Virgil
Posts:
8,833
Registered:
1/6/11


Re: UNCOUNTABILITY
Posted:
Dec 21, 2012 3:51 PM


In article <6cfff32fe65f468aae5ce71babb776d6@hf3g2000vbb.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 21 Dez., 17:36, Zuhair <zaljo...@gmail.com> wrote: > > On Dec 21, 2:04 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > > And they are not definable with parameters either. > > > > Do you mean that the bijection between the set of all naturals and the > > set of all finite words is NOT definable at all, whether with or > > without parameters? > > No, this bijection has obviously been defined. What is no definable is > a set with uncountably many elements, or, better: What is not > definable are uncountably many elements. (The set is definable, yet > not exististing free of contradictions.)
I ask again: what is WM's definition of countability of a set?
The usual definition, as used by Canter an everyone since, except possibly WM, is that there must be either a surjection from N to the set or an injection from the set to N in order to establish countability, and the provable inability to create either mapping makes a set uncountable.
WM seems to be using some different definition, but will not say what it is. > > Note finally: Every Cantor diagonal r differs from any other real > number by a finite initial segment n(r) of its string of digits. That > is not possible with the Binary Tree. A diagonal does not differ from > all finite paths
Every infinite path differs from every finite path at all but finitely many nodes. But even in WM's incomplete infinite binary tree there are absolutely no finite paths. every path has infinitely many nodes in it.
So any reference to "finite paths" is at best misleading, and more likely deliberately false.
> i.e., for every initial segment n(r) of every real > number r there exists a finite path of the Binary Tree that is n(r).
And uncountably many infinite sequnces that begin with n(r), as many as sequences as in the entire tree, as an easy bijection proves.
> You may consider actual infinity as well as uncountable languages, but > that does not change the fact that Cantor's argument does not apply.
It still applies quite nicely everywhere except possibly in WMythology. 

