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For me, the problem, when considered as purely abstract has a solution as long as the missing information is provided: how to takethat the state at noon is the limit of the previous states and limit in what sense. However, when thinking of this problem intuitively, the limit of the sequence of states is not something you can think in a single manner. Fundamentally, I think there is no way to answer.

For me, the problem, when considered as purely abstract has a solution as long as the missing information is provided: how to take the limit. However, when thinking of this problem intuitively, the limit of the sequence of states is not something you can think in a single manner. Fundamentally, I think there is no way to answer.

For me, the problem, when considered as purely abstract has a solution as long as the missing information is provided: that the state at noon is the limit of the previous states and limit in what sense. However, when thinking of this problem intuitively, the limit of the sequence of states is not something you can think in a single manner. Fundamentally, I think there is no way to answer.

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1 minute before midnightIt's worth reading amoeba's answer that is just excellent and clarifies the problem very much. I don't exactly disagree with his answer but want to point out that the solution of the problem is based on a certain convention. What is interesting is that this sort of problem shows that this convention, putwhile often used, is questionable.

Just as he says there is a technical point about proving that for each ball the probability to stay in the urn forever is 0. 1/2 minute before midnight remove itApart from this point, the problem is not about probabilities. 1/4 minute before midnight put it backA deterministic equivalent may be given. It is much easier to understand. The key idea is: since every ball is absent from the urn from some point in time, the urn at the end is empty. If you represent the presence in the urn of each ball by a sequence of zeros and ones, each sequence is 0 from a certain range, thus its limit is 0.

Now the problem can be simplified even more. I call the moments 1/8 minute before midnight remove it again, 2, 3... How many. for simplicity:

  • moment 1: put ball 1 in the urn
  • moment 2: remove it
  • moment 3: put ball 2 in the urn
  • moment 4: remove it
  • moment 5: put ball 3 in the urn
  • ...

What balls at midnightthe end (noon) ? ZeroWith the same idea, onesame answer: none.

But fundamentally, half a ballthere is no way to know, <0| + <1|because the problem does not say what happens at noon. Actually, it is possible that at the end of times, Pikachu comes suddenly in the urn. Or maybe balls? all suddenly collapse and merge into one big ball. Not meaning that this is meant to be realistic, it's just not specified.

Set theoryThe problem can give anonly be answered if a certain convention tells us how to go to the limit: a continuity assumption. The state of the urn at noon is the limit of its states before. Where should we look for a continuity assumption that would help us answer to the original problemquestion?

In physical laws? Physical laws ensure a certain continuity. I think of a simplistic classical model, not calling on to this slightly modified onereal modern physics. It just saysBut fundamentally, physical laws would bring exactly the same questions as the mathematical ones: the way we choose to describe continuity for physical laws relies on asking the question mathematically: what is continuous, how?

We have to look for a continuity assumption in a more abstract way. The usual idea is to define the state of the urn as a function from the set of balls has nointo $\{0;1\}$. 0 means absent, 1 means present. And to define continuity, we use product topology, aka pointwise convergence. We say that the state at noon, is the limit of the states before noon according to this topology. But both describeWith this topology, there is a process of puttinglimit, and removing ballsit is 0: an empty urn.

But now we modify the problem a little in order to challenge this topology:

  • moment 1: put ball 1 in the urn
  • moment 2: remove it
  • moment 3: put ball 1 in the urn
  • moment 4: remove it
  • moment 5: put ball 1 in the urn
  • ...

For the same topology, the sequence of states has no limit. That's where I start to see the paradox as a true paradox. For me this modified problem is essentially the same. Imagine you are the urn. You see balls coming and going. If you can't read the number on it, whether it is the same ball or another one does not change what's happening to you. Instead of putting only newseeing balls as individual distinct elements, you see them as a quantity of matter coming in and out. The continuity could naturally be defined by looking at variations of the urn inquantity of matter. And there is indeed no limit. In a way this problem is the same as the original problem, where you re-usedecide to ignore the ball you have already removedidentity, which providesthus leading to a partdifferent metric and a different notion of convergence. And even if you could see the incoming populationnumber on the balls, the state could be seen as just a flickering presence with a growing number. From

In one case, the urn pointlimit of view, if the balls are indistinguishablesequence of your states is "empty", you seein the same process goingother case the limit is undefined.

The formalization of the problem with the product topology fundamentally relies on separating what happens to each different ball, and thus creating a metric reflecting the "distinguishablitiy". ButOnly because of this changesseparation, a limit can be defined. The fact that this separation is so fundamental to the answer tobut not fundamental for describing "what's going on" in the problemurn (a point that is endlessly arguable), makes me think the solution is the consequence of a convention rather than a fundamental truth.

The answer depends onFor me, the formalizationproblem, when considered as purely abstract has a solution as long as the missing information is provided: how to take the limit. However, when thinking of this problem intuitively, the limit of the sequence of states is not something you can think in a single manner. Fundamentally, I think there is no way to answer.

1 minute before midnight, put a ball in. 1/2 minute before midnight remove it. 1/4 minute before midnight put it back in, 1/8 minute before midnight remove it again... How many balls at midnight? Zero, one, half a ball, <0| + <1| balls?

Set theory can give an answer to the original problem, not to this slightly modified one. It just says the set of balls has no limit. But both describe a process of putting and removing balls.

Instead of putting only new balls in the urn in the original problem, you re-use the ball you have already removed, which provides a part of the incoming population. From the urn point of view, if the balls are indistinguishable, you see the same process going on. But this changes the answer to the problem.

