$\begingroup$

This answer aims to do four things:

Review Ross's mathematical formulation of the problem, showing how it follows directly and unambiguously from the problem description. Defend the position that Ross's paradoxical solution is both mathematically sound and relevant to our understanding of the physical world, whether or not it is 100% physically realizable. Discuss certain fallacious arguments rooted in physical intuition, and show that the oft-stated "physical" solution of infinite balls at noon is not only in contradiction to mathematics, but to physics as well. Describe a physical implementation of the problem which may make Ross's solution more intuitive. Start here for the answer to Carlos's original question.

1. How to Describe the Problem Mathematically

We will unpack the initial "infinite process modeling" step of Ross's argument (p. 46). Here is the statement we will focus on justifying:

Define $E_n$ to be the event that ball number 1 is still in the urn after the first n withdrawals have been made... The event that ball number 1 is in the urn at 12 P.M. is just the event $\bigcap_{n=1}^\infty E_n$.

Before we unpack Ross's statement, let's consider how it is even possible to understand the urn's contents at noon, after an infinite sequence of operations. How could we possibly know what is in the urn? Well, let's think about a specific ball $b$; you can imagine $b=1$ or $1000$ or whatever you want. If ball $b$ was taken out at some stage of the process before noon, certainly it won't be in the urn at noon. And conversely, if a given ball was in the urn at every single stage of the process up until noon (after it was added), then it was in the urn at noon. Let's write these statements out formally:

A ball $b$ is in the urn at noon if and only if it was in the urn at every stage $n \in \{n_b, n_b + 1, n_b + 2, ...\}$ before noon, where $n_b$ is the stage the ball was added to the urn.

Now let's unpack Ross's statement - what does $\bigcap_{n=1}^\infty E_n$ mean in plain English? Let's take a single realization $x$ of the urn process and talk it out:

$x \in E_1$ means that ball 1 is in the urn after stage 1 of the process.

$x \in E_1 \bigcap E_2$ means that ball 1 is in the urn after stages 1 and 2 of the process.

$x \in E_1 \bigcap E_2 \bigcap E_3$ means that ball 1 is in the urn after stages 1, 2, and 3 of the process.

For any $k \in \{1, 2, 3, ...\}$, $x \in \bigcap_{k=1}^n E_k$ means that the ball is in the urn after stages $1$ thru $n$.

Clearly, then, $x \in \bigcap_{k \in \{1, 2, 3...\}} E_k$ means that, in realization $x$ of this urn process, ball 1 is in the urn after stages 1, 2, 3, et cetera: all finite stages $k$ before noon. The infinite intersection $\bigcap_{n = 1}^\infty E_n$ is just another way of writing that, so $\bigcap_{n = 1}^\infty E_n$ contains precisely the realizations of the process where ball 1 was in the urn at all stages before noon. An event is just a defined set of realizations of a process, so the last sentence is precisely equivalent to saying that $\bigcap_{n = 1}^\infty E_n$ is the event that ball 1 was in the urn at all stages before noon, for this random process.

Now, the punchline: by our "if and only if" statement above, this is exactly the same as saying that ball 1 was in the urn at noon! So $\bigcap_{n = 1}^\infty E_n$ is the event that ball 1 is in the urn at noon, just as Ross originally stated. QED

In the derivation above, everything we said is equally valid for both the deterministic and probabilistic versions, because deterministic modeling is a special case of probabilistic modeling in which the sample space has one element. No measure theoretic or probability concepts were even used, beyond the words "event" and "realization" (which are just jargon for "set" and "element").

2. The Paradoxical Solution is Mathematically Sound and Relevant to Physics

After this setup point, the deterministic and probabilistic variants diverge. In the deterministic variant (version 2 from amoeba's post), we know ball 1 is taken out on the first step, so $E_1 = \emptyset$ and the infinite intersection, of course, is also empty. Similarly, any other ball $b$ is taken out at stage $b$ and is not present at noon. Thus the urn cannot contain any numbered ball $b$ at noon and must therefore be empty.

