Kann man den Fall Mückenheim auf sich beruhen lassen?

23/11/2008 - 17:40 von Ralf Bader | Report spam
Das könnte man so sehen, solange Mückenheim seinen Unsinn privat àußert.
Kann man es angesichts des Postings in sci.math.
Message-ID:
<d3b30504-6c39-4f7b-9c8d-531731954348@f13g2000yqj.googlegroups.com>
immer noch so sehen? Aus diesem Beitrag:

[Zitatanfang]

http://www.hs-augsburg.de/~mueckenh/LgGU08H.pdf

Most of the questions are more than trivial, but I am bothered by
quesions 9, 10 and 11. Not only does he spout his nonsense about
binary trees in this newsgroup, he also requires his students to
regurgitate that for their exam scores.



In fact most students do easily understand that topic. I usually
explain it as follows:

If you know a bit of mathematics, then you know also that the limit of
a convergent sequence is not necessarily a member of that sequence.
The sequence 3, 3.1, 3.14, ... for example does not contain pi. You
need infinitely many members of the sequence to identify pi (unless
you have a formula identifying it).

In the binary tree there are only countably many convergent sequences
(paths). Therefore the real numbers (or what set theory tells us about
R) is too numerous to be identified by paths in the binary tree.

Proof: Remove all infinite paths from the binary tree, i.e., all paths
representing real numbers, such that only the finite paths remain.
What does this change? Nothing, as far as the nodes are concerned. All
nodes remain there, because each one belongs to a finite path.

Now take every finite path of tree that have a last node 1 and extend
it by zeros:
Example: The finite path 0.xxx...xxx1 (where each x denotes either 0
or 1) is extended to 0.xxx...xxx1000 The set of infinite paths
created in this way together with the path 0.000... covers all nodes
of the binary tree. (Of course the same could be accomplished withg
slightly adapted line of arguing by other extensions.)

This proof shows that the nodes of the binary tree are not capable of
identifying all infinite paths that thoughtlessly are expected to
reside in the binary tree. All nodes are already needed to identify
the infinite papths of the form 0.xxx...xxx1000 And transformed to
the binary or decimal system this proof shows that it is impossible to
uniquely identify uncountably many real numbers.

By the way, a related problem has just appered in de.sci.math. -
without solution. Cover all rational numbers between -oo and +oo by a
geometric sequence of closed intervals of total measure 2 (or
epsilon). Between these intervals there can only exist aleph_0
singular points.
Got it?
[Zitatende]

Was Mückenheim hier aus seiner Lehrpraxis berichtet, spricht m.E. für sich.
Nebenbei existiert das im letzten Absatz behauptete Problem nur in der
Phantasie des Mückenheim. Kann es sein, daß einer Analysis-Vorlesungen
hàlt, der über Jahre hinweg permanent gezeigt hat, daß er völlig unfàhig
ist, ein Minimum an Verstàndnis für die Struktur des Systems der reellen
Zahlen zu entwickeln?


Ralf
 

Lesen sie die antworten

#1 Roland Franzius
23/11/2008 - 18:08 | Warnen spam
Ralf Bader schrieb:
Das könnte man so sehen, solange Mückenheim seinen Unsinn privat àußert.
Kann man es angesichts des Postings in sci.math.
Message-ID:

immer noch so sehen? Aus diesem Beitrag:

[Zitatanfang]
http://www.hs-augsburg.de/~mueckenh/LgGU08H.pdf

Most of the questions are more than trivial, but I am bothered by
quesions 9, 10 and 11. Not only does he spout his nonsense about
binary trees in this newsgroup, he also requires his students to
regurgitate that for their exam scores.



In fact most students do easily understand that topic. I usually
explain it as follows:

If you know a bit of mathematics, then you know also that the limit of
a convergent sequence is not necessarily a member of that sequence.
The sequence 3, 3.1, 3.14, ... for example does not contain pi. You
need infinitely many members of the sequence to identify pi (unless
you have a formula identifying it).

In the binary tree there are only countably many convergent sequences
(paths). Therefore the real numbers (or what set theory tells us about
R) is too numerous to be identified by paths in the binary tree.

Proof: Remove all infinite paths from the binary tree, i.e., all paths
representing real numbers, such that only the finite paths remain.
What does this change? Nothing, as far as the nodes are concerned. All
nodes remain there, because each one belongs to a finite path.

Now take every finite path of tree that have a last node 1 and extend
it by zeros:
Example: The finite path 0.xxx...xxx1 (where each x denotes either 0
or 1) is extended to 0.xxx...xxx1000 The set of infinite paths
created in this way together with the path 0.000... covers all nodes
of the binary tree. (Of course the same could be accomplished withg
slightly adapted line of arguing by other extensions.)

This proof shows that the nodes of the binary tree are not capable of
identifying all infinite paths that thoughtlessly are expected to
reside in the binary tree. All nodes are already needed to identify
the infinite papths of the form 0.xxx...xxx1000 And transformed to
the binary or decimal system this proof shows that it is impossible to
uniquely identify uncountably many real numbers.

By the way, a related problem has just appered in de.sci.math. -
without solution. Cover all rational numbers between -oo and +oo by a
geometric sequence of closed intervals of total measure 2 (or
epsilon). Between these intervals there can only exist aleph_0
singular points.
Got it?
[Zitatende]

Was Mückenheim hier aus seiner Lehrpraxis berichtet, spricht m.E. für sich.
Nebenbei existiert das im letzten Absatz behauptete Problem nur in der
Phantasie des Mückenheim. Kann es sein, daß einer Analysis-Vorlesungen
hàlt, der über Jahre hinweg permanent gezeigt hat, daß er völlig unfàhig
ist, ein Minimum an Verstàndnis für die Struktur des Systems der reellen
Zahlen zu entwickeln?



Es ist absolut nichts Ungewöhnliches dabei, dass ein Physiker mit 0-2
Vordiplomscheinen in Mathematik Professor für Mathematik an einer
Fachhochschule wird. Früher war das ja ein Job, den man Igenieuren mit
Rechenschieberkenntnissen übertrug.

Die Frage ist nur, wie er Berufspraxis in Mathematik erworben haben
soll. Das ist aber nach der Berufung ohne Belang.

Von Leuten, die die Verhàltnisse kennen, denen er entwuchs, weiss man,
dass zu gewissen Zeiten wegen Überlaufs Scheine an Nebenfàchler verteilt
wurden, wenn sie versprachen, nicht in den Übungen aufzutauchen.

Dass so jemand unter dem Einfluss einer nicht zu stoppenden Entlohnung
und dem Wohlwollen kirchlicher Kreise immun gegen fachliche Einwànde
wird, ist wohl einzusehen. Was sollte er sonst deiner Meinung nach tun?
Die Wahrheit über sich einzusehen?

Erinnert in seiner Zielstebigkeit, an den Tatsachen vorbei zu
schwadronieren, ein wenig an BWL-Professoren, die nach gründlichem
Studium der Finanzmathematik zu dem Schluss kamen, dass Aktienkurse
naturgemàß nur steigen können.


Roland Franzius

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