Re: infinity

The original question:
Problem 1
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?

We can change the above question into an equivalent problem.
Problem 2.
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 minute 1 you put balls 1 to 10 in the vase and take out number 1.
At minute 2 you put balls 11 - 20 in the vase and take out number 2.
At minute 3 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 an infinite time from
now, how many ping pong balls are in the vase?

William Hughes wrote:
> > According to this problem, noon is not reachable.
>
> Wrong. The experiment described is quite well defined. We know
> which balls are added to the vase and when and we can consider
> the different outcomes by choosing which balls are removed from
> the vase and when. Noon is not reachable in any physical sense,
> but we know that the experiment described has no physical realization
> or approximation.
>
I hope you can see that 'noon' in problem 1 is equivalent to 'an
infinite time from now' in problem 2. They are un-reachable. Again I
have to emphasis that in order to prove the vase is empty, we have to
prove that the last ball is taken out from the vase.

Yes, you can claim from problem 2 that ball n is removed from the vase
at the nth minute. The vase can only be claimed to be empty if ball n
is the last ball in the vase. Obviously by the time the nth ball is
removed there are many more balls in the vase. So the vase cannot be
empty even though every ball currently in the vase will be removed
later in the future.

By mathematical induction. n=1 is true. n=n+1 is also true. It implies
that it is true for all n. The vase is not empty for all n > 0.

Mathematical induction still cannot tell you what happens exactly when
n is infinite which does not exist physically or mathematically. But
it can claim that the vase is not empty as n approaches to infinity.

> > In the vase example, if we only look at the withdrawal portion of the
> > argument then the claim is not strong. If both putting in and taken
> > out of balls are taken into account. In order to prove that the vase
> > is empty at noon, you must have to show at certain time the rate of
> > taking out is greater than the rate of putting in.
>
> No. The rate at which we add balls is infinite. So is the rate at
> which we remove them. We cannot conclude anything from this fact. True, > at any time before noon the rate at which we are adding balls is greater > than the rate at which we are removing them. This tells us nothing about
> what happens at noon.
> -William Hughes
In the equivalent problem (problem 2), the rate of adding the balls to
the vase is 10 balls/minute. The withdrawal rate is 1 ball/minute.

Assume f(x) is an increasing function and f(0) > 0. Even though f(oo)
is undefined but I am quite sure that f(oo) cannot be 0.

.

Relevant Pages

  • Re: infinity
    ... Physically: Infinity is undefined physically. ... >> to show that each ball is removed from the vase before noon and is not ... >> the vase is empty before noon. ... The operation of adding or removal of balls is undefined at noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... To become empty means there is a change of state in the vase, from having balls to not having balls. ... There are always a specific number of balls, if additions and removals occur instantaneously. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Since the vase was empty to start with, it cannot later "become" empty after once having been empty, at least according to that definition. ... Noon does not exist in the experiment, or else you have infinitely numbered balls. ... insertion or removal or location of balls is a function of time. ...
    (sci.math)
  • Re: infinity
    ... >>> Physically or mathematical it is not difficult to prove that the vase ... # of balls in the vase can be ... this statement alone is not sufficient to claim ... >>> the vase is empty before noon. ...
    (sci.math)
  • Re: infinity
    ... Number of balls in the vase at noon is f= OO. ... Unfortunately, if infinity gets involved, this statement alone is not sufficient to claim the vase is empty before noon. ... then the sum becomes 0. ...
    (sci.math)
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