In the probabilistic variant, the same phenomenon happens, just in a softer "in-expectation" sense. The probability of any given ball's being present dwindles to zero as we approach noon, and at the limiting time of noon, the ball is almost surely not present. Since each ball is present with probability zero, and the sum of infinitely many zeros is still zero, there are almost surely no balls in the urn at noon. All of this is shown completely rigorously by Ross; details can be filled in with a knowledge of graduate-level measure theory, as @ekvall's answer shows.

If you accept the standard arguments about mathematical objects expressed as infinite sequences, for example $0.999... = 1$, the argument here should be just as acceptable, as it relies on the exact same principles. The only question remaining is whether the mathematical solution applies to the real world, or just the platonic world of mathematics. This question is complex and is discussed further in section 4.

That said, there is no reason to presuppose that the infinite urn problem is unphysical, or to reject it as irrelevant even if it is unphysical. Many physical insights have been gained from studying infinite structures and processes, for example, infinite wires and percolation lattices. Not all of these systems are necessarily physically realizable, but their theory shapes the rest of physics. Calculus itself is "unphysical" in some ways, because we don't know if it is possible to physically realize the arbitrarily small distances and times that are often its subject of study. That doesn't stop us from putting calculus to incredibly good use in the theoretical and applied sciences.

3. The Unphysicality of Solutions Based on "Physical Intuition"

For those who still believe that Ross's math is wrong or physically inaccurate in the deterministic variant, and the true physical solution is infinitely many balls: regardless of what you think happens at noon, it is impossible to deny the situation before noon: every numbered ball added to the urn eventually gets removed. So if you think there are somehow still infinitely many balls in the urn at noon, you must admit that not one of those balls can be a ball added before noon. So those balls must have come from somewhere else: you are asserting that infinitely many balls, unrelated to the original problem process, suddenly pop into existence at precisely noon to rescue the continuity of cardinality from being violated. As unphysical as the "empty set" solution might seem intuitively, this alternative is objectively and demonstrably unphysical. Infinite collections of objects do not pop into being in an instant just to satisfy poor human intuitions about infinity.

The common fallacy here seems to be that we can just look at the number of balls as time approaches noon, and assume that the divergent trend yields infinitely many balls at noon, without regard to exactly which balls are being taken in and out. There has even been an attempt to justify this with the "principle of indifference", which states that the answer shouldn't depend on whether the balls are labeled or not.

Indeed, the answer does not depend on whether the balls are labeled or not, but that is an argument for Ross's solution, not against it. From the perspective of classical physics, the balls are effectively labeled whether you think of them as labeled or not. They have distinct, permanent identities which are equivalent to labels, and a truly physical analysis must account for this, whether or not numbers are literally written on the balls. The labels themselves do not directly affect how the solution comes out, but they are needed to describe exactly how the balls are moved around. Some procedures leave balls in the urn forever, others provably remove every ball that is added, and labels are needed to even describe the difference between these procedures. Attempting to ignore the labels is not "physical", it's just neglecting to understand the physical problem precisely enough to solve it. (The same goes for complicated variants that reshuffle the labels at each stage. What matters is which balls are in the urn, not the labels someone has placed or replaced on them. This can be determined by ignoring the complicated relabeling scheme entirely and simply using a single unchanging labeling scheme, the one of Ross's original problem.)

The only way distinguishability would fail to be true is if the "balls" were quantum mechanical particles. In this case, the indifference principle fails spectacularly. Quantum physics tells us that indistinguishable particles behave completely differently than distinguishable ones. This has incredibly fundamental consequences for the structure of our universe, such as the Pauli exclusion principle, which is perhaps the single most important principle of chemistry. No one has attempted to analyze a quantum version of this paradox yet.

4. Describing the Solution Physically

We have seen how vague "physical" intuitions can lead us astray on this problem. Conversely, it turns out that a more physically precise description of the problem helps us understand why the mathematical solution is actually the one that makes the most physical sense.

