Model theory is one of the central components of mathematical logic and has deep connections with several branches of mathematics, primarily with algebra and algebraic geometry. We will try to make this course rich in examples demonstrating the operation of logical methods in specific situations for various classes of structures. At the end of the course, a model-theoretic approach to combinatorial independent statements will be presented using the example of the Kanamori-McAloon principle in Ramsey theory.

The course is designed for students who have taken introductory courses in algebra and mathematical logic.

COURSE PROGRAM

1.  Predicate logic language, models, definability. Classic examples: elementary geometry, arithmetic, standard algebraic structures.
2.  Translations and interpretations. Internal models. Interpretations of theories.
3.  The compactness theorem and its applications. Elementary equivalence. Theorems of Levenheim-Skolem and Maltsev on elementary substructures and extensions. Non- standard models of arithmetic.
4.  Quantifier elimination. Sufficient conditions on the elimination of quantifiers. Classical theories with quantifier elimination: divisible torsion-free abelian groups, dense linear orders without endpoints.
5.  Completeness and categoricity of a theory in a given cardinality.
6.  Algebraically closed fields. Elimination of quantifiers and its consequences. Completeness and categoricity of the elementary theory of algebraically closed fields of a fixed characteristic in any uncountable cardinality.
7.  Ordered fields. Real closure. Seidenberg-Tarski theorem on the elimination of quantifiers in the elementary theory of the ordered field of reals.
8.  Types, type space. Isolated types and the omitting types theorem.
9.  Prime models and countably categorical theories. Ryll-Nardzewski's theorem.
10.  Indiscernible elements. Application of diagonally indiscernible elements to construct models of Peano arithmetic. The Kanamori-McAloon principle and its independence from the axioms of Peano arithmetic.

Lecturer
Beklemishev Lev Dmitrievich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center