Subsystems of Second Order Arithmetic

Author: Stephen George Simpson

Publisher: Cambridge University Press

ISBN: 052188439X

Category: Mathematics

Page: 444

View: 2523

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

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: 5693

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
Posted in Mathematics

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: 7712

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.
Posted in Mathematics

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: 9615

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
Posted in Mathematics

Metamathematics of First-Order Arithmetic

Author: Petr Hájek,Pavel Pudlák

Publisher: Cambridge University Press

ISBN: 1316739457

Category: Mathematics

Page: N.A

View: 2294

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).
Posted in Mathematics

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: 4705

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

Proof And Computation: Digitization In Mathematics, Computer Science And Philosophy

Author: Mainzer Klaus,Schwichtenberg Helmut,Schuster Peter Michael

Publisher: World Scientific

ISBN: 9813270950

Category: Mathematics

Page: 300

View: 5634

This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes Predicative Foundations, Constructive Mathematics and Type Theory, Computation in Higher Types, Extraction of Programs from Proofs, and Algorithmic Aspects in Financial Mathematics. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields. Contents: Proof and Computation (K Mainzer) Constructive Convex Programming (J Berger and G Svindland) Exploring Predicativity (L Crosilla) Constructive Functional Analysis: An Introduction (H Ishihara) Program Extraction (K Miyamoto) The Data Structures of the Lambda Terms (M Sato) Provable (and Unprovable) Computability (S Wainer) Introduction to Minlog (F Wiesnet) Readership: Graduate students, researchers, and professionals in Mathematics and Computer Science. Keywords: Proof Theory;Computability Theory;Program Extraction;Constructive Analysis;PredicativityReview: Key Features: This book gathers recent contributions of distinguished experts It makes emerging fields accessible to a wider audience, appealing to a broad readership with diverse backgrounds It fills a gap between (under-)graduate level textbooks and state-of-the-art research papers
Posted in Mathematics

Foundations of Mathematics

Author: Andrés Eduardo Caicedo,James Cummings,Peter Koellner,Paul B. Larson

Publisher: American Mathematical Soc.

ISBN: 1470422565

Category: Continuum hypothesis

Page: 322

View: 6422

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas.
Posted in Continuum hypothesis

Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

Author: Denis R. Hirschfeldt,Karen Lange,Richard A. Shore

Publisher: American Mathematical Soc.

ISBN: 1470426579

Category: Computable functions

Page: 101

View: 6880

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.
Posted in Computable functions

Logic and Structure

Author: Dirk van Dalen

Publisher: Springer Science & Business Media

ISBN: 3662029626

Category: Mathematics

Page: 220

View: 9665

New corrected printing of a well-established text on logic at the introductory level.
Posted in Mathematics

Agendas and Instability in American Politics, Second Edition

Author: Frank R. Baumgartner,Bryan D. Jones

Publisher: University of Chicago Press

ISBN: 9780226039534

Category: Political Science

Page: 368

View: 7399

When Agendas and Instability in American Politics appeared fifteen years ago, offering a profoundly original account of how policy issues rise and fall on the national agenda, the Journal of Politics predicted that it would “become a landmark study of public policy making and American politics.” That prediction proved true and, in this long-awaited second edition, Bryan Jones and Frank Baumgartner refine their influential argument and expand it to illuminate the workings of democracies beyond the United States. The authors retain all the substance of their contention that short-term, single-issue analyses cast public policy too narrowly as the result of cozy and dependable arrangements among politicians, interest groups, and the media. Jones and Baumgartner provide a different interpretation by taking the long view of several issues—including nuclear energy, urban affairs, smoking, and auto safety—to demonstrate that bursts of rapid, unpredictable policy change punctuate the patterns of stability more frequently associated with government. Featuring a new introduction and two additional chapters, this updated edition ensures that their findings will remain a touchstone of policy studies for many years to come.
Posted in Political Science

