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Third-Cycle Courses

Faculty of Engineering | Lund University

Details for the Course Syllabus for Course FMA280F valid from Spring 2017

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General
  • English
  • If sufficient demand
Aim
  • Set theory is a fundamental building block in modern mathematics. The aim of the course is to give an introduction to set theory and its relations to other parts of mathematics, thereby providing the participants with important knowledge for further studies in mathematics.
Contents
  • The aim of this course is to study some basic set-theoretic tools which entered many parts of mathematics without being questioned. E.g. the definition of a manifold, the existence of non-measurable sets and many other concepts involve non-constructive techniques like the axiom of choice. The course will on one hand enlighten the mathematical problems and paradoxes that occur if one takes these tools as granted and also indicate that those techniques enable to give (relatively short and simple) proofs of concrete statements in an abstract way.The axiom of choice and equivalent statements;

    Detailed contents of the course:
    - The concept of well-ordering;
    - Introduction to ordinal and cardinal numbers. The principle of transfinite induction;
    - Filters and ultrafilters;
    - Paradoxes (like Cantor, Vitali, Banach–Tarski) resulting from ''naive'' set theory;
    - The continuum hypotheses;
    - A first glance into set-theoretic topology. In particular we will touch the question how large the Stone–Czech compactification of the natural numbers is;
    -Some old and new set-theoretic proofs of statements in standard analysis, topology, measure theory or algebra;
    - Counter-intuitive examples like the Hydra Problem having a set-theoretic but no constructive proof.
Knowledge and Understanding
  • For a passing grade the doctoral student must
  • have a good knowledge of the basic concepts and results in set theory and understand how they relate to different parts of mathematics.
Competences and Skills
  • For a passing grade the doctoral student must
  • be able to use the tools from set theory to give proofs of some statements in different parts of mathematics.
Judgement and Approach
  • For a passing grade the doctoral student must
  • be able to judge if and how a basic result in mathematics depends on set theory.
Types of Instruction
  • Lectures
Examination Formats
  • Oral exam
  • Seminars given by participants
  • Failed, pass
Admission Requirements
  • No particular special knowledge in advanced mathematics is needed.
Assumed Prior Knowledge
  • No particular special knowledge in advanced mathematics is needed. A first glance at the basics of general topology may be of use in the second part.
Selection Criteria
Literature
  •  
  • Any introductory book on set theory or ordinal numbers or set-theoretic topology will do. F.e.:
    W. Sierpiński, Cardinal and ordinal numbers, PWN, Warsaw 1958
    (available as pdf online)
Further Information
Course code
  • FMA280F
Administrative Information
  • 2017-01-17
  • Professor Thomas Johansson

All Published Course Occasions for the Course Syllabus

2 course occasions.

Course code ▽ Course Name ▽ Division ▽ Established ▽ Course syllabus valid from ▽ Start Date ▽ End Date ▽ Published ▽
FMA280F An Introduction to Set Theory Mathematics Spring 2017 2020‑01‑13 (approximate) 2020‑03‑31
FMA280F An Introduction to Set Theory Mathematics 2020‑01‑28 Spring 2017 2020‑02‑11 (approximate) 2020‑04‑28 2020‑01‑28

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