Verbindung hergestellt.connected.
num: 29583
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GRUPPE: de.sci.mathematik
FROM : Moebius
DATE : Mon, 24 Nov 2025 18:41:14 +0100
TEMA : =?UTF-8?Q?Supertasks_-_Das_Ross=E2=80=93Littlewood-Paradox?=
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... auch bekannt als Vasenproblem. Daran hat sich auch der gute
Mückenheim schon (erfolglos) versucht.
Tatsächlich beruht das "Paradox" vor allem auf einem "psychologischen
Showeffekt":
The problem starts with an empty vase and an infinite supply of balls.
An infinite number of steps are then performed, such that at each step
10 balls are added to the vase and 1 ball removed from it. The question
is then posed: /How many balls are in the vase when the task is finished/?
To complete an infinite number of steps, it is assumed that the vase is
empty at one minute before noon, and that the following steps are performed:
- The first step is performed at 30 seconds before noon.
- The second step is performed at 15 seconds before noon.
- Each subsequent step is performed in half the time of the previous
step, i.e., step n is performed at (1/2)^n minutes before noon.
This guarantees that a countably infinite number of steps is performed
by noon. Since each subsequent step takes half as much time as the
previous step, an infinite number of steps is performed by the time one
minute has passed. The question is then: /How many balls are in the vase
at noon?/
Und weiter:
"The most intuitive answer seems to be that the vase contains an
infinite number of balls by noon, since at every step along the way more
balls are being added than removed. By definition, at each step, there
will be a greater number of balls than at the previous step. There is no
step, in fact, where the number of balls is decreased from the previous
step. If the number of balls increases each time, then after infinite
steps there will be an infinite number of balls."
"Intuitiv", aber (in dieser Form) falsch.
Man kann diesem "Deppenargument" den Wind aus den Segeln nehmen, indem
man auf den "psychologischen Showeffekt" verzichtet - nämlich dem, that
the number of balls increases each time.
Absolut überflüssig und irreführend. Man kann stattdessen von Anfang an
von einer Vase mit (abzählbar) unendlich vielen balls ausgehen!
Also:
The problem starts with an vase containing countably infinitely many
balls. An infinite number of steps are then performed, such that at each
step 1 ball is removed from the vase. The question is then posed: /How
many balls are in the vase when the task is finished/?
To complete an infinite number of steps, it is assumed that the vase
contains countably infinitely many balls one minute before noon, and
that the following steps are performed:
- The first step is performed at 30 seconds before noon.
- The second step is performed at 15 seconds before noon.
- Each subsequent step is performed in half the time of the previous
step, i.e., step n is performed at (1/2)^n minutes before noon.
This guarantees that a countably infinite number of steps is performed
by noon. Since each subsequent step takes half as much time as the
previous step, an infinite number of steps is performed by the time one
minute has passed. The question is then: /How many balls are in the vase
at noon?/
Damit fällt das "intuitive" Deppenargument schon mal auf die Nase.
Vermutlich würde ein Mückenheim nun so argumentieren: "Da zu jedem
Zeitpunkt nur endlich viele Bälle aus der Vase genommen wurden, aber
(immer noch) unendlich viele Bälle in der Vase verbleiben, müssen "am
Ende" auch (immer noch) unendlich viele Bälle in der Vase sein - WIE
KÖNNTE ES ANDERS SEIN?"
Yeah, wie könnte es in Mückenheims Welt auch anders sein. :-)
So vielleicht?
Suppose that the balls in the vase were numbered [starting with 1, 2, 3,
... and so on], and that at step 1 ball number 1 is removed. At step 2
ball 2 is removed. This means that by noon, every ball labeled n is
removed at step n. Hence, the vase is empty at noon. This is the
solution favored by [me].
Natürlich GILT AUCH:
The number of balls that one ends up with depends on the order in which
the balls are removed from the vase. As stated previously, the balls can
be removed in such a way that no balls will be left in the vase at noon.
However, if ball number 1 were removed from the vase at step 1, ball
number 3 at step 2, ball number 5 at step 3, and so forth, then it is
clear that there will be an infinite number of balls left in the vase at
noon (namely the balls numbered 2, 4, 6, ...). In fact, depending on
which ball is removed at the various steps, any chosen number of balls
can be placed in the vase by noon [...]
Jetzt muss natürlich -wie könnte es anders sein- ein Philosph
dazwischengrätschen:
Although the state of the balls and the vase is well-defined at every
moment in time prior to noon, no conclusion can be made about any moment
in time at or after noon. Thus, for all we know, at noon, the vase just
magically disappears, or something else happens to it. But we don't
know, as the problem statement says nothing about this. Hence, like the
previous solution, this solution states that the problem is
underspecified, but in a different way than the previous solution. This
solution is favored by philosopher of mathematics Paul Benacerraf.
WTF?!
Ja, gut, dann NEHMEN WIR halt FOR THE SAKE OF THE ARTGUMENT *explizit*
an, dass die Vase NICHT einfach so verschwindet "[before,] at or after
noon". Heiliges Blechle! Und dass auch die balls NICHT wieder in die
Vase zurückspringen, wenn sie einmal herausgenommen wurden. SO, sind wir
JETZT zufrieden, Herr Benacerraf?
Bleibt noch ein "Einwand" eines weiteren Philosophen, der auch
Mathematiker ist, auf den ich hier nicht näher eingehen möchte.
Lit.: https://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox
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