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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
- Predicate logic language, models, definability. Classic examples: elementary
geometry, arithmetic, standard algebraic structures.
- Translations and interpretations. Internal models. Interpretations of theories.
- The compactness theorem and its applications. Elementary equivalence. Theorems
of Levenheim-Skolem and Maltsev on elementary substructures and extensions. Non-
standard models of arithmetic.
- Quantifier elimination. Sufficient conditions on the elimination of quantifiers.
Classical theories with quantifier elimination: divisible torsion-free abelian groups,
dense linear orders without endpoints.
- Completeness and categoricity of a theory in a given cardinality.
- 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.
- Ordered fields. Real closure. Seidenberg-Tarski theorem on the elimination of
quantifiers in the elementary theory of the ordered field of reals.
- Types, type space. Isolated types and the omitting types theorem.
- Prime models and countably categorical theories. Ryll-Nardzewski's theorem.
- 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.
 RSS: Forthcoming seminars
 Lecturer
Beklemishev Lev Dmitrievich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
