Paul

What is unphysical is insisting one cannot complete an infinite sequence. For every second that passes, one must first pass a half second, then a quarter, then an eighth, and so on. Your math is not powerful enough to represent this simple and physical infinite process.

It's not something I need to ask people about. It is clearly physical. The only mildly unphysical part is the timing of moving the balls. It is true one cannot time motions with infinite precision in a quantum mechanical universe, but in a Newtonian one you can.

It's silly to insist on perfect conformity to quantum mechanics and relativity when your objections were not made on those grounds. Stay focused guys!

The infinite wire also has all parts exerting the same action (whatever that means).

Tedious

History shows that when mathematics demonstrates something rigorously, it's best to take it seriously even if it seems impractical and hard to understand with our finite minds.

The fact is that mathematics is full of strange and unphysical constructs which are very useful. The complex numbers were viciously attacked as pie-in-the-sky nonsense when first developed, but now they are indispensible not only to physicists but to engineers.

It is also practical to study this impractical system because the result is particularly elegant and simple, serving as an easy approximation for finite wires.

One can scoff at the practicality of such a wire, but for physicists it is just as valid a system to study as the more familiar finite ones we encounter in the real world.

If the balls move themselves in and out of the infinitely long Urn at their scheduled times, the problem is no less physical than various classical examples in physics, such as the infinite current carrying wire.

Section 4 of my answer explains in painstaking detail how this would work. The only unphysical part was having a robot move the balls. The balls must move themselves for it to completely work.

This took me a bit to realize, but there's really nothing unphysical about Ross-Littlewood's urn problem. It is perfectly realizable if the balls are in an infinite line and move back and forth at the scheduled times, perhaps using programmed internal motors.

Ross is an extremely distinguished statistician who has written many books and articles on probability and statistics, and I am a PhD mathematician who vouches for the rigor of his solution to the problem. The very least you could do is point out a single flaw in the actual original proof. All your attempts so far have failed. That ought to give you pause.

You have yet to point to an infinite sum in the proof whose convergence you question. I'm not going to relitigate the entire proof for you.

The $\infty$ in the sum is still just a symbol, not an actual mathematical object. It simply means that we should sum over all natural numbers. And no, you are absolutely 100% wrong.

$0 \times \infty$ is not a valid mathematical expression. $\infty$ is just a symbol that denotes growth without bound. It isn't a real mathematical object. In contrast, Ross's expression is the sum of an infinite collection of zeros. This is a valid mathematical expression and is equal to zero. Once again you, like cmaster, have substituted the valid and rigorous math of Ross's proof with your own invalid constructs, then declared the original constructs invalid on that basis.

You cite another answerer who ignored the actual text of the proof, substituted his own incorrect proof, pointed out the incorrect step in his proof, and declared the original proof incorrect. The rest of your text is too vague to even criticize. "Zeros at a discontinuity" ... sorry, no one knows what you're talking about. The sum of an infinite collection of zeros is always and everywhere zero, without any qualification whatsoever.

Most of the claims you have made about how Ross's solution proceeds are false. Infinity is not used as a number anywhere in the problem or the solution. Nor are multiple results obtained for the same problem. You would know this if you had actually read and understood Ross's solution.

You don't know that an infinite number of balls is impossible or untestable. Science still has not determined if the universe is infinite or not. My answer provides a plausibly physical thought experiment with an infinite urn. The problem is unphysical but not for the reasons you state.

Please excuse and allow me to clarify. Like most other incorrect answers in the series, this answer shows no awareness of the rigorous argument made by Ross provides a rough, informal argument that reaches either the wrong conclusion or no conclusion at all. As you said, $\infty - \infty$ has no definite answer, so evidently your analysis is not powerful enough to support or rule out any solution to this problem. In other words, it's not an answer.

... than human intuition, which is known to be especially vague, confused, and fallible on this topic.

I frequently change my views. But when I have a clear, rigorous mathematical proof of my view and the challenge to my view is unintelligible, I am not inclined to change my view. In this case it is your views that need to change, but your thoughts are too foggy to even be challenged and corrected. Hopefully in the future you will realize that modern mathematics provides a clearer and deeper understanding of infinity.

Limits have to prove themselves relevant by living inside the mathematics of an experimentally validated physical model. The 1/2 limit is not valid except in some very exotic contexts well outside this discussion. No such issue with the RL solution. It's all very standard Newtonian physics.

RL is as physically realizable as any number of other infinite structures studied in physics. See sections 2 and 4 of my answer. Regarding the energy transfer variant, the balls would have to get arbitrarily small for this to work. Momentum conservation would complicate things further. Potentially interesting but no clear paradoxes yet.

"that we should keep track of the balls ... is one of those implicit assumptions" and there's the difference of philosophy between us. I don't care about "what we should do" (who's to say?). I care about what would actually happen in the physical world. Physicists found their approach to the world on tracking what happens to the whole system using math. If you care about "what we should do", I have yet to hear why we should not follow their approach, whose results we rely on in every other context.

To be fair I will readily admit that, had I not heard of the paradox and its variations, I would probably agree with you that the number of balls is infinite in the limit. But that would be because I was ignoring what was actually going on in the physical system. The lesson from the more precisely described paradox variations is that many results are possible at noon, but which result happens is ultimately determined by what exactly happens to every single physical ball in the system.

