I think I really don’t see what the intuition behind the suggested strategy is.
The standard proof of Löb’s theorem as I understand it can be obtained by trying to formalize the paradoxical sentence “If this sentence is true, then C”. Intuitively, this sentence is clearly true, therefore C. (This of course doesn’t work formally, what is missing is exactly the assumption that you can conclude C from a proof of C).
With your suggested proof, I just wouldn’t believe it’s a proof in the first place, so (for me) there’s no reason to believe that it should work. If you can explain in more detail what exactly you would like to formalize, I would be curious.
I think I really don’t see what the intuition behind the suggested strategy is.
The standard proof of Löb’s theorem as I understand it can be obtained by trying to formalize the paradoxical sentence “If this sentence is true, then C”. Intuitively, this sentence is clearly true, therefore C. (This of course doesn’t work formally, what is missing is exactly the assumption that you can conclude C from a proof of C).
With your suggested proof, I just wouldn’t believe it’s a proof in the first place, so (for me) there’s no reason to believe that it should work. If you can explain in more detail what exactly you would like to formalize, I would be curious.