Re: Infinity : An interesting variant
- From: "Verdigris" <etho3681@xxxxxxxxxxxxxx>
- Date: Sun, 19 Mar 2006 00:42:02 GMT
apoorv
This is a succinct demonstration of fighting fire with fire ie of infinity
overcoming itself through a process of unlimited (mental) acceleration.
I see no problem in the result that is not already inherent in the Cantorian
theory of infinity. It seems to me that the final interval is:
[t(u-1), t(u)], where t (u-1) = (r =1to r = w-1)Sum 2^-r. (nb: u-1 stands
for penultimate)
The inherent difficulty lies in Cantor's idea that the set of natural
numbers is a satisfactory yardstick. The final ball takes 2^-w hours to
transfer from vase A
to vase B, but the inclusion of ball zero would exceed your neat figure of 1
hour, as would the inclusion of an arbitrary number of extra balls in vase
A. (try recasting the model with extra balls).
The weakness of this model is that it assumes that there is only on
infinity, which continues only as far as the smallest rational fraction
1/2^w. The model subverts the conventional mathematical sum to 'infinity'
used in evaluating a declining geometric progression. This infinity is
covertly absolute infinity, not the one cowering below the symbol w. Your
model works for any finite or limited but infinite number of balls: the
issue of a zero time interval does not arise.
It would be interesting to replicate the mental experiment with two jars,
one containing w black balls (odd numbers) and the other w white balls (even
numbers). The black and white balls being transferred alternately to the
empty jar.
Kind regards
Tony Thomas
"apoorv" <sudhir_sh@xxxxxxxxxxx> wrote in message
news:15199779.1142668883163.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
I am not sure whether this variant of the balls in vase problem been
discussed before in this forum; so, in any case here it is.
Consider a vase A with an infinite number of balls numbered from 1 onwards
with a ball numbered w (omega) as well.We have another vase B which is
empty.
Precisely at 00 hrs , we start transferring balls from vase A to vase B as
follows:
Time Interval-------------Ball Number
[00,1/2)----------------------1
[1/2,1/2+1/4)-----------------2
[3/4,3/4+1/8)-----------------3
and so on.(We can presume that the transfer of the ball is complete just
at the beginning of the corresponding interval.)
Precisely at 01hrs, we transfer the ball numbered w to the vase B.
For every initial interval of natural numbers, there is a precisely
defined sub-time interval of [00,01) during which the numbers on the balls
in vase B constitute that interval of natural numbers.The set in vase B
during any interval [01,t) is {1,2,3. . .w}.
Thus, there does not appear to be any instant (leave alone an interval)
when the contents of vase B are {1,2,3. . .}.
How does the transition from finite intervals to the infinite interval
{1,2,3...w} take place without ever going through {1,2,3. . .}?
-Apoorv
.
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