Re:Mueckenheim's Confusion

In article <1173430532.626624.124950@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

On 8 Mrz., 16:25, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1173299798.650965.55...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
mueck...@xxxxxxxxxxxxxxxxx writes:

> On 7 Mrz., 16:01, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > > Did you ask the referee? Or are you the arbiter yourself? C(oo) is
> > > a
> > > part of T(oo).
> >
> > What do you *mean* with "is a part of T(oo)". How do you *define*
> > C(oo). You continuously *fail* to give a proper definition of it.
> > And do not refer to "cross section" of T(oo), because that is
> > undefined
> > either. Neither do refere to limits, unless you *define* those
> > limits.
>
> I did define them by means of set theory:
>
> C(oo) = U(C(n)) where C(n) is the cardinal number |L(n)| of L(n) which
> is the last level of T(n) where T(n) is the binary tree with n levels
> where n is a natural number and the union has to be taken over all
> natural numbers.

Ok, so C(oo) is the cardinal number aleph-0.

> T(oo) = U(T(n)) according to the definiton I gave for the union of
> trees. This union can also be obtained by
> T(oo) = U(L(n)) with L(n) defined as above.

So T(oo) is a set of nodes.

> C(oo) c |T(oo)|. (Both are cardinal numbers, both are aleph_0.)

And you mean *that* with C(oo) is part of T(oo). A strange formulation,
but correct.

And how do you conclude from this that the number of paths is countable?

> > > No, ***. I know that the functions in set theory are not continuous
> > > and perform unpredictable jumps when leaving the finite and
> > > entering
> > > the infinite. One of these tricks is the set of all finite sets
> > > being
> > > countable and the set of all infinite sets being uncountable.
> > > Another
> > > trick is the famous observation:
> > > Forall n in N: |{2,4,6,...,2n}| < 2n
> > > while
> > > lim {n-->oo} |{2,4,6,...,2n}| > lim {n-->oo} 2n, and similar jokes.
> >
> > That is tricky of you. You have defined *neither* limit.
>
>
> I defined for several times:
> lim {n-->oo} {1,2,3,...,n} = N
> lim {n-->oo} |{1,2,3,...,n}| = aleph_0
>
> lim {n-->oo} {2,4,6,...,2n} = set of all even numbers
> lim {n-->oo} |{2,4,6,...,2n}| = aleph_0

Ah, you need *four* definitions here? Only two would be sufficient, I
think. Because none of the definitions follows from any of the other
definitions.

> The limit lim {n-->oo} n is a number which n comes as close as you
> like. This is not omega or any greater number. If it exists, then it
> can only be less.

So, according to *your* definitions it is >. Not according to common
definitions. Would it not be possible that your definitions are not
consistent? It is your last paragraph which is inconsistent with the
other definitions. See how you *did* define:
> lim {n-->oo} |{1,2,3,...,n}| = aleph_0
which means (as |{1,2,3,...,n}| = n:
lim {n-->oo} n = aleph_0?

> > > Have you really been blinded for the fact that most of the subsets
> > > of
> > > nodes of the complete cannot serve as paths?
> >
> > That does not matter. Some subsets are paths. Of these some are
> > infinite
> > and some are finite. The finite ones are a subset of a countable set,
> > the
> > infinite ones are a subset of an uncountable set. But there *do*
> > exist
> > uncountable subsets of uncountable sets.
>
> May be, but not in the tree --- as long as it stretches there is no
> uncountable set.

Why not?

> > At each finite level, the cross-section C(n) is the number of
> > path-bundles.
> > This does not tell us anything about the infinite tree.
>
> Infinite trees have infinite cross section, however, the cardinal
> number is closely connected to the length n. Do you claim that there
> are paths of uncountable length (number of nodes)? If not, then you
> will obtain from the countability of the |2^omega| nodes the
> countability of the |2^omega| paths.

Why? I see only that C(oo) is aleph-0, but I see no relation between
paths and aleph-0.


Isn't it difficult to look from this perspectice? The cross sections
*are* numbers of paths.

For finite trees, they are, but there is no such thing as a cross
section for an infinite tree, at least when using WM's definition of
cross section.

1) We know tha every digi of a real number stands on a finite place.
(We know that here is no digit the last one, but that is here
completely irrelevant.)
2) From (1) we obtain that every node of a path is at a finite place.
That means, there is no part of a path which would jut out f the tree.
3) All paths which are in the tree cross each cross section (that is
why it is named cross section).
There is no path or even splitting of paths outside of every cross
section.

Therefore, the limit processes for paths-lengths and cross sections
are identical.

It is not path lengths that become uncountably in the complete infinite
binary tree, but the cardinality of the set of paths, which is easily
seen to be the same as the cardinality of the set of all binary
sequences, and that Cantor showed to be uncountable.


Regards, WM
.

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