Author: Stephen George Simpson
Publisher: Cambridge University Press
ISBN: 052188439X
Category : Mathematics
Languages : en
Pages : 461
Book Description
This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Subsystems of Second Order Arithmetic
Author: Stephen George Simpson
Publisher: Cambridge University Press
ISBN: 052188439X
Category : Mathematics
Languages : en
Pages : 461
Book Description
This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Publisher: Cambridge University Press
ISBN: 052188439X
Category : Mathematics
Languages : en
Pages : 461
Book Description
This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Subsystems of Second Order Arithmetic
Author: Stephen G. Simpson
Publisher:
ISBN: 9783642599712
Category : Computer science
Languages : en
Pages : 444
Book Description
"From the point of view of the foundations of mathematics, this definitive work by Simpson is the most anxiously awaited monograph for over a decade. The "subsystems of second order arithmetic" provide the basic formal systems normally used in our current understanding of the logical structure of classical mathematics. Simpson provides an encyclopedic treatment of these systems with an emphasis on *Hilbert's program* (where infinitary mathematics is to be secured or reinterpreted by finitary mathematics), and the emerging *reverse mathematics* (where axioms necessary for providing theorems are determined by deriving axioms from theorems). The classical mathematical topics treated in these axiomatic terms are very diverse, and include standard topics in complete separable metric spaces and Banach spaces, countable groups, rings, fields, and vector spaces, ordinary differential equations, fixed points, infinite games, Ramsey theory, and many others. The material, with its many open problems and detailed references to the literature, is particularly valuable for proof theorists and recursion theorists. The book is both suitable for the beginning graduate student in mathematical logic, and encyclopedic for the expert." Harvey Friedman, Ohio State University.
Publisher:
ISBN: 9783642599712
Category : Computer science
Languages : en
Pages : 444
Book Description
"From the point of view of the foundations of mathematics, this definitive work by Simpson is the most anxiously awaited monograph for over a decade. The "subsystems of second order arithmetic" provide the basic formal systems normally used in our current understanding of the logical structure of classical mathematics. Simpson provides an encyclopedic treatment of these systems with an emphasis on *Hilbert's program* (where infinitary mathematics is to be secured or reinterpreted by finitary mathematics), and the emerging *reverse mathematics* (where axioms necessary for providing theorems are determined by deriving axioms from theorems). The classical mathematical topics treated in these axiomatic terms are very diverse, and include standard topics in complete separable metric spaces and Banach spaces, countable groups, rings, fields, and vector spaces, ordinary differential equations, fixed points, infinite games, Ramsey theory, and many others. The material, with its many open problems and detailed references to the literature, is particularly valuable for proof theorists and recursion theorists. The book is both suitable for the beginning graduate student in mathematical logic, and encyclopedic for the expert." Harvey Friedman, Ohio State University.
Subsystems of Second Order Arithmetic
Author: Stephen G. Simpson
Publisher: Cambridge University Press
ISBN: 1139478915
Category : Mathematics
Languages : en
Pages : 445
Book Description
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.
Publisher: Cambridge University Press
ISBN: 1139478915
Category : Mathematics
Languages : en
Pages : 445
Book Description
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.
Subsystems of Second-order Arithmetic, and Descriptive Set Theory Under the Axiom of Determinateness
Author: Robert Alan Van Wesep
Publisher:
ISBN:
Category :
Languages : en
Pages : 242
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 242
Book Description
Proof-theoretic Investigations of Subsystems of Second-order Arithmetic
Author: Jeremy David Avigad
Publisher:
ISBN:
Category :
Languages : en
Pages : 314
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 314
Book Description
Subsystems of Second Order Arithmetic
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Book Description
Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles
Author: Denis R Hirschfeldt
Publisher: World Scientific
ISBN: 9814612634
Category : Mathematics
Languages : en
Pages : 231
Book Description
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.
Publisher: World Scientific
ISBN: 9814612634
Category : Mathematics
Languages : en
Pages : 231
Book Description
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.
Subsystems of Second-order Arithmetic, and Descriptive Set Theory Under the
Author: Robert A. van Wesep
Publisher:
ISBN:
Category :
Languages : en
Pages : 116
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 116
Book Description
Metamathematics of First-Order Arithmetic
Author: Petr Hájek
Publisher: Cambridge University Press
ISBN: 1107168414
Category : Mathematics
Languages : en
Pages : 475
Book Description
A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
Publisher: Cambridge University Press
ISBN: 1107168414
Category : Mathematics
Languages : en
Pages : 475
Book Description
A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
Reverse Mathematics
Author: John Stillwell
Publisher: Princeton University Press
ISBN: 0691196419
Category : Mathematics
Languages : en
Pages : 198
Book Description
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.
Publisher: Princeton University Press
ISBN: 0691196419
Category : Mathematics
Languages : en
Pages : 198
Book Description
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.