Paradox!

[Wizardry]

And I'm still waiting for a proof of that.
The only thing you shown is that when the clock strike twelve your labeling scheme is fucked.
That has nothing to do with number of remaining balls.

The labeling scheme is literally as simple as it can be. Paint a number on each ball before you do anything. The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Every ball is added. To be explicit. Let k be the index of any ball. Then it is added at step ceiling(k/10). For example ball 157 is added at step 16, when balls 151-160 are added. This occurs 1/2^16 before 12.

Every ball is removed. Let k be the index of any ball. It is removed at step k, at time t = 12 - 1/2^k.

Every ball is removed after it is added and no ball is added after it is removed. 12 - 1/2^k > 12 - 1/2^ceiling(k/10) because k > ceiling(k/10).

0 balls remain at t = 12. A ball remains at t = 12 if it is added at some finite time before t = 12 and it is not removed at some finite time before t = 12. But every ball is removed at some finite time before t = 12, because every ball has some finite label k, and is removed at t = 12 - 1/2^k, which is a finite time before t = 12. So no balls remain.

I suggested an alternative proof in an earlier post. Let A be the set of balls at t = 12 and suppose that it is nonempty. Then because the natural numbers are well ordered, A has a least element. Let m be the smallest numbered ball remaining. Then m must have been added and not removed. But we know that m was removed at step m. So m is not contained in A, a contradiction.

QED.

 

[Wizardry]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

Well, you can subtract infinity from infinity and get a non-infinite result. Indeed, that is what infinity is. The distinguishing feature of an infinite set is that it has proper subsets that are in bijective correspondence with the whole set. So by subtracting one infinite set from another you can get any subset with an infinite complement. And it is trivially easy to construct a subset of any size with an infinite complement.

{1, 2, 3, 4,...} - {1, 2, 3, 4,...} = {} 0
{1, 2, 3, 4,...} - {2, 3, 4, 5,...} = {1} 1
{1, 2, 3, 4,...} - {3, 4, 5, 6,...} = {1, 2} 2
...
{1, 2, 3, 4,...} - {1, 3, 5, 7,...} = {2, 4, 6, 8,...} infinite

 

[bigfield]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

Well, you can subtract infinity from infinity and get a non-infinite result. Indeed, that is what infinity is. The distinguishing feature of an infinite set is that it has proper subsets that are in bijective correspondence with the whole set. So by subtracting one infinite set from another you can get any subset with an infinite complement. And it is trivially easy to construct a subset of any size with an infinite complement.

{1, 2, 3, 4,...} - {1, 2, 3, 4,...} = {} 0
{1, 2, 3, 4,...} - {2, 3, 4, 5,...} = {1} 1
{1, 2, 3, 4,...} - {3, 4, 5, 6,...} = {1, 2} 2
...
{1, 2, 3, 4,...} - {1, 3, 5, 7,...} = {2, 4, 6, 8,...} infinite

OK, that makes sense, and I think I've seen that done before.

Intuitively that still doesn't make sense in terms of the problem. The difference between the added set and subtracted set of balls also increases to infinity. It can't both go to infinity and go to zero. I understand that is why it is called a a paradox but it assumes that time ever actually reaches t=12:00.

As someone else stated, this is a Zeno's paradox - the time never actually reaches 12 noon; we can only say what the number of balls approaches as time approaches 12 noon. The number of balls can't go to zero any more than the adding and subtracting can cease.

 

[Wizardry]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

Well, you can subtract infinity from infinity and get a non-infinite result. Indeed, that is what infinity is. The distinguishing feature of an infinite set is that it has proper subsets that are in bijective correspondence with the whole set. So by subtracting one infinite set from another you can get any subset with an infinite complement. And it is trivially easy to construct a subset of any size with an infinite complement.

{1, 2, 3, 4,...} - {1, 2, 3, 4,...} = {} 0
{1, 2, 3, 4,...} - {2, 3, 4, 5,...} = {1} 1
{1, 2, 3, 4,...} - {3, 4, 5, 6,...} = {1, 2} 2
...
{1, 2, 3, 4,...} - {1, 3, 5, 7,...} = {2, 4, 6, 8,...} infinite

OK, that makes sense, and I think I've seen that done before.

Intuitively that still doesn't make sense in terms of the problem. The difference between the added set and subtracted set of balls also increases to infinity. It can't both go to infinity and go to zero. I understand that is why it is called a a paradox but it assumes that time ever actually reaches t=12:00.

