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?

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    $\begingroup$ Try this quora.com/… $\endgroup$ – Gerald Edgar Jun 27 '16 at 21:38
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    $\begingroup$ As Gerald Edgar has commented, the quora page gives full details. Interestingly, in true Gödel fashion, the argument he constructed was one of self-reference, involving the application of the terms of the articles to the articles themselves. Both Morgenstern and Einstein pleaded with Gödel not to bring it up at the ceremony, but their pleas fell on deaf ears apparently. Shrewdly, the judge cut him off before he could get into full stride. $\endgroup$ – Nick R Jun 27 '16 at 22:59
  • $\begingroup$ @NickR I read that, and it said that the author's theory as to what Godel had discovered. $\endgroup$ – PyRulez Jun 27 '16 at 23:01
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    $\begingroup$ The version I have read is in Palle Yourgrau's biography of Gödel, though he doesn't spell out the argument in detail. Yourgrau sites no source for the story, but adds that Morgenstern and Einstein's concerns were well founded since the FBI had been intercepting Gödel's correspondences with his mother for years. Yourgrau also states that years later, when asked for a legal analogy to his incompleteness theorem, Gödel restated his argument concerning the constitution - but again no source is given. $\endgroup$ – Nick R Jun 27 '16 at 23:08
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    $\begingroup$ Already discussed in this post. $\endgroup$ – Mauro ALLEGRANZA Jun 28 '16 at 7:45

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