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Map propositional letters to arithmetic sentences, map falsity to a refutable sentence, commute with implication and interpret $\square$ by an arithmetical provability predicate. Propositional axioms remain theorems. The K axiom uses the second of the derivability conditions, and the Löb axiom uses the Internal form of Löb's theorem. Modus ponens rule is preserved directly, and the necessitation rule uses the first derivability condition. Induction on a GL proof therefore proves the displayed soundness implication for Peano arithmetic with the stipulated provability predicate.

The necessitation rule permits boxing a theorem in a modal proof system. It does not permit boxing an arbitrary temporary assumption. Under an arithmetic interpretation, it uses the external first of the derivability conditions: an actual proof of $A$ supplies a proof that $A$ has a proof. This distinction matters when proving the Internal form of Löb's theorem.

Löb's theorem states that, for a first-order theory with the stated provability and diagonalization properties,where $BA=P_S(┌A┐)$. It does not require an extra consistency assumption.

By the diagonal lemma, choose $D$ with $S⊢D\leftrightarrow (BD\to A)$. necessitation rule and distribution applied to $D\to (BD\to A)$ giveUsing positive introspection $BD\to B(BD)$ gives the unconditional theorem $S⊢BD\to BA$. If $S⊢BA\to A$, it follows that $S⊢BD\to A$, and the reverse fixed-point implication gives $S⊢D$. Necessitating gives $S⊢BD$, hence $S⊢BA$ and finally $S⊢A$.

The same derivation also gives the Internal form of Löb's theorem, needed for modal soundness. Put $H=(BA\to A)$. From the unconditional $BD\to BA$, propositional reasoning and the fixed point give $S⊢H\to D$. necessitation rule and distribution imply $S⊢BH\to BD$, and henceThis internal statement was derived without necessitating a temporary assumption.

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:

Provability logic studies modal principles valid when $\square A$ is interpreted by an arithmetical provability predicate. The distribution axiom corresponds to formalized modus ponens rule, necessitation corresponds to verification of actual proofs, and the Löb axiom is the Internal form of Löb's theorem. The standard system capturing these principles is Gödel-Löb logic.