Assignment 3. Supertasks

For submission

1. A compass passes through a magnetic anomaly that leads its needle to rotate over the compass card faster and faster. It jumps in discrete steps through N, E, S, W, N, E, S, W, ... such that infinitely many jumps are completed by midnight. At midnight, where does the needle point?

2. The compass needle has two ends. Initially one points N and the other S. When the compass passes through the anomaly, the first end cycles through N, E, S, W, N, ... and the other end cycles through the opposite directions S, W, N E, S, ...

2a Where does each end of the needle point after the infinity of jumps is completed?

2b Do they then point in opposite directions?

3. Two compasses side by side pass through the same anomaly and undergo the same sequence of discrete steps.

3a. Where are their needles each pointing after the infinity of jumps is completed?

3b. Whatever your answer may have been to this first question, do they point in the same direction after the infinity of jumps is completed?

For discussion

Not for submission

A. Surely Thomson's lamp must be on or off at midnight, whether we can say which is the case. Does not that mean that the paradox persists?

B. Surely the marble of Black's transfer machine must be somewhere in space, whether we can say what that position is. Does not that mean that the paradox persists?

C. Consider the proposition "There are infinitely many stars." For it to be true, there simply have to be infinitely many stars. This is a factual condition about the world about us. Can the same analysis be given for the proposition in arithmetic: "there are infinitely many prime numbers"?

D. What is the time reversal of the Thomson lamp supertask? What is the schedule of on/off switchings? Can you devise a mechanism that implements it?

E. The chapter leaves open whether the reversal of the supertasks of Black's transfer machine and the many accelerating spaceships is admissible. Just what separates admissible from inadmissible supertasks?