```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 sayingYes, but it would not be correct to call it a paradox (i.e., somethingcontrary to intuition like the relativistic twin paradox) but anantinomy, because both results contradicting each othe can be obtainedformally.>> (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 ofthe paths of the Binary Tree:01, 23, 4, 5, 67, ...Regards, WM
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