So if you Gödelized the universe - mapped every conceivable state to a number (proving that that is possible left as an exercise for the reader) - then created mathematical operations on those numbers that transitioned the universe from one state to another 'physically possible' successor state.. I guess Gödel would be able to give you a number representing a universe such that you could not prove whether its state was possible or not?

Then all you have to do is demonstrate that we live in such a universe, and all the philosophers can retire because we've found the ultimate answer to the ultimate question.

I think we all know exactly what the Gödel number for our universe would be...