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
finally by Cory Taylor)
Cory,
If you're happy with my response... Great! But your message indicates an underlying dissatisfaction with the resolution.
________________
"...the math is irrelevant here."
No. It is relevant. Which is why if one misapplies concepts (as you have done), one might be led to erroneous results.
_________________
"...it is the interpretation of the problem itself from where the paradox is derived..."
Actually, there is no paradox, in the mathematical sense (even if Levik chose to place the problem in that category). There is a paradox if one doesn't understand what the problem is really asking, generally due to a lack of understanding of the concepts with which the problem deals (infinities).
_______________
"I have chosen... to ignore different constraints."
You have chosen to ignore constraints, but not conststraints imposed by the problem. You have chosen to ignore constraints imposed by the logic describing infinities. I have not ignored any constraints.
______________
"I don't agree that I can't create an infinite sum by infinitly adding a constant term. This is a basic tenet of calculus"
No. Calculus (well Riemann Sums anyway) does not add an infinite number of items. It deals with the limit of a summation as a variable approaches infinity (or zero depending on which variable you are looking at). This is an important distinction. (By, the way, ∑9 as n goes from 1 to infinity approaches infinity, no argument here.... it just doesn't apply to the problem at hand.)
_______________
"...this aleph null is defined as the smallest infinity - what heppens[sic] to this if you take away one?"
You question results from a misunderstanding of infinity. Aleph Null is not an integer. It describes a set ONTO which and FROM which I can provide a mapping from/to other sets. It is equivalent to the set of integers, or the set of positive even integers, or the set of prime numbers. There is no infinite set for which I cannot provide a mapping onto Aleph Null (for if there were, it would be an infinite set smaller than Aleph Null). And I am not going to include a proof here.
Aleph One, which is equivalent to the set containing the real numbers is larger and while there exists a mapping of the Aleph One set (e.g., real numbers) ONTO an Aleph Null set (the integers), there does not exist the reverse. Therefore Aleph One is larger than Aleph Null.
______________________
I encourage you to take the initiative and learn more about the subject.
--- SK
re: finally
