Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / A / 4 / d / Solution

Under the arithmetical interpretation $\square A=P_S(┌A┐)$, the Löb axiom says that a formal proof of the local reflection implication for $A$ already entails a formal proof of $A$:It is a formalized constraint on self-verification of a formal proof system. Externally, Löb's theorem says that if the system itself proves its own local reflection implication for $A$, it proves $A$. Distribution reflects closure of formal proofs under implication, necessitation reflects that a theorem has an arithmetically recognizable formal proof, and positive introspection reflects formal proof verification inside arithmetic.

Taking $A$ to be a refutable first-order sentence, especially a contradiction, recovers the obstruction behind the second incompleteness theorem: a first-order consistent sufficiently strong system cannot prove the corresponding claim that no formal proof of that first-order sentence exists. This does not identify provability with truth, nor grant the unrestricted factivity principle $\square A\to A$ as a theorem of the system. The Löb axiom is the characteristic additional scheme of Gödel–Löb provability logic.