Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights.

This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field.

Topics and features:

• Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction
• Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results
• Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments
• Includes a large number of exercises of varying levels of difficulty, supplementing each chapter

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

Table of Contents

Part I Computable mathematics

Computability theory

Instance-solution problems

Problem reducibilities

Part II Formalization and syntax

Second order arithmetic

Induction and bounding

Forcing

Part Ill Combinatorics

Ramsey's theorem

Other combinatorial principles

Part IV Other areas

Analysis and topology

Algebra

Set theory and beyond

About the Authors

Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut. He obtained his PhD from the University of Chicago, and has held postdoctoral positions at the University of Notre Dame and the University of California, Berkeley. He has held visiting positions at the National University of Singapore and Charles University, Prague. His research focuses on the computability theoretic and reverse mathematical aspects of of combinatorics, and on the interactions of reverse mathematics with computable analysis and other areas.

Carl Mummert is a Professor of Computer and Information Technology at Marshall Univeristy. He obtained his Ph.D. from Pennsylvania State University and held postdoctoral positions at Appalachian State University and the University of Michigan. His research has included the reverse mathematics of topology and combinatorics as well as higher order reverse mathematics.