Posed the question to AI and this may just be the loophole:

The contradiction Gödel may have identified here is that the Constitution appears to rely on self-consistency for democratic continuity but lacks any intrinsic safeguard against a self-referential modification that would negate its own premises. Thus, Gödel’s concern may have been that the Constitution logically allows for its own undoing in a way that defies the democratic principles it is meant to uphold. This, for a logician, would constitute a true contradiction: a system that both guarantees democracy and allows for the abolition of democracy through its own rules.

This paradox—known in logic as a "self-amendment paradox" or "self-referential inconsistency"—suggests that the Constitution could theoretically enable an amendment that invalidates its own democratic principles or amends away the amendment power itself.

In logical terms, Article V allows the Constitution to be self-modifying, meaning it could alter the very conditions of its own existence. For example:

Scenario 1: Congress or a convention could amend the Constitution to prohibit any further amendments, thus contradicting the very nature of a "living" document. Scenario 2: An amendment could be passed that eliminates all democratic processes, effectively installing a permanent authoritarian regime. In both cases, the Constitution would simultaneously be both democratic (because amendments are allowed by democratic process) and anti-democratic (because the amendments could eliminate democratic processes).