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(3): more by SilverKnight)
my definitions;
by saying that the value of infinity is not static, I mean that while adding one to a particular "instance" of infinity results in the same infinity (i.e. if you were allowed to represent infinity as a variable, then x+1=x), that infinity is not at all related to a different infinity (x<>x).
comments referring to the "value" of infinity referred to the number of elements. Yes this was fuzzy - I apologize. Now - while both the sets in question had an infinite size, it is not true that the subtraction of the two sets left null. This was the point of that comment, and right to the heart of my argument. We are dealing with completely different infinities with the addition of the ten marbles than we are with the subtraction of the marbles. When you quantize (which of course you can't do, but remember, we've suspended disbelief) the adding marbles infinity minus the subtracting marbles infinity you do not end with the empty set., hence the jar isn't empty...
The reason that I've mentioned that im not going to debate the intricacies or definitions of math is simply because I'm not familiar with them. I have no understanding of cardinality (if thats the term) nor can I give you the definition of a set. This does not mean than my approach is invalid - by all means if the reasoning is flawed im happy to leave it at that.
Your final comment about the sum of infinite 9's is a reference to the functional aspect of this problem. Reading the problem, it boils down to infinite iterations of a process. Each discreet iteration produces a net increase of 9 marbles in the jar. Without respect to labels, 10 go in and 1 comes out, resulting in my magical 9. This result is independent of the iteration number, and so the expression derived for the total number of marbles in the jar after any number of iterations is 9*n. As n approaches (and we've conceded, reached) infinity this number approaches 9*infinity. This expression greatly disputes the claim that there are no marbles left in the jar. Either one or the other must be wrong. I'm not here to tell you where the error lies in your method (I've stated that i don't have the proper background), simply that there is one.
To hazard a guess, I would say that the problem itself has asked the results of an impossible situation, which, of course, is why the problem is listed in paradoxes. We have all seen MANY "proofs" that 1=0, or some other similar fallacious statement. Very often I can find the place where the proof becomes invalid, and if not then I generally have an idea. In those rare cases where I can't, does that mean that the math is in fact valid? Obviously no. This is the case here - your math is flawed, as it produces a flawed result. From the first iteration, the number of marbles is positive, and it only increases. When you start with a positive value (after any iteration of your choice, the number of marbles in the jar is positive) and increase it for an infinite period of time this CANNOT produce a null value.
Edited on September 16, 2003, 4:15 am
re(4): more
