# On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018

This will be a talk for the Philosophy of Mathematics Seminar in Oxford, October 29, 2018, 4:30-6:30 in the Ryle Room of the Philosopher Centre.

Abstract. In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $\subseteq$ rather than the element-of relation $\in$.  How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $\subseteq$ in place of $\in$? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.

This is joint work with Makoto Kikuchi, and the talk is based on our joint articles:

The talk will also mention some related recent work with Ruizhi Yang (Shanghai).

# Parallels in universality between the universal algorithm and the universal finite set, Oxford Math Logic Seminar, October 2018

This will be a talk for the Logic Seminar in Oxford at the Mathematics Institute in the Andrew Wiles Building on October 9, 2018, at 4:00 pm, with tea at 3:30.

Abstract. The universal algorithm is a Turing machine program $e$ that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory.

Slides

# Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018

This will be a series of lectures on the philosophy of mathematics, given at Oxford University, Michaelmas term 2018. The lectures are mainly intended for undergraduate students preparing for exam paper 122, although all interested parties are welcome.

My approach to the philosophy of mathematics tends to be grounded in mathematical arguments and ideas, treating philosophical issues as they arise organically. The lectures will accordingly be organized around mathematical themes, in such a way that naturally brings various philosophical issues to light.

Here is a tentative list of topics, which may be updated as the term approaches.

Lecture 1. Numbers. Numbers are perhaps the essential mathematical idea, but what are numbers? Our various number systems — natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more —  provide a background for classical arguments on incommensurability, the irrationality of $\sqrt{2}$, the infinitude of primes and transcendental numbers, leading to discussions of Platonism, Frege’s number concept, Peano’s numbers, Dedekind’s arguments, and the philosophy of structuralism.

Lecture 2. Rigour. We shall treat the problem of mathematical rigour in the development of the calculus. Informal limit concepts and the use of infinitesimals ultimately led to formal concepts of limit and continuity, as well as a capacity for refined notions such as uniform continuity, accompanied by increasing abstraction in the function concept, which we shall illustrate with the Conway base 13 function and space-filling curves.

Lecture 3. Infinity. Beginning with Zeno’s paradox and classical ideas on potential versus actual infinity, we shall follow the allegory of Hilbert’s hotel to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity. Some arguments give rise to the distinction between constructive and non-constructive proof. Time permitting, we shall count into the transfinite ordinals.

Lecture 4. Geometry. We shall discuss classical Euclidean geometry, accompanied by the Euclidean proof concept and the ideal of straight-edge and compass construction, which leads to the concept of constructible numbers. The impossibility of certain constructions, such as duplicating the cube or trisecting the angle, serves as a counterpoint. The rise of non-Euclidean geometry, especially in light of scientific observations and theories suggesting that physical reality may not be Euclidean, challenges previous accounts of what geometry is about. We shall discuss formalizations of geometry, including Russell on Euclid, Russell and Tarski on betweenness, and the significance of Tarski’s decision procedure.

Lecture 5. Proof. What is proof? What is the relation between proof and truth? After clarifying the distinction between syntax and semantics, we shall introduce formal proof systems and highlight the importance of soundness, completeness and effectivity in any such system. We shall discuss the increasing significance of computer-verified proof, presenting the four-color theorem as an illustrative example.

Lecture 6. Computability. What does it mean for a function to be computable? While Gödel’s early approach via primitive recursive functions fell short, Turing’s machine concept proved extremely robust, forming a foundation ultimately for the contemporary computer era. We shall discuss the Church-Turing thesis, in both weak and strong forms, as well as the undecidability of the halting problem. Turing’s oracle concept leads to a mathematical theory of information content. Time and space permitting, we shall carry out a supertask computation, transcending the Turing limit.

Lecture 7. Incompleteness. The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the Rosser variation, the second incompleteness theorem, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.

Lecture 8. Set theory. We shall discuss the emergence of set theory as a foundation of mathematics. An initially naive theory, challenged fundamentally by the Russell paradox, grew into Zermelo’s formal set theory, founded on the idea of a cumulative universe of sets and providing a robust general context in which to undertake mathematics, while also enabling the clarification of fundamentally set-theoretic issues surrounding the axiom of choice, the continuum hypothesis and an increasingly diverse hierarchy of large cardinal concepts. The development of forcing solved many stubborn questions and illuminated a ubiquitous independence phenomenon, feeding into philosophical issues concerning the criteria by which one should add new axioms to mathematics and the question of pluralism in mathematical foundations.

# Oxford University, Professor of Logic & Sir Peter Strawson Fellow, University College Oxford

In September 2018, I took up a new position in Oxford:

I am looking forward to starting this new chapter in my life and academic career.

Wish me luck!