Reverse Mathematics
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Reverse Mathematics
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Author : John Stillwell
language : en
Publisher: Princeton University Press
Release Date : 2019-09-24
Reverse Mathematics written by John Stillwell and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-24 with Mathematics categories.
This volume presents reverse mathematics to a general mathematical audience for the first time. Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. to logic.
Reverse Mathematics
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Author : Damir D. Dzhafarov
language : en
Publisher: Springer Nature
Release Date : 2022-07-25
Reverse Mathematics written by Damir D. Dzhafarov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-25 with Computers categories.
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.
Slicing The Truth
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Author : Denis R Hirschfeldt
language : en
Publisher: World Scientific
Release Date : 2014-07-18
Slicing The Truth written by Denis R Hirschfeldt and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-18 with Mathematics categories.
This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions. Contents:Setting Off: An IntroductionGathering Our Tools: Basic Concepts and NotationFinding Our Path: König's Lemma and ComputabilityGauging Our Strength: Reverse MathematicsIn Defense of DisarrayAchieving Consensus: Ramsey's TheoremPreserving Our Power: ConservativityDrawing a Map: Five DiagramsExploring Our Surroundings: The World Below RT22Charging Ahead: Further TopicsLagniappe: A Proof of Liu's Theorem Readership: Graduates and researchers in mathematical logic. Key Features:This book assumes minimal background in mathematical logic and takes the reader all the way to current research in a highly active areaIt is the first detailed introduction to this particular approach to this area of researchThe combination of fully worked out arguments and exercises make this book well suited to self-study by graduate students and other researchers unfamiliar with the areaKeywords:Reverse Mathematics;Computability Theory;Computable Mathematics;Computable Combinatorics
Reverse Mathematics 2001
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Author : Stephen G. Ross
language : en
Publisher: CRC Press
Release Date : 2005-09-01
Reverse Mathematics 2001 written by Stephen G. Ross and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-09-01 with Mathematics categories.
Reverse Mathematics is a program of research in the foundations of mathematics, motivated by the foundational questions of what are appropriate axioms for mathematics, and what are the logical strengths of particular axioms and particular theorems. The book contains 24 original papers by leading researchers. These articles exhibit the exciting rece
Reverse Mathematics 2001
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Author : Stephen G. Simpson
language : en
Publisher: Cambridge University Press
Release Date : 2017-03-30
Reverse Mathematics 2001 written by Stephen G. Simpson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-30 with Mathematics categories.
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Reverse mathematics is a program of research in the foundations of mathematics, motivated by two foundational questions: 'what are appropriate axioms for mathematics?' and 'what are the logical strengths of particular axioms and particular theorems?' This volume, the twenty-first publication in the Lecture Notes in Logic series, contains twenty-four original research papers from respected authors that present exciting new developments in reverse mathematics and subsystems of second order arithmetic since 1998.
Reverse Mathematics And Ordered Groups
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Author : David Reed Solomon
language : en
Publisher:
Release Date : 1998
Reverse Mathematics And Ordered Groups written by David Reed Solomon and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.
Subsystems Of Second Order Arithmetic
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Author : Stephen G. Simpson
language : en
Publisher: Cambridge University Press
Release Date : 2009-05-29
Subsystems Of Second Order Arithmetic written by Stephen G. Simpson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-05-29 with Mathematics categories.
Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic.
Reversibility In Dynamics And Group Theory
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Author : Anthony G. O'Farrell
language : en
Publisher: Cambridge University Press
Release Date : 2015-05-28
Reversibility In Dynamics And Group Theory written by Anthony G. O'Farrell and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-28 with Mathematics categories.
An accessible yet systematic account of reversibility that demonstrates its impact throughout many diverse areas of mathematics.
Reverse Mathematics On Lattice Ordered Groups
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Author : Alexander S. Rogalski
language : en
Publisher:
Release Date : 2007
Reverse Mathematics On Lattice Ordered Groups written by Alexander S. Rogalski and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with categories.
Bounded Reverse Mathematics
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Author : Nicholas Henry
language : en
Publisher:
Release Date : 2019-03-31
Bounded Reverse Mathematics written by Nicholas Henry and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-31 with categories.
First we provide a unified framework for developing theories of Bounded Arithmetic that are associated with uniform classes inside polytime (P) in the same way that Buss's theory S 12 is associated with P. We obtain finitely axiomatized theories many of which turn out to be equivalent to a number of existing systems. By formalizing the proof of Barrington's Theorem (that the functions computable by polynomial-size bounded-width branching programs are precisely functions computable in ALogTime, or equivalently NC 1 ) we prove one such equivalence between the theories associated with ALogTime, solving a problem that remains open in [Ara00, Pit00]. Our theories demonstrate an advantage of the simplicity of Zambella's two-sorted setting for small theories of Bounded Arithmetic. Then we give the first definitions for the relativizations of small classes such as NC 1 , L, NL that preserve their inclusion order. Separating these relativized classes is shown to be as hard as separating the corresponding non-relativized classes. Our framework also allows us to obtain relativized theories that characterize the newly defined relativized classes. Finally we formalize and prove a number of mathematical theorems in our theories. In particular, we prove the discrete versions of the Jordan Curve Theorem in the theories V 0 and V 0 (2), and establish some facts about the distribution of prime numbers in the theory VTC 0 . Our V 0 - and V 0 (2)-proofs improve a number of existing upper bounds for the propositional complexity of combinatorial principles related to grid graphs. Overall, this thesis is a contribution to Bounded Reverse Mathematics, a theme iiwhose purpose is to formalize and prove (discrete versions of) mathematical theorems in the weakest possible theories of bounded arithmetic.