Master's in Mathematical Logic and Foundations of Computer Science

The preliminary course will take place from September 2 to 13.

• 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.).

• Ultraproducts, compactness.
• Elementary extensions, Lowenheim-Skolem theorems.
• Diagram method, preservation theorems.
• Back-and-forth arguments, elimination of quantifiers.
• The space of types, omitting types theorem, kappa-saturated models, atomic models.
• Omega-categorical theories, Ryll-Nardzewski's theorem. Decidability of some axiomatic theories.

• ordinals, transfinite recursion;
• axiom of choice and equivalent statements;
• arithmetic of infinite cardinals; cofinality, regular and singular cardinals, König's theorem;
• von Neumann's hierarchy, reflection theorems;
• filters and ultrafilters, stationary sets in $\omega_1$, Fodor's lemma;
• well-founded relationships and Mostowski's collapse;
• some elements of descriptive set theory.

• 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.

This course is a natural continuation of the first semester Model Theory course: in the first semester course, given an L-structure M, you will identify all the L-statements which are true in M (i.e. the theory of M). Conversely, in this course, given a complete L-theory T, we will classify its models up to isomorphism.

The asymptotic properties of finite fields, i.e. the properties that are true in every sufficiently large finite field, can be understood by looking at pseudo finite fields: the infinite models of set of statements that hold in every finite field. This class was defined by Ax and he gave an algebraic characterisation of it: they are exactly the perfect, pseudo- algebraically closed fields with exactly one extension of any given degree.

Pseudo finite structures have lately played an important role in the model theoretic study of certain questions in combinatorics, among others in results of Hrushovski in additive combinatorics. These results find some of their roots in the work of Chatzidakis, van den Dries and Macintyre that gave a precise description of sets definable in a pseudo-finite field by, among other things, exhibiting a pseudo-finite equivalent of of the counting measure. The goal of

The goal of this class will be to introduce the results of Ax and Chatzidakis-van den Dries-Macintyre as well as introduce the algebraic notions necessary to understand them. We will then consider questions related to geometric model theory like the study of definable groups, the imaginaries or classification questions.

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.

In classical descriptive set theory, we are interested in sets appearing naturally in various topics in mathematics, in particular in functional analysis, harmonic analysis, dynamical systems and group theory. One of the goals is to study the topological complexity of these sets. For example, one can classify the Borel subsets of the real line by considering the number of steps necessary to construct them, starting from the open sets, and allowing countable unions and taking complements.

We work in Polish topological spaces, where Baire’s theorem is a powerful tool. We will first be interested in the Borel subsets of the Polish spaces, and we will see that the countable ordinals define a natural hierarchy among them. Then we will study the direct images of the Borel sets by the Borel maps (the analytic sets) and their complement (the co-analytic sets). In particular, we will provide a method allowing us to prove that some sets are co-analytic but not Borel.

We will finish this course with an introduction to effective descriptive set theory and its appli- cations. One of its very powerful tool is the Gandy-Harrington topology, and we will establish its properties allowing to use it to prove many dichotomy results. We will provide the details for at least three examples: the Hurewicz dichotomy, the Silver dichotomy and the Kechris-Solecki-Todorčević dichotomy. We will state some more recent examples, and give some details if time permits.

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 programmes, 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 the developments since the 1960s and will present the classical tools for the study of the Curry-Howard correspondence. After some review and additions to the basic course, the course will focus on two fundamental concepts, the second order and linearity, and their development, particularly in an algebraic context. In particular, the results of the course will be applied to the study of PCF, an idealized programming language.

Basic Type Theory

• Pure Type Systems
• Martin-Löf's Type Theory
• Calculus of Inductive Constructions
• Proof/Program Correspondence
• Proof Irrelevance
• Inductive and Coinductive Types
• Extensionality in Type Theory

Homotopy Type Theory

• Type/Space and Equality/Path Correspondence
• Homotopic Concepts in Type Theory (contractible spaces, h-levels, univalence, weak factorisation systems, fibrations, cofibrations)
• Higher Inductive Types (spheres, quotients, truncations)
• Axiom of Choice and Classical Reasoning In Homotopy Type Theory

Models

• Categories with Families
• Cubical Type Theory
• Internal Translations
• ω-groupoïdes
• Kan complexes

The course take place during the last 6 weeks of the 1st semester and the first 6 weeks of the 2nd semester.

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.

The objective of the course is to present several points of view on complexity from logic, recursion theory or analysis. The common feature of these approaches is that they move away from the notion of the machine (and its associated measures such as time and space) in favour of a more descriptive view of the calculation. The course aims in particular to study logical formalisms in terms of their power of expression and to present multiple characterizations of the usual complexity classes.

These descriptive or implicit approaches to complexity have had important applications in database theory, programming languages and more recently in the analysis of differential equation systems, or in understanding the power of alternative computational models based on bioinformatics, or analog computation.

We will first aim to present results on classical complexity [8, 13], to extend to algebraic models such as Blum Shub and Smale's model [3, 2], to continuous space such as neural network/deep learning models [17], then to time and continuous space such as Shannon's model [16].

The purpose of this course is to give an overview of computer science basics of blokchains (communication protocols, games) and examples of blockchain protocols used in cryptocurrencies and smart contracts.

How is logic formal?

The course will be devoted to this question. In particular, we will examine three main reasons for declaring logic "formal": because it uses discursive resources that can be said to be formal (schematic); because it concerns forms (whose status is to be specified: "logical constants" for Russell, "forms derived from something in general" for Husserl, to mention two important examples); and because it aims at an independent validity of any particular content (logic as universal science).

These three main reasons are not necessarily compatible. Moreover, the examination of the question raised will of course imply taking into account the history of logic, and a reflection on the situation of logic between philosophy and mathematics. This will be an opportunity to examine the issue of "absolute generality", that is, the possibility of a theory covering absolutely everything in general.

Aim of the class: knowledge of the philosophical issues of the history of logic in the 20th century.