Sets, Models and Proofs

The textbook "Sets, Models and Proofs" offers a concise and independent introduction to mathematical logic, with a particular focus on the fundamental topics of predicate logic and model theory. It includes numerous examples from various areas of mathematics, including algebra, linear algebra, and analysis, highlighting the relevance and utility of logic in these fields. The authors begin with a presentation of set theory and the axiom of choice, which are used in everyday mathematics. At a gentle pace, the first important results of model theory are presented, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel's completeness theorem for predicate logic. The book also covers the formal axiomatic system of Zermelo and Fraenkel and concludes with an extensive list of suggestions for further study. It is primarily aimed at mathematics students who are already familiar with basic analysis, algebra, and linear algebra, and includes numerous exercises of varying difficulty, making it suitable for both self-study and as a textbook for a one-semester university course in the second or third year.