Das Kalenderblatt 091130

29/11/2009 - 13:13 von WM | Report spam
By now there are substantially different models of set theory,
satisfying one or another special axiom - the axiom of
constructibility, Martin's axiom, or
the axiom of determinateness. The striking result of these technical
developments is that different models of set theory give different
answers to specific mathematical problems. The continuum hypothesis is
true on the Gödel axiom of constructibility, but false in certain
Cohen models of set
theory. Whitehead's problem provides another striking example. He
considered a homomorphism f: A --> G of one abelian group A onto
another such group G, in the case when the kernel is just the
(additive) group of integers.
In case G is a free abelian group, the epimorphism splits (that is,
there is a homomorphism h: G --> A with fh = 1. Whitehead asked:
Conversely, if such an f always splits, is G free? It now turns out
that the answer may be yes or no, depending on the model of set
theory. This is one of many striking cases where explicit mathematical
problems have different answers, depending on the model used for set
theory. (See Eklof [2].) Mathematics, we hold, deals with multiple
models of the world. It is not subsumed in any one big model or by any
one grand system of axioms.
The idea that set theory is relative is not new; it was clearly stated
for axiomatic set theory by Skolem in 1922 [9]. We intend simply to
draw some of the philosophic consequences of that relativity. For the
Platonist, there is a real world of sets, existing forever, described
only approximately by the Zermelo-Fraenkel axioms or by their
modifications. It may be that some final insight will give a definite
axiom system, but the sets themselves are the underlying mathematical
In our view, such a Platonic world is speculative. It cannot be
clearly explained as a matter of fact (ontologically) or as an object
of human knowledge (epistemologically). Moreover, such ideal worlds
rapidly become too elaborate; they must display not only the sets but
all the other separate structures which mathematicians have described
or will discover. The real nature of these structures does not lie in
their often artificial construction from set theory, but in their
relation to simple mathematical ideas or to basic human activities.
Hence, we hold that mathematics is not the study of intangible
Platonic worlds, but of tangible formal systems which have arisen from
real human activities.

2. Paul C. Eklof, Whitehead's problem is undecidable, this MONTHLY, 83
(1976) 775-787.
9. Th. Skolem, Einige Bemerkungen zur axiomatische Begründung der
Mengenlehre. Fifth Congress of Scandinavian Mathematicians, 1922,
Helsingfors, 1923, pp. 2 17-232.

OF MATHEMATICS", The American Mathematical Monthly, Vol. 88, No. 7
(1981) 462-472.]

Gruß, WM

Lesen sie die antworten

#1 Carsten Schultz
29/11/2009 - 13:24 | Warnen spam
WM schrieb:
2. Paul C. Eklof, Whitehead's problem is undecidable, this MONTHLY, 83
(1976) 775-787.

Dazu auch:
AUTHOR = {Eklof, Paul C. and Mekler, Alan H.},
TITLE = {Almost Free Modules: Set-theoretic Methods},
YEAR = 1990,
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam}
Ich habe das als gut lesbar in Erinnerung.



Carsten Schultz (2:38, 33:47)
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