[Gödel's Constructible Universe, L][1], is the fundamental example of [inner model theory][2]. L was the first inner model, and is the minimal one. One could argue that the point of inner model theory is to build L-like models that do things L cannot.

  [1]: http://en.wikipedia.org/wiki/G%F6del%27s%20constructible%20universe
  [2]: http://en.wikipedia.org/wiki/Inner%20model