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

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology � 190
Posted: Jan 13, 2013 6:12 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 13 Jan., 22:59, Virgil <vir...@ligriv.com> wrote:
> > In article
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > On 13 Jan., 00:26, Virgil <vir...@ligriv.com> wrote:
> > > > In article

> >
> > > >  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.
--