Two Applications Of Logic To Mathematics

Author : Gaisi Takeuti
Publisher : Princeton University Press
Page : 148 pages
File Size : 55,5 Mb
Release : 2015-03-08
Category : Mathematics
ISBN : 9781400871346

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Two Applications of Logic to Mathematics by Gaisi Takeuti Pdf

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Author : Richard W. Kaye
Publisher : Cambridge University Press
Page : 12 pages
File Size : 51,8 Mb
Release : 2007-07-12
Category : Mathematics
ISBN : 9781139467216

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The Mathematics of Logic by Richard W. Kaye Pdf

This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis.

Author : Ira Rosenbaum
Publisher : Unknown
Page : 110 pages
File Size : 45,7 Mb
Release : 1950
Category : Logic, Symbolic and mathematical
ISBN : UOM:39015017298277

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Introduction to Mathematical Logic and Its Applications by Ira Rosenbaum Pdf

Author : Andreas Blass,Yi Zhang
Publisher : American Mathematical Soc.
Page : 314 pages
File Size : 51,7 Mb
Release : 2005
Category : Algebraic geometry
ISBN : 9780821834749

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Logic and Its Applications by Andreas Blass,Yi Zhang Pdf

Two conferences, Logic and Its Applications in Algebra and Geometry and Combinatorial Set Theory, Excellent Classes, and Schanuel Conjecture, were held at the University of Michigan (Ann Arbor). These events brought together model theorists and set theorists working in these areas. This volume is the result of those meetings. It is suitable for graduate students and researchers working in mathematical logic.

Author : Serge Lang
Publisher : Unknown
Page : 475 pages
File Size : 41,5 Mb
Release : 1988-01
Category : Mathematics
ISBN : 3540967877

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Basic Mathematics by Serge Lang Pdf

Author : Heinz-Dieter Ebbinghaus
Publisher : Springer Science & Business Media
Page : 653 pages
File Size : 44,6 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662090589

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Ω-Bibliography of Mathematical Logic by Heinz-Dieter Ebbinghaus Pdf

Gert H. Müller The growth of the number of publications in almost all scientific areas, as in the area of (mathematical) logic, is taken as a sign of our scientifically minded culture, but it also has a terrifying aspect. In addition, given the rapidly growing sophistica tion, specialization and hence subdivision of logic, researchers, students and teachers may have a hard time getting an overview of the existing literature, partic ularly if they do not have an extensive library available in their neighbourhood: they simply do not even know what to ask for! More specifically, if someone vaguely knows that something vaguely connected with his interests exists some where in the literature, he may not be able to find it even by searching through the publications scattered in the review journals. Answering this challenge was and is the central motivation for compiling this Bibliography. The Bibliography comprises (presently) the following six volumes (listed with the corresponding Editors): I. Classical Logic W. Rautenberg 11. Non-classical Logics W. Rautenberg 111. Model Theory H.-D. Ebbinghaus IV. Recursion Theory P.G. Hinman V. Set Theory A.R. Blass VI. ProofTheory; Constructive Mathematics J.E. Kister; D. van Dalen & A.S. Troelstra.

Author : Neil Jones,Masami Hagiya,Masahiko Sato
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 47,7 Mb
Release : 1994-03-30
Category : Mathematics
ISBN : 3540579354

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Logic, Language and Computation by Neil Jones,Masami Hagiya,Masahiko Sato Pdf

This volume contains 15 papers from research areas where Japanese theoretical computer science is particularly strong. Many are about logic, and its realization and applications to computer science; others concern synthesis, transformation and implementation of programming languages, and complexity and coding theory. Not coincidentally, all the authors are either former students or close colleagues of Satoru Takasu, professor and director at the Research Institute of Mathematical Sciences at the University of Kyoto. The purpose of this volume is to celebrate Professor Takasu's influence on theoretical computer science in Japan and worldwide by his research, his philosophy, and his advising of students. The breadth, depth and quality of the papers are characteristic of his interests and activities.

Author : Michael L. O'Leary
Publisher : John Wiley & Sons
Page : 464 pages
File Size : 46,8 Mb
Release : 2015-09-08
Category : Mathematics
ISBN : 9780470905883

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A First Course in Mathematical Logic and Set Theory by Michael L. O'Leary Pdf

A mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs Highlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, A First Course in Mathematical Logic and Set Theory introduces how logic is used to prepare and structure proofs and solve more complex problems. The book begins with propositional logic, including two-column proofs and truth table applications, followed by first-order logic, which provides the structure for writing mathematical proofs. Set theory is then introduced and serves as the basis for defining relations, functions, numbers, mathematical induction, ordinals, and cardinals. The book concludes with a primer on basic model theory with applications to abstract algebra. A First Course in Mathematical Logic and Set Theory also includes: Section exercises designed to show the interactions between topics and reinforce the presented ideas and concepts Numerous examples that illustrate theorems and employ basic concepts such as Euclid’s lemma, the Fibonacci sequence, and unique factorization Coverage of important theorems including the well-ordering theorem, completeness theorem, compactness theorem, as well as the theorems of Löwenheim–Skolem, Burali-Forti, Hartogs, Cantor–Schröder–Bernstein, and König An excellent textbook for students studying the foundations of mathematics and mathematical proofs, A First Course in Mathematical Logic and Set Theory is also appropriate for readers preparing for careers in mathematics education or computer science. In addition, the book is ideal for introductory courses on mathematical logic and/or set theory and appropriate for upper-undergraduate transition courses with rigorous mathematical reasoning involving algebra, number theory, or analysis.

Author : Anil Nerode,Richard A. Shore
Publisher : Springer Science & Business Media
Page : 383 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781468402117

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Logic for Applications by Anil Nerode,Richard A. Shore Pdf

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 recent dramatic growth in the applications of logic to computer science. Thus our choice of topics has been heavily influenced by such applications. Of course, we cover the basic traditional topics - syntax, semantics, soundness, completeness and compactness - as well as a few more advanced results such as the theorems of Skolem-Lowenheim and Herbrand. Much of our 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 PROLOG. We deal extensively with the mathematical foundations of all three of these subjects. In addition, we include two chapters on nonclassical logic- modal and intuitionistic - that are becoming increasingly important in computer science. We develop the basic material on the syntax and se mantics (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 introduced for classical logic. We indicate how it can easily be adapted to various other special types of modal log ics. A number of more advanced topics (including nonmonotonic logic) are also briefly introduced both in the nonclassical logic chapters and in the material on Logic Programming and PROLOG.

Author : Jaakko Hintikka,I. Niiniluoto,Esa. Saarinen
Publisher : Springer Science & Business Media
Page : 459 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789400998254

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Essays on Mathematical and Philosophical Logic by Jaakko Hintikka,I. Niiniluoto,Esa. Saarinen Pdf

The Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference were held in JyvaskyIa, Finland, June 29-July 6, 1976. The Conferences were organized by a committee which consisted of the editors of the present volume. The Conferences were supported financially by the Ministry of Education of Finland, by the Academy of Finland, and by the Division of Logic, Methodology, and Philosophy of Science of the International Union of History of Science. The Philosophical Society of Finland and the Jyvaskyla Summer Festival gave valuable help in various practicalities. 35 papers by authors representing 10 countries were presented at the two meetings. Of those papers 24 appear here. THE EDITORS v TABLE OF CONTENTS PREFACE v PART 1/ PROOF THEORY GEORG KREISEL / Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory 3 DAG PRAWITZ / Proofs and the Meaning and Completeness of the Logical Constants 25 v. A. SMIRNOV / Theory of Quantification and tff-calculi 41 LARS SVENONIUS/Two Kinds of Extensions of Primitive Recursive Arithmetic 49 DIRK VAN DALEN and R. STATMAN / Equality in the Presence of Apartness 95 PART II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N oo~.

Author : Yves Nievergelt
Publisher : Springer Science & Business Media
Page : 425 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461201250

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Foundations of Logic and Mathematics by Yves Nievergelt Pdf

This modern introduction to the foundations of logic and mathematics not only takes theory into account, but also treats in some detail applications that have a substantial impact on everyday life (loans and mortgages, bar codes, public-key cryptography). A first college-level introduction to logic, proofs, sets, number theory, and graph theory, and an excellent self-study reference and resource for instructors.

Author : Feng Ye
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 48,8 Mb
Release : 2011-07-06
Category : Science
ISBN : 9789400713475

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Strict Finitism and the Logic of Mathematical Applications by Feng Ye Pdf

This book intends to show that radical naturalism (or physicalism), nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book.

Author : Stefan Bilaniuk
Publisher : Orange Groove Books
Page : 166 pages
File Size : 44,6 Mb
Release : 2009-09-01
Category : Mathematics
ISBN : 1616100060

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A Problem Course in Mathematical Logic by Stefan Bilaniuk Pdf

Author : Wilfried Sieg
Publisher : Oxford University Press
Page : 452 pages
File Size : 52,6 Mb
Release : 2013-03-07
Category : Computers
ISBN : 9780195372229

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Hilbert's Programs and Beyond by Wilfried Sieg Pdf

David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
Page : 555 pages
File Size : 44,5 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9789400959910

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Encyclopaedia of Mathematics by Michiel Hazewinkel Pdf

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.