Another question about infinity

[Steven G]

Suppose you have a HUGE box and a countable infinite number of balls numbered 1,2,3,4,...

Step 1: Put into the box balls numbered 1-10 and then remove ball # 1.
Step 2: Put into the box balls numbered 11-20 and then remove ball # 2.
...
Step n: Put into the box balls numbered (10(n-1) + 1)-10n and then remove ball # n.

Now I agree that if you let go to infinity that there will be no balls left.

If instead of removing ball i in step i, we remove that last ball we put in the box during step i.
Now there will be an infinite number of ball.

Now back to the 1st scenario. Lets define B_n to be the number of balls after step n. Clearly B_n = 9n.

Now here is where I get lost. [math] lim x-->\infty (9n) = \infty.[/math] But this limit should be 0 as indicated above.

This is troubling me as I can't seem to justify what is going on to my satisfaction. The only thing that I can come up with is that at step n there are many balls in the box with a higher number than n. That is the ball n that is removed never is the last ball in the box. Is that what is happening. Infinity can be strange but is still fun to look at!

 

[lev888]

Now I agree that if you let go to infinity that there will be no balls left.

Please explain. If at each step you add 9 balls, how can the amount decrease?
Also, any box holds a finite number of balls, assuming their size is not zero.

 

[Dr.Peterson]

Here is a discussion about this problem: Infinite Sums and Empty Sets.

In the course of that discussion, it is mentioned that this is called the Ross-Littlewood paradox. You will observe that there is no one right answer given in Wikipedia.

 

[lev888]

Also, any box holds a finite number of balls, assuming their size is not zero.

Hmm, maybe not, if the size decreases fast enough.

 

[Steven G]

Please explain. If at each step you add 9 balls, how can the amount decrease?
Also, any box holds a finite number of balls, assuming their size is not zero.

Ball number n was removed at step n and this is true for any n. So at the end there are no balls in the infinite sized box. If you do not believe this then all I ask is that you tell me the number of any ball that is left. For the record my answer would be that the ball i that you claim is left was actually removed in step i.

 

[lev888]

Ball number n was removed at step n and this is true for any n. So at the end there are no balls in the infinite sized box. If you do not believe this then all I ask is that you tell me the number of any ball that is left. For the record my answer would be that the ball i that you claim is left was actually removed in step i.

We don't care about any particular ball. All that matters is the number of balls in the box at step n.

 

[lev888]

We don't care about any particular ball. All that matters is the number of balls in the box at step n.

Claim: the Amazon river is empty at infinity (let's assume the Earth is still around then).
Proof: pick any H₂O molecule. "At the end" it's in the ocean. QED.

 

[Steven G]

We don't care about any particular ball. All that matters is the number of balls in the box at step n.

Mathematicians agree that there are no balls in the end. I do not mind at all that you disagree with me but I am peasant in terms of being a mathematician. I think this time you should discuss this with a mathematician.

Please read this here

 

[JeffM]

Mathematicians agree that there are no balls in the end. I do not mind at all that you disagree with me but I am peasant in terms of being a mathematician. I think this time you should discuss this with a mathematician.

Please read this here

Actually, if you read the wiki article cited by Dr. P, there is no agreement among mathematicians. The positions range from no balls, an infinite number of balls, any finite number of balls, the problem is underspecified, and the problem is misspecified. I think that covers the range pretty well.