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From: Virgil on 1 Dec 2006 15:07
In article <1164974083.860766.111290(a)16g2000cwy.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> cbrown(a)cbrownsystems.com schrieb:
>
> > > The binary tree
> > >
> > > Consider a binary tree which has (no finite paths but only) infinite
> > > paths representing the real numbers between 0 and 1 as binary strings.
> > > The edges (like a, b, and c below) connect the nodes, i.e., the binary
> > > digits 0 or 1.
> > >
> > > 0.
> > > /a \
> > > 0 1
> > > /b \c / \
> > > 0 1 0 1
> > > ..........................
> > >
> > > The set of edges is countable, because we can enumerate them. Now we
> > > set up a relation between paths and edges.
> >
> > > Relate edge a to all paths
> > > which begin with 0.0. Relate edge b to all paths which begin with 0.00
> > > and relate edge c to all paths which begin with 0.01.
> >
> > So your relation is supposed to be a function mapping edges -> sets of
> > paths, correct?
>
> Yes. But the mapping is not the usual one (one edge --> one path). The
> edges are subdivided in shares.

Then it does not establish a one-to-one relationship. In one-to-one you
mast math whole with whole and not fractionate.
From: Virgil on 1 Dec 2006 15:13
In article <1164974976.838760.48250(a)j44g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>
> > In article <1164878462.838895.276420(a)n67g2000cwd.googlegroups.com>
> > mueckenh(a)rz.fh-augsburg.de writes:
> > > Dik T. Winter schrieb:
> > ...
> > > > > How can you judge about that without the slightest idea of what
> > > > > they
> > > > > wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic
> > > > > point
> > > > > of view is to look at the universe of all sets not as a fixed
> > > > > entity
> > > > > but as an entity capable of "growing", i.e., we are able to
> > > > > "produce"
> > > > > bigger and bigger sets." So a set (like the set of all sets) is not
> > > > > a
> > > > > fixed entity.
> > > >
> > > > There is nothing in that that shows that a set is not a fixed entity.
> > >
> > > The universe of all sets can grow. Define: "The universe of all sets is
> > > called the set of all sets", and you see it.
> >
> > And that is the confusion. If it is a set it cannot grow, but as Fraenkel
> > et al. do not define it as a set it is allowed to grow.
>
> They talk about the set of all sets, but cannot use it for the well
> known reasons.
>
> > They do not state
> > that a set can grow because they do not state that the universe is a set.
>
> Here is a book on ZF set theory: Karel Hrbacek and Thomas Jech:
> "Introduction to Set Theory"
> Marcel Dekker Inc., New York, 1984, 2nd edition. they write: The
> letters X and Y in these expressions are variables; they stand for
> (denote) unspecified, arbitrary sets.
>
> Therefore we can denote a set by X and we can say that the set X grows.

When one calls X a variable, that does not mean that its value varies
but that its value can be any member of some set of allowable values.


> > That is not "the set of states". You can talk about "the current set of
> > states" or about "the set of states in 1957" or whatever. At least
> > mathematically. In mathematics, by definition, a set can not grow.
>
> Wrong. Read my explanation above.

WRONG! I read it and found that it is a misinterpretation.
>
> > You are, of course, entitled to use another definition, but that will
> > not clarify the discussion at all (and you are not using standard set
> > theory).
>
> Hrbacek and Jech teach standard set theory including the fact that in
> ZF everything is a set.

And WM misinterprets, as usual.
From: Virgil on 1 Dec 2006 15:23
In article <1164976004.827617.95770(a)80g2000cwy.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> stephen(a)nomail.com schrieb:
>
> > mueckenh(a)rz.fh-augsburg.de wrote:
> >
> > > Dik T. Winter schrieb:
> >
> > >> >
> > >> > In my tree there are no finite (terminating) paths!
> > >>
> > >> Sorry, as far as I see your tree has also finite paths. I see paths
> > >> from the root to the first nodes below the roots.
> >
> > > That is an edge. But no path does stop there. Every path is continued,
> > > infinitely.
> >
> > So you do not know what a path is, or a tree apparently.
>
> First you should know: There is no final judgement what a path is or a
> tree.
>
> > In a tree, there exists exactly one path between any two nodes.
>
> That may be your definition and it may be useful for some purposes. But
> here it would confuse the matter. In the binary tree as I use it there
> is no finite path and there is no path which connects nodes in
> horizontal direction.
>
> Regards, WM

The difficulty arises from using one word, "path" in two meanings.
Let us reserve the word "path" for what starts at the root node and
either does not end or ends at a leaf node (connecteded to other parts
of the tree by a single edge to a parent node)

For a sequence of nodes and edges necessarily starting at some node and
ending at another, use the word "track"

We can then say that between any two nodes there is a unique track, but
that in a tree, every path must start at the root node and be of maximal
length".
From: Virgil on 1 Dec 2006 15:32
In article <1164981837.640099.23940(a)n67g2000cwd.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> William Hughes schrieb:
>
> > > > Extending the concept of bijection from sets to potentially
> > > > infinite sets is trivial.
> > >
> > > May be if you apply your personal definition of potentially infinity,
> > > but not if you apply the generally accepted definition.
> >
> >
> > Informally we have that a potentially infinite set is a set
> > which is always finite, but to which we can add an element
> > whenever we want. We say that x is an element of
> > the potentially infinite set if we can add enough elements
> > to get to x.

Then a "potentially infinite set" is not one set but a sequence of sets,
each containing all its predecessors as proper subsets.

It is, essentially, like the set of members of omega considered as a
sequence.
>
> Exactly. Therefore the potentially infinite sequence of digits of a
> real number is always representing a rational number, even a rational
> number with a finite sequence of digits.
> >
> > More formally.
> >
> > Let a generalized set be a function
> > on sets which takes the value true or false.

What is the domain of that function? It must be a set, since the domain
of every function is a set, and must contain at least all the sets, S,
for which A(S) is true. So Domain(A) must be an actually infinite set.
From: Virgil on 1 Dec 2006 15:35
In article <1164982199.959381.134510(a)j72g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Eckard Blumschein schrieb:
>
>
> > I recall being a little boy wondering when I was told that while there
> > is no evidence proving the existence of god there is also no evidence
> > showing his non-existence. Are those crippled who don't believer in CH?
> > I consider the background of CH given in the difference between number
> > and continuum. This might be crippled down to the truth? Do you agree?
> >
> No, I am sorry, I do not. The continuum is nothing but our failure to
> look closely enough. In physics it lasted 2000 years to settle the idea
> of the atom and to supplement and complete it by the uncertainty
> relations. The majority of matematicians is not yet far sighted enough
> to recognize the same situation in their realm.


We do not yet /know/ that the physical universe is not continuous, so
why should we reject a mathematically continuous real number system?
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