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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 10:49 AM
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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
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