Hm I think your substitution isn't right, and the more correct one is "it is provable that (it is not provable that False) implies (it is provable that False)", ala .

I'm again not following well, but here are some other thoughts that might be relevant:

It's provable for any $P$ that $⊥ \to P$ , i.e. from "False" anything follows. This is how we give "False" grounding: if the system proves False, then it proves everything, and distinguishes nothing.
There are two levels at which we can apply Löb's Theorem, which I'll call "outer Löb's Theorem" and "inner Löb's Theorem".
- Outer Löb's Theorem says that whenever PA proves $□ P \to P$ , then PA also proves $P$ . It constructs the proof of $P$ using the proof of $□ P \to P$ .
- Inner Löb's Theorem is the same, formalized in PA. It proves $□ (□ P \to P) \to □ P$ . The logic is the same, but it shows that PA can translate an inner proof of $□ P \to P$ into an inner proof of $P$ .
- Notably, the outer version is not $(□ P \to P) \to P$ . We need to have available the proof of $□ P \to P$ in order to prove $P$ .