Courses

Elements of Naive Set Theory. The Boolean algebra of subsets. Binomial relations. Order relations. Functions. Introduction to axiomatic set theory. Antinomies. Foundation of natural, integer and rational numbers. Foundation of real numbers with Dedekind sections and with Cauchy sequences of rational numbers. Operations of addition and multiplication between natural, integer, rational and real numbers. Basic properties of these. Order in the sets of natural, integer, rational and real numbers. Countable and uncountable sets. Polynomials. The Cantor-Berstein theorem. Operations on polynomials. Order of polynomials. Continuum hypothesis. Ordinal types and Ordinal numbers. Elementary theory of ordinal types and ordinal numbers. Operations between ordinal types and ordinal numbers. Order between them. Hyperfinite induction. Axiom of choice. Corollaries of the axiom. Zorn and Zermelo lemmas. Trichotomy principle. Notable subsets of the real numbers: Cantor set, Borel sets, Baire sets, etc.