"... provided that no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article; and that no state, without its consent, shall be deprived of its equal suffrage in the Senate."
Unless I'm misunderstanding some part of this paper, this seems more like a feature, than a bug. It also reflects practical realities: if a legislature overwhelmingly supports a change, any roadblocks or restrictions will most likely be ineffective in the long term.
Exactly. In the worst case, if an obstructionist minority of states were to block an overwhelmingly popular and desperately needed amendment, a sufficiently large number of states could simply write a new Constitution. After all, that's exactly how the current U.S. Constitution got put into place.
The Articles of Confederation could only be amended by unanimous vote of all 13 states. And of course, they needed unanimity to amend the unanimity clause. So it was an amendment clause without loopholes.
But only 12 states showed up at the Constitutional Convention! Unanimity was obviously not going to happen.
So the delegates wrote a new Constitution anyway, and specified a ratification threshold of 9 states. Once this threshold had been reached, the other 4 states could have fun governing their rump country under the old Articles. But the ratifying states would be in a customs union and have a strong central government, and they could make do just fine without the holdouts.
In the end, only two states held out. Eventually, both gave in. North Carolina didn't even last a year on its own. Rhode Island lasted 15 months.
The same people who pulled off this stunt also wrote the Constitution. So they were well-aware that a watertight amendment process was no bar to the truly determined. Might as well leave it leaky.
Interesting to note that something like this was tried back in 1861. In the end, these things boil down to what people are willing to do and what they are willing to put up with.
Hopefully this doesn't result in another civil war over the issue of gun control. (The current stance seems to be to allow individual states great latitude in how they interpret the 2nd amendment.)
I agree with you that it is not "one weird trick" or a "hack" or something that is unknown and easily exploitable, which is why I think Einstein et al. dismissed the concern, supposing this is the actual concern that Godel was proposing.
If anything the adoption of the 22nd Amendment - using the process that the 5th allows - somewhat shows that the system works the opposite direction by placing more restrictions on potential tyrannies.
What would happen if 3/4s of the states passed an amendment repealing this clause?
The prohibition of changes to representation to the Senate doesn't work as an inescapable barrier and isn't intended, it works as a red-flag warning.
The enforcement mechanism it relies on isn't law-as-mystic-amulet, is the same one that enforced the view American colonists had of their proper freedoms when it clashed with the policies of the British Crown.
Some programmers have a hard time with this.
Not necessarily. Consider that originally the US Senate was elected by the state legislatures. Representing the people in the federal government was the role of the House, the Senate was to represent the states. So the concern may not have been that 3/4ths of the states would want to disenfranchise the remainder (which was fairly implausible given that the states benefiting from disproportionate representation outnumber the more populous states), but rather that a populist movement could remove the Senate as a check by the state legislatures on federal power. Which is exactly what happened when the Seventeenth Amendment brought about direct election of US Senators.
2. No, but not only because of the "loophole". Godel feared that Article V permitted a legally sanctioned dictatorship, but this clause would be irrelevant in that case. It would certainly be possible to reduce the Senate to obsolescence by limiting the powers of the Senate via permissible amendments, while maintaining the original suffrage in the Senate as required.
Edit: You have edited your comment, so my replies are no longer clear. I am responding to two questions: what was the intent of the suffrage clause, and is it effective?
A little bit of a tangent, but amendment process shouldn't be confused with the theory of the Constitution as a living document. In fact, they're almost opposites. The living document doctrine holds that the Constitution can change -- without formal amendment -- based on the evolving standards of a society.
E.g., what is "cruel & unusual" is not based on what the authors had in mind when they wrote the phrase, but rather what society thinks that means today.
If law-making is a game, then it is a game in which changing the rules is a move.
As the National Archives describes: “Some continued to argue that the delegates in Philadelphia had exceeded their congressional authority by replacing the Articles of Confederation with an illegal new document.” 
Because (i)the Articles of Confederation required a unanimous vote to alter it, (ii) the Constitutional Convention was only empowered to propose alterations to the Articles and (iii) the Constitution, as adopted, broke the “unanimous vote required for alterations” rule, these anti-Federalists, and perhaps Gödel, felt the US Constitution was not established on firm legal footing. [2, 3, 4]
The obvious answer — that the government derives its just consent from the governed, and should they choose to alter it, accept it in democratic votes in each of the states, and abide by it, it becomes both de jury and de facto law — is the superior one, IMHO.
You raised a very interesting meta-question. Not that it's of any practical significance 200 years later, but the Articles of Confederation are very much an important part of US history.
If the government goes off track from the Constitution with the willingness of the populace there's nothing in the Constitution or the world that can stop it.
Consider the birth of the Constitution itself: there was no mandate for an entirely new baseline document to replace the Articles of Confederation. But one was created and the states adopted it. It came into power because the citizens of the United States gave it power. That's the only way governments are formed and changed. The founders of the US understood that very well.
Theoretically possible, sure. Fascinating, absolutely. Likely? Not really.
A similar compact for approving amendments could be an issue for instance.
edit: I mixed up my EU facts below. The European Parliament is filled by direct election. It's the Council that is elected by the executive branches of member states, and is thus more analogous to the US Senate before switching to direct election. The EU is complicated for my American brain to keep straight! I still might have some of this wrong.
Compare the electoral college to the European Parliamentary elections, which are decided not by all EU citizens but by the Council, or the President of the European Convention, who is elected by the European Parliament.
