I recently encountered Godel's incompleteness theorem (there's two technically). What it basically states is that there are some statements in mathematics, many of them very simple, which are impossible to actually prove using mathematics.
As physics is based upon mathematics, is the objective world equally flawed in its logic?
Do some logic systems become too complex to be predicted logically? Is determinism therefore flawed?
Is it impossible to simulate neural activity and hence consciousness? Would this allow for free will?
If you are an advanced mathematician then your contributions would be greatly appreciated, of course. I personally know nothing of set theory so I can't say anything at all about the technical details.
Discuss! :P