Re: infinity

Jesse F. Hughes said:
>> Let's change the problem slightly. Again, we have an infinite set of
>> ping pong balls, each ball labeled with a natural number. But instead
>> of the old procedure, let's put *all* of the balls into the vase at
>> 11:59 and remove the first one. At 11:59:45, we remove the second,
>> and so on.
>> [...]
>>
>> in both examples (the ten-at-a-time example and the all-at-once
>> example), we put the same number of balls in the vase: one ball for
>> each natural number. And in both examples, we removed the balls in
>> exactly the same way. How can the outcome be any different?
>
Tony Orlow (aeo6) wrote:
> You didn't remove the balls the same way. In the original example, every
> time you removed a ball you added ten. How can that be ignored? If you
> started with an infinite number, then removed one and added ten, you would
> NOT have zero at noon. Or, would you?

Let's pose a slightly different problem:
1. At one minute before noon, add 10 balls to the vase.
2. 30 seconds later, add 10 balls to the vase.
3. Repeat at each moment 1/2 as close to noon as the previous step,
adding 10 balls to the vase at each step, until noon is reached.

So far, so good? At noon, there should be an infinite number of balls
in the vase, right?

Now, continue:
5. At one minute before 1:00, remove 1 ball.
6. 30 seconds later, remove 1 ball.
7. Repeat at each moment 1/2 as close to 1:00 as the previous step,
removing 1 ball from the vase at each step, until 1:00 is reached.

Now at 1:00, how many balls are left in the vase?

This problem is somewhat simpler than the orginal, since the
balls do not need to be labeled.


It's obvious that by noon there are an infinite number of balls,
since 10 balls were added at each step for an infinite number of
steps.

It should also be obvious that by 1:00 there are zero balls left
in the vase, since 1 ball was removed at each step for an infinite
number of steps.

I expect some to claim that there are an infinite number (or
perhaps 9 x infinity) balls left in the vase at 1:00; if so,
please explain how that is the case.

.

Relevant Pages

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