Subsystems of Second Order Arithmetic

Author: Stephen George Simpson

Publisher: Cambridge University Press

ISBN: 052188439X

Category: Mathematics

Page: 444

View: 7206

This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.

Kurt Gödel

Essays for his Centennial

Author: Solomon Feferman,Charles Parsons,Stephen G. Simpson

Publisher: Cambridge University Press

ISBN: 1139487752

Category: Mathematics

Page: N.A

View: 4442

Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

Slicing the Truth

On the Computable and Reverse Mathematics of Combinatorial Principles

Author: Denis R Hirschfeldt

Publisher: World Scientific

ISBN: 9814612634

Category: Mathematics

Page: 232

View: 3344

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

Author: Stephen George Simpson

Publisher: A K Peters Ltd

ISBN: 9781568812632

Category: Mathematics

Page: 401

View: 9925

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 recent developments in reverse mathematics and subsystems of second order arithmetic.

Effective Mathematics of the Uncountable

Author: Noam Greenberg,Denis Hirschfeldt,Joel David Hamkins,Russell Miller

Publisher: Cambridge University Press

ISBN: 1107014514

Category: Mathematics

Page: 204

View: 4994

A comprehensive introduction to eight major approaches to computation on uncountable mathematical domains.

Metamathematics of First-Order Arithmetic

Author: Petr Hájek,Pavel Pudlák

Publisher: Cambridge University Press

ISBN: 1316739457

Category: Mathematics

Page: N.A

View: 2441

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. This volume, the third publication in the Perspectives in Logic series, is a much-needed monograph on the metamathematics of first-order arithmetic. The authors pay particular attention to subsystems (fragments) of Peano arithmetic and give the reader a deeper understanding of the role of the axiom schema of induction and of the phenomenon of incompleteness. The reader is only assumed to know the basics of mathematical logic, which are reviewed in the preliminaries. Part I develops parts of mathematics and logic in various fragments. Part II is devoted to incompleteness. Finally, Part III studies systems that have the induction schema restricted to bounded formulas (bounded arithmetic).

Lambda Calculus with Types

Author: Henk Barendregt,Wil Dekkers,Richard Statman

Publisher: Cambridge University Press

ISBN: 1107276349

Category: Mathematics

Page: N.A

View: 5454

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Reverse Mathematics

Proofs from the Inside Out

Author: John Stillwell

Publisher: Princeton University Press

ISBN: 1400889030

Category: Mathematics

Page: 200

View: 4727

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John 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. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the “right axiom” to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Goodbye, Descartes

the end of logic and the search for a new cosmology of the mind

Author: Keith J. Devlin

Publisher: John Wiley & Sons Inc

ISBN: N.A

Category: Mathematics

Page: 301

View: 8888

A narrative that traces how the concept of the mind as a logic machine developed over time asserts that logic is an inadequate model for the human mind's functionality and concludes that current efforts at replicating human thought will most likely fail. 15,000 first printing. $25,000 ad/promo.

Graph Structure and Monadic Second-Order Logic

A Language-Theoretic Approach

Author: Bruno Courcelle,Joost Engelfriet

Publisher: Cambridge University Press

ISBN: 1139644009

Category: Mathematics

Page: N.A

View: 1422

The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The authors not only provide a thorough description of the theory, but also detail its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.

