Assignment 3. Supertasks

For submission

1. The pointer on the device shown can rotate. When the pointer rotates, it jumps over the settings indicated, N, E, S, W, .... Each jump taking it from one setting to the next. Imagine that the pointer is made to move faster and faster, so that it completes infinitely many jumps by midnight. At midnight, what is its setting?

2. The pointer has two tips, a black and white tip.

2a Before the pointer is set in motion, to what does each tip point?

2b After the infinitely many jumps are completed, to what does each tip point?

2c After the infinitely many jumps are completed, assuming nothing else is done to the device, do the two tips point to opposite settings?

3. Two identical devices are set up side by side. They complete exactly the same infinity of jumps by midnight and nothing else is done to them.

3a. At midnight, what are the settings on the two devices?

3b. At midnight, are the settings on the two devices the same?

For discussion

Not for submission

A. Surely Thomson's lamp must be on or off at midnight, whether or not 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. The number π, written in decimal notation, has an infinite, non-repeating sequence of digits:

3.14159265358979323846264338327950288419716939937510...

Here is an infinite calculation designed to compute the last digit of π. At stages 1, 2, 3, ... we compute successive parts of the infinite decimal expansion 3.14159... of π. We write the last digit of each part in a single box that we have set aside for the last digit.

Stage 1. Compute 3.1 Write "1" in the box.      1

Stage 2. Compute 3.14     Write "4" in the box.         4

Stage 3. Compute 3.141 Write "1" in the box.     1

Stage 4. Compute 3.1415 Write "5" in the box.   5
...

Once all the infinitely many stages are completed, all we need to do is look in the box to learn which is the last digit of π. Right? Or is there a problem?

D. 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"?

E. 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?

F. 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?