Introduction to Model Theory

Author: Philipp Rothmaler

Publisher: CRC Press

ISBN: 9789056992873

Category: Mathematics

Page: 324

View: 1862

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.
Posted in Mathematics

Model Theory and the Philosophy of Mathematical Practice

Formalization without Foundationalism

Author: John T. Baldwin

Publisher: Cambridge University Press

ISBN: 110810021X

Category: Science

Page: N.A

View: 952

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.
Posted in Science

Enzyklopädie Philosophie und Wissenschaftstheorie

Bd. 2: C–F

Author: Jürgen Mittelstraß

Publisher: Springer-Verlag

ISBN: 3476001342

Category: Philosophy

Page: 601

View: 1198

Die Enzyklopädie Philosophie- und Wissenschaftstheorie, das größte allgemeine Nachschlagewerk zur Philosophie im deutschsprachigen Raum, erscheint in einer aktualisierten und erweiterten kompletten Neuauflage. Sie umfasst in Sach- und Personenartikeln nicht nur den klassischen Bestand des philosophischen Wissens, sondern auch die neuere Entwicklung der Philosophie, insbesondere in den Bereichen Logik, Erkenntnis- und Wissenschaftstheorie sowie Sprachphilosophie. Ausführlich berücksichtigt sind auch Grundlagenreflexionen in den Wissenschaften und deren Geschichte. Zu den ca. 400 neu aufgenommenen Artikeln gehören z.B. Bioethik, Chaostheorie, Dekonstruktivismus, angewandte Ethik, Fundamentalismus, Genetik, Intelligenz und Komplexitätstheorie sowie zahlreiche Personenartikel. Die umfassenden Bibliografien und vollständigen Werkverzeichnisse wurden bei allen Artikeln auf den neuesten Stand gebracht.
Posted in Philosophy

Rechnen mit dem Unendlichen

Beiträge zur Entwicklung eines kontroversen Gegenstandes

Author: SPALT

Publisher: Springer-Verlag

ISBN: 3034852428

Category: Juvenile Nonfiction

Page: 243

View: 944

"Alle Einsender haben es versäumt zu erklären, wie so zahlreiche richtige Lehrsätze aus einer widerspruchsvollen Voraussetzung her geleitet werden können, wie es die einer unendlichen Größe ist. Alle haben sie mehr oder weniger die erforderten [Qualitäten der] Ein fachheit und Klarheit und über allem der Strenge außer acht ge lassen. Die meisten von ihnen haben nicht einmal gesehen, daß das gesuchte Prinzip nicht auf den Infinitesimalkalkül beschränkt sein sollte, sondern auf Algebra und auf Geometrie, wie sie in der Weise der Alten gehandhabt wird, auszudehnen war. Nach Ansicht der Akademie ist daher die Frage nicht in vollem Umfang gelöst. "2 Heute, im Abstand von zwei Jahrhunderten sehen wir, daß diese Preisaufgabe der Akademie die Qualität einer Forschungsaufgabe für viele Generationen hatte - und daß sie nach den Maßstäben der Akademie bis auf den heutigen Tag nicht gelöst ist - vielleicht, weil sie in dieser Form tatsächlich unlösbar ist. Gefragt wurde nach einem einzigen Mathematischen Prinzip des Unendlichen, welches, ohne widerspruchsvoll zu sein, hinreicht, sämtliche wahren mathema tischen Lehrsätze in einfacher, klarer und strenger Weise zu deduzieren - und zwar in allen mathematischen Gebieten (ausdrücklich genannt wurden neben der Infinitesimalrechnung die Geometrie und die Algebra). In heutiger Sicht unerfüllbar scheint jedenfalls die Forderung der Einzigkeit; Es ist bisher nicht zu sehen, wie ein einziges solches Prinzip für die gesamte Mathematik formulierbar sein könnte. Die Entwicklung der Geometrie im frühen 19. Jahrhundert verlief noch am ehesten in den von der Preisaufgabe gewünschten Bahnen.
Posted in Juvenile Nonfiction

