[For a review of the German original (1978) see Zbl 0399.03001.]
Rigorous introduction to the first-order logic and to some of its extensions is the main topic of the book. It covers the standard material up to the Completness Theorem. The concept of proof is based on a sequent calculus. In order to illustrate the expressive power of first-order logic, a short excursion to model theory is included. The compactness theorem, Löwenheim-Skolem theorems and some basic facts about elementary classes of models are proved. A carefully chosen sequence of examples should convince the reader that the means of first-order logic are not sufficient to describe categorically some basic mathematical structures unless set-theoretic terms are used. The subtle relation between logic and set theory is discussed in detail and some extensions of the first-order logic are introduced, namely the weak and strong second-order logics, \(L_{\omega_ 1\omega}\) and L(Q). The limitations of the formal method come out quite naturally from the presentation of the Gödel’s incompleteness theorems. The Trakhtenbrot’s theorem on satisfiable sentences in finite structures, Fraissé’s theorem on algebraic characterization of elementary equivalence and Lindström’s characterization of first-order logic among logical systems are the highlights of the book.
The text is selfcontained and gives a concise presentation of some central topics of mathematical logic from contemporary perspective.