TimS comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion
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Scientifically, I suggest most of psychology and psychiatry (I'm looking at you, Law & Order: SVU). In a similar vein is programming/hacking (e.g. 24, any crime drama).
In popular culture? What misunderstandings of it have you seen?
Hmm, not popular culture. Certainly arguing with people purveying nonsense as a form of the argument "you can't be certain therefore I might be right."