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


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 counterintuitive 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

