Imagine a universe empty but for two gods, the Creator and the Destroyer. Suddenly, the Creator begins her creative activity: she creates two quirks every second and never stops. As soon as there are quirks to destroy, the Destroyer also begins to act: he destroys one quirk every second and never stops. (Let a quirk be the ultimate sub-atomic particle; a quirk looks like a beach ball in 11-dimensional space but not quite as diagonal.) After ₀ seconds, does the universe contain any quirks?

While they create and destroy, the gods talk about what the universe will look like after ₀ seconds.

Destroyer:

There will be no quirks; I will destroy them all. Add up the number of quirks you will create, and the number of quirks I will destroy, and you will see that they are equal: 2 · ₀ = ₀.

Creator:

There will still be quirks. Your acts of destruction can be put into one-to-one correspondence with my acts of creation. Hence, each of your acts of destruction will leave at least one quirk undestroyed, namely, one of the two that I created at the same time. So at any second (now that creation has begun) there will be at least one quirk in the universe. Note that this is true even if the quirks disappear of their own accord after two seconds of life so that they do not accumulate. Hence the universe will never again be empty of quirks.

Two questions. (1) Who is right? (2) Where is the mistake in the argument for the incorrect conclusion?

Don't say they are both right, for the contradiction between them would spread quickly. If we cannot eliminate one of the answers as mistaken, then there would be a conflict between the transfinite arithmetic (in the Destroyer's argument) and one-to-one correspondence as applied to infinite sets (in the Creator's argument). But if so, then there is an internal conflict within transfinite arithmetic, since many of its fundamental theorems (e.g. Cantor's theorem) were proved by appeal to one-to-one correspondence applied to infinite sets. So until we detoxify it, the quirk problem does not simply illustrate one of the counter-intuitive results of transfinite arithmetic; it is a paradox which threatens the foundations of transfinite arithmetic. Help!

This is a variation of a problem suggested to me by Hal Hanes.

This file is an electronic hand-out for the course, Logical Systems.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.