Re: Review of Mueckenheims book.

In article <1172236723.076787.19820@xxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 23 Feb., 04:45, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1172133316.431221.52...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
> On 21 Feb., 04:26, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > > > Upto this point
> > > of the book the "potential infinite" set of natural numbers
> > > has not yet been defined, so the meaning is pretty unclear.
> > >
> > WM: Didn't you read chapter 8?
> >
> > I thought so. Can you point me to the page where you define it?
>
> Cantor definies it at p. 99.

Yes, Cantor defines it. Does this reply mean that every term defined in a
quote is the definition you actually use? If so, you should have stated
that.

No. But his definitions are the best definitions of infinities I have
ever seen.

So how would I have known that you did use *that* definition? You did
not define the term, but you used the definition from a quote, while you
in general do not use the definitions from the quotes. Is that not a
bit confusing?

Perhaps. But that explanation is *not* in the book. In the book you
use: lim{n -> oo} | { 2, 4, 6, ..., 2n } | = aleph-0
without defining that limit anywhere.

That *is* the definition:
"und mit dem Grenzwert der Abschnitte
lim{n -> oo} | { 2, 4, 6, ..., 2n } | = aleph_0"

Is *that* a definition?

So you use undefined terminology in your book.

Compare the etymology of the name Limesordinalzahl.

That is irrelevant. The *limit* has not been defined. Moreover, aleph-0
is not an ordinal number but a cardinal number.

But it is also common knowledge that aleph_0 is the limit of
the cardinalities of the finite initial segments of N. I quote "Es ist
sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu
denken" (and elsewhere is given omega = aleph_0).

You may think about it as such, but it is *not* a definition of the limi
above.

> The limit lim{n -> oo} 2n remains finite while the limit
> lim{n -> oo} | { 2, 4, 6, ..., 2n } | is aleph_0.

Again, without defining that second limit.

Again, this is the definition.

You have a strange way of defining things. Anyway is your statement:
lim{n -> oo} 2n remains finite
a definition of that limit?

I have no idea how that should show such a thing. Once we have a proper
definition of such limits, we might state that, indeed,
lim{n -> oo} | { 2, 4, 6, ..., 2n } | = aleph-0
and
lim{n -> oo} 2n = aleph-0.

Oh no! How should that be possible if every natural number 2n is less
than aleph_0? Do you believe in infinite natural numbers suddenly?

Eh? lim{n -> oo} | { 2, 4, 6, ..., 2n } | = lim{n -> oo} n = aleph-0.
According to your summary definitions. I do not state that when I give
a definition for those limits that the limits are natural numbers. Why
do you think so? We might define that (improper) limit as follows:
Given a sequence a_i, such that for each natural number n there
can be found a natural number n_0 such that for all i > n0, a_i > n,
we state:
lim{n -> oo} a_i = aleph-0.
Not stating anything about aleph-0 being a natural number (it that were
the case there would be a contradiction). With this definition, both
limits are aleph-0.

Oh. I am trying to read chapter 10 (which purportedly goes about
existence), but it is heavy going.

Indeed? Nothing simpler than that.

You think so. I do not think so.

> > So it is not an equivalence relation amongst sets.
>
> It is an equivalence relation amongst sets with elements which are
> finite numbers.

So it is not an equivalence relation amongst sets.

Amongst existing sets. I did not state otherwise.

So there are no sets of functions? News to me.

> It does not raise the wrong impression that there were infinite
> numbers.

Just opinion.

No, it really does not raise this impression.

The *wrong* was just opinion. Also it does not raise the impression that
sets of points, sets of functions and so on do exist. And according to
your writings above, the set of countries in the EU does not exist either,
something that you would allow in earlier writings.

So half an edge is that part of an edge that stops half-way, where the
other half starts at that point and continues to the next node?

It is no defined what part of the edge is taken. If you get half a
cake, you will not ask whether you get the right or left one.

No, but with half edges I become quite interested. As in mathematics such
things are not defined, I wonder how you define them. Half edges, quarter
edges, the mind boggles.

> Try to construct a path which does not split off by an edge.

Be precise. Each path splits off any other path at a specific edge.
There is *no* edge such that a particular path splits off that edge
from all other paths.

Why should it? As long as the path is defined, it splits off by edges
(and is defined by edges or nodes, which is the same). That is enough.
Either the path gets undefined somewhere, or it continues to split off
by edges.

It continues to split off.

In no case a path can be defined by edges but not split off
by edges. IIn no case there can be more existing paths than existing
edges.

Proof, please.

> How many paths must a bundle have to be considered a bundle?
> But you believe in the empty set, don't you?

Pray define your terms. But each path-bundle "has" infinitely many
paths.

If paths exist, then they are singletons of path bundels. Perhaps they
do not exist?

What the heck *is* a singleton of path bundles? I would think that that
is a set containing a single path bundle. In what way is a path a set
containing a single path bundle?

> Perhaps this is so because they do not have existence at all?

No. It is because they are not terminating.

The number of edges is not terminating too. There are no problems from
this side. I did not say that paths should stop.

The number of edges is not terminating, but there are countably many.
The number of edges in a path is also not terminating, so there are
countably many edges in a path. This says *nothing* about the number
of paths, which is uncountable.

I have no idea. I do not think that bijection is the answer to
everything, but I think it is a good answer when comparing set sizes.
And with that in mind, aleph_0 < 2^aleph_0.

You compare infinite sets by bijection, but you refuse to compare them
by split-off positions?

I can not compare arbitrary sets by split-off positions. How do I
compare the sets {1, 5} and {2, 6} by split-off positions?

> It is impossible that there are more splitting results than splitting
> positions. Do you agree???

No. When going to the infinite many things that are valid in the finite
are not necessarily valid anymore.

Like the bijection of n and 2n, for instance? Why do you believe so
firmly in this stuff?

Because it is well-defined, in contrast to what you are trying to do.

> Not even by tunnel effect the tree can create or contain more paths
> than nodes.

You think so.

I know it. After all, every split-off happens at a finite distance
from the root. S there is no more infinity involved than in he
bijection of n wih 2n.

Again, making no sense at all. When you are talking about split-offs,
you should look at path-bundles, and both the split-offs and the
path-bundles come in as countably many. For a path there is *no*
specific split-off.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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