I found that people citing Gödel almost invariably think that he has proven the epistemological impotence of mathematics, in their hearts even freeing them from the stifling dictates of logic and probability. (Instead, Gödel has disproven one of the ways of defining truth.)
For most people Gödel's Undecidability Theorem means: if some genius mathematicians has proven in ways that mere mortals cannot fully comprehend that mathematics cannot grasp the world as well as my intuition, then that's good enough for me.