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A mathematician associated with set theory and the foundations of mathematics.

Foundations of mathematics asks what our objects mean, what we assume about them, and why an argument establishes its conclusion. Set theory gives a common language for collections and functions; mathematical logic makes the structure of statements and deductions precise. A mathematical definition fixes a meaning, an axiom supplies an assumption, and a mathematical proof explains what follows. Separating these roles makes results reusable: once the group axioms are enough for a conclusion, that conclusion applies to every group, regardless of how its members are represented.

The assumptions can change the answer to a familiar calculation. Over the real numbers, $x^2=x$ factors as $x(x-1)=0$, so $x$ is zero or one. The same reasoning holds in any integral domain, because a product can be zero only when a factor is zero. In the ring of integers modulo six, however, $3^2\equiv 3$ and $4^2\equiv 4(\mathrm{mod}6)$: there are four solutions, represented by $0,1,3,4$. The extra solutions are idempotents, and the failed step involves zero divisors. Writing the domain is part of explaining the answer.

This precision also reveals surprising possibilities. A two-dimensional grid of natural numbers is a countable set, while all infinite binary sequences form an uncountable set. The Cantor pairing function supplies a complete enumeration of the first; Cantor's diagonal argument defeats every proposed enumeration of the second. Both arguments are short, yet distinguish two sizes of infinity. The subject's appeal is this ability to turn a vague question into one whose assumptions, construction, and limits can all be inspected.

Official course description.

Basic-option schedule. The central description gives indicative choices including algebraic topology, category theory, information theory and quantum processes and computation; annual availability follows the handbook. 2026–27 handbook. Advanced-option schedule. Indicative choices include algebraic geometry, automata, logic and games, and axiomatic set theory. At least five lecture courses are required overall. Compulsory dissertation on a topic related to the taught material, undertaken in Trinity term and the summer.

This University of Oxford Part C paper is cross-listed for Mathematics, Mathematics and Computer Science, and Mathematics and Philosophy. Section A concerns topology, and section B concerns set theory. ^[ref] The archive contains the complete editable four-page transcription, with the source closure bars checked against the scan. Bounded official-source searches found no advertised answers to this exact paper. All source leaves and unnumbered preliminary tasks are covered below.

Source release: Oxford examination archive release thread, Part C bundle pages 305–308. This combined PDF covers both topology and set theory. The cover cross-lists Mathematics, Mathematics and Computer Science, and Mathematics and Philosophy, all Part C, Monday 7 June 2010, 2.30pm–5.30pm.

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

This University of Oxford Part B paper is cross-listed for Mathematics, Mathematics and Computer Science, Mathematics and Philosophy, and Mathematics and Statistics. Section A concerns mathematical logic and section B concerns set theory. ^[ref] The archive contains a complete editable five-page transcription. Bounded official-source searches found no advertised answers to this exact paper. All 29 source leaves and the attached unnumbered tasks are covered below. In Q7(v), the convergence condition is retained exactly as printed, without adding absolute-value bars.

This University of Oxford Part B paper is cross-listed for Mathematics, Mathematics and Computer Science, Mathematics and Philosophy, and Mathematics and Statistics. It contains mathematical logic in section A and set theory in section B. ^[ref] The archive contains the full editable five-page transcription. Bounded official-source searches found no advertised answers for this exact 2008 paper. All source leaves are solved below, including all five optional arguments in A1(e) and the unnumbered preliminary tasks in B5, B6 and B7.

Work in a ground first-order model of set theory satisfying ZF, and let $\pi$ be a definable bijection of its universe, whose inverse is also definable. Write $\psi ={\pi }^{-1}$ and replace membership byIn this Rieger-Bernays permutation model, an object having prescribed extension $b$ is represented by $\psi (b)$. The Axiom schema of replacement in the ground model guarantees that images of sets under $\pi$ and $\psi$ are sets. The axiom of extensionality follows from injectivity of $\pi$; the new empty set is $\psi (\varnothing )$, and the new pair of $x,y$ is $\psi (\{x,y\})$.

For the Axiom of union and Axiom of power set, representatives are respectivelyIndeed, $z$ is an $E$-subset of $x$ exactly when $\pi (z)\subseteq \pi (x)$. Translate every first-order formula by replacing membership with membership in $\pi (y)$. Ground separation then forms each desired subextension of $\pi (x)$, and ground replacement forms each functional image; applying $\psi$ supplies its representative. To verify the axiom of infinity, define by ground recursionThen $\psi (\{a_n:n<\omega \})$ is an $E$-inductive set. Thus every ZF axiom except possibly the Axiom of foundation survives.

