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: circles
by Cory Taylor)
Let me quote you again, with responses:
"Further, having stopped processing your marbles, there must have been a last iteration"
No, your statement is a non-sequitor. You must differentiate between "stopped processing" and "no longer processing". The first implies there was a last. There is no last. And the flaw in your reasoning is that you assume there is/was.
"you've made no attempts to ever dispute the reasoning Iíve come up with "
I certainly have, right here where I explain what I see as flaws in your logic...
and so has Brian Wainscott right here where he points to things you have said and explains what he sees as flaws.
"I have not violated or misused the term infinity in my method."
Yes, you have. here you wrote that two particular infinities are not the same size (when they are), and here you refer, again, to the last element of a set of infinite elements.
This is somewhat analogous as saying "a mile north of the north pole..." In either case, the terminology applied in this situation simply implies a misunderstanding of the issue. North of the north pole has no meaning. The "highest integer" also has no meaning (which is essentially what you're saying).
"You have considered that infinity is a number that can be worked with. Not so. Infinity is not a number, it is an idea."
Neither Brian, nor I, nor did Levik (in the solution) suggest that infinity is a real number (see definition of a real number). But you are correct in that it is an idea. What's more it is an idea that is useful in mathematics. And is generally dealt with by utilizing set theory and mappings--more about this later.
"There are an infinite numbers in fact, both greater than and less than infinity, and wouldn't you know it - there all called infinity."
But there are different cardinalities of infinities. Now, if you so choose not to learn about them... well fine.
"If you wish to convince me, then find the hole in my reasoning."
I believe I have. You are misapplying terms (such as "last marble") and concepts (such as requiring an infinite time to deal with infinite marbles, when we don't need infinite time, we only need infinite iterations), and of course, if we are allowed to misapply concepts we can show anything we wish. If you wish to talk about infinities that apply the "legal operations" and proper terminology in discussing them.
"Saying that youíre labeling marbles one to infinity (which would then include infinity, as marbles 1 to 5 include the marble 5) defies the definition of the term infinity."
By "labeling" them, we attempt to simply the problem (wow... imagine that :-). I agree that Levik could have used a more rigourous statement than that, but I think it is clear that what is meant by the statement is: Each marble has a unique positive integer on it and every integer is represented. If this restatement pleases you, then please use this instead, and analyze the problem again.
"If you finish, there is a last. If there is a last, there was never an infinity. "
No... another non-sequitor. Again, I agree that Levik could have used more rigorous language, but I think the meaning is clear.... the nth iteration took place 1/n minutes before the end of the minute. Period. There IS NO LAST. If you wish to ask about any given iteration, I can tell you when it occurred. And yes, of course, we could not really do this in practice.
After all this (and I think I have DEFINITELY addressed your major points), it seems clear that the only useful way to talk about infinity is to UNDERSTAND what infinity is.
As you point out... it is NOT a real number. So, let's STOP talking about the LAST marble.
As I've pointed out it is generally dealt with by referring to elements. That is how we've answered the problem, and set theory and operations is a natural way to describe that. You use inapplicable language and concepts. If you CHOOSE not to use applicable language to describe the situation, then what more can I say? Perhaps the best thing I can suggest is that you read up on the subject to better understand what the problem is asking.
P.S. You're right... there ARE an infinite number of infinities greater than Aleph Null (the lowest infinity).
Edited on September 17, 2003, 4:46 am