Date: Dec 27, 2012 7:08 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Distinguishability of paths of the Infinite Binary tree???

On 26 Dez., 20:14, Zuhair <zaljo...@gmail.com> wrote:
> On Dec 26, 6:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>

> > Cantor's and Hessenberg's "proofs" simply show that infinity is never
> > finished and a complete infinite set is not part of sober thinking.

>
> > Regards, WM
>
> To make the discussion fruitful, lets take all possibilities available
> and see what is the response to each.
>
> (1) To say that the formal proof of Cantor is clear and exact in
> formal terms, but the distinguishability argument is clear on
> intuitive level but has not been verified in formal terms, so
> accordingly we have the option of saying that Infinity do not copy
> intuitions derived from the finite world, and deem the result as just
> counter-intuitive but not paradoxical. I think this is the standard
> approach.
>
> (2) To say that the distinguishability argument is so clear and to
> accept it as a proved result despite the possibilities of verifying it
> at formal level or not, and also maintaining that Cantor's proof is
> very clear and valid, and so we deduce that we have a genuine paradox
> that resulted from assuming having completed infinity, and thus we
> must reject having completed infinity. That's what WM is saying


Yes, but it would not be correct to call it a paradox (i.e., something
contrary to intuition like the relativistic twin paradox) but an
antinomy, because both results contradicting each othe can be obtained
formally.
>
> (3) To consider countability of the finite initial segments FALSE,
> i.e. to say that we have uncountably many finite initial segments of
> reals and as well we have uncountably many reals. This clearly
> preserves congruity of the argument, but it requires justification,
> and the justification can be based on the principle of "parameter free
> definability of sets", since the alleged bijection between the finite
> initial segments of the reals and the set N of all naturals is NOT
> parameter free definable, then this bijection does not exist, and it
> is false to say that it is. This claim only accepts infinite sets to
> exist if there is a parameter free formula after which membership of
> those sets is determined, so if there is non then it doesn't accept
> the existence sets that are not parameter free definable.


Here is a parameter free enumeration of all finite initial segments of
the paths of the Binary Tree:

0
1, 2
3, 4, 5, 6
7, ...

Regards, WM