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: