Here's something cool I found out about on Wikipedia. Goodstein's theorem, which is that every Goodstein sequence converges to 0, has a pretty simple proof using infinite ordinals, but it can be shown to be unprovable within Peano arithmetic. It's a really natural-looking theorem as examples of undecidability go, and the proof using ordinals is easy to follow. The proof that it's unprovable in PA is trickier, unfortunately.

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The unity sound a lot like Vicki from "Small Wonder" in my head, but I'm not sure if that's even remotely canonical, so don't start writing fanfic with that presupposition.