The Balls and Urns Paradox

Start with an empty urn having an unlimited capacity, and a infinite number of balls, labeled “1”, “2”, “3”, etc.

  • At time T minus 1 second, put balls #1-10 into the urn, then take ball #1 out.
  • At time T minus 1/2 second, put balls #11-20 into the urn, then take ball #2 out.
  • At time T minus 1/4 second, put balls #21-30 into the urn, then take ball #3 out.
  • At time T minus 1/8 second, put balls #31-40 into the urn, then take ball #4 out.
  • And so on. At each step, take out the lowest-numbered remaining ball.

At time T, how many balls are in the urn?

This thought experiment is known as the Balls and Urn Paradox, the Balls and Vase Paradox, the Ross–Littlewood paradox, and probably other names. It belongs to a class of thought experiments called supertasks. Obviously, it violates the laws of physics in this universe; but it could still be possible to reason about it mathematically.

The technically correct answer is that there is no answer. This task is not physically possible, and there’s no unique way to imagine it being possible, so we can stop right there.

The orthodox answer, though, is “zero”. Every ball (seemingly) is removed at some point before time T, so therefore (seemingly) there cannot be any balls left at time T. Oddly, this implies that the number of balls left depends on which balls you remove. If you always remove, say, the highest-numbered ball instead of the lowest-numbered ball, then the urn will end up containing an infinite number of balls.

Often, when somebody writes about this, they give “zero” as the correct answer.

I object. It’s not that it’s wrong, but it’s not the one and only correct answer. Personally, I don’t even think it’s the best answer.

I accept that the limit of the ball count does not necessarily have to equal the ball count of the limit. In general, it is possible for a limit to have properties that are quite distinct from the properties of any intermediate step. Again, I don’t think the “zero” answer is wrong, and I definitely don’t think it’s wrong for this reason in particular.

Still, I think the best answer is that the urn will end up with an infinite number of balls, no matter which balls you remove. The “zero” answer relies on an unproven and unprovable assertion:

The Every Ball’s Number Is Finite Axiom: If an urn contains no balls that have a finite label, then the urn contains no balls.

This statement is true in the context of finite tasks, but as this is not a finite task, I see no particular justification for it.

What I envision that the balls remaining in the urn would all have infinite labels. Something has to give, and that’s the sacrifice I would choose to make in order to answer the puzzle. Infinite numbers are not that exotic. The ordinal numbers are one example of such a thing. Surreal numbers are another.

To be clear, I did not invent the idea that “infinity” is a possible answer. Some descriptions of the paradox do acknowledge this possibility. But others do not.

The Balls and Urn Paradox is much like Bertrand Russell’s paradox of Tristram Shandy’s diary (sometimes called Tristram Shandy’s autobiography). Supposedly, it took Tristram Shandy one year to write about the first day of his life. If he continues at that pace, and he lives an infinitely long time, will he ever finish his diary? Some argue that he does, reasoning that one cannot point to a specific day of his life that he never wrote about. As with the Balls and Urns Paradox, I personally don’t think that’s the best answer.

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