Course detail

Mathematical Logic

FSI-SMLAcad. year: 2014/2015

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. Finally, some further important logics will be discussed which have applications in mathematics and computer science.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

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.

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 active presence at exercises and passing of a written mid-term test. A written exam will be organized at the end of semester.

Course curriculum

Not applicable.

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 exercises will be checked.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. Nerode, R.A. Shore, Logic for Applications, Springer-Verlag 1993 (EN)
E.Mendelson, Introduction to Mathematical Logic, Chapman&Hall, 2001 (EN)

Recommended reading

J.Rachůnek, Logika, skriptum PřF UP Olomouc, 1986 (CS)

Classification of course in study plans

  • Programme N3901-2 Master's

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

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Basics of set theory and cardinal arithmetics
2. Language, formulas and semantics of propositional calculus
3. Formal theory of the propositional logic
4. Provability in propositional logic, completeness theorem
5. Language of the (first-order) predicate logic, terms and formulas
6. Semantic of predicate logics
7. Axiomatic theory of the first-order predicate logic
8. Provability in predicate logic
9. Theorems on compactness and completeness, prenex normal forms
10.First-order theories and their models
11.Undecidabilitry of first-order theories, Gödel's incompleteness theorems
12.Second-order theories (monadic logic, SkS and WSkS)
13.Some further logics (intuitionistic logic, modal and temporal logics, Presburger arithmetic)

Exercise

26 hod., compulsory

Teacher / Lecturer

Mgr. Jan Pavlík, Ph.D.

Syllabus

Relational systems and universal algebras
1. Sets, cardinal numbers and cardinal arithmetic
2. Sentences, propositional connectives, truth tables,tautologies and contradictions
3. Independence of propositional connectives, axioms of propositional logic
4. Deduction theorem and proving formulas of propositional logic
5. Terms and formulas of predicate logics
6. Interpretation, satisfiability and truth
7. Axioms and rules of inference of predicate logic
8. Deduction theorem and proving formulas of predicate logic
9. Transforming formulas into prenex normal forms
10.First-order theories and some of their models
11.Monadic logics SkS and WSkS
12.Intuitionistic, modal and temporal logics, Presburger arithmetics