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Lecturer(s)
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Course content
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1. Mathematical logic: Elementary means of the statement logic, statement logic laws. 2. Determination of truth values of statement forms. Basic theorems on tautologies. Principle of duality, total systems and bases of logical conjunctions. 3. Normal logical forms. 4. Basic terms of the predicate logic. 5. Set theory: Zermelo-Fraenkel axiom system. Cartesian product and its properties, relation of equivalence, relation of order. 6. Functions and their properties, the Zermelo theorem on order function. 7. Equivalent sets and their cardinal numbers, cardinal arithmetic and inequality. 8. The Cantor-Bernstein theorem, the Cantor theorem and its consequences. Tarski and Dedekind definitions of infinite sets, the Dedekind theorem. Uncountable and countable sets, their examples and properties. 9. Uncountable sets, properties of transfinite cardinal numbers. The Peano model of arithmetic of the set No, mathematical induction. 10. Similarity of sets, well-ordered sets, transfinite induction. Ordinal numbers, ordinal arithmetic and inequalities. 11. Relation between cardinal and ordinal numbers.
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training)
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Learning outcomes
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Deepen knowledges in mathematical logic and set theory.
1. Knowledge Students define and describe basic elements of the logic and the set theory and recognise the relationships between them.
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Prerequisites
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Knowledge of basic mathematical concepts is assumed.
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Assessment methods and criteria
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Oral exam
Credit: presentation of knowledge. Exam: understand the subject and prove the major theorems.
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Recommended literature
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Balcar B., Štěpánek P. (2001). Teorie množin. Academia Praha.
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Bukovský L. (2005). Množiny a všelico okolo nich. UPJŠ v Košiciach.
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Cunningham D. W. (2016). Set Theory A First Course. Cambridge University Press.
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Manin Yu. I. (2010). A Course in Mathematical Logic for Mathematicians. Springer.
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Švejdar V. (2002). Logika, neúplnost a složitost. Academia, Praha.
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Vopěnka P. (2015). Úvod do klasické teorie množin. Fragment.
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