Das Kalenderblatt 090921

20/09/2009 - 14:53 von WM | Report spam
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted
as the appropriate foundation for modern mathematics, proof theorists
have known for decades that virtually all mainstream mathematics can
actually be formalized in much weaker systems which are essentially
number-theoretic in nature. [...] not only is it possible to formalize
core mathematics in these weaker systems, they are in important ways
better suited to the task than ZFC [...] most if not all of the
already rare examples of mainstream theorems whose proofs are
currently thought to require metaphysically substantial set-theoretic
principles actually do not; and set theory itself, as it is actually
practiced, is best understood in formalist, not platonic, terms, so
that in a real sense set theory is not even indispensable for set
theory. [...] set theory should not be considered central to
mathematics.

Probably most mathematicians are more willing to be platonists
about number theory than about set theory, in the “truth platonism”
sense that they firmly believe every sentence of first order number
theory has a definite truth value, but are less certain this is the
case for set theory. Those mathematicians who are unwilling to affirm
that the twin primes conjecture, for example, is objectively true or
false are undoubtedly in a small minority; in contrast, suspicion that
questions like the continuum hypothesis or the existence of measurable
cardinals may have no genuine truth value seems fairly widespread.
Some possible reasons for this difference in attitudes towards
number theory and set theory are (1) a sense that natural numbers are
evident and accessible in a way that arbitrary sets are not; (2)
suspicion that sets are philosophically dubious in a way that numbers
are not; (3) the existence of truly basic set-theoretic questions such
as the continuum hypothesis which are known to be undecidable on the
basis of the standard axioms of set theory, and the absence of
comparable cases in number theory; and (4) the fact that naive set
theory is inconsistent. The classical paradoxes of naive set theory
particularly cast doubt on the idea of a well-defined canonical
universe of sets in which all set-theoretic questions have definite
answers.
One philosophically important way in which numbers and sets, as
they are naively understood, differ is that numbers are physically
instantiated in a way that sets are not. Five apples are an instance
of the number 5 and a pair of shoes is an instance of the number 2,
but there is nothing obvious that we can analogously point to as an
instance of, say, the set {{0/}}. [...]
Unfortunately, the philosophical difficulties with set-theoretic
objects platonism are extremely severe. First, there is the
ontological problem of saying just what sets are.[...]
Perhaps the most influential philosophical defense of set theory is
the Quine-Putnam indispensability argument. According to this
argument, mathematics is indispensable for various established
scientific theories, and therefore any evidence that confirms these
theories also confirms the received foundation for mathematics, namely
set theory. But as a result of work of many people going back to
Hermann Weyl, we now know that the kind of mathematics that is used in
scientific applications is not inherently set-theoretic, and indeed
can be developed along purely number-theoretic lines. This point has
been especially emphasized by Feferman. Consequently, contrary to
Quine and Putnam, the confirmation of present-day scientific theories
provides no special support for set theory. [...]
This raises the possibility that the use of set theory as a
foundation for mathematics may be an historical aberration. We may
ultimately find that ZFC really has no compelling justification and is
completely irrelevant to ordinary mathematical practice.

[NIK WEAVER: "IS SET THEORY INDISPENSABLE?"]
http://www.math.wustl.edu/~nweaver/indisp.pdf

Den Hinweis auf diesen lesenswerten Artikel verdanke ich Albrecht
Storz.

Gruß, WM
 

Lesen sie die antworten

#1 Cary Crank
20/09/2009 - 16:30 | Warnen spam
On 20 Sep., 14:53, WM wrote:
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted
as the appropriate foundation for modern mathematics, proof theorists
have known for decades that virtually all mainstream mathematics can
actually be formalized in much weaker systems which are essentially
number-theoretic in nature. [...] not only is it possible to formalize
core mathematics in these weaker systems, they are in important ways
better suited to the task than ZFC [...] most if not all of the
already rare examples of mainstream theorems whose proofs are
currently thought to require metaphysically substantial set-theoretic
principles actually do not; and set theory itself, as it is actually
practiced, is best understood in formalist, not platonic, terms, so
that in a real sense set theory is not even indispensable for set
theory. [...] set theory should not be considered central to
mathematics.

   Probably most mathematicians are more willing to be platonists
about number theory than about set theory, in the “truth platonism”
sense that they firmly believe every sentence of first order number
theory has a definite truth value, but are less certain this is the
case for set theory. Those mathematicians who are unwilling to affirm
that the twin primes conjecture, for example, is objectively true or
false are undoubtedly in a small minority; in contrast, suspicion that
questions like the continuum hypothesis or the existence of measurable
cardinals may have no genuine truth value seems fairly widespread.
   Some possible reasons for this difference in attitudes towards
number theory and set theory are (1) a sense that natural numbers are
evident and accessible in a way that arbitrary sets are not; (2)
suspicion that sets are philosophically dubious in a way that numbers
are not; (3) the existence of truly basic set-theoretic questions such
as the continuum hypothesis which are known to be undecidable on the
basis of the standard axioms of set theory, and the absence of
comparable cases in number theory; and (4) the fact that naive set
theory is inconsistent. The classical paradoxes of naive set theory
particularly cast doubt on the idea of a well-defined canonical
universe of sets in which all set-theoretic questions have definite
answers.
   One philosophically important way in which numbers and sets, as
they are naively understood, differ is that numbers are physically
instantiated in a way that sets are not. Five apples are an instance
of the number 5 and a pair of shoes is an instance of the number 2,
but there is nothing obvious that we can analogously point to as an
instance of, say, the set {{0/}}. [...]
   Unfortunately, the philosophical difficulties with set-theoretic
objects platonism are extremely severe. First, there is the
ontological problem of saying just what sets are.[...]
   Perhaps the most influential philosophical defense of set theory is
the Quine-Putnam indispensability argument. According to this
argument, mathematics is indispensable for various established
scientific theories, and therefore any evidence that confirms these
theories also confirms the received foundation for mathematics, namely
set theory. But as a result of work of many people going back to
Hermann Weyl, we now know that the kind of mathematics that is used in
scientific applications is not inherently set-theoretic, and indeed
can be developed along purely number-theoretic lines. This point has
been especially emphasized by Feferman. Consequently, contrary to
Quine and Putnam, the confirmation of present-day scientific theories
provides no special support for set theory. [...]
   This raises the possibility that the use of set theory as a
foundation for mathematics may be an historical aberration. We may
ultimately find that ZFC really has no compelling justification and is
completely irrelevant to ordinary mathematical practice.

[NIK WEAVER: "IS SET THEORY INDISPENSABLE?"]http://www.math.wustl.edu/~nweaver/indisp.pdf

Den Hinweis auf diesen lesenswerten Artikel verdanke ich Albrecht
Storz.

Gruß, WM



Unter einer „Menge“ verstehen wir jede Zusammenfassung M von
bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder
unseres Denkens (welche die „Elemente“ von M genannt werden) zu einem
Ganzen.

( Georg Cantor um 1800 )

Man braucht das Zeug, um Beweise mit reellen Zahlen R führen zu
können.

Aber wohl weiß hier keiner, wieso man R braucht, oder ?

..


Die Existenz von R baut sich auf die " Dedekind'schen Schnitte " auf -
kurz gesagt :

Wenn eine ein 10 cm mal 10 cm großes Papierstück diagonal
durchschneidet ( +++ der Physiker WEISS, daß man das NICHT KANN +++ )
und dann die Flàche eines der beiden Teile in Quadratzentimeter
ausdrücken will, dann
kann man den Wert NICHT IN Q FINDEN !

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