(i) Elements of Naive Set Theory. Algebra of Boole of subsets.  Binary relations. Order relations, Functions. (ii) Introduction to axiomatic theory of sets. Contradictions. Foundation of natural, integer and rational numbers. Foundation of real numbers using Dedekind cuts and Cauchy sequences of rational numbers. The operations of addition and multiplication between natural, integer, rational and real numbers. Basic properties of these numbers. Order on the sets of natural, integer, rational and real numbers. (iii) Countable and uncountable sets. Cardinalities. The theorem of Cantor-Berstein. Operations of cardinalities. Order on cardinalities.  Continuum hypothesis. (iv) Imperative types and imperative numbers. Basic theory of imperative types and imperative numbers. Operations between imperative types and imperative numbers. Order between them. Transfinite induction. (v) Axiom of choice. Consequences of this axiom. Lemmas of Zorn and Zermelo. Trisect authority. (vi) Remarkable subsets of real numbers: Cantor set, Borel sets, Baire sets e.t.c.