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Lecturer(s)
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Course content
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1. Propositional logic, formulas and semantics of propositional logic, conjunctive and disjunctive normal forms of formulas. 2. Hilbert's propositional calculus, the concept of proof, the deduction theorem, the completeness theorem. 3. Predicate logic, languages, terms and formulas, structures, the definition of validity of a formula in a structure (Tarski's definition of truth). 4. Hilbert's predicate calculus, the completeness theorem, Gödel's incompleteness theorems. 5. Multi-valued extensions of classical logic. 6. The concept of infinity in mathematics, paradoxes of infinity and set theory, principles for creating sets. 7. Real and natural numbers in set theory, relations, mappings, and their properties. 8. Cardinality of sets, the Cantor-Bernstein theorem and its consequences. 9. Finite, countable, and uncountable sets (various examples and basic assertions concerning these types of sets). 10. Cantor's theorem on the cardinality of the power set. 11. Basic Arithmetic Operations on Cardinality.
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training), Dialogic Lecture (Discussion, Dialog, Brainstorming), Group work
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Learning outcomes
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The aim of the course is to introduce students to the fundamentals of mathematical logic and naive set theory. Students: - become familiar with the basics of propositional logic (truth values, logical consequence, normal forms, Hilbert calculus, the completeness theorem) and with multi-valued propositional logic, - become familiar with the basics of predicate logic (structure for a given language, logical consequence, Hilbert calculus, the completeness theorem, Gödel's incompleteness theorems), - understand the paradoxes of set theory and learn the principles of forming sets (an alternative to axiomatics in naive set theory), - learn to compare the cardinalities of sets, - learn the basic theorems (the Cantor-Bernstein theorem, Cantor's power set theorem), - acquire a basic understanding of countable and uncountable sets, - become familiar with the basic operations of cardinal arithmetic.
After completing the course, the student: Knowledge: - understands the basic concepts of mathematical logic and naive set theory, - is familiar with the semantic and syntactic aspects of propositional and predicate logic, - understands the principle of comparing the cardinalities of sets using bijective and injective mappings. Skills: - can apply the completeness theorem, e.g., determine whether a finite set of formulas is consistent, - is able to determine the cardinality of various subsets of the real numbers, - can sum, multiply, or exponentiate certain infinite cardinalities, Competencies: - is able to identify deductive methods of mathematical logic within the context of standard mathematical practice, - obtain a deeper understanding of the application of set theory and its methods in other branches of mathematics (e.g., algebra, discrete mathematics, topology).
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Prerequisites
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A knowledge of basic mathematical concepts and various methods of mathematical proofs is assumed.
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Assessment methods and criteria
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Oral exam, Written exam
To successfully complete the course, the student must: - learn the basic concepts of mathematical logic and set theory, - understand the completeness theorem and be able to apply it to specific examples, - be able to compare the cardinalities of sets using mappings, - solve practical and theoretical problems, - complete the course requirements (credit tasks/tests), - pass the final exam.
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Recommended literature
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Balcar B., Štěpánek P. (2001). Teorie množin. 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.
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Manin Yu. I. (2010). A Course in Mathematical Logic for Mathematicians.
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Švejdar V. (2002). Logika, neúplnost a složitost. Praha.
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Vopěnka P. (2015). Úvod do klasické teorie množin. Fragment.
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