Course: Model Theory

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Course title Model Theory
Course code KMI/PGSTM
Organizational form of instruction Lecture
Level of course Doctoral
Year of study not specified
Semester Winter and summer
Number of ECTS credits 12
Language of instruction Czech, English
Status of course unspecified
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Vychodil Vilém, doc. RNDr. Ph.D.
Course content
Introduction: Language, formulas, structural inductions, theories. Structures and models of theories, expansions and reducts of structures. Cardinality of sets, well-order, ordinal numbers, transfinite induction. Models constructed from constants: Models constructed from constants, canonical structures. Completion of theories. Compactness theorem and its applications. Omitting of types. Elementary chains: Diagrams of structures. Elementary equivalence. Elementary substructures. Chains of structures and their unions. Elementary chains of structures. Applications. Löwenheim-Skolem theorem: Löwenheim-Skolem theorem: ``downward direction'' and its consequences. Skolem's ``illusive paradox''. Löwenheim-Skolem theorem: ``upward direction'' and its consequences for predicate logic. Reduced products and ultraproducts: Centered system of sets, filters, ultrafilters, existence of nontrivial ultrafilters. Reduced products, ultraproducts, theorem of Los'. Compactness as an application of ultraproducts. Elementary classes of structures. Applications of ultraproducts for characterization of implicationally defined classes of algebras.

Learning activities and teaching methods
  • Preparation for the Exam - 120 hours per semester
Learning outcomes
The students become familiar with basic concepts of model theory.
1. Knowledge Describe and understand comprehensively principles and methods of model theory.

Assessment methods and criteria
Oral exam, Written exam

Active participation in class. Completion of assigned homeworks. Passing the oral (or written) exam.
Recommended literature
  • Burris S., Sankappanavar H. P. (1981). A Course in Universal Algebra. Springer-Verlag, New York.
  • Ebbinghaus H. D., Flum J. (1999). Finite Model Theory. Springer-Verlag Berlin Heidelberg (druhé vydání).
  • Gratzer G. (1979). Universal Algebra. Springer-Verlag Berlin Heidelberg (druhé vydání).
  • Chang C. C., Keisler H. J. (1990). Model Theory. North Holland.
  • Ježek J. (1976). Univerzální algebra a teorie modelů. SNTL Praha.
  • Mendelson E. (1997). Introduction to Mathematical Logic. Chapman & Hall, UK (fourth edition).
  • Poizat B. (2000). A Course in Model Theory. Springer.
  • Sochor A. (2001). Klasická matematická logika. Karolinum, Praha.
  • Švejdar V. (2002). Logika: neúplnost, složitost a nutnost. Academia, Praha.
  • Wechler W. (1992). Universal Algebra for Computer Scientists. Springer-Verlag Berlin Heidelberg.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester