```Date: Dec 9, 2012 12:40 PM
Author: Zaljohar@gmail.com
Subject: Re: Mathematics in brief

On Dec 9, 1:14 pm, WM <mueck...@rz.fh-augsburg.de> wrote:> On 9 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote:>>>>>>>>>> > On Dec 9, 11:35 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 9 Dez., 05:55, Zuhair <zaljo...@gmail.com> wrote:>> > > > You need to prove that the set of all paths is countable, and so far> > > > you didn't present a proof of that.>> > > The set of all finite paths is countable. Therefore it is not possible> > > to define an infinite path by adding nodes to any finite path. All> > > nodes to be added are already in finite paths. Therefore, by following> > > the nodes of a path, you never define an infinite path. It is> > > interesting that practically everybody not yet brainwashed can> > > understand that.>> > Every node is reachable by a finite path, that's correct. But that is> > irrelevant> > here, we are speaking here about the number of all "path"s in the> > Binary Tree> > and not about the number of all nodes. we know that the number of all> > nodes> > is countable, the question is: is the number of all paths (finite and> > infinite)> > is countable?>> So it has become obvious now, that is not possible to define "all> paths" by nodes. Only the finite paths can be defined by their nodes.> How can you define all paths if not by nodes?>>Simply there are non finitely definable paths.Anyhow what is the proof that ALL reals can be represented by paths ofan infinite Binary Tree (actually two trees). It looks that only acountable subset of reals can be represented in that way. I'm not surereally.Anyway the diagonal argument of Cantor is provable in very weaksystems of ZFC which are proved to be consistent. So uncountability ofreals is a possibility.Of course also Countability of reals is a possibility! since we canhave countable models of ZFC or any theory that can define all thereals.>> > > Hence, there remains only the possibility to define the infinite path> > > by a finite definition. But there are only countably many.>> > Yes, correct.>> > What is that contradiction, nobody is assuming that all infinite paths> > are definable?>> The contradiction is that an undefinable path cannot be put into any> order or well-order.>>>> > > But there is not even a contradiction. More precisely: There would be> > > a contradiction, if there wer uncountably many diagonal possible. But> > > in fact, no infinite sequence of paths defines a diagonal-path unless> > > the sequence has a finite definition such that every path is known.>> > No that is not necessary really, the infinite sequence of paths can be> > indefinable>> Again: A real number is a quantity and must be in trichotomy with all> other reals. An undefinable number can neither be in trichotomy nor> can it be pit into a well-ordering. It simply is not a number. Butr> Cantor's ideas are too nice to be given up when faced with reality of> real numbers.>> It is a shame that someone defends the concept of "undefinable> number", unthinkable thought, anusable use, - and nevertheless claims> to be a logican and mathematician!>This includes the main bulk of experts on foundations of mathematics.Actually I see this statement of yours really unsubstantiated. Theconcept of uncountability is not unthinkable thought, nor it isunusable use there is no such a thing, it might have applications,this is for the future to tell, not for the wimps of yours to dictate.Zuhair
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