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Set theory (introductory)

lecture

master

Mathematics MSc

2

Semesters 1-4

Autumn/Spring semester

Naive and axiomatic set theory. Subset, union, intersection, power set. Pair, ordered pair, Cartesian product, function. Cardinals, their comparison. Equivalence theorem. Operations with sets and cardinals. Identities, monotonicity. Cantor’s theorem. Russell’s paradox. Examples.  Ordered sets, order types. Well ordered sets, ordinals. Examples. Segments. Ordinal comparison. Axiom of replacement. Successor, limit ordinals. Theorems on transfinite induction, recursion. Well ordering theorem. Trichotomy of cardinal comparison. Hamel basis, applications. Zorn lemma, Kuratowski lemma, Teichmüller-Tukey lemma. Alephs, collapse of cardinal arithmetic. Cofinality. Hausdorff’s theorem.  Kőnig inequality. Properties of the power function. Axiom of foundation, the cumulative hierarchy. Stationary set, Fodor’s theorem. Ramsey’s theorem, generalizations. The theorem of  de Bruijn and Erdős. Delta systems.

A. Hajnal, P. Hamburger: Set Theory. Cambridge University Press, 1999.