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