Jeffrey Kegler writes:
The story of Gödel's citizenship hearing had been much repeated over the years. What was known was that on 5 December 1947, Kurt Gödel went to his citizenship hearing in Trenton, New Jersey. The examiner was Judge Philip Forman. As his witnesses, Gödel brought his two closest friends, Oskar Morgenstern and Albert Einstein. Gödel was granted citizenship, and took his oath on 2 April 1948. Those were the reliably established facts.
Afterwards, Morgenstern told many people that he and Einstein had had their hands full preventing the brilliant, but politically naive, Gödel from derailing his citizenship chances. No account directly from Morgenstern or anyone else at the hearing had survived, but hearsay versions circulated widely. The hearsay versions show considerable variation, but their burden is something like the following:
Gödel, in his usual manner, had read extensively in preparing for the hearing. In the course of his studies, Gödel decided that he had discovered a flaw in the U.S. Constitution -- a contradiction which would allow the U.S. to be turned into a dictatorship. Gödel, usually quite reticent, seemed to feel a need to make this known. Morgenstern and Einstein warned Gödel that it would be a disaster to confront his citizenship examiner with visions of a Constitutional flaw leading to an American dictatorship.
Arriving in Princeton, the trio had no idea who the examiner would be. They happened to run into Judge Forman. Forman was a friend of Einstein's -- when Einstein became a citizen, Forman had administered the oath. How lucky this was became apparent almost immediately during the questioning. Forman happened to remark how fortunate it was that the US was not a dictatorship, which Gödel took as a cue to explain his discovery. A surprised Forman exchanged glances with Einstein and Morgenstern, cut Gödel off, and forced-marched the hearing through to a successful conclusion.
The History, and the Legend
Nobody seems to know what Gödel's proof was.
Note: Morgenstern's write up of the affair has been discovered: here.
(via @PaulHalpern and @TakingHayekSeriously)