Level of mathematical proof is considerably higher in some parts of economics -- particularly in the use of fixed-point theorems, notably the Kakutani fixed point theorem, a generalization of the classical Brouwer fixed point theory from real analysis -- in general equilibrium models. One rarely if ever sees anything comparable in political science.
Godel's theorem, by the way, states that there are some propositions in any non-trivial formal system that are unprovable, not that all are unprovable. "1+1 = 2" is proveable (albeit in multiple ways, depending on where you want to start axiomatically). For arguments about systems where all propositions are unprovable, see the qualitative methods thread...