M4155 Set Theory

Faculty of Science
Spring 2025
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Giuseppe Leoncini, M.Sc. (seminar tutor)
Guaranteed by
prof. RNDr. Jiří Rosický, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
Knowledge of basic set theoretical concepts is welcome.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course introduces basic set theory and its significance for mathematics. In particular, with the theory of cardinal and ordinal numbers, their arithmetics and with the axiom of choice.
Learning outcomes
Understanding of basic set theoretical concepts;
mastering the set theoretical way of thinking;
ability to analyze set theoretical context of mathematical concepts and results;
awareness of possibilities and limitations of the formalization of mathematics.
Syllabus
  • 1. Set theory: origin of set theory, set theory as a fundament of mathematics, concept of infinity, the construction of natural and real numbers. 2. Cardinal numbers: cardinal numbers, ordering of cardinal numbers, Cantor-Bernstein theorem, operations with cardinal numbers. 3. Well-ordered sets: well-ordered sets, transfinite induction, operations with well-ordered sets. 4. Ordinal numbers: ordinal numbers, ordering of ordinal numbers, ordinal arithmetic, countable ordinal numbers. 5. Axiom of choice: axiom of choice, well-ordering principle, maximality principle, applications of the axiom of choice to cardinal arithmetics. 6. Elements of axiomatic set theory: axiom of regularity, cumulative hierarchy, axiom sheme of replacement, permutation model of set theory. 7. Set theory in algebra and analysis: measures, filters, measurable cardinal numbers, Konig's theorem, weakly compact cardinal numbers, infinitary logics, compact cardinal numbers.
Literature
  • J. Rosický, Teorie množin II., http://www.math.muni.cz/~rosicky/
  • KOLÁŘ, Josef, Olga ŠTĚPÁNKOVÁ and Michal CHYTIL. Logika, algebry a grafy. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1989, 434 s. info
  • BALCAR, Bohuslav and Petr ŠTĚPÁNEK. Teorie množin. 1. vyd. Praha: Academia, 1986, 412 s. info
  • FUCHS, Eduard. Teorie množin. Vyd. 1. Brno: Rektorát UJEP, 1974, 176 s. info
Teaching methods
The course presents required knowledge and ways of thinking; shows their applications; provides a feeling about axiomatic set theory; stimulates a discussion about its subject.
The exercises: solving theoretical problems focused on practising basic concepts and theorems.
Assessment methods
Course ends by an oral exam. Presence at the course recommended. Homeworks are given, handed in exercises. Credits from exercises are necessary for an exam.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
1. Understanding of basic set-theoretical concepts 2. Mastering the theory of well ordered sets, ordinal and cardinal numbers 3. Understanding the axiom of choice.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
  • Enrolment Statistics (Spring 2025, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2025/M4155