Proof Theory

Second Edition

Author: Gaisi Takeuti

Publisher: Courier Corporation

ISBN: 0486320677

Category: Mathematics

Page: 512

View: 8497

This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

Refactoring

Improving the Design of Existing Code

Author: Martin Fowler,Kent Beck,John Brant,William Opdyke,Don Roberts

Publisher: Addison-Wesley

ISBN: 013306526X

Category: Computers

Page: 455

View: 4960

As the application of object technology--particularly the Java programming language--has become commonplace, a new problem has emerged to confront the software development community. Significant numbers of poorly designed programs have been created by less-experienced developers, resulting in applications that are inefficient and hard to maintain and extend. Increasingly, software system professionals are discovering just how difficult it is to work with these inherited, "non-optimal" applications. For several years, expert-level object programmers have employed a growing collection of techniques to improve the structural integrity and performance of such existing software programs. Referred to as "refactoring," these practices have remained in the domain of experts because no attempt has been made to transcribe the lore into a form that all developers could use. . .until now. In Refactoring: Improving the Design of Existing Code, renowned object technology mentor Martin Fowler breaks new ground, demystifying these master practices and demonstrating how software practitioners can realize the significant benefits of this new process. With proper training a skilled system designer can take a bad design and rework it into well-designed, robust code. In this book, Martin Fowler shows you where opportunities for refactoring typically can be found, and how to go about reworking a bad design into a good one. Each refactoring step is simple--seemingly too simple to be worth doing. Refactoring may involve moving a field from one class to another, or pulling some code out of a method to turn it into its own method, or even pushing some code up or down a hierarchy. While these individual steps may seem elementary, the cumulative effect of such small changes can radically improve the design. Refactoring is a proven way to prevent software decay. In addition to discussing the various techniques of refactoring, the author provides a detailed catalog of more than seventy proven refactorings with helpful pointers that teach you when to apply them; step-by-step instructions for applying each refactoring; and an example illustrating how the refactoring works. The illustrative examples are written in Java, but the ideas are applicable to any object-oriented programming language.

The Autonomy of Mathematical Knowledge

Hilbert's Program Revisited

Author: Curtis Franks

Publisher: Cambridge University Press

ISBN: 0521514371

Category: Mathematics

Page: 213

View: 5334

This study reconstructs, analyses and re-evaluates the programme of influential mathematical thinker David Hilbert, presenting it in a new light.

Structure and Interpretation of Computer Programs

Author: Harold Abelson

Publisher: Mit Press

ISBN: 9780262011532

Category: Computers

Page: 657

View: 3338

Structure and Interpretation of Computer Programs has had a dramatic impact on computer science curricula over the past decade. This long-awaited revision contains changes throughout the text. There are new implementations of most of the major programming systems in the book, including the interpreters and compilers, and the authors have incorporated many small changes that reflect their experience teaching the course at MIT since the first edition was published. A new theme has been introduced that emphasizes the central role played by different approaches to dealing with time in computational models: objects with state, concurrent programming, functional programming and lazy evaluation, and nondeterministic programming. There are new example sections on higher-order procedures in graphics and on applications of stream processing in numerical programming, and many new exercises. In addition, all the programs have been reworked to run in any Scheme implementation that adheres to the IEEE standard.

Principia Mathematica

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 4452

Fundamentals of Mathematical Logic

Author: Peter G. Hinman

Publisher: A K Peters/CRC Press

ISBN: 9781568812625

Category: Mathematics

Page: 896

View: 9834

This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.

Pure Inductive Logic

Author: Jeffrey Paris,Alena Vencovská

Publisher: Cambridge University Press

ISBN: 1107042305

Category: Computers

Page: 354

View: 307

A self-contained guide to pure inductive logic, the study of rational probability treated as a branch of mathematical logic.

Logic

A God-Centered Approach to the Foundation of Western Thought

Author: Vern S. Poythress

Publisher: Crossway

ISBN: 1433532328

Category: Religion

Page: 736

View: 2207

For the well-rounded Christian looking to improve their critical thinking skills, here is an accessible introduction to the study of logic (parts 1 & 2) as well as an in-depth treatment of the discipline (parts 3 & 4) from a professor with 6 academic degrees and over 30 years experience teaching. Questions for further reflection are included at the end of each chapter as well as helpful diagrams and charts that are appropriate for use in high school, home school, college, and graduate-level classrooms. Overall, Vern Poythress has undertaken a radical recasting of the study of logic in this revolutionary work from a Christian worldview.

Foundational Adventures

Author: Neil Tennant

Publisher: N.A

ISBN: 9781848901179

Category: Philosophy

Page: 314

View: 705

This volume is a tribute by his peers, and by younger scholars of the next generation, to Harvey M. Friedman, perhaps the most profound foundationalist since Kurt Godel. Friedman's researches, beginning precociously in his mid-teens, have fundamentally shaped our contemporary understanding of set theory, recursion theory, model theory, proof theory and metamathematics. His achievements in concept formation and theory formulation have also renewed the standard set by Godel and Alfred Tarski for the general intellectual interest and importance of technical work in foundations. Friedman pioneered the now well-established and flourishing field of Reverse Mathematics, whose aim is to calibrate the intrinsic logico-mathematical consistency-strength of all the important theorems of mathematics. He has relentlessly pursued the full extent of the incompleteness phenomena into which Godel provided the first revealing glimpse. The Godel--Friedman program, as it is now deservingly called, seeks to find simple, natural and elegant mathematical statements of a combinatorial nature, that can be proved to be independent of set theory even when extended by powerful large-cardinal existence axioms.