Valid from: Spring 2017
Decided by: Professor Thomas Johansson
Date of establishment: 2017-01-17
Course type: Third-cycle course
Teaching language: English
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.
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.
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.
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)
Type of instruction: Lectures
Examination formats: Oral exam, seminars given by participants
Grading scale: 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.