Anyone interested in this would immensely enjoy Meta Math! by Gregory Chaitin. Basically, he demonstrates that there are theorems which are randomly true: they're true for no particular reason. It's, of course, deeply connected with Gödel's work.

If anyone's working in this space, I'd love to chat with people familiar this stuff.

Edit: arXiv PDF preprint is at http://arxiv.org/pdf/math/0404335.pdf