From the always-excellent Futility Closet:
In 1947 Einstein and economist Oskar Morgenstern accompanied Gödel to his U.S. citizenship exam because they were concerned about his unpredictable behavior: During his voluminous preparation for the exam, Gödel said, he had uncovered a flaw in the U.S. constitution that could lead to a dictatorship. Einstein and Morgenstern told him that the exam would really be quite simple and urged him not to prepare so extensively.
At the hearing, judge Phillip Forman asked Gödel:
“Now, Mr. Gödel, where do you come from?”
“Where I come from? Austria.”
“What kind of government did you have in Austria?”
“It was a republic, but the constitution was such that it finally was changed into a dictatorship.”
“Oh! That is very bad. This could not happen in this country.”
“Oh, yes,” Gödel said. “I can prove it.”
“So of all the possible questions, just that critical one was asked by the Examinor,” Morgenstern wrote later. “Einstein and I were horrified during this exchange; the Examinor was intelligent enough to quickly quieten Gödel and say, ‘Oh, God, let’s not go into this.'”
The logician got his citizenship and the friends returned to Princeton. What was the flaw that Gödel had found? There’s no record of it in Morgenstern’s account, so we don’t know. Stephen Hawking suggests that it involved the president’s power to fill vacancies during Senate recesses, and Barry University law professor F.E. Guerra-Pujol conjectures that it might involve the constitution’s power to amend itself. Maybe it’s best if we never discover it. [emphases mine]
Oh, but... we've re-discovered it. We have indeed. Not only did Gödel know about it, but it seems to have been no surprise to Forman.
I would say, it's related to the Paradox of Tolerance. And to the fact that the Constitution was actually designed not for all the people of the United States, but for the white male landowner faction alone.