In this project you will investigate several paradoxes that arise when studying sets. These paradoxes can be resolved by applying the techniques of modern set theory as set forth by Zermelo and Fränkel (among others). However, we're not interested in getting that carried away.

1. Define the word paradox.
2. For any given set A, the power set of A is the set of all subsets of A. For instance, if A = {1,2}, then the power set of A is the set {Æ,{1},{2},{1,2}}. Find the power set of B = {1,2,3} and C = {1,2,3,4}.
3. Argue that the power set of a set A cannot be equivalent to A. Georg Cantor was the first to prove this for general sets, finite or infinite.
4. (Cantor's Paradox) Consider the set of all sets. Explain why its power set must be a subset of itself. How does this contradict what you've said above? How (or why) does this contradiction arise?
5. Write a one or two paragraph biography of Georg Cantor.
6. Give an example of a set that is a member of itself. (Hint: Such a set is mentioned somewhere on this page.)
7. (Russell's Paradox) Let R be the set of all sets that are not members of themselves. Then R is neither a member of itself nor not a member of itself. Explain.
8. Write a one or two paragraph biography of Bertrand Russell.
9. Russell's Paradox has been stated in many other ways. Here is one of them:

   (Catalogue Paradox) Consider a library that compiles a bibliographic catalog of all (and only those) catalogs that do not list themselves. Then does the library's catalog list itself?

   Find another version of Russell's Paradox.