On 9 Dez., 20:22, Zuhair <zaljo...@gmail.com> wrote: > On Dec 9, 9:45 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 9 Dez., 18:40, Zuhair <zaljo...@gmail.com> wrote: > > > > 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. > > > 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. > > > > 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??? 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 > > Cantor's argument is about REALs whether definable after some finite > parameter free formula or not.
Undefinable reals can have any desired property. We cannot prove it because we cannot define the number. Cantor's argument concerns definable numbers. He was very disappointed when he heard that Koenig had proved the countability of definable numbers. Cantor did not consider undefinable numbers as anything sensible.
> So you are incorrect when you say that > Cantor's proof concerns definable numbers only! this is just another > assumption of yours that is alien to Cantor's argument.
Hahaha. You know Cantor very well, I persume.
> Everyone knows > that definable reals are countable, this is pretty much standard. > However Cantor proved that those are not all the reals.
Every Cantor-list delievers every digit of the diagonal. Hence the diagonal is definable.