Das Kalenderblatt 100206

05/02/2010 - 15:56 von WM | Report spam
The reaction to Skolem's results was split. For example, Fraenkel was
not sure about the correctness of the proof of the Löwenheim-Skolem
theorem, and he seems to have had difficulties in analysing the role
of logic with sufficient rigour to understand Skolem's paradox [...]

A "War" Against Skolem

What about Zermelo? When faced with the existence of countable models
of first-order set theory, his first reaction was not the natural one,
namely to check Skolem's proof and evaluate the result - it was
immediate rejection {{da hat sich also nichts geàndert}}. Apparently,
the motivation of ensuring "the valuable parts of set theory" which
had led his axiomatizations from 1908 and from the Grenzzahlen paper
had not only meant allowing the deduction of important set-theoretic
facts, but had included the goal of describing adequately the set-
theoretic universe with its variety of infinite cardinalities. Now it
was clear that Skolem's system, like perhaps his own, failed in this
respect. Moreover, Skolem's method together with the epistemological
consequences Skolem had drawn from his results, could mean a real
danger for mathematics {{aber nein, ganz im Gegenteil}} like that
caused by the intuitionists: In his Warsaw notes W3 he had clearly
stated that "true mathematics is indispensably based on the assumption
of infinite domains," among these domains, for instance, the
uncountable continuum of the real numbers. Hence, following Skolem,
"already the problem of the power of the continuum loses its true
meaning".
Henceforth Zermelo's foundational work centred around the aim of
overcoming Skolem's relativism and providing a framework in which to
treat set theory and mathematics adequately. Baer speaks of a real war
Zermelo had started, wishing him "Heil und Sieg und fette Beute", at
the same time pleading for peace [...]

However, peace was not to come. [...] a vivid impression of Zermelo's
uncomprising engagement, at the same time also revealing his worries:
It is well known that inconsistent premises can prove anything one
wants; however, even the strangest consequences that Skolem and others
have drawn from their basic assumption, for instance the relativity of
the notion of subset or equicardinality, still seem to be insufficient
to raise doubts about a doctrine that, for various people, already won
the power of a dogma that is beyond all criticism. [...] His remedy
consisted of infinitary languages {{Rettung von Unsinn durch Unsinn}}.
[...] Skolem had considered such a possibility, too, but had discarded
it because of a vicious circle (Skolem 1923, p. 224):

In order to get something absolutely uncountable either the axioms
themselves would have to be present in an absolutely uncountably
infinite number or one would have to have an axiom which could provide
an absolutely uncountable set of first-order sentences. However, in
all cases this leads to a circular introduction of higher infinities,
that means, on an axiomatic basis higher infinities exist only in a
relative sense.

[Heinz-Dieter Ebbinghaus,Volker Peckhaus: "Ernst Zermelo: an approach
to his life and work " Springer (2007) p. 199 ff]

http://www.springer.com/math/histor...ringer-_-0

Gruß, WM
 

Lesen sie die antworten

#1 Playboy Number 9
05/02/2010 - 16:21 | Warnen spam
On 5 Feb., 15:56, WM wrote:
The reaction to Skolem's results was split. For example, Fraenkel was
not sure about the correctness of the proof of the Löwenheim-Skolem
theorem, and he seems to have had difficulties in analysing the role
of logic with sufficient rigour to understand Skolem's paradox [...]

A "War" Against Skolem

What about Zermelo? When faced with the existence of countable models
of first-order set theory, his first reaction was not the natural one,
namely to check Skolem's proof and evaluate the result - it was
immediate rejection {{da hat sich also nichts geàndert}}. Apparently,
the motivation of ensuring "the valuable parts of set theory" which
had led his axiomatizations from 1908 and from the Grenzzahlen paper
had not only meant allowing the deduction of important set-theoretic
facts, but had included the goal of describing adequately the set-
theoretic universe with its variety of infinite cardinalities. Now it
was clear that Skolem's system, like perhaps his own, failed in this
respect. Moreover, Skolem's method together with the epistemological
consequences Skolem had drawn from his results, could mean a real
danger for mathematics {{aber nein, ganz im Gegenteil}} like that
caused by the intuitionists: In his Warsaw notes W3 he had clearly
stated that "true mathematics is indispensably based on the assumption
of infinite domains," among these domains, for instance, the
uncountable continuum of the real numbers. Hence, following Skolem,
"already the problem of the power of the continuum loses its true
meaning".
        Henceforth Zermelo's foundational work centred around the aim of
overcoming Skolem's relativism and providing a framework in which to
treat set theory and mathematics adequately. Baer speaks of a real war
Zermelo had started, wishing him "Heil und Sieg und fette Beute", at
the same time pleading for peace [...]

However, peace was not to come. [...] a vivid impression of Zermelo's
uncomprising engagement, at the same time also revealing his worries:
It is well known that inconsistent premises can prove anything one
wants; however, even the strangest consequences that Skolem and others
have drawn from their basic assumption, for instance the relativity of
the notion of subset or equicardinality, still seem to be insufficient
to raise doubts about a doctrine that, for various people, already won
the power of a dogma that is beyond all criticism. [...] His remedy
consisted of infinitary languages {{Rettung von Unsinn durch Unsinn}}.
[...] Skolem had considered such a possibility, too, but had discarded
it because of a vicious circle (Skolem 1923, p. 224):

In order to get something absolutely uncountable either the axioms
themselves would have to be present in an absolutely uncountably
infinite number or one would have to have an axiom which could provide
an absolutely uncountable set of first-order sentences. However, in
all cases this leads to a circular introduction of higher infinities,
that means, on an axiomatic basis higher infinities exist only in a
relative sense.

[Heinz-Dieter Ebbinghaus,Volker Peckhaus: "Ernst Zermelo: an approach
to his life and work " Springer  (2007) p. 199 ff]

http://www.springer.com/math/histor...-540-49...

Gruß, WM




Ja dieser Zermelo soll ja die Mengenlehre begründet haben
( etwa : R ist offen UND abgeschlossen ! ), aber ist sie noch
zeitgemàß ?

1) PCs laufen aber auf endlichen Mengen, und alle Berechnungen
werden heute mit PCs durchgeführt.

2) Die QM zeigt uns die Orbitale der Atome auf.
Für Feinheiten wird aber ausschließlich die " Störungstheorie "
verwendet,
welche NUR ENDLICH VIELE Glieder der Taylorreihe benutzt.

3 ) Um 1800 wollten alle den Hauptsatz der
Analysis ( die von Leibniz und Newton begründet worden ist ) beweisen.
Mit Newton allein wird aber heute keine Physik mehr gemacht.

Ähnliche fragen