Das Kalenderblatt 091126

25/11/2009 - 12:24 von WM | Report spam
As a foundation for mathematics, then, set theory is far less firm
than what is founded upon it; for common sense in set theory is
discredited by the paradoxes. Clearly we must not look to the set-
theoretic foundation of mathematics as a way of allaying misgivings
regarding the soundness of classical mathematics. Such misgivings are
scarce anyway, once such offenses against reason as the infinitesimal
have been set right. [...]
For the one thing we insist on, as we sort through the various
possible plans for passable set theories, is that our set theory be
such as to reproduce, in the eventual superstructure, the accepted
laws of classical mathematics. This requirement is even useful as a
partial guide when in devising a set theory we have to choose among
intuitively undecidable alternatives. We may look upon set theory, or
its notation, as just a conveniently restricted vocabulary in which to
formulate a general axiom system for classical mathematics - let the
sets fall where they may. {{Prioritàt hat also in jedem Falle die
Mathematik. Und wenn die Mengenlehre nicht in der Lage ist, die
einfache und eindeutige Tatsache zu reproduzieren, dass der binàre
Baum nicht mehr Pfade als Knoten aufweisen kann, dann ist sie
ungeeignet.}}
[Willard V. O. Quine: "The ways of paradox and other essays", Harvard
University Press (1966) p. 31f
http://books.google.de/books?id=YRe...navlinks_s

Gruß, WM
 

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#1 Bobo
25/11/2009 - 17:28 | Warnen spam
WM wrote:

Und wenn die Mengenlehre nicht in der Lage ist, [...] zu
reproduzieren, dass der binàre Baum nicht mehr Pfade als Knoten
aufweisen kann, dann ist sie ungeeignet.}}



Nee, gerade dann ist z.B. ZF ja widerspruchsfrei. ;-)


Bobo

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