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Mathematical logic studies formal languages, formal proofs, model theory, computability theory, and the foundations of mathematics. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of mathematics. Proof theory studies deductions; model theory studies interpretations in which statements hold.

Three questions organize the subject: is an expression a well-formed first-order formula, is it true in a specified first-order structure, and does it have a formal proof from specified axioms? These questions are distinct. For example, $\exists n(n+n=1)$ is a first-order sentence in an appropriate first-order language; it is false in the natural numbers but true in the rational numbers, with $n=1/2$.

The beauty of mathematical logic is that the same logical connectives and rules of universal quantification and existential quantification organize arguments across mathematics. Classical propositional logic uses two truth values; Intuitionistic propositional logic interprets logical implication through constructions on proofs, without unrestricted law of excluded middle. This proof-based framework is developed in Frank Pfenning, Natural Deduction, sections 1–3.

Model theory studies mathematical first-order structures through the first-order formulas they satisfy. A theory specifies sentences, while a model of a first-order theory gives a domain and interpretations of the nonlogical symbols making those sentences true.

For example, dense total orders without endpoints have a common complete first-order theory, although a countable such order and the real line cannot be isomorphic. The Löwenheim-Skolem theorem and compactness theorem explain why first-order truth can leave cardinality and other global structure undetermined. Quantifier elimination, complete types, and elementary embeddings provide tools for describing exactly what a theory can distinguish.

Official course description.

Six to eight taught units from the Part C mathematics and statistics menus. Shared branches include Category theory course of the University of Oxford, Algebraic topology course of the University of Oxford, model theory and elliptic curves. Current departmental catalogue. Compulsory dissertation worth two units.

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).

This University of Oxford Part C paper covers Gödel's Incompleteness Theorems in section A and Model theory in section B, cross-listed for Mathematics, Mathematics and Computer Science, and Mathematics and Philosophy. ^[ref] The archive contains the complete editable five-page transcription. Bounded official-source searches found no advertised answers for this exact paper. A2(c) is answered with a counterexample to its literal semantic hypothesis and a proof of the intended canonical case. The vector-space comparison keeps infinitude of the underlying set distinct from infinite dimension.

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.

Richard Kaye works on mathematical logic and arithmetic model theory. His exposition of Tennenbaum's theorem studies why countable nonstandard arithmetic cannot have an ordinary computable presentation with computable operations.^[ref]

The proof encodes a standard nonrecursive set inside a nonstandard model, using recursively inseparable sets and overspill. If the model's arithmetic were computable in the required way, the coded set could be recovered algorithmically, giving a contradiction. Kaye's account makes the interaction between model theory and computability theory explicit.^[ref]