Course detail

# Mathematical Logic

In the course, the basics of propositional and predicate logics will be taught. First, the students will get acquainted with the syntax and semantics of the logics, then the logics will be studied as formal theories with an emphasis on formula proving. The classical theorems on correctness, completeness and compactness will also be dealt with. After discussing the prenex forms of formulas, some properties and models of first-order theories will be studied. We will also deal with the undecidability of first-order theories resulting from the well-known Gödel incompleteness theorems.

Learning outcomes of the course unit

The students will acquire the ability of understanding the principles of axiomatic mathematical theories and the ability of exact (formal) mathematical expression. They will also learn how to deduct, in a formal way, new formulas and to prove given ones. They will realize the efficiency of formal reasonong and also its limits.

Prerequisites

Students are expected to have knowledge of the subjects General algebra and Methods of discrete mathematics taught in the bachelor's study programme.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

E.Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001 (EN)
J.Rachůnek, Logika, skriptum PřF UP Olomouc, 1986 (CS)
A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993 (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

The course unit credit is awarded on condition of having attended the seminars actively and passed a written test. The exam has a written and an oral part. The written part tests student's ability to deal with various problems using the knowledge and skills acquired in the course. In the oral part, the student has ro prove that he or she has mastered the related theory.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The aim of the course is to acquaint students with the basic methods of reasoning in mathematics. The students should learn about general principles of predicate logic and, consequently, acquire the ability of exact mathematical reasoning and expression. They should also get familiar with some other important formal theories utilizied in mathematics and informatics, too.

Specification of controlled education, way of implementation and compensation for absences

The attendance at seminars is required and will be checked regularly by the teacher supervising a seminar. If a student misses a seminar due to excused absence, he or she will receive problems to work on at home and catch up with the lessons missed.

Classification of course in study plans

• Programme M2A-P Master's

branch M-MAI , 1. year of study, summer semester, 5 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Introduction to mathematical logic
2. Propositions and their truth, logic operations
3. Language, formulas and semantics of propositional calculus
4. Principle of duality, applications of propositional logic
5. Formal theory of the propositional logic
6. Provability in propositional logic, completeness theorem
7. Language of the (first-order) predicate logic, terms and formulas
8. Semantic of predicate logics
9. Axiomatic theory of the first-order predicate logic
10.Provability in predicate logic
11.Prenex normal forms, first-order theories and their models
12. Theorems on compactness and completeness
13.Undecidability of first-order theories, Gödel's incompleteness theorems

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Relational systems and universal algebras
1. Sentences, propositional connectives, truth tables,tautologies and contradictions
2. Duality principle, applications of propositional logic
3. Complete systems and bases of propositional connectives
4. Independence of propositional connectives, axioms of propositional logic
5. Deduction theorem and proving formulas of propositional logic
6. Terms and formulas of predicate logics
7. Interpretation, satisfiability and truth
8. Axioms and rules of inference of predicate logic
9. Deduction theorem and proving formulas of predicate logic
10. Transforming formulas into prenex normal forms
11.First-order theories and some of their models
12.Theorems on completeness and compactness
13. Undecidability of first-order theories, Gödel's incompleteness theorems

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