Date: Jan 13, 2013 9:54 PM Author: ross.finlayson@gmail.com Subject: Re: Matheology § 190 On Jan 13, 3:12 pm, Virgil <vir...@ligriv.com> wrote:

> In article

> <fa843976-e6be-411c-977c-85cced20f...@w8g2000yqm.googlegroups.com>,

>

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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