Model theory

Quick Info. Lectures of 90 minutes each.

Syllabus. First order languages. Theories and their models. Loweinheim-Skolem theorems. Ultraproducts and compactsness. Model-completeness and quantifier elimination. Types. Saturated models. Minimality. Prime models and a glance into categoricity.

Course design. The course is structured into two parts (4.5 + 3 credits). For the first five weeks, there will be two (old fashioned, frontal) lectures per week (10 lectures in total). At the end of the sixth week there will be a written (sit-down) exam (4 credits). Since then and untill the end of the course, the students will work on their project (3 credits). The whole lenght of the course is 8 weeks.

Description. Model theory is a branch of mathematical logic that initially emerged from the humus of abstract logics and later found applications in many areas of pure mathematics. During the lectures of the course will concentrate on the most established theory, covering the main results from the 1930s and 40s. Every student will also do a project aimed at giving the chance to to get it touch with more up-to-date topics and experience a more research-oriented part of the subject.

Audience and Prerequisites. The only real presequisite is some basic exposure to mathematical structures, even only in practise. It is although highly encouraged to have attended the course Logical theories (or equivalent) from the Master in Logic.

Lecture Content Reference Additional Material
1 Historical introduction. Structure of the course. Examples of structures. Recap on first order languages and their (Tarskian) semantics. Chap. 1 and 2 of [M] or Chap 1 and 2 of [K].
2 Recap on Completeness and compactness. Lowenheim-Skolem theorems. Chap. 3 (recap) and 6 of [M].
3 Ultrafilters and ultraproducts. Chap. 5.5-6 of [M] or Chap. 4 of [P].
4 Quantifiers elimination (I). Relation to Model completeness. Characterization via primitive formulas. Introducing back and forth. Scott's theorem Approx half of Chap. 7 of [M]. Chap. 7 of [H].
5 Quantifiers elimination (II). Usingback and foth to eliminate quantifiers. Finishing Chap. 7 of [M]. Chap. 7 of [H].
6 Types and saturation (I). The notion of type and examples. The space of types and its properties. Chap. 5 of [P]. Chap. 5 of [K].
7 Types and saturation (II). Saturated models. Every model embeds in a saturated one. Homogeneity. Universality. Selected sections of Chap. 9 of [P]. Chap. 13 of [K].
8 A case study in algebra: Fields. Selected sections of Chap. 6 of [P].
9 Prime models and categoricity (I). Prime models. Omitting types theorem. Definition of atomic model. Every atomic model is prime. Countable atomic models are isomorphic. Chap. 10 of [P].
10 Prime models and categoricity (II). A model is prime if and only if it is atomic and countable. A theory has an atomic model if and only if isolated types are dense. Categoricity and isolated types. Chap. 10 of [P].

Bibliography (for the lectures). For the first entry of the bibliography you can click on the to access the lecture notes. By a total coincidence, this course is incredibly similar to these lectures notes.

  • Notes on Model Theory [G] by Geschke.

  • Model Theory [M] by Manzano.

  • A course in Model Theory [P] by Poizat.

  • Model Theory for Beginners [K] by Roman Kossak.

Other suggested readings connected to the course.

  • A shorter model theory [H] by Hodges.

  • Sets, Models and Proofs by Moerdijk and van Oosten.

  • Beginning Mathematical Logic: A Study Guide by Peter Smith.

Lecture notes. In the additional material you will sometimes find some lecture notes. Those are kindly offered and taken live by Alessio Zaninotto. While they don't cover every single statement and comments that gets said during the lecture, they offer a very good representation of the general content of the lecture.

Description of the project. You will find below a list of keywords/topics/labels. Each student will choose precisely one of the items in the list and self-study the topic. Every student will have to create a document of 3 to 5 pages presenting the topic, stressing on its relevance in the scientific landscape in which it emerged. This includes: the main problems that originated the subjects, its initial historical developments, its main techniques, its main results, a list of the main contributors, papers, books (or surveys) and an extensive list of references. An expected output (or a model, or a fac-simile) of the project could be the following document . The document above is the project of a former student from 2024, in that year students were not supposed to provide any proof as it was part of their job to appropriately find references and work material. This year, since I am suggesting references, it is expected from the student to present at least a couple of relevant proofs in their project.

Projects list.

  • Abstract elementary classes.

    Baldwin, Categoricity.

  • Infinitary model theory.

    Marker, Lectures in infinitary Model Theory

  • Continuous model theory.

    Flum, Topological Model Theory. Hart, An Introduction to continuous model theory.

  • Finite Model Theory.

    Flum, Finite model Theory. Libkin, Finite model theory. Kolaitis, Finite Model Theory and its Applications.

  • Geometric stability theory.

    Pillay, Geometric Stability Theory.

  • Lascar group.

    Ziegler, Introduction to the Lascar group. Pelaez, About the Lascar Group

  • Minimal theories.

    Marker, Strongly Minimal Sets.

  • o-minimal theories and tame topology.

    van den Dries, Tame Topology and o-minimal structures.