
Re: Mathematics in brief
Posted:
Dec 9, 2012 4:41 AM


On Dec 9, 11:35 am, WM <mueck...@rz.fhaugsburg.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?
> Hence, there remains only the possibility to define the infinite path > by a finite definition. But there are only countably many. > Yes, correct.
> >I think using Cantor's argument > > one can prove that for any countably many paths of the Binary Tree > > there is a path that is not among them and thus establish a proof of > > uncountability of those paths. > > Then there is a contradiction. But you dismiss every contradiction? > Therefore my proof mus be invalid??? > What is that contradiction, nobody is assuming that all infinite paths are definable?
> 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 diagonalpath 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 or infinitely definable! all of what is necessary for Cantor's argument to be carried on is that we can identify the i_th node of the i_th path, neither paths nor the sequence of them needs to be finitely definable. This requirement is alien to Cantor's argument.
As > you know an infinite sequence cannot be defined by listing its terms. > You need a finite definition to define the sequence and its diagonal. > Again we reach the conclusion: There are only countably many finite > definitions. > Yes correct and it follows that there are only countably many "finitely definable reals", I agree, this is trivial actually, everyone knows that.
Cantor's argument is not about "finitely definable reals" or simply what is called "definable reals". Cantor's argument is about countability of all "reals" whether those reals are definable or not, there might be a real that requires an infinite definition and therefore not finitely definable, there might be a real that is not definable at all.
Reals are defined in a clear manner without any mentioning of whether they are definable or not, and Cantor's argument is about those entities.
You keep confusing the reals for "definable" reals.
Zuhair > > > > Math is discourse about "possible" form. Uncountability of the reals > > is PROVED in very weak fragments of ZFC, actually in PREDICATIVE > > fragments of second order arithemtic, which are PROVED to be > > consistent. This provide a discourse about form, thus uncountability > > of the reals is a possibility! thus it is mathematical, since reals > > can be interpreted as forms in the set hierarchy. > > See above. People have not taken into account that only a *defined* > sequence gives a defined diagonal. There are only countably many > finite definitions. > > Regards, WM

