| Author | : Stephen George Simpson |
| Publisher | : Cambridge University Press |
| Total Pages | : 461 |
| Release | : 2009-05-29 |
| Genre | : Mathematics |
| ISBN | : 052188439X |
This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Subsystems of Second Order Arithmetic
| Author | : Stephen G. Simpson |
| Publisher | : Cambridge University Press |
| Total Pages | : 445 |
| Release | : 2009-05-29 |
| Genre | : Mathematics |
| ISBN | : 1139478915 |
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.
Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles
| Author | : Denis R Hirschfeldt |
| Publisher | : World Scientific |
| Total Pages | : 231 |
| Release | : 2014-07-18 |
| Genre | : Mathematics |
| ISBN | : 9814612634 |
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.
Reverse Mathematics
| Author | : John Stillwell |
| Publisher | : Princeton University Press |
| Total Pages | : 198 |
| Release | : 2019-09-24 |
| Genre | : Mathematics |
| ISBN | : 0691196419 |
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.
Harvey Friedman's Research on the Foundations of Mathematics
| Author | : L.A. Harrington |
| Publisher | : Elsevier |
| Total Pages | : 407 |
| Release | : 1985-11-01 |
| Genre | : Mathematics |
| ISBN | : 9780080960401 |
This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.
Metamathematics of First-Order Arithmetic
| Author | : Petr Hájek |
| Publisher | : Cambridge University Press |
| Total Pages | : 475 |
| Release | : 2017-03-02 |
| Genre | : Mathematics |
| ISBN | : 1107168414 |
A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
Handbook of Proof Theory
| Author | : S.R. Buss |
| Publisher | : Elsevier |
| Total Pages | : 823 |
| Release | : 1998-07-09 |
| Genre | : Mathematics |
| ISBN | : 0080533183 |
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth.The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.
Applied Proof Theory: Proof Interpretations and their Use in Mathematics
| Author | : Ulrich Kohlenbach |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 539 |
| Release | : 2008-05-23 |
| Genre | : Mathematics |
| ISBN | : 3540775331 |
This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
Foundations without Foundationalism
| Author | : Stewart Shapiro |
| Publisher | : Clarendon Press |
| Total Pages | : 302 |
| Release | : 1991-09-19 |
| Genre | : Mathematics |
| ISBN | : 0191524018 |
The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.
