|2 + 0||2 + 0||exam||pure math.||mm1c1he4m||4||compulsory|
Syllabus designed by:
- K. Hrbacek, T. Jech: Introduction to Set Theory. Marcel Dekker, 1999.
- P. Komjáth, V. Totik: Problems and Theorems in Classical Set Theory. Springer, 2006.
- Naive and axiomatic set theory. Subset, union, intersection, power set. Pair, ordered pair. Cartesian product, function.
- Cardinality, comparison of cardinalities. Cantor–Bernstein–Schroeder theorem. Operations on sets and cardinalities. Identities, monotonicity. Cantor's theorem, Russel's paradox. Axiom of Choice, its uses. Examples of cardinalities.
- Ordered sets, order type. Well-ordered sets, ordinals. Examples. Initial segment, comparison of ordinals. Axiom of Replacement. Successor and limit ordinal. Transfinite induction, the theorem of transfinite recursion. Well-ordering theorem.
- Trichotomy of comparison of cardinals. Hamel-basis, applications. Zorn's lemma, Kuratowski's lemma, Teichmüller-Tukey-lemma. Alephs, a+b=ab=max(a,b) for infinite cardinals. Cofinality, Hausdorff's theorem. König's inequality. Properties of the power function. Axiom of Regularity, cumulative hierarchy.
- Stationary sets, theorems of Neumer and Fodor. Ramsey's theorem, generalizations. De Bruijn-Erdős theorem. Delta-systems.