Course Syllabus for

An Introduction to Set Theory Introduktion till mängdlära

FMA280F, 5 credits

Valid from: Spring 2017
Decided by: Professor Thomas Johansson
Date of establishment: 2017-01-17

General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English

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.

Goals

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.

Course 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.

Course 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)

Instruction Details

Type of instruction: Lectures

Examination Details

Examination formats: Oral exam, seminars given by participants