Gödel's Incompleteness, very roughly, says that not every true statement (in a formal system) can be proven. In particular, it says that for a logic equipped with arithmetic the statement holds; how you get to security I do not follow.
[–] Balgair 3 days ago
[–] Tryk 2 days ago
I think perhaps where I would diverge is that we should separate the language from the program. Generally, we say that a programming language is Turing complete.
Gödel's incompleteness is analogous to a paradox, e.g the Liar paradox, that is one useful way to understand the statement. But it seems as if you're saying it's a Contradiction (from where we can derive anything) something like EFQ: https://en.wikipedia.org/wiki/Principle_of_explosion
That does not follow, not all paradoxes are contradictions. For instance, just because Gödel's incompleteness pertains to first-order Logic does not mean that we can derive anything in first-order logic.
The idea is that since you can make every security program turing complete, I thiiiiink?, then you can have it be a formal system. So, you can then get the code, in very tortured ways, to execute a paradox, thereby breaking it. Likely in enough ways to let you gain access, though not always.