Yes, I understand that, and am well aware of Godel's position on its application. Hence my statement that I extrapolate it further than intended.
My extrapolation is influenced by the Curry-Howard isomorphism, equating computer programs and mathematical proofs. Extrapolation to "reality" and philosophy, to me, comes from my belief of computing as universal and that consciousness is not so separate from computing.
That's kind of ridiculous, though. You completely lose the sense of the theorem once you expand the axioms beyond those explicitly defined. It would be like saying we don't know whether a boulder will stop rolling down a hill or not...because of the Halting Problem. That's just not how mathematics works.
I'm not expanding the axioms, I'm comparing the paradoxical nature of mathematics to other systems.
I'm not saying that reality is encompassed in the theorem because "mathematics", I'm positing that if mathematics, a man-made formal system, is self-consistent but requires a true, unprovable statement, then perhaps other man-made systems might reveal similar paradoxes, when put under scrutiny.
Mathematics is unique in that it is systematic and, as Godel discovered, able to be self referential and explicitly outline this paradox. Yet one can recreate the sense of the theorem in English with the phrase "this sentence is false".
This doesn't invalidate mathematics, nor English, nor any system. It simply demonstrates that what we may consider the "sound" logic of mathematics is paridoxical, and I believe that this paradox, this strange loop, is not unique to Godel's mathematical theory alone.
Considering the Godel sentence is a purely syntactical construction, I think even Godel himself would have a very difficult time even imagining what an analog would be in some higher system (much less actually constructing one). This is why he was so critical of attempts by others (e.g. Wittgenstein) to knock down the theorem by discussing its philosophical implications.
We are also free to point out the obvious nonsense that is extrapolating the axioms of arithmetic, on which the theorem depends, to whatever we want them to.
Philosophical insights can start from all kinds of empirical facts and theoretical proofs -- this includes a mathematical theorem like Godel's.
Philosophers don't (or don't all) use Godel's theorem to _prove_ something beyond that in an axiomatic way. They don't extrapolate, they use it as a raw information / material for further thinking.
In that role, the limited conditions under which the proof is true doesn't really matter. Only the fact that under such limited conditions, this (Godel's) result can occur matters -- which is philosophical far from evident, Hilbert/Russel etc thought they could axiomatize math without such contradictions.
My extrapolation is influenced by the Curry-Howard isomorphism, equating computer programs and mathematical proofs. Extrapolation to "reality" and philosophy, to me, comes from my belief of computing as universal and that consciousness is not so separate from computing.