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Topic: Matheology § 190
Replies: 15   Last Post: Jan 14, 2013 3:35 PM

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 1,973 Registered: 2/15/09
Re: Matheology § 190
Posted: Jan 13, 2013 9:54 PM

On Jan 13, 3:12 pm, Virgil <vir...@ligriv.com> wrote:
> In article
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>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 13 Jan., 22:59, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <fcbf94db-8b22-4a16-b837-ccd23e176...@k6g2000yqf.googlegroups.com>,

>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 13 Jan., 00:26, Virgil <vir...@ligriv.com> wrote:
> > > > > In article
> > > > > <4bffb7f3-9bfa-4dae-9108-da5e24389...@f4g2000yqh.googlegroups.com>,

>
> > > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > > On 12 Jan., 22:00, Virgil <vir...@ligriv.com> wrote:
> > > > > > > In article
> > > > > > > <c0615860-6190-4c10-9185-78ed2f6a2...@x10g2000yqx.googlegroups.com>,

>
> > > > > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > > > > Matheology 190
>
> > > > > > > > The Binary Tree can be constructed by aleph_0 finite paths.
>
> > > > > > > >         0
> > > > > > > >       1, 2
> > > > > > > >   3, 4, 5, 6
> > > > > > > > 7, ...

>
> > > > > > > Finite trees can be built having finitely many finite paths.
> > > > > > > A Complete Infinite Binary Tree cannot be built with only finite
> > > > > > > paths,
> > > > > > > as none of its paths can be finite.

>
> > > > > > Then the complete infinite set |N cannot be built with only finite
> > > > > > initial segments {1, 2, 3, ..., n} and not with ony finite numbers 1,
> > > > > > 2, 3, ...? Like Zuhair you are claiming infinite naturals!

>
> > > > > A finite initial segment of |N is not a path in the unary tree |N.
>
> > > > > And neither |N  as a unary tree nor any Complete Infinite Binary Tree
> > > > > has any finite paths.

>
> > > > >    "A Complete Infinite Binary Tree cannot be built with only
> > > > >    finite paths, as none of its paths can be finite."

>
> > > > > Means the same as
>
> > > > >    "A Complete Infinite Binary Tree cannot be built HAVING only
> > > > >    finite paths, as none of its paths can be finite."

>
> > > > > WM has this CRAZY notion that a path in a COMPLETE INFINITE BINARY TREE
> > > > > can refer to certain finite sets of nodes.

>
> > > > > And no one is claiming any infinite naturals, only infinitely many
> > > > > finite naturals.

>
> > > > So each n belongs to a finite initial segment (1,2,3,...,n).
> > > > Same is valid for the nodes of the Binary Tree: Each node belongs to a
> > > > finite initial segment of a path, the natural numbers (1,2,3,...,n)
> > > > denoting the levels which the nodes belong to.

>
> > > Since every binary path has a node at every "level" (distance from the
> > > root), it can only be represented by an infinite set of naturals in this
> > > way.

>
> > Irrelevant. Every distance is finite. There is no distance that is
> > larger than every finite distance. All finite distances are countable.

>
> Irrelevant! Any (complete) path in a Complete Infinite Binary Tree
> contains infinitely many nodes and infinitely many branchings, and
> cannot be distinguished from all other (complete) paths without
> specifying at least infinitely many of its nodes or its entire sequence
> of branchings.
>
>
>

> > Note: There is no natural number larger than every natural number. And
> > the number of natural numbers is completely irrelevant in this
> > context.

>
> Only in WMytheology.
>
> In reality it is relevant, since any (complete) path in a Complete
> Infinite Binary Tree contains infinitely many nodes and infinitely many
> branchings, and cannot be distinguished from all other (complete) paths
> without specifying at least infinitely many of its nodes or its entire
> sequence of branchings.
>
> Note that while a finite number of nodes can separate one path from any
> finite set of other paths and from some infinite sets of other paths, it
> takes infinitely many nodes to separate it from EVERY other path, since
> two infinite paths can share any finite number of nodes.
> --

Then, any two paths, as expansions, share any finite number of nodes,
and only finitely many, and have an order in the bread-first ordering,
and in the course-of-passage of that ordering of paths, regardless of
what the value of the path is from one to the next, the antidiagonal
result doesn't follow.

Find a result for transfinite cardinals in application: science may
find it of use.

Regards,

Ross Finlayson