Course : Set Theory

Intuitive introduction. Countable and uncountable sets. Cantor's theorem and the Schroeder-Bernstein theorem. Zermelo's six axioms. Cartesian product of sets, relations and functions. Natural numbers and the recursion theorem. Well-ordered spaces, transfinite recursion and induction. Comparability of well-ordered spaces, Hartogs' theorem and fixed-point theorem. Axiom of choice and equivalent statements. Cardinal arithmetic, Koenig's theorem, cofinality, regular cardinals. Axiom of replacement, ordinal numbers and their arithmetic, cardinal numbers. Cumulative hierarchy of pure founded sets.