Search Results for "foundations-of-set-theory-studies-in-logic-and-the-foundations-of-mathematics"

The Foundations of Mathematics

The Foundations of Mathematics

  • Author: Kenneth Kunen
  • Publisher: N.A
  • ISBN: 9781904987147
  • Category: Mathematics
  • Page: 251
  • View: 708
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Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.

Handbook of Mathematical Logic

Handbook of Mathematical Logic

  • Author: J. Barwise
  • Publisher: Elsevier
  • ISBN: 9780080933641
  • Category: Mathematics
  • Page: 1164
  • View: 9245
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The handbook is divided into four parts: model theory, set theory, recursion theory and proof theory. Each of the four parts begins with a short guide to the chapters that follow. Each chapter is written for non-specialists in the field in question. Mathematicians will find that this book provides them with a unique opportunity to apprise themselves of developments in areas other than their own.

Set theory

Set theory

  • Author: Kazimierz Kuratowski,Andrzej Mostowski
  • Publisher: N.A
  • ISBN: N.A
  • Category: Descriptive set theory
  • Page: 417
  • View: 5655
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Handbook of Proof Theory

Handbook of Proof Theory

  • Author: S.R. Buss
  • Publisher: Elsevier
  • ISBN: 9780080533186
  • Category: Mathematics
  • Page: 810
  • View: 5861
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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.

Handbook of Computability Theory

Handbook of Computability Theory

  • Author: E.R. Griffor
  • Publisher: Elsevier
  • ISBN: 9780080533049
  • Category: Mathematics
  • Page: 724
  • View: 699
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The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.

Categorical Logic and Type Theory

Categorical Logic and Type Theory

  • Author: Bart Jacobs
  • Publisher: Gulf Professional Publishing
  • ISBN: 9780444508539
  • Category: Mathematics
  • Page: 760
  • View: 5187
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This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

Classical Recursion Theory

Classical Recursion Theory

The Theory of Functions and Sets of Natural Numbers

  • Author: P. Odifreddi
  • Publisher: Elsevier
  • ISBN: 9780080886596
  • Category: Computers
  • Page: 667
  • View: 7820
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1988 marked the first centenary of Recursion Theory, since Dedekind's 1888 paper on the nature of number. Now available in paperback, this book is both a comprehensive reference for the subject and a textbook starting from first principles. Among the subjects covered are: various equivalent approaches to effective computability and their relations with computers and programming languages; a discussion of Church's thesis; a modern solution to Post's problem; global properties of Turing degrees; and a complete algebraic characterization of many-one degrees. Included are a number of applications to logic (in particular Gödel's theorems) and to computer science, for which Recursion Theory provides the theoretical foundation.

Undecidable Theories

Undecidable Theories

  • Author: Alfred Tarski,Andrzej Mostowski,Raphael Mitchel Robinson
  • Publisher: Elsevier
  • ISBN: 0444533788
  • Category: Decidability (Mathematical logic)
  • Page: 98
  • View: 4807
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Foundations of Set Theory

Foundations of Set Theory

  • Author: A.A. Fraenkel,Y. Bar-Hillel,A. Levy
  • Publisher: Elsevier
  • ISBN: 9780080887050
  • Category: Computers
  • Page: 412
  • View: 9880
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Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians.

Proof Theory

Proof Theory

  • Author: K. Schütte
  • Publisher: Springer Science & Business Media
  • ISBN: 3642664733
  • Category: Mathematics
  • Page: 302
  • View: 3767
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The Foundations of Mathematics and Other Logical Essays

The Foundations of Mathematics and Other Logical Essays

  • Author: Frank Plumpton Ramsey,Richard Bevan Braithwaite
  • Publisher: Psychology Press
  • ISBN: 9780415225465
  • Category: Philosophy
  • Page: 292
  • View: 9681
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First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

The Logical Foundations of Scientific Theories

The Logical Foundations of Scientific Theories

Languages, Structures, and Models

  • Author: Decio Krause,Jonas R.B. Arenhart
  • Publisher: Routledge
  • ISBN: 1315535203
  • Category: Mathematics
  • Page: 162
  • View: 5380
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This book addresses the logical aspects of the foundations of scientific theories. Even though the relevance of formal methods in the study of scientific theories is now widely recognized and regaining prominence, the issues covered here are still not generally discussed in philosophy of science. The authors focus mainly on the role played by the underlying formal apparatuses employed in the construction of the models of scientific theories, relating the discussion with the so-called semantic approach to scientific theories. The book describes the role played by this metamathematical framework in three main aspects: considerations of formal languages employed to axiomatize scientific theories, the role of the axiomatic method itself, and the way set-theoretical structures, which play the role of the models of theories, are developed. The authors also discuss the differences and philosophical relevance of the two basic ways of aximoatizing a scientific theory, namely Patrick Suppes’ set theoretical predicates and the "da Costa and Chuaqui" approach. This book engages with important discussions of the nature of scientific theories and will be a useful resource for researchers and upper-level students working in philosophy of science.