Lambda Calculus with Types

Author: Henk Barendregt,Wil Dekkers,Richard Statman

Publisher: Cambridge University Press

ISBN: 1107276349

Category: Mathematics

Page: N.A

View: 7299

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.
Posted in Mathematics

The Bulletin of Symbolic Logic

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Electronic journals

Page: N.A

View: 6172

Posted in Electronic journals

Structure and Interpretation of Computer Programs

Author: Harold Abelson

Publisher: Mit Press

ISBN: 9780262011532

Category: Computers

Page: 657

View: 3682

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.
Posted in Computers

The Autonomy of Mathematical Knowledge

Hilbert's Program Revisited

Author: Curtis Franks

Publisher: Cambridge University Press

ISBN: 0521514371

Category: Mathematics

Page: 213

View: 5155

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

The Logic of Scientific Discovery

Author: Karl Popper

Publisher: Routledge

ISBN: 1134470029

Category: Philosophy

Page: 480

View: 9252

Described by the philosopher A.J. Ayer as a work of 'great originality and power', this book revolutionized contemporary thinking on science and knowledge. Ideas such as the now legendary doctrine of 'falsificationism' electrified the scientific community, influencing even working scientists, as well as post-war philosophy. This astonishing work ranks alongside The Open Society and Its Enemies as one of Popper's most enduring books and contains insights and arguments that demand to be read to this day.
Posted in Philosophy

The Mathematics of Language

Author: Marcus Kracht

Publisher: Walter de Gruyter

ISBN: 3110895668

Category: Language Arts & Disciplines

Page: 605

View: 3633

This book studies language(s) and linguistic theories from a mathematical point of view. Starting with ideas already contained in Montague's work, it develops the mathematical foundations of present day linguistics. It equips the reader with all the background necessary to understand and evaluate theories as diverse as Montague Grammar, Categorial Grammar, HPSG and GB. The mathematical tools are mainly from universal algebra and logic, but no particular knowledge is presupposed beyond a certain mathematical sophistication that is in any case needed in order to fruitfully work within these theories. The presentation focuses on abstract mathematical structures and their computational properties, but plenty of examples from different natural languages are provided to illustrate the main concepts and results. In contrast to books devoted to so-called formal language theory, languages are seen here as semiotic systems, that is, as systems of signs. A language sign correlates form with meaning. Using the principle of compositionality it is possible to gain substantial insight into the interaction between form and meaning in natural languages.
Posted in Language Arts & Disciplines

Logic for Applications

Author: Anil Nerode,Richard A. Shore

Publisher: Springer Science & Business Media

ISBN: 1461206499

Category: Computers

Page: 456

View: 2321

In writing this book, our goal was to produce a text suitable for a first course in mathematical logic more attuned than the traditional textbooks to the re cent dramatic growth in the applications oflogic to computer science. Thus, our choice oftopics has been heavily influenced by such applications. Of course, we cover the basic traditional topics: syntax, semantics, soundnes5, completeness and compactness as well as a few more advanced results such as the theorems of Skolem-Lowenheim and Herbrand. Much ofour book, however, deals with other less traditional topics. Resolution theorem proving plays a major role in our treatment of logic especially in its application to Logic Programming and PRO LOG. We deal extensively with the mathematical foundations ofall three ofthese subjects. In addition, we include two chapters on nonclassical logics - modal and intuitionistic - that are becoming increasingly important in computer sci ence. We develop the basic material on the syntax and semantics (via Kripke frames) for each of these logics. In both cases, our approach to formal proofs, soundness and completeness uses modifications of the same tableau method in troduced for classical logic. We indicate how it can easily be adapted to various other special types of modal logics. A number of more advanced topics (includ ing nonmonotonic logic) are also briefly introduced both in the nonclassical logic chapters and in the material on Logic Programming and PROLOG.
Posted in Computers

Analysis with Ultrasmall Numbers

Author: Karel Hrbacek,Olivier Lessmann,Richard O'Donovan

Publisher: CRC Press

ISBN: 1498702651

Category: Mathematics

Page: 316

View: 1814

Analysis with Ultrasmall Numbers presents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilon–delta methods. Students can fully prove fundamental results, such as the Extreme Value Theorem, from the axioms immediately, without needing to master notions of supremum or compactness. The book is suitable for a calculus course at the undergraduate or high school level or for self-study with an emphasis on nonstandard methods. The first part of the text offers material for an elementary calculus course while the second part covers more advanced calculus topics. The text provides straightforward definitions of basic concepts, enabling students to form good intuition and actually prove things by themselves. It does not require any additional "black boxes" once the initial axioms have been presented. The text also includes numerous exercises throughout and at the end of each chapter.
Posted in Mathematics

The Oxford Handbook of Philosophy of Mathematics and Logic

Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 0190287535

Category: Mathematics

Page: 856

View: 2738

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.
Posted in Mathematics