Gödel’s Loophole really isn’t anything like Fermat’s Last Theorem, because constitutional scholars are pretty sure of what Gödel had in mind. It’s a constitutional version of the idea that, if a genie wafts out of an oil lamp and offers you three wishes, you should begin by wishing for more wishes. In what amounts to a genuine oversight, Article V, the amendment provision, does not prohibit amending Article V. It’s very hard to ratify a constitutional amendment, but if a President could amass enough power and accrue enough blindly loyal followers he could get an amendment ratified that revised the mechanism of amendment itself. If a revised Article V made it possible for a President to amend the Constitution by fiat (e.g., “The President, whenever he shall deem it necessary, shall make amendments to this Constitution, which shall be valid to all intents and purposes, as part of this Constitution”), he could turn a democracy into a dictatorship without ever having done anything unconstitutional.
Constitutional incompleteness may mean that there are rights that cannot be insured by legal means alone. To secure democracy and justice - social, racial, economic - we must be prepared to win those rights not in constitutional court but through political action on the streets and at the ballot box.