In Joel David Hamkin’s paper Supertasks and Computation, he relates the following puzzle: Suppose that you have a countable infinity of dollar bills, and one day you meet the devil, who offers you the following bargain: In the first half minute from now, the devil will give you two dollar bills, and take one from you in return. In the quarter minute after that, the devil again gives you two dollar bills, and takes one from you in return. And so on, in the eighth of a minute after that, and the sixteenth of a minute after that, etc. After a minute, the whole transaction is complete. Should you take this bargain?
The answer is “no” and the reason is that the devil could do the following: Think of the bills you have at the start as being numbered 1, 3, 5, etc. and imagine that the devil has an initial pile of bills numbered 2, 4, 6, etc. Then on the nth transaction, the devil gives you the two lowest-numbered bills from his initial pile and takes bill n from you (one can easily show that you have bill n in your possession at this point). Since the devil takes bill n from you on the nth transaction, he gets all the bills in the end and you end up with nothing.
So, even though you start with infinitely many bills and each transaction produces a net gain of one bill for you, after all the transactions are done you have nothing.
In that puzzle, the devil was able to use a tricky strategy to give you more than he took at each stage and still end up with everything. In the following puzzle, which made the rounds when I was a graduate student, no matter what the devil does, he takes everything from you!
You and the devil are taking a train ride together. The train stops at each ordinal. At stop 0, you have countably infinitely many dollar bills. At each stop, the devil does the following two things (in order):
- If you have nonzero number of dollar bills, the devil takes one and destroys it.
- The devil gives you countably infinitely many dollar bills.
Prove that no matter what the devil does, when the train reaches stop (the first uncountable ordinal), you will have no money.
For each stop which is before , let be the first stop by which all the dollar bills which you had at stop which will be destroyed by the devil before stop have already been destroyed by the devil.
Since you have only countably many dollars, is a supremum of countably many countable ordinals, and is therefore itself countable.
Now, let . As a supremum of countably many countable ordinals, is itself countable.
Lemma. For any , at stop , you have no money.
Proof. Suppose that you have at least one dollar bill at stop . Then one of your dollar bills will be destroyed by the devil at that stop. Let that be dollar bill . Then you must have had at all stops for sufficiently large . In particular, there is an ordinal such that you had both at stop and stop . But, by the definition of , this is only possible if is not destroyed at any countable ordinal. But is destroyed at stop , which is a contradiction.
Corollary. At stop , you have no money.
Proof. Suppose not. Any dollar bill that you have at stop you must have had at some countable ordinal . But then, by the above corollary, that dollar was destroyed by stop .