You have an empty container, and an infinite number of marbles, each numbered with an integer from 1 to infinity.
At the start of the minute, you put marbles 1 - 10 into the container, then remove one of the marbles and throw it away. You do this again after 30 seconds, then again in 15 seconds, and again in 7.5 seconds. You continuosly repeat this process, each time after half as long an interval as the time before, until the minute is over.
Since this means that you repeated the process an infinite number of times, you have "processed" all your marbles.
How many marbles are in the container at the end of the minute if for every repetition (numbered N)
A. You remove the marble
numbered (10 * N)
B. You remove the marble numbered (N)
(In reply to
re: more by Brian Wainscott)
but my point exactly was that a marble is a marble is a marble... It is the labels themselves that cause the problem. Reading carefully you've put all the marbles into the bag, and then you take one out. I can see your argument in the case where for each iteration you put in the 9 smallest numnered marbles remaining in your stock that have a remainder when divided by ten, but throw out the lowest numbered remaining marble that is divisible by ten, but this is fundamentally different than what is happening. Each iteration results in 10 balls in and then one ball out. An infinite number of iterations will not allow you to reduce the number of marbles to zero. The mathematics that supports this (and it is valid once the nomenclature has been derived) is flawed because of a numbering scheme that we have imposed on the situation - the actual sitation does not support the solution.
You may ask what possible naming convention could be derived to sidestep this discrepancy and that would be an excellent question for which I certainly dont have an answer (there may not be any such convention), but that doesn't mean that you can ignore the flaw.
The basic problem is that you're considering all infinities to be equal, which is not the case. Infinity minus infinity is not zero - it is some undefined quantity that can only be derived by knowing where the infinities came from in the first place. In this case the knowledge of where the infinities comes from leads to a simple mathematical solution to the problem, not some philosophical debate.
As an example, consider the process of normailizing number (if you're familiar with it). Physicist use this slution method quite frequently to remove infinities form equations - they do so by dividing equations by infinity. When they do so, they must choose their used infinity VERY cartefuly so that they produce an answer with relevance - simply dividing by the first infinity that comes along will only multiply their problems.
Lastly, as counter to the argument you've given of picking a marble "n", well what if I pick the marble infinity? The core of the problem - which infinity did I pick? You've limited your set size by forcing me to pick from within a limited set of numbers, and it is this exact limiting which is clouding your solution.
re(2): more