As someone else stated, this is a Zeno's paradox - the time never actually reaches 12 noon; we can only say what the number of balls approaches as time approaches 12 noon. The number of balls can't go to zero any more than the adding and subtracting can cease.

The number of balls in the vase a step n is 9n, which goes to infinity as n does. But that doesn't guarantee anything. It just happens to be the case that the limit of the number of balls in the vase is different from the number of balls in the limit.

And Zeno's paradox hasn't really been a paradox since calculus was invented. The reason it seems paradoxical is that it accepts the notion that you can divide up a finite distance into an infinite number of pieces, but then implicitly fails to do the same for the corresponding time. When Zeno's paradox is described, each distance step is described in the same way, giving the impression that each of the infinite steps takes the same amount of time, and of course if you break the distance up like 1/2 + 1/4 + 1/8 + ... = 1, but the time up like 1 + 1 + 1 + 1 + ... = infinity, you're going to find that motion is impossible because it takes infinite time to traverse any finite distance. But of course, the time also breaks up like 1/2 + 1/4 + 1/8 + ... = 1, and finite distances are traversed in finite time, just as you've always observed that they do.

 

[Juma]

And I'm still waiting for a proof of that.
The only thing you shown is that when the clock strike twelve your labeling scheme is fucked.
That has nothing to do with number of remaining balls.

The labeling scheme is literally as simple as it can be.
Paint a number on each ball before you do anything.

Who said it was was complicated? The important thing is that it is arbitrary.

The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Thus the highest numbered ball has number 10n and the smallest numbered ball has number n+1 making the number of balls in each step N = 10n- n+1 = 9n -1.
Dont see how that could be zero if n is infinite...

A ball remains at t = 12 if it is added at some finite time before t = 12 and it is not removed at some finite time before t = 12. But every ball is removed at some finite time before t = 12, because every ball has some finite label k, and is removed at t = 12 - 1/2^k, which is a finite time before t = 12.

Number of balls at noon is not depending on wether a specific ball remains. You forget that new balls are added before the old ones are removed...

I suggested an alternative proof in an earlier post. Let A be the set of balls at t = 12 and suppose that it is nonempty. Then because the natural numbers are well ordered, A has a least element. Let m be the smallest numbered ball remaining. Then m must have been added and not removed. But we know that m was removed at step m. So m is not contained in A, a contradiction.
QED.

But m, the smallest order number of the remaining balls is infinite, thus not necessarily removed, thus no contradiction.

What you describe is better described as putting balls in an ordered line, adding 10 at the end and remove 1 from the start.
If we assume that each ball has a fixed position this means that as the process continues the line will move out of any area but the line will ever increase in length and number of contained balls.

Thus the line is now infinitely long but nowhere.

Thus you are wrong: you have not shown that the number of balls is zero: only that your numbering scheme leads to infinite indices.

The balls "remaining" does not care less that their ordernumber are offset by an infinite number.

 

[Deepak]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

Well, you can subtract infinity from infinity and get a non-infinite result. Indeed, that is what infinity is. The distinguishing feature of an infinite set is that it has proper subsets that are in bijective correspondence with the whole set. So by subtracting one infinite set from another you can get any subset with an infinite complement. And it is trivially easy to construct a subset of any size with an infinite complement.

{1, 2, 3, 4,...} - {1, 2, 3, 4,...} = {} 0
{1, 2, 3, 4,...} - {2, 3, 4, 5,...} = {1} 1
{1, 2, 3, 4,...} - {3, 4, 5, 6,...} = {1, 2} 2
...
{1, 2, 3, 4,...} - {1, 3, 5, 7,...} = {2, 4, 6, 8,...} infinite

OK, that makes sense, and I think I've seen that done before.

Intuitively that still doesn't make sense in terms of the problem. The difference between the added set and subtracted set of balls also increases to infinity. It can't both go to infinity and go to zero. I understand that is why it is called a a paradox but it assumes that time ever actually reaches t=12:00.

As someone else stated, this is a Zeno's paradox - the time never actually reaches 12 noon; we can only say what the number of balls approaches as time approaches 12 noon. The number of balls can't go to zero any more than the adding and subtracting can cease.

The intuitive reasoning doesn't work, unless you're also arguing that the sum of the geometric series where a=.5 and r=.5 is not actually 1. Intuitively that doesn't seem terribly controversial since the value is convergent, and we all know through everyday experience that Achilles will eventually pass the tortoise. For some reason, here, since the value is divergent we want to treat it as a series of iterations with a 'last' step before noon that's ever so close but not quite there.

