
Re: Mathematics in brief
Posted:
Dec 9, 2012 2:37 PM


On Dec 9, 9:45 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > On 9 Dez., 18:40, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > > > > > On Dec 9, 1:14 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > On 9 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote: > > > > > 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? > > > > 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. > > No they are not anywhere. Your assertion is simpky false. I construct > one path through each node such that every node has its own path. (It > is irrelevant, which and how many other nodes belong to that path.) By > this construction every node is covered by its own path. And there is > no chance to define any further path by further nodes. There are no > further nodes available. > > > > > Anyhow what is the proof that ALL reals can be represented by paths of > > an infinite Binary Tree (actually two trees). It looks that only a > > countable subset of reals can be represented in that way. I'm not sure > > really. > > Only a countable subset can be represented by the Binary Tree. The > reason is that no path is really actually infinite. > > Then you are not addressing what Cantor was speaking about, he is speaking about reals represented by ACTUALLY infinite sequences (paths in your case). It is clear that the set of all reals represented by FINITE sequences is countable, but those are just a very small subset of the set of all reals.
If one assumes Actual infinity, then it is easy to recover the diagonal path from any bijection between the reals and the set of all paths of the infinite binary tree, and this will be a path that is not present in the tree of course. You will need uncountably many infinite binary trees to recover all the reals. And again you simply failed to demonstrate a clear contradiction with Cantor's argument.
What you are not getting is that uncountability of the reals is a PROVED issue, it is proved in very weak fragments of second order arithmetic that are PROVED to be consistent. I don't know if you really get what I'm saying here.
However on the other hand still you can get countable models of those theories where the set of all reals can be defined. So both countability of the reals and uncountability of reals are open possibilities and can be spoken about by consistent discourses. So both are pieces of mathematics. Everything depends on the model you are working in.
> > > Anyway the diagonal argument of Cantor is provable in very weak > > systems of ZFC which are proved to be consistent. So uncountability of > > reals is a possibility. > > And it leads to a contradiction with the fact that all real numbers > that are paths in the Binary Tree form a countable set. > > > > > Of course also Countability of reals is a possibility! since we can > > have countable models of ZFC or any theory that can define all the > > reals. > > I talk about *the* real numbers, which Cantor proved uncountable. > > > > 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. > > The main bulk of experts on foundations of astrology is by far more > trustworthy. > > > Actually I see this statement of yours really unsubstantiated. > > You believe in undefinable numbers. But what should that belief be > good for???
They are good for letting you know that you cannot place the reals with the naturals in oneone correspondence in some models.
We can believe anything we like of undefinable numbers > like of unicorns. Cantor's proof concerns definable numbers only! So > undefinable numbers do not support your standpoint anyhow. > > Regards, WM

