Model theory 24

Quick Info. Lectures of 90 minutes each. Second semester of 2024. Meeting twice a week. The lectures will be delivered at Humanisten every Tuesday and Thursday from 10.15 to 12 in room C250. We start on the 3rd of September.

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.

Exam. Each of these parts of the exam is graded from 1 to 10. The student's final mark will be U if the weighted mean is strictly below 5, G if the weighted mean is between 5 and 7, VG if the weighted mean is strictly above 7.

Sit-down exam. The sit-down exam will happen in Viktoriagatan 30 in the following dates: 14 October 8:30–12:30 and 20 November 12:00–16:00 (retake). You can find more information about the sit-down exam by clicking here. You can find a fac-simile of the exam by clicking on the sticky note .

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. The general aim of the project is to familiarise with an important task in research, which is building a bibliography on a topic autonomously, and being able to frame correctly a collection of keywords in a scientific discourse. There is no expectation that the student masters the topic, or penetrates the techniques of the topic, this is a preliminary work aimed at collecting the material that in a second stage of a research process would be studied. Sometimes this process goes under the name of literature review. After doing their research, 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 page of the Stanford Encyclopedia of Philosophy on Modal Logics .

Projects list.

Abstract elementary classes.
Infinitary model theory.
Continuous model theory.
Finite Model Theory.
Morley's theorem and Shelah categoricity conjecture.
Geometric stability theory.
Laskar group.
Minimal theories.
o-minimal theories and tame topology.
Ehrenfeucht-Fraïssé games.
Fraïssé classes.