Entry requirements

120 credits with the courses Algebraic Structures and Logic II. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• construct models using ultra-products;
• apply the compactness theorem in the construction of theories and models;
• determine whether two models are elementarily equivalent;
• use methods from the course to show that some theories have quantifier elimination;
• characterise theories with a unique infinite countable model;
• given a model theoretic property, decide whether a concrete structure has this property or not, and be able to justify why;
• construct concrete examples to illustrate important model theoretic notions;
• outline proofs of important theorems of the course and explain the main ideas of the proofs;
• give examples of algebraic applications of model theory.

Content

Ultra products. The compactness theorem. Elementary substructures and extensions, categoricity, elimination of quantifiers, types, Stone spaces, algebraic closure in structures. Saturated structures, atomic structures, prime models, omitting types, characterisations of theories with a unique infinite countable model, theories with finitely many infinite countable models, minimal theories, dimension, total categoricity, Steinitz' theorem and its model theoretic version. Introduction to model theoretic stability theory. Applications to algebra.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination combined with assignments given during the course in accordance with instructions at course start.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.