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Re: Mathematics in brief
Posted:
Dec 9, 2012 3:19 PM
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On Dec 9, 10:59 pm, WM <mueck...@rz.fh-augsburg.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.
Zuhair
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