Mathematics BSc
Course description
2013. Set theory
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 0 2 + 0 exam pure math. mm1c1he4m 4 compulsory
Course coordinator
Strong Weak Prerequisites
Lecture
Strong:
Analysis1L (mm1c1an1) or
Foundations of analysisL (mm1c1ap2)
Strong:
Algebra1L (mm1c1al1)
Literature
• K. Hrbacek, T. Jech: Introduction to Set Theory. Marcel Dekker, 1999.
• P. Komjáth, V. Totik: Problems and Theorems in Classical Set Theory. Springer, 2006.
Syllabus
• 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.