Please stop massacring Gödel. Seriously, please stop.
The basic idea of the theorem is this:
Given an axiom based system conforming to certain conditions there exists statements which are both true and unproveable within that system.
And things you can prove true that say they are false, making them true and false at the same time, which is what Godel actually did. This forces you to pick from inconsistancy, where things are true and false at the same time, a disaster for math, or admitting your axiomatic system is incomplete, and some things will always remain unknowble to your axiomatic system. Yeah, perhaps some are true, but all he directly proved is that they are unknowable.
This theorem has absolutely no application outside of the formal systems where it applies. It doesnt apply to quantum mechanics, it doesnt apply to science and it doesnt apply to the big steaming pile of gosa in the OP.
Steven Hawking in post 1 may be the most famous scientist on earth. He's not the greatest but he's certainly on that top level. What about "Godel and the End of Physics" did you find complicated? Normal humans can't follow that kind of math, but it's a world class scientist flat out saying it applies to physics and that it's been proven with math.
Greg Chaitin in post 1 who invented Algorithmic Information Theory, information theory for digital information probably is the greatest living mathematician, he's sure the greatest living expert on Godel. And he says the same damn thing.
It's damn obvious if you understand it, that mathematical science is eternally limited, but what both are talking about, is the educated, informed argument against Godel. What about if you use an infinite number of axioms, what if you take an infinite number of things on pure faith, can you then plug all the gaps? What they are talking about in those two peer reviewed papers, is that even with an infinite number of axioms, even if you take an infinite number of things on pure faith, and are right about all of them, even then, the math is incomplete, and science will never describe the universe, much less make it work.