Virgil
Posts:
4,482
Registered:
1/6/11
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Re: On the infinite binary Tree
Posted:
Dec 13, 2012 5:06 PM
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In article
<b7cf8590-2265-4643-8964-5bd97e0bcc39@c14g2000vbd.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 13 Dez., 21:23, Virgil <vir...@ligriv.com> wrote:
> > In article
>
> > > MY Binary Tree contains the paths of real numbers of the unit
> > > interval.
> >
> > Provably not all of them.
>
> My Binary Tree contains all nodes.
{0,1,2,3,4,5,6,7,8,9} contains all decimal digits, but not all strings
of them.
> We could also use a decimal tree.
> That contains all digits at all finite positions. Provably. And it is
> constructed by a countable set of decimal paths.
Not so!, The set of paths in decimal tree is no more countable that the
set of paths in a binary tree.
A set being countable means, by definition, that one can prove
existence of a surjection from N to that set.
Whereas for the set of paths of a complete infinite binary, or decimal,
tree is provably not surjectible from N, via the Cantor argument.
> >
> > > > where is that proof? please show us
> >
> > > I will it show it to you for all the paths that I used to construct
> > > the above tree and, in addition, for all the paths that you can
> > > identify as beeing missing there.
> >
> > Promises, promises!
>
> No, I stand by my offer. Tell me which paths you can identify as
> missing in the tree that I constructed by countably many infinite
> paths. I will show the bijection.
Show us your listing and we will show you just as many more that it
missed.
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