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I'm a senior pure math student, and we have to do a final project for my real analysis class this semester. Right now I have two topics that interest me, but they seem too broad. The two topics are "non-standard analysis" and "ordinals and cardinals." The former is by far my favourite, but I don't know how to approach it. I.e. I need an interesting question in regards to it, or a fun theorem or unexpected result in non-standard analysis that my project can be centered on; I don't want to just start defining what infinitesimals are, without having a clear objective.

Does anyone have any suggestions? I'd greatly appreciate it.

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I suggest something different! $(H^1)^* = \text{BMO}$. –  Jonas Teuwen Jan 8 '12 at 18:15
    
We're only now getting into Lebesgue measure, so that's probably a bit too advanced. Thanks, though. –  FPP Jan 8 '12 at 18:25
    
How about giving a rigorous discussion of the Banach-Tarksi paradox and explaining why it doesn't violate conservation of area. Also might discuss why it is impossible to construct a set function on the power set of $\mathbb{R}$ satisfying the four properties: (i) It should be defined for all sets in the power set; (ii) for an interval it gives its length; (iii) it is countable additive; (iv) it is translation invariant. (It isn't known whether there is one satisfying (i)--(iii), but continuum hypothesis implies that there isn't). –  William Jan 8 '12 at 18:48
    
PS. I only mention these because I did a little presentation on that when I was an undergrad :). –  William Jan 8 '12 at 18:49
    
Hah, it's funny that you mention Banach-Tarski. Unfortunately, someone already grabbed that topic before I even had a chance to do some research on it :(. Good suggestion, though, thanks. That's the kind of thing I'm looking for (but maybe related to non-standard analysis or ordinals/cardinals). –  FPP Jan 8 '12 at 19:07
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