Consider an infinite Newtonian Universe governed by the laws of classical mechanics. This Universe contains two objects: an infinite Shelf and an infinite Urn, which start at the Origin of the Universe and run alongside one another, one feet apart, forever and ever. The Shelf lies on the line $y = 0$ feet, while the Urn lies on the line $y = 1$ feet. Along the Shelf are laid infinitely many identical balls, evenly spaced one foot apart, the first being one foot from the Origin (so ball $n$ is on the line $x = n$ feet). The Urn - which is really just like the Shelf, but a bit more ornate, closed over, and generally Urnish - is empty.

An Aisle connects the Shelf and Urn at the bottom, and on top of the Aisle, at the Origin, sits an Endeavor robot with an infinite power supply. Beginning at 11 AM, Endeavor activates and begins zooming back and forth in the Aisle, transferring balls between Urn and Shelf according to Ross-Littlewood's programmed instructions:

When the program commands ball $n$ to be inserted into the Urn, the ball $n$ feet from the Origin is transferred from the Shelf to the Urn.

When the program commands ball $n$ to be removed from the Urn, the ball $n$ feet from the Origin is transferred from the Urn to the Shelf.

In either case, the transfer is made straight across, so the ball remains $n$ feet from the Origin. The process unfolds as specified in the Ross-Littlewood problem:

At 11:00 AM, Endeavor transfers balls 1-10 from Shelf to Urn, then moves one of the Urn balls back to Shelf.

At 11:30 AM, Endeavor transfers balls 11-20 from Shelf to Urn, then moves one of the Urn balls back to Shelf.

At 11:45 AM, Endeavor transfers balls 21-30 from Shelf to Urn, then moves one of the Urn balls back to Shelf.

et cetera...

As the process continues, each new step requires longer trips up and down the Aisle, and only half the time to make the trips. Thus, Endeavor must move up and down the Aisle exponentially faster as noon closes in. But it always keeps up with the program, because it has an infinite power supply and can move as fast as needed. Eventually, noon arrives.

What happens in this more vividly imagined version of the paradox? Watched from above, the approach towards noon is truly spectacular. Within the Urn, a Wave of balls appears to propagate outward from the Origin. The Wave's size and speed grow without bound as noon approaches. If we were to take pictures immediately after each step what would the layout of balls would look like? In the deterministic case, they would look exactly like the step functions in amoeba's answer. The ball positions $(x, y)$ would follow precisely the curves he has plotted. In the probabilistic case, it would look roughly similar, but with more straggling near the Origin.

When noon arrives, we take stock of what has happened. In the deterministic version, each ball was transferred from the Shelf to the Urn exactly once, then moved back at a later step, with both transfers happening before noon. At noon, the Universe must be back to its original 11 AM state. The Wave is no more. Each ball is back exactly where it started. Nothing has changed. The Urn is empty. In the probabilistic version the same thing happens, except now the result is only almost sure rather than sure.

In either case, "physical objections" and complaints about infinity seem to vanish into thin air. Of course the Urn is empty at noon. How could we have imagined otherwise?

The only remaining mystery is the fate of Endeavor. Its displacement from the Origin and its velocity became arbitrarily large as noon approached, so at noon, Endeavor is nowhere to be found in our infinite Newtonian Universe. The loss of Endeavor is the only violation of physics which has occurred during the process.

At this point, one could object that Endeavor is not physically possible, since its speed grows without bound and would eventually violate the relativistic limit, the speed of light. However, we can change the scenario slightly to resolve this issue. Instead of a single robot, we could have infinitely many robots, each responsible for a single ball. We could program them beforehand to ensure perfect coordination and timing according to Ross's instructions.

Is this variation 100% physical? Probably not, because the robots would have to operate with arbitrarily precise timing. As we approach noon, the precision demanded would eventually fall below the Planck time and create quantum mechanical issues. But ultimately, an infinite wire and an infinite percolation lattice might not be all that physical either. That doesn't stop us from studying infinite systems and processes and determining what would happen if the obstructing physical constraints were suspended.

4a. Why Count Monotonicity is Violated

A number of Ross skeptics have questioned how it is possible that the number of balls in the urn increases without bound as we approach noon, then is zero at noon. Ultimately we must believe in rigorous analysis over our own intuition, which is often wrong, but there is a variation of the paradox that helps illuminate this mystery.