For the record I won't even attempt to evaluate incoherent statements like "the set $1/x$ which has an empty limit". There's only so much time in the day.

It's not just set theory. It's actually keeping track of what's going on in the physical system. Once again, in this new variation #3498343, we have vagueness about which balls are being removed, so the problem is ill-posed. Only Ross's problem is well-posed.

The problem with these solutions that try to just count the balls and ignore which balls they are is, they have no idea which balls are present in the limit. They're actual balls! They had to come from somewhere! Tell me about a single one - which one is it and where did it come from?

You have shown no "ignoratio elenchi" here. The mathematics of Ross's original problem is relatively simple in the world of graduate-level probability theory and stochastic processes, and the answer is 100% rigorously backed by the analysis, as ekvall has painstakingly demonstrated (not that it was really necessary).

The 100 rule has absolutely no implications for the probabilistic version since it is never ever activated beyond step 10 in any realization of the process. As for the Achilles-Turtle flag paradox, I don't have a copy of it to read, but I am extremely skeptical of any analysis that throws away the simple result from the set-theoretic limit and attempts to substitute some human intuition backed by zero mathematics or physics.

Relabeling exercises have no physical relevance. The balls have permanent identities which are equivalent to a single, unchanging labeling system. Attempting to determine a result based on elaborate relabeling systems amounts to spinning yourself around and trying to convince yourself that the world is spinning.

This is a clever variation though because it does cause the reasoning in section 4a of my answer to fail. The finite-ball case no longer converges to the infinite-ball case with respect to ball count. But the answer remains the same, even if that nice property is lost.

In the deterministic version, the urn is empty at noon regardless of the new 100 ball rule. After the first 10 stages, where no ball is removed, the steps proceed exactly as before because there are always at least 100 balls and the rule is never enforced. So the same reasoning applies and the urn is empty notwithstanding your attempt to make it not be so. The same reasoning applies to the probabilistic version, with the same result as in the original problem.

I strongly suggest that if we propose a variation, we go back to the deterministic version and figure out what it says about that version first before bothering with the probabilistic version. The probabilistic version is just an average over realizations of the deterministic version.

To question that assumption only in this problem, because one has an intuitive issue with the result derived from it, is special pleading & unphilosophical.

I don't have access to the A&K paper but would be interested to see it. As it is, I do have an issue with this: "solutions to the problem have to make implicit assumptions, which is in this case that we can use the principle of continuity on the set of balls inside the urn to state what happens at infinity". Yes, this is an assumption, but it is an assumption that would be accepted without dispute in any other context. If I take a ball out of a box, it stays out of the box unless I put it back in. That is all that is needed for Ross's solution to go through.

Each ball is almost surely not in the urn at noon, so there are almost surely no balls in the urn at noon. That's all it is.

I don't think anyone should be arguing that P(empty) = 1 if the empty set is not a possible outcome of the Allis and Koetsier variation.

If P(empty) = 1, the other one is also true...

I don't have a text that says exactly this but it's not hard to see. My sample space for coin flips is {heads, tails, biscuit}. Biscuit has probability zero. This sample space is functionally equivalent to the simpler {heads, tails} space - you can study any coin flip problem with either space and get the same answer.

Probability theory doesn't require every member of the sample space to be a realizable possibility in the process. I can add COW and PINBALL to the sample space of possible outcomes and it doesn't affect anything.

Either way works.

Formally, Ross identifies the probability of an empty set at noon as the measure of the inverse image of the empty set under this mapping. From this foundation, I assure you it is possible to fill in the gory details quite rigorously.

Importantly, the noon state is not a separate part of the sample space. Instead, it is defined by the mapping sending each subset-sequence $(A_1, A_2, A_3, ....) $ to its infinite intersection $\cap_{i=1}^\infty A_i$. The justification for modeling the problem this way is explained in my answer.

In the standard formalization of this problem, there is no transition between a pre-noon and noon sample space. The problem has only one sample space: the set of all possible sequences of pre-noon states. Each pre-noon time corresponds to a subset of the natural numbers, the balls in the urn during that time, so the sample space is the set of all sequences of subsets of the naturals: $(2^\mathbb{N})^\mathbb{N}$.

@NeilG at this point you're just repeating points that my answer refutes in detail. I will leave it at that unless you have something new to say.

One last point - the "infinity balls at noon" answer is actually more unphysical than the "empty set at noon" answer. See my answer for why.

So, to conclude: if we believe in math (or a consistent subset thereof) as the model of physical reality, and math provides us a SINGLE UNIQUE solution, and we reject this solution as unphysical, the only option is that the math used is not in our physical subset, the problem itself is unphysical, and it has no physical solution. It doesn't have a different solution, depending on your intuition or how you look at it - it has NO solution.

Furthermore, to define your new set-theoretic limit, you require the existence of the original limit, since you have to check the original limit's cardinality to determine if the new limit exists. So really, your new limit is just a domain restriction of the traditional mathematical operation, and you are declaring this restriction to be the correct model for the problem based on physical intuition. Which in the end, brings you full-circle to my exact position on the problem: the math solution exists and is unique but it isn't physical. We're in agreement! :-D