Master's in Mathematical Logic and Foundations of Computer Science

• Propositional calculus: truth tables, tautologies, normal forms, compactness.
• Predicate calculus: first-order languages, terms, formulas, models; satisfaction of a formula in a model; substructures; isomorphisms; elementary equivalence.
• Set theory: axioms of Zermelo-Frænkel; cardinals; Cantor and Cantor-Bernstein theorems; finite sets, countable sets.
• Introduction to programming: introduction to Ocaml functional programming; connection to lambda-calculus, recursivity, ML typing; common data structures (Boolean, integers, lists, options, trees, etc.).

• 1st-order languages, structures, theories
• Ultraproducts, compactness.
• Elementary extensions, Lowenheim-Skolem theorems, elementary chains.
• Preservation theorems.
• Back-and-forth arguments.
• Quantifier elimination, model completeness
• The space of types.
• (If time allows it) Realized and omitted types, atomic models.

• Axioms of ZF
• Ordinals, cardinals, transfinite recursion
• Ordinal and cardinal arithmetic
• The Axiom of Choice and equivalents, filters and ultrafilters
• Cofinality, regular/singular cardinals, König's theorem
• Stationary and club sets, Fodor's lemma
• Absoluteness and reflection theorems
• The constructible universe

• Completeness theorem of the LK sequent calculus with equality by Henkin's witnesses.
• Sequent calculus: cut elimination of and median sequent theorem in LK. Herbrand's theorem. LJ sub-calculation: intuitionist logic and its BHK interpretation. Properties of the sub-formula and existential witnesses in LJ.
• Natural deduction: NK and NJ systems. Cut elimination in NJ. Properties of the sub-formula and existential witnesses in NJ, then in HA (intuitionist arithmetic).
• Lambda-calculus: Confluence and standardization properties. Representation of recursive functions. T system. Curry-Howard correspondence. Realizability, strong standardization and program correctness.

• Computability: recursive functions and functions computable by machines; logical characterization of computable functions; s-m-n theorem and fixed point theorems; the concept of reduction and undecidable problems.
• Introduction to complexity: time and space complexity classes, hierarchy theorems, reductions, completeness, Boolean circuits, introduction to algebraic complexity.
• Formal arithmetic: Peano axioms and weak subsystems; arithmetization of logic; undecidability theorems; Gödel's incompleteness theorems.

The course presents the fundamental concepts of category theory, accompanied by numerous examples. The main goal is to pave the way towards the modern applications of category theory in logic, theoretical computer science and homotopy theory.

One half of this module will consist of course work, the other half will consist of practical work on a machine. The course will finish with a project to be carried out in Coq. The first part of this course is a prerequisite for the Homotopy Type Theory course.

This course is a natural continuation of the first semester Model Theory course. It will seek to understand and classify the models of a given 1st order theory through the types that can be realized or omitted.

In model-theoretic terms, an expansion M of the real ordered field is o-minimal if all M-definable subsets of the reals have finitely many connected components. This can also be formulated in purely geometric terms, as a property of a collection of real sets, stable under the boolean set-operations, Cartesian products and linear projections. The sets definable in an o-minimal structure share many topological tameness properties with real algebraic and real analytic sets (good dimension theory, uniform finiteness, stratification), which makes o-minimal geometry relevant to problems in Diophantine and arithmetic geometry, non-oscillatory dynamical systems and asymptotic analysis. I will give an overview of the main results about o-minimal structures and then I will concentrate on illustrating the main methods for proving that a collection of real functions generates an o-minimal structure. There are essentially no prerequisites for this course, other than the basic undergraduate notions of algebra and analysis: the model-theoretic background needed is minimal and self-contained references will be provided to those who might need them.

On 8 August 1900, at the Second International Congress of Mathematicians in Paris, David Hilbert set out a list of 23 mathematical problems which, in his opinion, should serve as a guide for future research in the new century. The first problem in this list, Cantor's continuum hypothesis, was solved in two stages: by Gödel (1938) who constructed an internal model of the generalized continuum hypothesis, and by Paul Cohen (1963), who invented a model construction for the negation of Cantor's hypothesis. This course will mainly cover the two model constructions of set theory introduced by Gödel and Cohen.

The module will explore the ways that logic interact with the theory of games. Some examples where game theory enters set theory are strategic closure and determinacy, for model theory Ehrenfeucht-Fraïssé games, for descriptive complexity the pebble game, for automata theory certain decision arguments. Jouko Väänänen states that there are essentially three kinds of games in logic, and that there are essentially connected, forming a « strategic balance of logic ». We shall explore that balance. Lectures will be based on « Models and games » by Jouko Väänänen, with some additional explorations into descriptive set theory, large cardinals and decidability.

Proof theory has undergone at least two major developments over the past century as a result of Gödel's incompleteness theorems. The first took place in the 1930s, immediately after the results on incompleteness, with the introduction and study of natural deduction and sequent calculus s by Gentzen and lambda-calculus by Church. Church then showed the undecidability of predicate calculus via lambda-calculus while introducing a universal computation model while Gentzen deduced the consistency of various logical systems as a corollary of cut elimination of breaks in sequent calculus.

The second stage took place in the 1960s with the gradual highlighting, through the Curry-Howard correspondence, of the profound links between proofs and programs, from the correspondence between simply typed lambda-calculus and minimal propositional natural deduction to the various extensions of this correspondence to the second order, to classical logic and to the emergence of the notion of linearity in proof theory. Linear logic has profoundly renewed the links between the formal semantics of programming languages on one hand and proof theory on the other. Linear algebra is the third pole of this correspondence, focusing on the notion of computational resource.

The basic course covered the first step. This course will be devoted to some more recent developments.

Homotopy theory, which is devoted to the study of spaces up to deformation, has given rise to a branch of algebra called homotopical algebra, in which tools are developed for dealing with structures in which laws like associativity do not hold exactly like in classical algebra, but up to homotopy, these homotopies being themselves subject to coherences, etc.

Homotopy theory has also a logical side, in which types, proofs of equality, and proofs of equality of proofs of equality are interpreted as spaces, paths, and homotopies between paths, respectively. The notion of fibration, that plays an essential rôle in homotopy theory, is tightly related with substitution in dependent type theory. This interplay has led to a new version of type theory called homotopy type theory.

The course is a follow-up of that on category theory taught in the first semester, but can be followed by students who have already some basic background in category theory. We shall introduce the important notions underlying the subject: enriched categories, model categories, as well as different approaches to the definition of higher catégories, notably via simplicial sets. We shall also hint at connected subjects such as operads and ∞-operads, that also have arisen from topology. The lectures will partly follow the flow of exposition found in recent books (Categorical homotopy theory by Emily Riehl, The homotopy theory of (∞, 1)-categories by Julia Bergner, From categories to homotopy theory by Birgit Richter, Higher categories and homotopical algebra by Denis-Charles Cisinski, Simplicial methods for higher categories by Simona Paoli, which all offer opportunities to the interested students for learning more), with an eye on the links with homotopy type theory.

The forms of the general