 Solution

Induct on a derivation in Gödel-Löb logic. Propositional axiom instances translate to theorems of Peano arithmetic. The image of $\perp$ is a refutable sentence, so it behaves as falsity in those translated propositional arguments.

The translation of axiom $\mathrm{K}$ is the distribution clause of the Hilbert-Bernays-Löb derivability conditions. The translation of axiom $\mathrm{L}$ isthe Internal form of Löb's theorem, whose derivation was given above and applies to the stipulated arithmetical provability predicate for Peano arithmetic. Thus every axiom has a provable translation.

The modus ponens rule is preserved because the interpretation commutes with implication. The necessitation rule is preserved by the first derivability condition: if $\mathrm{PA}⊢{\mathcal{I}}_i(A)$, thenHere the equality denotes the definition of the translated formula, not an arithmetic equality of truth values. A propositional substitution merely changes the arithmetic sentences assigned to the variables. The induction proves the Arithmetical soundness of GL: