is an infinite set, each of whose elements are finite

> (viz. the sequence of initial segments {{1}, {1,2}, {1,2,3}, ...}; and

> the set of all countable subsets of a countable set is uncountable. The

> only thing this shows is that intuition is sometimes insufficient to

> grasp complex things.

No, it shows that there is no nonsense great enough for matheologians

not to believe in (and to call their thinking "complex" and a simple

and clear contradiction "intuition"). No set of finite subsets of |N

exists, that was uncountable. Only a subset containing uncountably

many infinite subsets is uncountable. However, it is impossible to

define infinite subsets by themselves. You need always a finite

definition. The set of all finite definitions however is countable. No

intuition requred.

Cantor's and Hessenberg's "proofs" simply show that infinity is never

finished and a complete infinite set is not part of sober thinking.

You could be excused perhaps if the CIBT was the only contradiction of

your belief. But there is a lot more, for instance this one:

http://planetmath.org/?op=getobj&from=objects&id=12607

Regards, WM