The problem with this sort of thinking is that the exact same process is used to show that, for example, the set of even positive integers is in a one-to-one correspondence with the positive integers. If we say we add a 'next' term to either set then we treat the operation atomically. That is to say we add to both in a single step, not to one then the other. The sequencing of this problem sticks in our minds as different because it's specifically asking to do one step then the other.

But once you know that the set of even positive integers is actually in one-to-one correspondence with the positive integers then we have no problem conceiving of them reaching to infinity separately, and if I asked you what the difference between a set the size of the positive even numbers and a set the size of the positive numbers I'd guess that you wouldn't have the same qualms that we can only approximate because we can never actually count that high.

What the problem above is asking, when high noon strikes, is what is the difference between the set of 'positive integers x where x mod 10 != 0' and the set of 'positive integers x where x mod 10 == 0'.

Does that non-rigorous explanation seem less controversial?

 

[bigfield]

The number of balls in the vase a step n is 9n, which goes to infinity as n does. But that doesn't guarantee anything. It just happens to be the case that the limit of the number of balls in the vase is different from the number of balls in the limit.

I'm not sure I understand what you are referring to by "the number of balls in the limit": It seems obvious to me that if 9n goes to infinity, then number of balls in the vase also goes to infinity.

And Zeno's paradox hasn't really been a paradox since calculus was invented. The reason it seems paradoxical is that it accepts the notion that you can divide up a finite distance into an infinite number of pieces, but then implicitly fails to do the same for the corresponding time. When Zeno's paradox is described, each distance step is described in the same way, giving the impression that each of the infinite steps takes the same amount of time, and of course if you break the distance up like 1/2 + 1/4 + 1/8 + ... = 1, but the time up like 1 + 1 + 1 + 1 + ... = infinity, you're going to find that motion is impossible because it takes infinite time to traverse any finite distance. But of course, the time also breaks up like 1/2 + 1/4 + 1/8 + ... = 1, and finite distances are traversed in finite time, just as you've always observed that they do.

I don't see how the distance can be expressed as 1/2 + 1/4 + 1/8 + ... = 1. The sum never equals 1, it just approaches 1.

While I agree that it must take a finite length of time to traverse a finite distance, I think that is missing the point. The distance (i.e. between Achilles and the tortoise) is never closed completely. If the distance is never traversed completely then there is no solution for the length of time taken to do so.

 

[bigfield]

The intuitive reasoning doesn't work, unless you're also arguing that the sum of the geometric series where a=.5 and r=.5 is not actually 1. Intuitively that doesn't seem terribly controversial since the value is convergent, and we all know through everyday experience that Achilles will eventually pass the tortoise. For some reason, here, since the value is divergent we want to treat it as a series of iterations with a 'last' step before noon that's ever so close but not quite there.

How do we know that Achilles will eventually pass the tortoise? If he halves the distances with every step in time, and continues to do so without fail, then he should never reach the tortoise. They only thing I think we can assert from lived experience is that is not physically possible to accurately and precisely continue to halve the distance between himself and the tortoise.

I understand that the sum converges on 1, but I don't see how that also makes it equal to 1.

The problem with this sort of thinking is that the exact same process is used to show that, for example, the set of even positive integers is in a one-to-one correspondence with the positive integers. If we say we add a 'next' term to either set then we treat the operation atomically. That is to say we add to both in a single step, not to one then the other. The sequencing of this problem sticks in our minds as different because it's specifically asking to do one step then the other.

But once you know that the set of even positive integers is actually in one-to-one correspondence with the positive integers then we have no problem conceiving of them reaching to infinity separately, and if I asked you what the difference between a set the size of the positive even numbers and a set the size of the positive numbers I'd guess that you wouldn't have the same qualms that we can only approximate because we can never actually count that high.

I do have qualms with that, but that is perhaps because my understanding of infinity is limited. To me it doesn't make sense to say two infinite sets are the same size.

What the problem above is asking, when high noon strikes, is what is the difference between the set of 'positive integers x where x mod 10 != 0' and the set of 'positive integers x where x mod 10 == 0'.

I would think that there is either no solution, or that there is a infinite difference: the set of odd numbers.

Does that non-rigorous explanation seem less controversial?

Nope, but I do appreciate you trying. I'm learning something!