Suppose that instead of infinitely many balls, we have $10N$ balls labeled 1, 2, 3, up to $10N$, and we issue the following addition to the rules for the ball mover:

If the instructions ask you to move a ball that does not exist, ignore that instruction.

Note that the original problem is unchanged if we add to it this instruction, since the instruction will never be activated with infinitely many balls. Thus, we can think of the original problem and this new family of problems to be part of the same family, with the same rules. Examining the finite $N$ family, especially for very large $N$, can help us to understand the "N = $\infty$" case.

In this variation, the balls accumulate 9 per step as before, but only up to step $N$ of the process. Then the numbers for balls to be added no longer correspond to actual balls, and we can only comply with the instruction to remove balls, and the process stops after $9N$ additional steps, for a total of $10N$ steps. If $N$ is very large, the removal-only phase occurs very close to noon, when the tasks are being done very rapidly, and the urn is emptied out very quickly.

Now suppose we do this variation of the experiment for each value of $N$ and graph the ball count over time, $f_N(t)$, where $t$ ranges from 0 to 1 hour after 11AM (i.e. 11AM to noon). Typically $f_N(t)$ rises for a while, then falls back to zero at or before $t=1$. In the limit as $N$ approaches infinity, the graph rises ever higher and the fall is ever more rapid. By noon the urn is always empty: $f_N(1) = 0$. In the limiting graph, $f(t) = \lim_{N \rightarrow \infty} f_N(t)$, the curve approaches infinity for $t < 1$ but $f(1) = 0$. This is precisely the result derived in Ross's proof: the ball count diverges to infinity before noon, but is zero at noon. In other words, Ross's solution preserves continuity with respect to N: the pointwise limit of the ball count as $N \rightarrow \infty$ matches the ball count in the infinite ball case.

I do not consider this a primary argument for Ross's solution, but it may be helpful for those who are puzzled about why the ball count goes up forever, than crashes to zero at noon. While strange, it is the limiting behavior of the finite version of the problem as $N \rightarrow \infty$, and thus does not come as a "sudden shock" in the infinite case.

A Final Reflection

Why has this problem proven to be such a tar-pit for so many? My speculation is that our physical intuition is much vaguer than we think it is, and we often draw conclusions based on imprecise and incomplete mental conceptions. For example, if I ask you to think of a square that is also a circle, you may imagine something squarish and circlish, but it won't be precisely both of those things - that would be impossible. The human mind can easily mash together vague, contradictory concepts into a single mental picture. If the concepts are less familiar, like the Infinite, we can convince ourselves that these vague mental mashups are actually conceptions of the Real Thing.

This is precisely what happens in the urn problem. We do not really conceive of the whole thing at once; we think about bits and pieces of it, like how many balls there are over time. We wave away supposedly irrelevant technicalities, like what happens to each humble little ball over time, or how exactly an "urn" can hold infinitely many balls. We neglect to set out all the details precisely, not realizing that the result is a mashup of inconsistent, incompatible mental models.

Mathematics is designed to rescue us from this condition. It disciplines and steels us in the face of the unfamiliar and the exotic. It demands that we think twice about the things that "must" be true... right? It reminds us that no matter how strange things get, one and one is still two, a ball is either in an urn or it is not, and a statement is either true or false. If we persevere, these principles eventually bring clarity to most of our problems.

Those who subordinate mathematical analysis to "physical" or "common-sense" intuitions do so at their peril. Hand-waving about intuitions is only the start of physics. Historically, all successful branches of physics have eventually founded themselves on rigorous mathematics, which culls away incorrect physical intuitions, strengthens correct ones, and enables the rigorous study of ideal systems, such as the infinite current-carrying wire, which illuminate the behavior of the more complicated, messy real world. Ross-Littlewood is a physical problem, typically interpreted as one of classical mechanics, and classical mechanics has a completely mature and rigorous mathematical foundation. We should rely upon mathematical modeling and analysis for our intuitions about the world of classical physics, not the other way around.