Simultaneously Prove Every Ball Was Removed Yet Number of Balls Goes to Infinity



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This is a very short article showing how we can set up a mathematical process of balls in a vase and then simultaneously prove two facts:

This contradictory situation is known as the Ross-Littlewood paradox.



Setting Up the Mathematical Process

We start with an empty vase \( V \) at time \( t = 0 \). Time advances step by step: \( t = 0, 1, 2, \dots \) At every time step \( t > 0 \), two things happens:

Below is an illustration of the first six steps of this process. The integers in the set \( V \) stands for the numbered balls within the vase, while the integers in set \( R \) represents the balls removed.



Number of Balls Goes to Infinity

Based on the illustration above, we can see that the number of balls in vase \( V \) grows by one every time step. So, it is reasonable to assume that the number of balls in the vase will go to infinity

More rigorously, the number of balls in \( V \) at time \( t \) is \( f(t) = 2t - t = t \). The function \( f(t) = t \) is monotonically increasing and goes to infinity as \( t \) goes to infinity.



Every Ball Was Removed

The most surprising part of this paradox is that we are also able to prove that every single ball placed within the vase must have been removed at some point in the process. The proof is extremely simple:

Any ball in \( V \) that is marked with the number \( n \) must have been removed at \( t = n \)!



Doing This in Finite Time

This paradox can be easily reformulated to finish in finite time instead of requiring time \( t \) to go to infinity. We simply change the time steps to \( t = 0, \frac{1}{2}, \frac{1}{2} + \frac{1}{4}, \frac{1}{2} + \frac{1}{4} + \frac{1}{8}, \dots \)

So, the first four steps of the process now look like this.

Notice that we wait less and less time before performing the next step. The entire process now takes place in just \( \sum_{n=1}^{\infty} \frac{1}{2^n} = 1 \) second!

The astute reader would probably have already noticed that this is just a sleight of hand: the infinity is just shifted into that \(1\) second. Instead of an infinite end point, we have infinite steps within a finite interval. These kinds of processes are known as supertasks.



Sum of Infinite Sequences

The Ross-Littlewood paradox is analogous to what happened with the Grandi’s series:


\[ 1 - 1 + 1 - 1 + 1 - 1 + \dots \]

By bracketing the terms in two different ways, one can make a false argument that it must be equal to both \( 0 \) and \( 1 \).


\[ 1 = 1 + (- 1 + 1) + (- 1 + 1) + \dots \]\[ 0 = (1 - 1) + (1 - 1) + (1 - 1) + \dots \]

The solution to this apparent contradiction is the rule that we can only bracket terms in an infinite sequence if the sequence is convergent. There is also a way to “prove” that the series is equal to \( \frac{1}{2} \).


\[ \begin{align} S &= 1 - 1 + 1 - 1 + 1 - 1 + \dots \\[0.5em] &= 1 - (1 + 1 - 1 + 1 - 1 + \dots) \\[0.5em] &= 1 - S \\[0.5em] &= \frac{1}{2}. \end{align} \]

The solution to this one is the rule that we can only assign a value \( S \) to infinite sequences if and only if it converges.



Possible Resolution

The resolution to the Ross-Littlewood paradox that I personally believe in is the same as that for Grandi’s series. I see both the number of balls going to infinity and that every ball was removed as true statements. The paradox only happens if we want to know what happens at the end of the process, similar to wanting to know the sum of non-convergent series.

There is no paradox if we view the process as non-convergent and the number of balls in the vase to be forever in flux.