The greatest explanation I've ever seen for the Incompleteness Theorems comes from Palle Yourgrau in his book A World Without Time. He gives an excellent introduction to his explanation: "To appreciate Godel's theorem is your birthright; let no one, including the mathematical police, deprive you of what you have a right to enjoy."
A rhetorical and shadowy group who discourage ordinary people from reading about or becoming interested in mathematics. A lot of popular science authors invoke this underground organisation, so it must exist.
