Virgil
Posts:
9,012
Registered:
1/6/11


Re: Mathematics in brief
Posted:
Dec 9, 2012 9:56 PM


In article <a2c2def8209244c1b0cf251f613682a1@x10g2000yqx.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 9 Dez., 21:19, Zuhair <zaljo...@gmail.com> wrote: > > On Dec 9, 10:59 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > > On 9 Dez., 20:37, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > 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. > > > > > Then you are wrong from the scratch. Every real number has a > > > representation by an infinite sequence (= infinite path of nodes in > > > the tree). But as my proff shows I construct the whole Binary Tree by > > > countably many paths. There are not more nodes available to add > > > further paths. > > > > > >You will need uncountably many infinite > > > > binary trees to recover all the reals. > > > > > That is purest nonsense. And it has nothing to do with Cantor's > > > diagonal which is of course an infinite sequence of digits > > > corresponding to a path in the Binary Tree. > > > > Yes corresponding to an ACTUAL infinite path in the Binary Tree, which > > is something that you already refuse to address. > > No, you misunderstood. I construct the actually infinite Binary Tree > by actually infinite paths like "every finite path which is appended > by an actually infinite sequence of 000..." or "every finite path > which is appended by an actually infinite sequence of 111..." or ...
Unless EVERY node in WM's tree has two child nodes, it cannot be a complete tree. and if every node does have two child nodes, the set of all the (necessarily infinite) strings of nodes which we call paths will be, as Cantor proved, well beyond countability. > > You cannot, by any means, find out from the constructed tree what kind > of paths I have used. Unless each node has two child nodes, I can tell that it is not complete.
> Therefore your claim that both trees are > different is provably false with respect to nodes. But the real > numbers are defined by their nodes and only by their nodes  at least > as far as Cantor's argument is concerned.
On the contrary, it is plainly the parentchild linkages between nodes, and not merely the nodes themselves, which determines what sort of tree one has. And the critical quality for completeness is that every node have two child nodes.
But this can not happen in WM's trees, so that they cannot be complete. 

