Das Kalenderblatt 091008

07/10/2009 - 13:34 von WM | Report spam
The induction principle is this: if a property holds for 0, and if
whenever it holds for a number n it also holds for n + 1, then the
property holds for all numbers. For example, let t(n) be the property
that there exists a number m such that 2 * m = n * (n + 1). Then t(0)
(let m = 0). Suppose 2 * m = n * (n + 1). Then 2 * (m + n + 1) = (n +
1) * ((n + 1) + 1), and thus if t(n) then t(n + 1). The induction
principle allows us to conclude t(n) for all numbers n. As a second
example, let p(n) be the property that there exists a non-zero number
m that is divisible by all numbers from 1 to n. Then p(0) (let m = 1).
Suppose m is a non-zero number that is divisible by all numbers from 1
to n. Then m * (n + 1) is a non-zero number that is divisible by all
numbers from 1 to n + 1, and thus if p(n) then p(n + 1). The induction
principle would allow us to conclude p(n) for all numbers n.
The reason for mistrusting the induction principle is that it
involves an impredicative concept of number. lt is not correct to
argue that induction only involves the numbers from 0 to n; the
property of n being established may be a formula with bound variables
that are thought of as ranging over all numbers. That is, the
induction principle assumes that the natural number system is given. A
number is conceived to be an object satisfying every inductive
formula; for a particular inductive formula, therefore, the bound
variables are conceived to range over objects satisfying every
inductive formula, including the one in question.
In the first example, at least one can say in advance how big is
the number m whose existence is asserted by t(n): it is no bigger than
n * (n + 1). This induction is bounded, and one can hope that a
predicative treatment of numbers can be constructed that yields the
result t(n). In the second example, the number rn whose existence is
asserted by p(n) cannot be bounded in terms of the data of the
problem.
lt appears to be universally taken for granted by mathematicians,
whatever their views on foundational questions may be, that the
impredicativity inherent in the induction principle is harmless—that
there is a concept of number given in advance of all mathematical
constructions, that discourse within the domain of numbers is
meaningful. But numbers are symbolic constructions; a construction
does not exist until it is made; when something new is made, it is
something new and not a selection from a pre-existing collection.
There is no map of the world because the world is coming into being.
[Edward Nelson: "Predicative arithmetic", Princeton University Press,
Princeton (1986)]
http://www.math.princeton.edu/~nelson/books/pa.pdf
http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT
Folie 63

Gruß, WM
 

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#1 Rudolf Sponsel
07/10/2009 - 15:01 | Warnen spam
WM schrieb:
The induction principle is this: if a property holds for 0, and if


...
http://www.math.princeton.edu/~nelson/books/pa.pdf



Danke für den Quellen-Hinweis, habe das Werk mal runtergeladen.

http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT
Folie 63



In dem Zusammenhang
http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT (Folie 64)
interessiert mich a) die Quelle des Zitates von William Thurston (*1946)
Topologe, Tràger der Fields-Medaille.

b) Erweitert: vor zwei Tagen rief mich einer an, der gegenwàrtige MATHEMATIKER
sucht - also nicht Klassiker wie Gauß, Poincare oder Weyl - die sich explizit
gegen die contradictio in adjecto Definition der "aktual" unendlichen Menge
wenden. Ich habe ihn an Dich verwiesen, bin aber selber an der Frage
interessiert. Nik Weaver scheint nach Deiner Seite (auch) dazuzugehören. Gibt
es irgendwo eine Liste (die ich gerne in meine Grundlagenstreitseite einbauen
würde)?

Gruß, WM



Vielen Dank, Gruß aus Erlangen
Rudolf Sponsel

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