 

[beero1000]

How do we know that Achilles will eventually pass the tortoise? If he halves the distances with every step in time, and continues to do so without fail, then he should never reach the tortoise. They only thing I think we can assert from lived experience is that is not physically possible to accurately and precisely continue to halve the distance between himself and the tortoise.

I understand that the sum converges on 1, but I don't see how that also makes it equal to 1.

That is exactly what convergence means. When we assign a number to the infinite sum, the only possible number to choose is exactly 1. Between any pair of distinct real numbers there is a non-zero distance and convergence means that with enough terms the sum is closer to 1 than any other real number x, so the sum cannot be x. The only logical choice is that the sum is equal to 1.

The problem with this sort of thinking is that the exact same process is used to show that, for example, the set of even positive integers is in a one-to-one correspondence with the positive integers. If we say we add a 'next' term to either set then we treat the operation atomically. That is to say we add to both in a single step, not to one then the other. The sequencing of this problem sticks in our minds as different because it's specifically asking to do one step then the other.

But once you know that the set of even positive integers is actually in one-to-one correspondence with the positive integers then we have no problem conceiving of them reaching to infinity separately, and if I asked you what the difference between a set the size of the positive even numbers and a set the size of the positive numbers I'd guess that you wouldn't have the same qualms that we can only approximate because we can never actually count that high.

I do have qualms with that, but that is perhaps because my understanding of infinity is limited. To me it doesn't make sense to say two infinite sets are the same size.

Does it make sense to say that two different finite sets are the same size? How do you go about testing that? What happens if we perform the same procedure for infinite sets? The notion of a one-to-one correspondence underlies all of these, and it is just that peoples' intuition break down what we move to infinite sets.

 

[beero1000]

And I'm still waiting for a proof of that.
The only thing you shown is that when the clock strike twelve your labeling scheme is fucked.

That has nothing to do with number of remaining balls.

The labeling scheme is literally as simple as it can be.
Paint a number on each ball before you do anything.

Who said it was was complicated? The important thing is that it is arbitrary.

So what? Every labeling scheme is arbitrary, and there are many different possible choices each giving different results. That is why it's called a paradox.

The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Thus the highest numbered ball has number 10n and the smallest numbered ball has number n+1 making the number of balls in each step N = 10n- n+1 = 9n -1.
Dont see how that could be zero if n is infinite...

Then can you specify a ball that remains?

A ball remains at t = 12 if it is added at some finite time before t = 12 and it is not removed at some finite time before t = 12. But every ball is removed at some finite time before t = 12, because every ball has some finite label k, and is removed at t = 12 - 1/2^k, which is a finite time before t = 12.

Number of balls at noon is not depending on wether a specific ball remains. You forget that new balls are added before the old ones are removed...

Every ball placed has a finite label, so if there is a ball remaining it will have a label. What is the label of even one ball that remains?

I suggested an alternative proof in an earlier post. Let A be the set of balls at t = 12 and suppose that it is nonempty. Then because the natural numbers are well ordered, A has a least element. Let m be the smallest numbered ball remaining. Then m must have been added and not removed. But we know that m was removed at step m. So m is not contained in A, a contradiction.
QED.

But m, the smallest order number of the remaining balls is infinite, thus not necessarily removed, thus no contradiction.

What you describe is better described as putting balls in an ordered line, adding 10 at the end and remove 1 from the start.
If we assume that each ball has a fixed position this means that as the process continues the line will move out of any area but the line will ever increase in length and number of contained balls.

Thus the line is now infinitely long but nowhere.

Thus you are wrong: you have not shown that the number of balls is zero: only that your numbering scheme leads to infinite indices.

The balls "remaining" does not care less that their ordernumber are offset by an infinite number.

Every ball placed has a finite label, there are no infinite indices.

 

[beero1000]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

What if the problem was changed so that you place 1 ball and remove 1 ball at each step? You still subtract infinity from infinity, but how many balls remain?

 

[Jimmy Higgins]

The answer is ERR as you never reach Noon. In order for the Infinite number of times of the process to cancel each other out, Noon isn't possible. Otherwise, it would be a finite process and there would be plenty of balls in the jar.

The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Thus the highest numbered ball has number 10n and the smallest numbered ball has number n+1 making the number of balls in each step N = 10n- n+1 = 9n -1.
Dont see how that could be zero if n is infinite...

Then can you specify a ball that remains?

Not specifically, but I also can not recite the 300,000,001 decimal value of Pi... therefore Pi doesn't exist.

