Das Kalenderblatt 091003

02/10/2009 - 11:36 von WM | Report spam
Several examples are used to illustrate how we deal cavalierly with
infinities and unphysical
systems in physics.
[...] At this point, we can ask what all this {{Cantors
Unendlichkeiten}} implies for physics and physical systems. There are
significant implications for physical systems if we accept the major
premise that rather than being merely potential, real infinities do
exist. We shall now cite some examples to illustrate some of these
The case of an infinitely long line charge distribution is a simple
example. In this case both the length and the electric charge are of
cardinality C for the continuum and consequently the ratio lambda
representing the charge density is finite. The case of the
thermodynamic limit invoked in statistical mechanics is a fascinating
counter-example. If we treat them as real infinities, then the ratio N/
V involves the number of atoms which is countable and thus of
cardinality aleph_0 whereas the volume of the system is of cardinality
C x C x C = C. For this ratio to be finite we must require that C must
equal a finite number times aleph_0. This last statement is however,
not true for real infinities. Hence it follows that the finiteness of
the number density N/V would not have been possible if N and V were
real infinities. Indeed the finiteness of the number density follows
from the fact that each of these quantities is finite to begin with,
as was discussed earlier.
[...] We conclude by making a few general observations.
Thermodynamic systems are devoid of infinities and are inherently
finite. N is countable and
large and V is measurable and macroscopically large but all physical
parameters are finite and
measurable and finite, including the number density. There is a class
of physical systems containing infinities but which can be re-examined
by using methods which have successfully prevailed in thermodynamics
and statistical mechanics, with a view to resolve the problem
infinities in these systems.
[...] Finally there may be physical systems containing real
infinities which cannot be transformed
away. These systems may perhaps be understood only by a re-examination
based on Cantor’s
transfinite mathematics. {{May or may not.}} In this context, it is
useful to remember that Cantor’s theory is well grounded in physical
reality {{aha, oder besser noch: oho! Wie das?}}: it is based on
arithmetic and set theory. {{Das gilt ebenso für Astrologie,
Numerologie und Traumdeutung, zum Beispiel bei Traumspinnen. Aber ein
Argument für den Eintritt in eine der Sekten bietet diese Arbeit
sicher nicht.}}

{{Anmerkung 9}} An interesting point in the history of mathematics is
the continuum hypothesis: that there is no cardinality bigger than
aleph_0 and smaller than C. This notion is neither proved nor
disproved. {{Meanwhile it has been proved: there is no cardinality
bigger than aleph_0. See, for instance, Das Kalenderblatt 090923.}}

[P. Narayana Swamy: "Infinities in Physics and Transfinite numbers in
Mathematics" (1999)]

Gruß, WM

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#1 Mengenlehrer
03/10/2009 - 00:26 | Warnen spam
WM schrieb:
Meanwhile it has been proved: there is no cardinality
bigger than aleph_0. See, for instance, Das Kalenderblatt 090923.

Da können wir ja froh sein, dass wir
die Große-Kardinalzahl-Axiome haben.
Damit gehen uns kleingeistige "Beweise" am Ar*** vorbei.

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