Das Kalenderblatt 100113

12/01/2010 - 12:58 von WM | Report spam
Meine FOM-Episode (13)

{{Als Reaktion auf meinen Beitrag
Who was the first to accept undefinable individuals in mathematics?
http://www.cs.nyu.edu/pipermail/fom...13464.html
korrigierte Bill Tait am 13. 3. 2009 seinen ersten Beitrag:}}

My reference to Bolzano and Cauchy in the message (from March 11) {{s.
Meine FOM-Episode (12)}} isn't really sound (although the main point,
that definability is relative to a language, is). The Cauchy sequence
defining a zero of the continuous real-valued function f defined on
[a,b] with f(a)<0<f(b) is defined (by a nested interval construction)
in terms of f. So in that sense, the zero is defined in terms of f.

A better example is from Cantor's original paper on the non-
denumerability of the reals. He concluded that there is a
transcendental number between a and b (when a<b) and made a point of
the fact that this result is significant even if it does not yield a
definition of such a transcendental. {{Genau den point machte er
nicht!}} The transcendental can in fact be defined in terms of an
enumeration of all the algebraic numbers between a and b; but in
Cantor's time, before Sturm's Theorem, no such enumeration could be
defined.

{{Meine Antwort ging direkt an Bill Tait (und an das FOM-Forum. Dort
wurde sie allerdings abgelehnt.)}}
It is not a matter of a theorem but a matter of practical ability
whether a definition can be given for some purpose. (With "definition"
here I mean "definition using a finite number of bits", which meaning
is independent of the language used.) The transcendental in the above
example cannot be defined.
First, there is no chance to give a complete but finite enumeration of
all algebraic numbers of an interval between a and b as individuals as
it would be required to isolate a remaining transcendental.
Second, there are many transcendental numbers in that interval such
that even a complete enumeration of all algebraics would not help to
identify one of the transcendentals.

Gruß, WM
 

Lesen sie die antworten

#1 Kick Em Off
13/01/2010 - 11:43 | Warnen spam
On 12 Jan., 12:58, WM wrote:
Meine FOM-Episode (13)

{{Als Reaktion auf meinen Beitrag
Who was the first to accept undefinable individuals in mathematics?http://www.cs.nyu.edu/pipermail/fom...13464.html
korrigierte  Bill Tait am 13. 3. 2009 seinen ersten Beitrag:}}

My reference to Bolzano and Cauchy in the message (from March 11) {{s.
Meine FOM-Episode (12)}} isn't really sound (although the main point,
that definability is relative to a language, is). The Cauchy sequence
defining a zero of the continuous real-valued function f defined on
[a,b] with f(a)<0<f(b) is  defined (by a nested interval construction)
in terms of f. So in that sense, the zero is defined in terms of f.

A better example is from Cantor's original paper on the non-
denumerability of the reals. He concluded that there is a
transcendental number between a and b (when a<b) and made a point of
the fact that this result is significant even if it does not yield a
definition of such a transcendental. {{Genau den point machte er
nicht!}} The transcendental can in fact be defined in terms of an
enumeration of all the algebraic numbers between a and b; but in
Cantor's time, before Sturm's Theorem, no such  enumeration could be
defined.

{{Meine Antwort ging direkt an Bill Tait (und an das FOM-Forum. Dort
wurde sie allerdings abgelehnt.)}}
It is not a matter of a theorem but a matter of practical ability
whether a definition can be given for some purpose. (With "definition"
here I mean "definition using a finite number of bits", which meaning
is independent of the language used.)  The transcendental in the above
example cannot be defined.
First, there is no chance to give a complete but finite enumeration of
all algebraic numbers of an interval between a and b as individuals as
it would be required to isolate a remaining transcendental.
Second, there are many transcendental numbers in that interval such
that even a complete enumeration of all algebraics would not help to
identify one of the transcendentals.

Gruß, WM



das erzeugt bei mir wie beim atom eine induzierte emission.

sonst hab ich immer eine spontane emission.

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