Ph103: Discussion Questions 4

Short Answer Questions

1. Given an example of a supertask, and explain how the example satisfies the definition.
2. Explain one way to "complete" the description of Thomson's lamp so as to avoid the paradox.
3. If we assume that the balls in the jar in Ross' Paradox have continuous worldlines, then how many balls are in the jar at the end of Ross' procedure?
4. Provide and explain an example in which a limit being approached does not actually obtain in the physical system that completes a supertask.
   
   
   
   

For Further Discussion

• Hilbert's Hotel. We have seen how, as the manager of Hilbert's hotel, you can make room for a new occupant even when all the rooms are full. (a) Is it possible to make room for 10 new occupants without removing any of the existing ones? (b) Is it possible to make room for infinitely many new occupants without removing any of the existing ones? If you answer no to either question, then explain why. If you answer yes, then explain what procedure the occupants should follow to make enough room.
• The infinite marketing scheme. Suppose two people purchase a house together, and each pay £1000. Each of them finds two more people, and gets each of those two people to buy their portion of the house for £1000. In this way, the two original people earn £1000, and the four new people go into debt of £2000 each. Suppose the procedure continues in this way. At each stage, the previous purchasers recuperate their loss and earn £1000, while the new purchasers pile up larger and larger debt.
  • Suppose this procedure is carried out infinitely many times. Is this a supertask according to our definition? Explain.
  • After infinitely many steps in the procedure have been completed, how much money will each person have earned?
  • Is there any group of people who won't have earned money on this infinite procedure?
  • The total debt is obviously increasing at each stage of the procedure. What is the mathematical limit of this debt as the procedure continues? How does this compare to the actual final amount of debt at the end of the procedure?
• More supertasks? We have seen a great number of supertasks. Can you invent any of your own? They may be similar to the ones we have seen in class. For each example you come up with, ask the following questions.
  • Is this a supertask according to the official definition?
  • Is there a sense in which this supertask is or is not a "complete description" of a system?
  • Is this supertask logically possible?
  • Is this supertask physically possible?