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A modal formula is built from propositional variables and logical constants using Boolean connectives and modal operators, such as the box operator. In provability logic, arithmetic realization replaces propositional letters by arithmetic sentences and interprets a box as the corresponding arithmetic provability assertion. Modal validity, theoremhood and truth under a particular arithmetic realization are distinct notions.

A modal logic extends propositional connectives by operators such as $\square$ and its dual $◊A=\neg \square \neg A$. An interpretation determines their meaning: necessity, knowledge and formal provability lead to different axiom systems. In a relational semantics, $\square A$ holds at a world when $A$ holds at every accessible world. Provability logic interprets the box through arithmetically coded proofs.

A modal operator forms a new formula from a formula, such as $A\mapsto \square A$. In provability logic, it represents an assertion that a proof of $A$ exists, rather than an assertion that $A$ is true. A box is not interchangeable with an unboxed formula.

The Gödel-Löb logic $\mathrm{GL}$ uses propositional variables, $\perp$, implication, and a unary modal operator $\square$. Other Boolean connectives are abbreviations; for example $\neg A$ means $A\to \perp$ and $\top$ means $\perp \to \perp$.

Its axioms are all classical propositional tautologies and the two schemesIts rules are the modus ponens rule and the necessitation rule, $\mathrm{GL}⊢A\Rightarrow \mathrm{GL}⊢\square A$. Uniform propositional substitution is allowed, or equivalently built into the axiom schemes. necessitation rule applies to theorems; a temporary assumption cannot simply be boxed. In provability logic, $\square A$ is read as the formal assertion that $A$ is provable.

This University of Oxford Part C paper is cross-listed for Mathematics and Mathematics and Philosophy. Section A concerns Gödel second incompleteness theorem and provability logic; section B concerns model theory. ^[ref] The archive contains the complete editable four-page transcription. Bounded official-source searches found no advertised answers to this exact paper. All 31 source leaves are answered below. The enumeration and function-definability conventions are stated explicitly, and the canonical-model construction uses the function-free setting permitted by the source.

This University of Oxford Part C paper is cross-listed for Mathematics and Mathematics and Philosophy. Section A concerns Gödel second incompleteness theorem and provability logic; section B concerns model theory. ^[ref] The archive contains a complete editable six-page transcription. Bounded official-source searches found no advertised solutions to this exact paper. All 37 source leaves are answered below. The solutions state the proof-numbering convention in A2(d), the naming convention for diagram inclusions in B6(ii), and the literal printed stability definition in B8(ii)(e).

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.

Source release: Oxford examination archive release thread, Part C bundle pages 300–304. This is one combined paper covering both courses: section A concerns Gödel second incompleteness theorem and provability logic, and section B concerns model theory. The cover cross-lists Mathematics and Mathematics and Philosophy, both Part C, Thursday 10 June 2010, 2.30pm–5.30pm.

Answer availability checked on 28 October 2026: bounded official Oxford searches by exact year, paper label and both subjects found no advertised exact-paper answers. The official Part C syllabus for the 2010 examination confirms the paired course identity. No later-year answer-release policy was assumed applicable.

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.