 

[Jimmy Higgins]

Though I don't buy that infinity cancels each other out. At no point are more than balls being removed that being added. If the reality of the answer doesn't fit with the logic, the logic is wrong! Increments of 9x can not lead to zero! The problem is including the (-1) as part of the equation. So relabel it as the incremental change per step. This is legit because the question supposes that both operations happen at the same time, therefore you never have added ten balls then taken one out of the jar. You just changed the quantity in the jar by Δx (or nine in this hypothetical).

The appropriate equation is: total = Δx/(1/n)

Where Δx is the incremental change in balls in the jar per step and n is the number of steps. The answer is infinity.

 

[Kharakov]

A bit of a rehashing:

Assuming .1 meters to travel into and out of the jar, at some point one would have to exceed the speed of light (although the jar would go thermonuclear way before this). So at the point where one has to do the deed in under 1/2,997,924,580 of a second (this right?) you've reached the light speed cut off.
1800 seconds / 2^42 iterations = 4.0927261579781771 *10^-10
.1 meters / 299,792,458 meters per second = 3.3356409519815207 * 10^-10

Any more iterations would take faster than light speed. I picked .1 meters "randomly" to use a round number. Nice that we ended up with 2^42 * 9 balls. Keeps the 42 from the original answer. This "random" coincidence was lobbed up for us like a little league pitch, but you don't have to believe it.

Assuming average human response time, one would get 108, which is one of those special numbers.

Assuming the mathematical limit approach, infinity...

 

[beero1000]

The answer is ERR as you never reach Noon. In order for the Infinite number of times of the process to cancel each other out, Noon isn't possible. Otherwise, it would be a finite process and there would be plenty of balls in the jar.

The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Thus the highest numbered ball has number 10n and the smallest numbered ball has number n+1 making the number of balls in each step N = 10n- n+1 = 9n -1.
Dont see how that could be zero if n is infinite...

Then can you specify a ball that remains?

Not specifically, but I also can not recite the 300,000,001 decimal value of Pi... therefore Pi doesn't exist.

So you're saying that even though every ball has been removed, there are still balls in the vase, but you can't tell me which ones. I guess next you'll say I'm just supposed to believe, and all will be revealed...

And the 300,000,001th digit of pi is 1.

 

[ryan]

Oh, I think that I have made some kind of intuitive sense out of this, at least for myself.

Because infinity is not a fixed number like 30 is, this question is essentially very vague. It is like asking if I start with no balls in the vase and I put in 10 and take 1 out, but I can do either as many times as I want, how many balls will be in the vase. So there could be any number equal to or greater than 0. So the answer to the OP is: x balls, where 0 ≤ x ≤ ∞.

 

[Jimmy Higgins]

The answer is ERR as you never reach Noon. In order for the Infinite number of times of the process to cancel each other out, Noon isn't possible. Otherwise, it would be a finite process and there would be plenty of balls in the jar.

The only thing that's complicated is that at step n, which occurs at t = 12 - 1/2^n you add balls in the interval (10(n-1)..10n] and remove ball n.

Thus the highest numbered ball has number 10n and the smallest numbered ball has number n+1 making the number of balls in each step N = 10n- n+1 = 9n -1.
Dont see how that could be zero if n is infinite...

Then can you specify a ball that remains?

Not specifically, but I also can not recite the 300,000,001 decimal value of Pi... therefore Pi doesn't exist.

So you're saying that even though every ball has been removed, there are still balls in the vase, but you can't tell me which ones. I guess next you'll say I'm just supposed to believe, and all will be revealed...

1) Noon never happens in your hypothetical.
2) Even if Noon does come, your logic doesn't line up with reality, so your math is wrong. You can not add more than take away and end up with nothing. If your math says otherwise, the math is bogus and you have not modeled the problem correctly. The last time the vase ever has 0 balls is right before 11:30.

And the 300,000,001th digit of pi is 1.

Actually it is 0.7.

 

[Juma]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

 

[beero1000]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon. You are trying to argue that the value at a limiting point in time is the limiting value at a point in time, which is not necessarily true.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

 

[Juma]

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon. You are trying to argue that the value at a limiting point in time is the limiting value at a point in time, which is not necessarily true.

The net influx if balls is Fb= 9/DT where DT is time since previous entry. Since DT->0 when t->12 it is obvious that the net influx is greater than any finite value.

If one can show that every molecule that has ever entered the river is gone at noon

You have not done that.