[ultrafilter]

There's no relation to his incompleteness theorems. Gödel proved that every universe V of sets that satisfies the (ZFC) axioms has a subuniverse L that satisfies both the axioms and the continuum hypothesis (CH). L is constructed by transfinite recursion. At each stage of the construction, the only new sets you admit are the definable subsets of the set of all sets you admitted in earlier stages.

http://en.wikipedia.org/wiki/Godel%27s_constructible_univers...

Unfortunately, proving that L satisfies ZFC and CH is a very delicate and technical matter. On the other hand, it can be fun because it's extremely meta. One of the steps of the argument is proving that L is the L of L. I learned the proof from a graduate textbook chapter titled "defining definability."