Interchange two distinct objects $a$ and $\{a\}$, fixing everything else. Then $\pi (a)=\{a\}$, so $a$ is a Quine atom: internally $a=\{a\}$. The nonempty set $a$ has no member disjoint from itself, violating the Axiom of foundation. The original membership model satisfies that axiom. Foundation is therefore relatively independent of the other ZF axioms.

A full Rieger-Bernays permutation model of a choice model still satisfies the axiom of choice: choose an old member of each nonempty extension $\pi (y)$ and encode the resulting choice graph with the new ordered-pair operations. To obtain failure of choice, an additional restriction by symmetries is necessary.

Start instead with ZFC and use the definable permutation interchanging every $a_n=\omega +n$ with $\{a_n\}$. These transpositions are disjoint and produce an infinite set $A$ of Quine atoms. Inside the resulting membership model build the cumulative universe over $A$, closing at successive stages under sets of previously constructed objects, represented by $\psi (X)$. Atomic objects are kept as the designated base objects; $\psi (\{a\})=a$ for $a\in A$. Every permutation of $A$ extends recursively to an automorphism of this universe, with the atomic self-loops fixed by the recursive convention.

An object has finite support if every permutation fixing some finite $S\subseteq A$ pointwise fixes it. Retain the objects which have finite support and whose members recursively also have finite support, interpreting the atomic self-loops as the base case. This is the hereditarily finite-supported Quine-atom model. It is transitive for the new membership relation. Extensionality is inherited. Pairs and unions have supports contained in the union of the finitely many relevant supports. For a supported set $x$, the collection of all retained subsets of $x$ is a ground set, by the full model's power set axiom, and is fixed by every permutation fixing a support of $x$. Its members are retained, so this collection supplies the internal power set. A first-order formula with retained parameters is invariant under their common stabilizer. Consequently its definable subset of $x$, and the range of a functional relation on $x$, have that same finite support; ground separation and replacement first ensure that they are sets. The pure natural-number hierarchy supplies infinity. These arguments verify all of ZF except Axiom of foundation, which fails at the Quine atoms.

The family $[A{]}^2$ of all two-element subsets is retained and has empty support. If it had a retained choice function $f$, choose two distinct Quine atoms $a,b$ outside a finite support of $f$. The transposition of $a,b$ fixes $f$ and fixes the unordered pair $\{a,b\}$, but moves either possible value $f(\{a,b\})$. This is impossible. Thus ZF without Foundation has both choice models and models in which Choice fails, relative to the usual consistency assumption. The construction also explains why twisting membership alone was insufficient.

Skolem's paradox is the apparent conflict between a countable model of a sufficiently strong set theory and that model's theorem asserting the existence of uncountable sets. Downward Löwenheim-Skolem theorem gives a countable elementary model whenever an appropriate infinite model exists; internally the same model still proves, for example, that its real-number set is uncountable.

The two uses of “countable” quantify over different collections of functions. Externally the whole model, and therefore the elements it regards as members of its real-number set, can be enumerated. Internally a set is countable only if an enumerating function exists among the sets of the model. The external enumeration need not be an element of the model.

Thus there is no contradiction: the model's assertion is that no function it contains is the required enumeration. Cantor's theorem remains valid inside the model. The model need not be transitive or have the actual standard natural numbers for this explanation; the essential issue is the restriction of internal quantifiers to its own domain.

A countable model of set theory can internally assert that an uncountable set exists. There is no contradiction because “there is a bijection” ranges only over functions in the model. An external enumeration of its domain need not itself be an element of that domain. Löwenheim-Skolem theorem controls external cardinality; Cantor's theorem remains valid for the sets and functions that the model contains.

A standard model of set theory interprets membership by the ambient membership relation restricted to its domain. A transitive model additionally has a transitive set or class as its domain. Standardness alone does not impose transitivity.

The usual set theory with extensionality, empty set, pairing, union, power set, infinity and full separation, before adjoining replacement. Modern presentations often include foundation; the convention should be specified when comparing theories. Class-theoretic variants add classes and class comprehension, rather than treating proper classes as sets.