
Re: Mathematics in brief
Posted:
Dec 9, 2012 2:39 PM


On Dec 9, 10:37 pm, Zuhair <zaljo...@gmail.com> wrote: > 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.
Sorry I meant between the naturals and the .....
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

