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Lecturer(s)
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Bělohlávek Radim, prof. RNDr. Ph.D., DSc.
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Course content
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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.
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Learning activities and teaching methods
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Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook)
- Preparation for the Exam
- 120 hours per semester
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Learning outcomes
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The students become familiar with basic concepts of model theory.
1. Knowledge Describe and understand comprehensively principles and methods of model theory.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam
Completing the assignments. Passing the exam.
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Recommended literature
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Burris S., Sankappanavar H. P. (1981). A Course in Universal Algebra. Springer-Verlag, New York.
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Ebbinghaus H. D., Flum J. (1999). Finite Model Theory. Springer-Verlag Berlin Heidelberg (druhé vydání).
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Gratzer G. (1979). Universal Algebra. Springer-Verlag Berlin Heidelberg (druhé vydání).
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Chang C. C., Keisler H. J. (1990). Model Theory. North Holland.
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Ježek J. (1976). Univerzální algebra a teorie modelů. SNTL Praha.
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Mendelson E. (1997). Introduction to Mathematical Logic. Chapman & Hall, UK (fourth edition).
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Poizat B. (2000). A Course in Model Theory. Springer.
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Sochor A. (2001). Klasická matematická logika. Karolinum, Praha.
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Švejdar V. (2002). Logika: neúplnost, složitost a nutnost. Academia, Praha.
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Wechler W. (1992). Universal Algebra for Computer Scientists. Springer-Verlag Berlin Heidelberg.
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