In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
Not within any common axiom system. And WM's axioms, those things he claims but does not and cannot prove, are not truths in, for example, ZFC, in which the axioms are at least explicit. > > > > (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
While that may be an incomplete listing of nodes, it is certainly not a listing of paths or of initial segments of paths, other than possible the root node as a one element path.