Re: infinity

On Wed, 03 Aug 2005 12:20:06 +0200, Jeroen Boschma
<jeroen.boschma@xxxxxx> wrote:

>Theo Jacobs wrote:
>>
>> Hello everyone,
>>
>> I'm having an argument with a friend about the following problem:
>>
>> Suppose you have a giant vase and a bunch of ping pong balls with an
>> integer written on each one, e.g. just like the lottery, so the balls
>> are numbered 1, 2, 3, ... and so on. At one minute to noon you put
>> balls 1 to 10 in the vase and take out number 1. At half a minute to
>> noon you put balls 11 - 20 in the vase and take out number 2. At one
>> quarter minute to noon you put balls 21 - 30 in the vase and take out
>> number 3. Continue in this fashion. Obviously this is physically
>> impossible, but you get the idea. Now the question is this: At noon,
>> how many ping pong balls are in the vase?
>
>Each time that you do something with the balls, you add 10 to the vase and take one out, so the
>number of balls in the vase increases by 9.
>
>Let T(n) be the time left until it's noon at the n-th time that you add balls to the vase, then T(n)
>= 2^(1-n). This function never reaches noon (T=0), but only approaches it for n->infinity. So the
>number of balls in the vase keep growing to infinity as you approach noon.

Your formula for T(n) is wrong. But that doesn't matter. It's true
that the limit of T(n) as n -> infinity is infinite. And that
doesn't change the fact that the vase is empty at noon.

(You're argument would be correct if cardinality were "continuous",
in the sense that the cardinality of the limit of a sequence of sets
is the limit of the cardinailities. But cardinality is simply not
continuous in this sense.)

>> An 2nd experiment goes as follows:
>> put no. 1 - 10 in, take no. 10 out, put no. 11-20 in, take no. 20 out, put
>> no. 21-30 in, take no. 30 out, etc.
>> (of course at the same moments as above)
>
>Same experiment, who cares what the number is on the ball that's taken out of the vase: each time
>that you do something with the balls, you add 10 to the vase and take one out, so the number of
>balls in the vase increases by 9.
>
>>
>> My friend claims both experiments end up with an empty vase.
>
>So each time you add more balls to the vase (10) then you take out (1), and that gives him an empty
>vase? Not in my universe...

In the first experiment, in your universe, can you name a ball that is
still in the vase at noon?

>> I think however the 2nd experiment ends up with a vase with an infinite
>> number of balls:
>> 1-9, 11-19, 21-29 etc. are definitely in the vase.
>>
>> He says it all has to do with Cantor's set theory, cardinality etc..., but
>> browsing the internet didn't really help me much.
>
>For somebody who wants to deal with these subjects, he's giving quite a 'naive' (I'm just being
>nice...)

Uh, you're also being wrong about the answer.

>answer to the simple experiments above.
>
>> Any information or relevant links are very welcome,
>>
>> Thanks, Theo


************************

David C. Ullrich
.

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