You have an infinite collection of balls in a storeroom, labeled with the natural numbers (1, 2, 3, etc.) and a vase that can hold any number, or all, of them.
At each integer time T, starting with T=1, you take the 10 lowest numbered balls out of the storeroom and put them in the vase, and then take the lowest numbered ball out of the vase and destroy it. So at any finite time T, there are 9T balls in the vase and all the balls labeled with a number less than or equal to T have been destroyed.
Now, because the number of balls at any given time T is given by 9T, in the limit as T approaches infinity, there are infinitely many balls in the vase. On the other hand, because every ball has a time T at which it will be destroyed, the limit of the set of balls in the vase as T approaches infinity is the empty set. So at T = infinity, you have an empty vase that contains infinitely many balls.
The moral of the story is to be careful what limits you take, because taking two different limits can give two different answers even if they seem like they’re measuring the same thing.
(Can this be used as an argument for the existence of nonstandard numbers?)
This is actually a good example of the difference between pointwise and uniform convergence. Consider the characteristic function of the vase at time t, g_t : N -> {0, 1}. Then g_t(n) = 1 if and only if ball n is in the vase at time t. It will actually help to convert g_t to a function on the real numbers, by making f_t(x) = g_t(floor(x)).
Now, for each ball n, there is a time when it will be destroyed, and therefore will never be in the vase after that time. So the characteristic function f_t(x) converges pointwise to f(x) = 0. This is presumably what you mean by the limit of the vase being the empty set.
But the criterion of uniform convergence is that for any epsilon>0 there is a t such that f_t is within epsilon of the limit everywhere. Which is obviously not true, because at any time t there are some balls in the vase, and so the characteristic function is 1 somewhere. So f_t(x) does not uniformly converge to anything.
As it happens, without uniform convergence, the limit of the integrals of f_t(x) (which just so happens to be the number of balls in the vase, by my setup) is not generally equal to the integral of the limiting function f(x). So, in a way it is not really true that you can say
in the limit as T approaches infinity, there are infinitely many balls in the vase
I think the answer to “how many balls did you put in the vase as T->\infty” and “How many balls have been destroyed as T->\infty” both have well defined answers. It’s just a fallacy to assume that the “total number of balls in the vase as T->\infty” is equal to the difference between these quantities in their limits.
The Ross-Littlewood Paradox is amusing.
You have an infinite collection of balls in a storeroom, labeled with the natural numbers (1, 2, 3, etc.) and a vase that can hold any number, or all, of them.
At each integer time T, starting with T=1, you take the 10 lowest numbered balls out of the storeroom and put them in the vase, and then take the lowest numbered ball out of the vase and destroy it. So at any finite time T, there are 9T balls in the vase and all the balls labeled with a number less than or equal to T have been destroyed.
Now, because the number of balls at any given time T is given by 9T, in the limit as T approaches infinity, there are infinitely many balls in the vase. On the other hand, because every ball has a time T at which it will be destroyed, the limit of the set of balls in the vase as T approaches infinity is the empty set. So at T = infinity, you have an empty vase that contains infinitely many balls.
The moral of the story is to be careful what limits you take, because taking two different limits can give two different answers even if they seem like they’re measuring the same thing.
(Can this be used as an argument for the existence of nonstandard numbers?)
This is actually a good example of the difference between pointwise and uniform convergence. Consider the characteristic function of the vase at time
t,g_t : N -> {0, 1}. Theng_t(n) = 1if and only if ballnis in the vase at timet. It will actually help to convertg_tto a function on the real numbers, by makingf_t(x) = g_t(floor(x)).Now, for each ball
n, there is a time when it will be destroyed, and therefore will never be in the vase after that time. So the characteristic functionf_t(x)converges pointwise tof(x) = 0. This is presumably what you mean by the limit of the vase being the empty set.But the criterion of uniform convergence is that for any
epsilon>0there is atsuch thatf_tis withinepsilonof the limit everywhere. Which is obviously not true, because at any timetthere are some balls in the vase, and so the characteristic function is 1 somewhere. Sof_t(x)does not uniformly converge to anything.As it happens, without uniform convergence, the limit of the integrals of
f_t(x)(which just so happens to be the number of balls in the vase, by my setup) is not generally equal to the integral of the limiting functionf(x). So, in a way it is not really true that you can sayas the integral does not transfer to the limit.
Great problem, thanks for mentioning it!
I think the answer to “how many balls did you put in the vase as T->\infty” and “How many balls have been destroyed as T->\infty” both have well defined answers. It’s just a fallacy to assume that the “total number of balls in the vase as T->\infty” is equal to the difference between these quantities in their limits.