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