In other words, they are states -- as opposed to nations (as per your EU example).
We had a flurry of amendments to the US constitution back in the 1960s. Since then, in 1971 we gave 18 year olds the right to vote. There was only one other amendment since then, in 1992, and it was "housekeeping". It delayed salary changes for legislators until after the next election.
I'm old enough to remember the contentious proposed Equal Rights Amendment.  It kicked around for years in the 1970s but was never ratified.
Since then, nothing. And, given the political climate in the USA (the "red state" vs "blue state" dichotomy) it's unlikely that any new amendment would pass anytime soon, unless it was necessary to solve a very very critical problem. It's hard to believe that an amendment to change Article V would ever pass both houses of Congress then get ratified by 38 states.
Not. Gonna. Happen.
It's definitely a wildcard.
The last time it was "threatened" is over 100 years ago. According to Wiki, Congress then passed  the 17th amendment:
for fear that such a convention—if permitted
to assemble—might stray to include issues above
and beyond just the direct election of U.S. Senators
- Einstein & Morgensten understood that the constitution is enforced by a set of institutions, beliefs, incentives and interpretations, not by an algorithmic execution of its rules. As such, Godel's point was irrelevant: of course the constitution didn't preclude the US from becoming a de facto dictatorship, and Godel's loophole would probably not be the practical way to go about it.
In the context of politics, it's a curiosity.
"The flap of the wing of a butterfly affects interstate commerce, therefore it can be regulated by the federal government"
A & !A - which is a contradiction.
therefore, our assumption was wrong, and thus B is true.
this means a single contradiction in a legal system renders the whole thing meaningless. everything becomes both legal and illegal at the same time.
a single semicolon can break an entire program because the program _must_ be consistent. that doesn't happen in the law because you have a judge who, at some point, will get pissed off at you. it doesn't matter if two conflicting rules - one dealing with the importing of flowers, and the other dealing with navigating riverways - are in conflict with each other. the judge isn't going to accept that as an argument for why you should be allowed to jaywalk - even if a technically correct interpretation of the law would say you should.
all of this is to say that the claim 'we have a nation of laws, not a nation of men' is entirely fatuous. it's just not true. we have a nation of men interpreting rules that are in a very convoluted language, designed to be 'unambiguous' - and yet still vague enough for an education person to find and exploit this ambiguity.
anyone reasonable would agree with me. if you disagree, you probably think 'reasonable cause' is reasonably clear.
First, the law is not a formal logical system (its a signalling mechanism between members of a community), and, second, there is no actual contradiction.
The fact that the law provides a mechanism for changing itself that applies to the whole of the law, including the part specifying the mechanism for change, is not a "logical contradiction". It is -- even viewing the law mechanistically (which, as just explained, is a somewhat limited view) -- just means that the law in the future may not be the same as the law now in any of its particulars. But that's not a contradiction, its a choice (and its a choice made largely because such a guarantee would be empty, and providing a formal mechanism which is not impractical makes it more likely that change will be governed by the framework -- even if it changes the framework in the process -- rather than simply abandoning the framework [something that Framers were keenly aware of since they were in a very real sense engaged in exactly the process of abandoning the framework of the Articles of Confederation precisely because of its impractical unanimity requirement.])
> all of this is to say that the claim 'we have a nation of laws, not a nation of men' is entirely fatuous.
Its only fatuous if you think law is meant to be mechanistic, rather than a communication mechanism; the "nation of laws, not a nation of men" is a relative description in reference to purely arbitrary authority. Yes, the distinction is not one of crisp binary categories, but then crisp binary categories usually are abstractions that fall apart when applied to the real world.
"Or suppose that there is a contradiction in the statutes of a particular country. There might be a statute that on feast days the vice-president had to sit next to the president, and another statute that he had to sit between two ladies. This contradiction may remain unnoticed for some time, if he is constantly ill on feast-days. But one day a feast comes and he is not ill. Then what do we do? I may say 'We must get rid of this contradiction.' All right, but does that vitiate what we did before? Not at all.
When a contradiction appears, then there is time to eliminate it. We may even put a ring round the second rule and say 'This is obsolete.'"
The 'normal' resolution of later legislation with earlier legislation would be that the later wins. However in this case, the House of Lords (at that time the UK's supreme court) ruled that Parliament had not intended that making fisheries rules should eject the UK from the EU and that the earlier European Communities Act had priority in this case and all future similar cases.
(Law also spends a lot of time and effort working out what category a particular "thing" actually is - businesses,goods,property - so it can be determined which law applies)
I believe this has been solved in intuitionistic logic (http://en.wikipedia.org/wiki/Intuitionistic_logic). If you don't except the law of the excluded middle and require constructive proofs you avoid many of these shenanigans.
A ∨ ¬A
A ∧ ¬A
------ (Assumption) ------ (∧E2)
A ∧ ¬A ¬A
------ (∧E1) ------ (Definition of ¬)
A A ⇒ ⊥
--------------- (⇒I, discharge assumption)
A ∧ ¬A ⇒ ⊥
--------------- (Definition of ¬)
¬(A ∧ ¬A)
(Note that the De Morgan's law ¬(A ∧ ¬A) ⇒ A ∨ ¬A does not hold without the LEM, so the above proof cannot be simplified in terms of it!)
The law is not a formal logical system.