This is a misconception of what is being said in GEB.
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.
Again, not a CS person.
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.
[dead]