Logic of Mathematics: A Modern Course of Classical Logic

A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic

Logic of Mathematics combines a full–scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems:
∗ Gödel′s theorems of completeness and incompleteness
∗ The independence of Goodstein′s theorem from Peano arithmetic
∗ Tarski′s theorem on real closed fields
∗ Matiyasevich′s theorem on diophantine formulas

Logic of Mathematics also features:
∗ Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types
∗ Clear, concise explanations of all key concepts, from Boolean algebras to Skolem–Löwenheim constructions and other topics
∗ Carefully chosen exercises for each chapter, plus helpful solution hints

At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic–requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms.

Part I contains a thorough introduction to mathematical logic and model theory–including a full discussion of terms, formulas, and other fundamentals, plus detailed coverage of relational structures and Boolean algebras, Gödel′s completeness theorem, models of Peano arithmetic, and much more.

Part II focuses on a number of advanced theorems that are central to the field, such as Gödel′s first and second theorems of incompleteness, the independence proof of Goodstein′s theorem from Peano arithmetic, Tarski′s theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems.

With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use–the perfect textbook for advanced students of mathematics, computer science, and logic.

Partial table of contents:

MATHEMATICAL STRUCTURES AND THEIR THEORIES.

Relational Systems.

Boolean Algebras.

Terms and Formulas.

Substitution of Terms.

Theorems and Proofs.

Generalization Rule and Elimination of Constants.

Peano Arithmetic.

Ultraproducts.

Supplementary Questions.

SELECTED TOPICS.

Total Functions.

Incompleteness of Arithmetic.

Tarski′s Theorem.

Matiyasevich′s Theorem.

Guide to Further Reading.

References.

Index.

ZOFIA ADAMOWICZ, PhD, is a professor at the Institute of Mathematics of the Polish Academy of Sciences in Warsaw.

PAWEL ZBIERSKI, PhD, is a professor at the Department of Mathematics at Warsaw University and the coauthor of Hausdorff Gaps and Limits.