Balls In BarrelsYou are currentlybrowsing as guest. Click here to log in 

When dealing with infinity, there are things that go wrong if you rely on your intuition. This page describes one of them. You may also want to see the page where Cantor visits Hilbert's Hotel.
I wonder if he indulges in the sport of StackingBlocks?
This gives us lots of specific times, all of them before midnight. If you think of any large number, there are more times than that. Loosely speaking, we would say that there are infinitely many times.
Suppose we have a barrel with lots of balls in it, and they're labelled 1, 2, 3, 4, etc. You'll notice that for any number you think of, there are more balls than that in the barrel.
We work to the following timetable. These actions take place at successive times as defined above.
Step number  Time  Action  Balls outside 
   
1  23:00:00  Remove 1 & 2  1, 2 
2  23:30:00  Return ball 2  1 
3  23:45:00  Remove 3 & 4  1, 3, 4 
4  23:52:30  Return ball 4  1, 3 
5  23:56:15  Remove 5 & 6  1, 3, 5, 6 
6  23:58:07.5  Return ball 6  1, 3, 5 
7  23:59:03.75  Remove 7 & 8  1, 3, 5, 7, 8 
7  23:59:03.75  Remove 7 & 8  1, 3, 5, 7, 8 
8  23:59:31.875  Return ball 8  1, 3, 5, 7 
9  23:59:45.9375  Remove 9 & 10  1, 3, 5, 7, 9, 10 
10  23:59:52.96875  Return ball 10  1, 3, 5, 7, 9 
11  23:59:56.484375  Remove 11 & 12  1, 3, 5, 7, 9, 11, 12 
12  23:59:58.2421875  Return ball 12  1, 3, 5, 7, 9, 11 
13  23:59:59.560546875  etc.  ... 
At each odd time step we take out two balls, and at each even time step we replace the highest numbered ball.
At any given moment we know exactly which balls are in the barrel and which balls are outside. At 1 minute to midnight balls 1 through 6 have been removed, and balls 2, 4 and 6 have been replaced. At 1 second to midnight balls 1 through 12 have been removed and balls 2,4,6,8,10 and 12 have been replaced. At 1 nanosecond to midnight balls 1 through 42 have been removed and the even balls up to 42 have been replaced. And so on.
So what is the situation at one minute past midnight?
Once again we take balls out and put them back. This time:
Step number  Time  Action  Balls outside 
   
1  23:00:00  Remove 1 & 2  1, 2 
2  23:30:00  Return ball 1  2 
3  23:45:00  Remove 3 & 4  2, 3, 4 
4  23:52:30  Return ball 2  3, 4 
5  23:56:15  Remove 5 & 6  3, 4, 5, 6 
6  23:58:07.5  Return ball 3  4, 5, 6 
7  23:59:03.75  Remove 7 & 8  4, 5, 6, 7, 8 
8  etc.  Return ball 4  5, 6, 7, 8 
Again, at any given moment we know exactly which balls are in the barrel and which balls are outside. At 1 minute to midnight balls 1 through 6 have again been removed, but now balls 1 to 3 have been replaced. At 1 second to midnight balls 1 through 12 have been removed and balls 1 through 6 have been replaced. At 1 nanosecond to midnight balls 1 through 42 have been removed and balls 1 through 21 have been replaced. And so on.
So what is the situation at one minute past midnight? You name any number and I can tell you both when it was taken out and when it was put back.
By the time midnight comes around every ball has been put back in the barrel.
This doesn't sound right. Leading up to midnight the number of balls outside the barrel is increasing, and yet after midnight they're all back in the barrel? Yes, it's very definitely unexpected, definitely not what your intuition tells you, but it's true.
So now let's rub off the numbers from the balls, and at each of our points in time we remove two balls and then put one back in. At 1 minute to midnight there are 6 balls outside. At 1 second to midnight there are 12 balls outside. At 1 nanosecond to midnight there are 21 balls outside.
How many are outside of the barrel after midnight? We can't tell !!
Without being explicit, without actually naming the balls and saying which ones are being removed and replaced we simply cannot tell. Case 1 says it might be infinitely many, case two says it might be none. We simply don't know because we don't have enough information.
Weird  but that's the strange world of infinity.
Last change to this page Full Page history Links to this page 
Edit this page (with sufficient authority) Change password 
Recent changes All pages Search 