I dunno how a 12 years old will respond to this explanation of the second incompleteness theorem, but in my experience non mathematicians follow it just fine:

1. Math needs postulates. Without assuming anything you can't infer anything
2. During the late 19th and early 20th centuries, mathematicians and philosophers collaborated to find appropriate postulates
3. One of the things you'd expect a system of such postulate is to be consistent, that is, not to contradict itself
4. People thought that it should be mathematically possible to prove that the postulates are consistent.
5. Since the postulates are the only initial assumptions of mathematics, this means that the quest was after a system that could prove the consistency of itself
6. Another thing that people expect from such a system, is that it would be expressive enough to describe mathematical structures. A consistent system which doesn't give you the tools to describe, say, the naturals and their arithmetic, is useless
7. For some time, people believed that such a system exists, and its only a matter of finding a proper way to define it
8. In 1934, Godel shattered this vision when he proved his (second) incompleteness theorem: Any system expressive enough to describe natural arithmetic and to prove its own consistency is inconsistent
9. Therefor, the only way a system of postulates will be strong enough to be a basis for math is if we accept that math could never prove its own consistency

Suggestion: This is a very bare bone explanation. You could spice it up with some history. I know its a bit unorthodox, but you might want to take a peak at Logicomix. It is a very fine example of a great balance between accuracy and accessibility for laymen, and as luck would have it -- it's topic is pretty much the development of analytic logic from Cantor through Russel to Godel.

Disclaimer: This is a very heuristic example, and it omits technical details (for example, the statement of the incompleteness theorem itself is actually considerably stronger than the one stated here), but I find these details overbearing and unimportant for laypersons.