The answer depends on the formalization.

It's worth reading amoeba's answer that is just excellent and clarifies the problem very much. I don't exactly disagree with his answer but want to point out that the solution of the problem is based on a certain convention. What is interesting is that this sort of problem shows that this convention, while often used, is questionable.

Just as he says there is a technical point about proving that for each ball the probability to stay in the urn forever is 0. Apart from this point, the problem is not about probabilities. A deterministic equivalent may be given. It is much easier to understand. The key idea is: since every ball is absent from the urn from some point in time, the urn at the end is empty. If you represent the presence in the urn of each ball by a sequence of zeros and ones, each sequence is 0 from a certain range, thus its limit is 0.

Now the problem can be simplified even more. I call the moments 1, 2, 3.... for simplicity:

  • moment 1: put ball 1 in the urn
  • moment 2: remove it
  • moment 3: put ball 2 in the urn
  • moment 4: remove it
  • moment 5: put ball 3 in the urn
  • ...

What balls at the end (noon) ? With the same idea, same answer: none.

But fundamentally, there is no way to know, because the problem does not say what happens at noon. Actually, it is possible that at the end of times, Pikachu comes suddenly in the urn. Or maybe balls all suddenly collapse and merge into one big ball. Not meaning that this is meant to be realistic, it's just not specified.

The problem can only be answered if a certain convention tells us how to go to the limit: a continuity assumption. The state of the urn at noon is the limit of its states before. Where should we look for a continuity assumption that would help us answer to the question?

In physical laws? Physical laws ensure a certain continuity. I think of a simplistic classical model, not calling on to real modern physics. But fundamentally, physical laws would bring exactly the same questions as the mathematical ones: the way we choose to describe continuity for physical laws relies on asking the question mathematically: what is continuous, how?

We have to look for a continuity assumption in a more abstract way. The usual idea is to define the state of the urn as a function from the set of balls into $\{0;1\}$. 0 means absent, 1 means present. And to define continuity, we use product topology, aka pointwise convergence. We say that the state at noon, is the limit of the states before noon according to this topology. With this topology, there is a limit, and it is 0: an empty urn.

But now we modify the problem a little in order to challenge this topology:

  • moment 1: put ball 1 in the urn
  • moment 2: remove it
  • moment 3: put ball 1 in the urn
  • moment 4: remove it
  • moment 5: put ball 1 in the urn
  • ...

For the same topology, the sequence of states has no limit. That's where I start to see the paradox as a true paradox. For me this modified problem is essentially the same. Imagine you are the urn. You see balls coming and going. If you can't read the number on it, whether it is the same ball or another one does not change what's happening to you. Instead of seeing balls as individual distinct elements, you see them as a quantity of matter coming in and out. The continuity could naturally be defined by looking at variations of the quantity of matter. And there is indeed no limit. In a way this problem is the same as the original problem where you decide to ignore the ball identity, thus leading to a different metric and a different notion of convergence. And even if you could see the number on the balls, the state could be seen as just a flickering presence with a growing number.

In one case, the limit of the sequence of your states is "empty", in the other case the limit is undefined.

The formalization of the problem with the product topology fundamentally relies on separating what happens to each different ball, and thus creating a metric reflecting the "distinguishablitiy". Only because of this separation, a limit can be defined. The fact that this separation is so fundamental to the answer but not fundamental for describing "what's going on" in the urn (a point that is endlessly arguable), makes me think the solution is the consequence of a convention rather than a fundamental truth.

For me, the problem, when considered as purely abstract has a solution as long as the missing information is provided: how to take the limit. However, when thinking of this problem intuitively, the limit of the sequence of states is not something you can think in a single manner. Fundamentally, I think there is no way to answer.

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1 minute before midnight, put a ball in. 1/2 minute before midnight remove it. 1/4 minute before midnight put it back in, 1/8 minute before midnight remove it again... How many balls at midnight? Zero, one, half a ball, <0| + <1| balls?

Set theory can give an answer to the original problem, not to this slightly modified one. It just says the set of balls has no limit. But both describe a process of putting and removing balls.

Instead of putting only new balls in the urn in the original problem, you re-use the ballsball you have already removed, which provides a part of the incoming population. From the urn point of view, if the balls are indistinguishable, you see the same process going on. But this changes the answer to the problem.

The answer depends on the formalization.

1 minute before midnight, put a ball in. 1/2 minute before midnight remove it. 1/4 minute before midnight put it back in, 1/8 minute before midnight remove it again... How many balls at midnight? Zero, one, half a ball, <0| + <1| balls?

Set theory can give an answer to the original problem, not to this slightly modified one. It just says the set of balls has no limit. But both describe a process of putting and removing balls.

Instead of putting only new balls in the urn in the original problem, you re-use the balls you have already removed, which provides a part of the incoming population. From the urn point of view, if the balls are indistinguishable, you see the same process going on. But this changes the answer to the problem.

The answer depends on the formalization.

1 minute before midnight, put a ball in. 1/2 minute before midnight remove it. 1/4 minute before midnight put it back in, 1/8 minute before midnight remove it again... How many balls at midnight? Zero, one, half a ball, <0| + <1| balls?

Set theory can give an answer to the original problem, not to this slightly modified one. It just says the set of balls has no limit. But both describe a process of putting and removing balls.

Instead of putting only new balls in the urn in the original problem, you re-use the ball you have already removed, which provides a part of the incoming population. From the urn point of view, if the balls are indistinguishable, you see the same process going on. But this changes the answer to the problem.

The answer depends on the formalization.

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