The Ross-Littlewood paradox involves an infinitely large empty vase and an infinite number of balls. At each step, 10 balls are put in the vase and then one ball is taken out. Each step takes half the amount of time as the previous one, ensuring that the task is completed in a finite amount of time.
At the end, how many balls does the vase contain?
The Intuitive Answer: Infinitely Many
The intuitive answer appears to be infinitely many. Each step involves a net increase of nine balls in the vase, so the number of balls just grows toward ∞.
The Surprising Answer: Zero
However, there is a sense in which the vase may be empty. Imagine that at the start, all of the balls are numbered. At step one, you put in balls 1 through 10 and remove ball 1. At step two, you put in balls 11 through 20 and remove ball 2, and so on.
For each step number n, the ball numbered n is removed from the vase. Eventually, ball number 1,000 will be removed. Then ball number 1,000,000, and so on. Since there are no balls that do not eventually get removed by the end, all of the balls must have come out.
An Unresolved Paradox
There is no general agreement on the solution to this paradox.
Why Does This Matter?
The Ross-Littlewood paradox highlights deep tensions in how we reason about infinity and completed infinite processes, known as supertasks. It challenges the assumption that intuitive arithmetic reasoning (adding 9 balls per step) can be safely extended to infinite sequences. The paradox also illustrates how the final result depends entirely on how the problem is formulated. If the balls are numbered and removed in a specific order, the vase is empty. If balls are removed randomly, the answer becomes indeterminate. This sensitivity to setup makes the Ross-Littlewood paradox a favorite example in set theory and the philosophy of mathematics for demonstrating that infinity does not always behave the way we expect.
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