Infinity in ethics

4. Infinitely divisible bounded time

The previous puzzle arose in part because time was unbounded (infinitely long). Puzzles can also arise when time is bounded (finitely long) but infinitely divisible (or dense). Suppose that time is dense, which is to say that between any two points in time there is at least one other point in time, and consider the points in time between any two arbitrarily chosen points in time – between 0 and 1, say, as measured in complete days. Suppose further that time has a metric (which measures the length of temporal intervals). For example, halfway between 0 and 1 there is 0.5 (half a day), halfway between 0.5 and 1 there is 0.75, and so on. Suppose that at time 0 there are infinitely many one-dollar bills that have been numbered using the natural numbers (1, 2, 3 and so on) and that no dollar bills later come into existence. Suppose that at time 0 God has possession of all the dollar bills, but offers to transfer them to you by this scheme: At each of the times 1/2, 3/4, 7/8, …, (2ⁿ−1)/2ⁿ, …, God will give you two arbitrarily chosen bills from those not yet given to you and will also then destroy the lowest numbered bill in your possession since the last transaction. All bills in your possession between time 0 and time 1 remain in your possession, unless destroyed by the above process at a time prior to time 1. You are not permitted to use these bills until time 1 and nothing else of value is affected by this scheme. Is God’s offer worth accepting? Does it increase the amount of money that you have?

It may seem that God’s offer will increase the amount of money that you have at time 1, but it will not. For when time 1 arrives, God will have destroyed every single dollar bill. This is so because (1) every dollar bill has a number and (2) for any given dollar bill, no matter what its number, at some point prior to time 1 that number will be the lowest of the numbers of the bills in your possession. Hence the bill will be destroyed prior to time 1. The specification of the problem entails that all the dollar bills are destroyed by time 1.

Still, this result is puzzling. Each transaction prior to time 1 increases the number of bills you have by one. Nonetheless, at time 1 you have no bills. Furthermore, very different results can be generated by seemingly trivial differences in the specification. Suppose, for example, that at each of the specified times God arbitrarily chooses two bills in his possession, destroys the bill with the smaller number of the two bills and then gives you the remaining one. In this case, you will have infinitely many bills at time 1. It seems quite strange, however, that merely changing which bill is destroyed at each of the given times should have any effect on how much money you end up with. Depending on the background assumption, however, it can.