According to this document, Kurt Gödel had discovered logical inconsistiencies in the U.S. constitution that could theoretically allow it to become a dictatorship. Morgenstern told him not to bring it up during his U.S. citizenship exam, but:
And then [the examiner] turned to Gödel and said, "Now Mr. Gödel, where do you come from?"
Gödel: "Where I come from? Austria."
The Examinor: "What kind of government did you have in Austria?"
Gödel: "It was a republic, but the constitution was such that it finally changed into a dictatorship."
The Examinor: "Oh! This is very bad. This could not happen in this country."
Gödel: "Oh, yes, I can prove it."
Luckily, the examiner changed the subject and Gödel could become a U.S. citizen.
What was his proof?