Here I will list interesting tasks that may be taken up as final projects. You can add your own provided that they are sufficiently amusing and non-trivial (read hard).

1. Understand what it is that the theorem says

2. Understand the proof and present it at the level suitable to the general level of the class.

3. Understand how Axion of Choice is involved.

4. Why is this a paradox? How does this make you feel?

Good start to see if this is something for you (wikipedia): http://en.wikipedia.org/wiki/Banach–Tarski_paradox

Group size 2 (maybe 3)

This axiom, a part of the currently most widely accepted set theory, ZF(C), is an excellent study case on how we go about accepting or rejecting mathematical principles in particular and other theoreticall principles in general.

1. What is the general statement of the axiom, i.e., what is the content of the axiom?

2. What is The Axiom of Countable Choice? Why is it that almost no one has difficulties with the countable version? What is the significant difference?

3. What are some forms (i.e. equivalent statements) of the axiom of choice that you find particularly counterintuitive?

4. What can our intuitions teach us about the infinite? Do you think the AC is an instructive case study here?

Start with Wikipedia:" http://en.wikipedia.org/wiki/Axiom_of_choice

Group 2 (perhaps 3)

So according to Goedel the formal arithmetic is incomplete. And according to the second incompleteness the arithmetic can't probe its own consistency.

1. So what? What are some of the significant consequences of this result?

2. Why don't we just add more rules and axioms and simply make it complete? Is it possible to do some such thing in any interesting sense?

3. For consistency, why don't we just add a claim asserting that the arithmetic is consistent to the system itself?

http://en.wikipedia.org/wiki/Goedel_second_incompleteness_theorem

Group 2--3.

Berry paradox: http://en.wikipedia.org/wiki/Berry%27s_paradox

Russell's Paradox: http://en.wikipedia.org/wiki/Russell%27s_paradox

Set of all sets: http://en.wikipedia.org/wiki/Set_of_all_sets

Burali-Forti paradox: http://en.wikipedia.org/wiki/Burali-Forti_paradox

Barber: http://en.wikipedia.org/wiki/Barber_paradox

Curry: http://en.wikipedia.org/wiki/Curry%27s_paradox

1. What connects all these paradoxes?

2. Do you think that it is reasonable to have a set theory that allows for sets X, Y , Z, where X is an element of X, Y is an element of Z, and Z is an element of Y.

3. Take a look at: http://plato.stanford.edu/entries/nonwellfounded-set-theory/ What are your thoughts?

4. The next sentence is false.

5. 1-- 5 are all true.

Group of 4 -- 5.