Einführung in die Modelltheorie

Vorlesungen

Author: Philipp Rothmaler

Publisher: Spektrum Akademischer Verlag

ISBN: 9783860254615

Category: Model theory

Page: 331

View: 1886

Posted in Model theory

Grundlagen der modernen Mathematik

Author: Herbert Meschkowski

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 371

View: 4113

Posted in Mathematics

National Union Catalog

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Catalogs, Union

Page: N.A

View: 3739

Includes entries for maps and atlases.
Posted in Catalogs, Union

Constructivism in Mathematics

Author: A.S. Troelstra,D. van Dalen

Publisher: Elsevier

ISBN: 0080570887

Category: Mathematics

Page: 355

View: 5938

These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.
Posted in Mathematics

The Princeton Companion to Mathematics

Author: Timothy Gowers,June Barrow-Green,Imre Leader

Publisher: Princeton University Press

ISBN: 9781400830398

Category: Mathematics

Page: 1056

View: 3312

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more. Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties. Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors Presents major ideas and branches of pure mathematics in a clear, accessible style Defines and explains important mathematical concepts, methods, theorems, and open problems Introduces the language of mathematics and the goals of mathematical research Covers number theory, algebra, analysis, geometry, logic, probability, and more Traces the history and development of modern mathematics Profiles more than ninety-five mathematicians who influenced those working today Explores the influence of mathematics on other disciplines Includes bibliographies, cross-references, and a comprehensive index Contributors incude: Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger
Posted in Mathematics

Classical Mathematical Logic

The Semantic Foundations of Logic

Author: Richard L. Epstein

Publisher: Princeton University Press

ISBN: 1400841550

Category: Mathematics

Page: 544

View: 5334

In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations. The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference. Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.
Posted in Mathematics

Monographic Series

Author: Library of Congress

Publisher: N.A

ISBN: N.A

Category: Monographic series

Page: N.A

View: 6157

Posted in Monographic series

Higher-Order Algebra, Logic, and Term Rewriting

First International Workshop, HOA '93, Amsterdam, The Netherlands, September 23 - 24, 1993. Selected Papers

Author: International Workshop on Higher-Order Algebra, Logic and Term Rewriting

Publisher: Springer Science & Business Media

ISBN: 9783540582335

Category: Computers

Page: 344

View: 1120

This volume contains the final revised versions of the best papers presented at the First International Workshop on Higher-Order Algebra, Logic, and Term Rewriting (HOA '93), held in Amsterdam in September 1993. Higher-Order methods are increasingly applied in functional and logic programming languages, as well as in specification and verification of programs and hardware. The 15 full papers in this volume are devoted to the algebra and model theory of higher-order languages, computational logic techniques including resolution and term rewriting, and specification and verification case studies; in total they provide a competently written overview of current research and suggest new research directions in this vigourous area.
Posted in Computers

Provability, Computability and Reflection

Author: Lev D. Beklemishev

Publisher: Elsevier

ISBN: 9780080954783

Category: Mathematics

Page: 493

View: 4583

Provability, Computability and Reflection
Posted in Mathematics

Mathematical Handbook for Scientists and Engineers

Definitions, Theorems, and Formulas for Reference and Review

Author: Granino A. Korn,Theresa M. Korn

Publisher: Courier Corporation

ISBN: 0486320235

Category: Technology & Engineering

Page: 1152

View: 6407

Convenient access to information from every area of mathematics: Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, game theory, much more.
Posted in Technology & Engineering

Kurt Gödel

Wahrheit & Beweisbarkeit

Author: Kurt Gödel,Eckehart Köhler,Bernd Buldt

Publisher: N.A

ISBN: 9783209038340

Category: Logicians

Page: 448

View: 4041

Posted in Logicians

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: 167

View: 2184

Posted in Logic, Symbolic and mathematical

Foundations of Constructive Mathematics

Metamathematical Studies

Author: M.J. Beeson

Publisher: Springer Science & Business Media

ISBN: 3642689523

Category: Mathematics

Page: 466

View: 9161

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.
Posted in Mathematics

Grundlagen der Mathematik

Author: Peter Schreiber

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: 239

View: 2326

Posted in Logic, Symbolic and mathematical