Reverse Mathematics: Problems, Reductions, and Proofs

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. It combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems. Topics and features include a complete introduction to reverse mathematics, a comprehensive treatment of the reverse mathematics of combinatorics, central results and methods, and a large number of exercises.

Format: Hardback
Length: 488 pages
Publication date: 26 July 2022
Publisher: Springer International Publishing AG

Reverse mathematics is a fascinating field that delves into the complexity of proving mathematical theorems and solving mathematical problems. It explores the intricacies of mathematical reasoning and offers insights into the foundations of mathematics. Typical questions in reverse mathematics include whether a result can be proven without first proving a related result, whether a particular problem can be solved by a computer, and the computational complexity of various mathematical tasks.

Reverse mathematics is a highly active area of study that intersects with mathematical logic and computability theory. It provides beautiful results and foundational insights into various branches of mathematics. The subject combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems.

This text offers a comprehensive treatment of reverse mathematics, covering topics such as combinatorics, graph theory, number theory, and algebra. It provides a detailed introduction to the field, including necessary background from computability theory, second-order arithmetic, forcing, induction, and model construction. The book also offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results.

Central results and methods from the past two decades are included in the book, appearing in book form for the first time. These results include preservation techniques, probabilistic arguments, and applications to other areas of mathematics. The book also includes a large number of exercises of varying levels of difficulty, supplementing each chapter.

The text is designed to be accessible to students with a standard first-year course in mathematical logic. It will also be a valuable reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.

Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Mathematics at the University of Erlangen-Nuremberg, Germany.

Weight: 922g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031113666
Edition number: 1st ed. 2022