NEW EDITION 2021: more and better motivations, chapter about tableaux method added, improved treatment of resolution method. [[[[[]]]]]Textbook for students in mathematical logic. Part 1. CONTENTS. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem. [[[[[[]]]]]] For Part 2 see my book at https://www.researchgate.net/publication/306112247_What_is_Mathematics_Godel's_Theorem_and_Around