Short Answer Questions

1. What is the difference between the ordinary and the "regressive" form of Zeno's Dichotomy Paradox? What did Zeno conclude from these arguments?
2. Explain the definition of a limit in your own words.
3. How does Salmon argue that Zeno's Paradox of the Arrow can be overcome?
4. Explain Weyl's "tile argument" that our space is note discrete. Draw diagram to illustrate.

For Further Discussion

• Solving Zeno's Dichotomy Paradox. The concept of a limit can be used to solve Zeno's Dichotomy Paradox.
  1. Write down an infinite sequence of sums corresponding to the distance Achilles travels in the "regressive" form of the paradox. (Hint: This is actually no different than the sums for the ordinary form of the paradox.)
  2. Identify how the terms in the sequence can be interpreted in terms of a distance that Achilles has run.
  3. Use the definition of a limit to argue that this sequence converges in the limit to 1. (Hint: Choose any fixed number positive number, like 1/8. Then argue that there is a term in the sequence beyond which every later term is within 1/8 of 1.)
  4. What does this imply about the distance that Achilles runs?
  5. What does it imply about Zeno's "regressive" Dichotomy Paradox?
  6. Why would this argument have been difficult before the concept of a limit was invented?
• Paradoxes of Discrete Space. Salmon's version of the Stadium Paradox establishes a sense in which if space is discrete, then one can set describe bodies that move past each other, but for which there is no position at which the passing occurs.
  • Is this really a paradox? What exactly, if anything, is problematic about it?
  You have seen how Weyl tried to establish that the space that we live in is not discrete.
  • Do you agree with Weyl's argument? How might someone challenge it?
• Completing Infinite Tasks. One might try to draw a lesson from all this about completing infinite tasks. What examples (besides a race) can you think of in which an infinite task can be completed? Can you think of examples of infinite tasks that cannot be completed? What is the difference between the two?

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