Reverse Mathematics
Problems, Reductions, and Proofs
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Beschreibung
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.
Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.
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.
Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.
Offers a comprehensive treatment of the reverse mathematics of combinatorics Includes a large number of exercises of varying levels of difficulty, supplementing each chapter Provides central results and methods from the past two decades, appearing in book form for the first time
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Damir D. Dzhafarov
Themen in »Reverse Mathematics«
Reverse mathematics Computability theory Second-order arithmetic Continuous mathematics Sequence coding
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“This book takes a broad approach to reverse mathematics, emphasizing connections to computability theory and Weihrauch analysis. It will serve well as a text for courses in reverse mathematics and as a reference for researchers in the area.” (Jeffry L. Hirst, zbMATH 1555.03002, 2025)
“Dzhafarov and Mummert's book provides a systematic introduction to new developments in reverse mathematics based on a computability-theoretic approach … . This book is a valuable reference for students and researchers working on reverse mathematics as well as other
branches of mathematical logic.” (Huishan Wu, Mathematical Reviews, September, 2023)
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