The argument would go something like this: "You say you are a non-believer, or that you cannot understand how an intellectual could have a spiritual life. But not only are there things that the most hardcore atheist must take on faith every day, but it is provable that there is are some things that are true but not provable, no matter how hard you try. So you might as well relax your objective of pure reason behind all your beliefs."
And then I would bring up the Incompleteness Theorem. But if you weren't a well-read philosopher, mathematician, or computer scientist, you probably wouldn't get it. So I tried finding a short form. I would use: "This sentence is true but not provable."
The form of the conversational proof was to assume the opposite is true, and find the contradiction. There is a question of whether this is a proof or merely a paradox, and the whole question probably would fall under the kind of word problem that justifiably bugged Wittgenstein, but it does give a sense of the Incompleteness Theorem without going into the diagonalization proof.
The real question is is there anything interesting that is true but not provable? That's a lot harder to show. Nevertheless, there was a time where I used Godel to give some pedigree to my principled, anti-intellectual position, or in other words, the smart Viennese guy says it's ok to believe. I have outgrown this argument; it wasn't convincing, but it was fun enough.
