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.