Set Theory An Introduction To Independence Proofs

Set Theory An Introduction To Independence Proofs

  • Author: K. Kunen
  • Publisher: Elsevier
  • ISBN: 0080570585
  • Category: Mathematics
  • Page: 330
  • View: 5332
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Studies in Logic and the Foundations of Mathematics, Volume 102: Set Theory: An Introduction to Independence Proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the Suslin problem, Martin's axiom, almost disjoint and quasi-disjoint sets, trees, extensionality and comprehension, relations, functions, and well-ordering, ordinals, cardinals, and real numbers. The manuscript then ponders on well-founded sets and easy consistency proofs, including relativization, absoluteness, reflection theorems, properties of well-founded sets, and induction and recursion on well-founded relations. The publication examines constructible sets, forcing, and iterated forcing. Topics include Easton forcing, general iterated forcing, Cohen model, forcing with partial functions of larger cardinality, forcing with finite partial functions, and general extensions. The manuscript is a dependable source of information for mathematicians and researchers interested in set theory.

Principia Mathematica to *56

Principia Mathematica to *56

  • Author: Alfred North Whitehead,Bertrand Russell
  • Publisher: Cambridge University Press
  • ISBN: 9780521626064
  • Category: Mathematics
  • Page: 410
  • View: 7526
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The great three-volume Principia Mathematica (CUP 1927) is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premises and primitive ideas, establishing that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will of course wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part I (in which the logical properties of propositions, propositional functions, classes and relations are established); section A of Part II (dealing with unit classes and couples); and Appendices A and C (which give further developments of the argument on the theory of deduction and truth functions).

A First Course in Mathematical Logic and Set Theory

A First Course in Mathematical Logic and Set Theory

  • Author: Michael L. O'Leary
  • Publisher: John Wiley & Sons
  • ISBN: 0470905883
  • Category: Mathematics
  • Page: 464
  • View: 7172
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Rather than teach mathematics and the structure of proofssimultaneously, this book first introduces logic as the foundationof proofs and then demonstrates how logic applies to mathematicaltopics. This method ensures that readers gain a firmunderstanding of how logic interacts with mathematics and empowersthem to solve more complex problems. The study of logic andapplications is used throughout to prepare readers for further workin proof writing. Readers are first introduced tomathematical proof-writing, and then the book provides anoverview of symbolic logic that includes two-column logicproofs. Readers are then transitioned to set theory andinduction, and applications of number theory, relations, functions,groups, and topology are provided to further aid incomprehension. Topical coverage includes propositional logic,predicate logic, set theory, mathematical induction, number theory,relations, functions, group theory, and topology.

Practical Foundations of Mathematics

Practical Foundations of Mathematics

  • Author: Paul Taylor
  • Publisher: Cambridge University Press
  • ISBN: 9780521631075
  • Category: Mathematics
  • Page: 572
  • View: 3191
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Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

The Search for Mathematical Roots, 1870-1940

The Search for Mathematical Roots, 1870-1940

Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel

  • Author: I. Grattan-Guinness
  • Publisher: Princeton University Press
  • ISBN: 9781400824045
  • Category: Mathematics
  • Page: 624
  • View: 5572
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While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.

Classification Theory and the Number of Non-isomorphic Models

Classification Theory and the Number of Non-isomorphic Models

  • Author: Saharon Shelah
  • Publisher: North Holland
  • ISBN: 9780444702609
  • Category: Mathematics
  • Page: 705
  • View: 9831
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Hardbound. In this research monograph, the author's work on classification and related topics are presented. This revised edition brings the book up to date with the addition of four new chapters as well as various corrections to the 1978 text.The additional chapters X - XIII present the solution to countable first order T of what the author sees as the main test of the theory. In Chapter X the Dimensional Order Property is introduced and it is shown to be a meaningful dividing line for superstable theories. In Chapter XI there is a proof of the decomposition theorems. Chapter XII is the crux of the matter: there is proof that the negation of the assumption used in Chapter XI implies that in models of T a relation can be defined which orders a large subset of mM. This theorem is also the subject of Chapter XIII.

Foundations of Constructive Mathematics

Foundations of Constructive Mathematics

Metamathematical Studies

  • Author: M.J. Beeson
  • Publisher: Springer Science & Business Media
  • ISBN: 3642689523
  • Category: Mathematics
  • Page: 466
  • View: 6437
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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.

Introduction to the Foundations of Mathematics

Introduction to the Foundations of Mathematics

Second Edition

  • Author: Raymond L. Wilder
  • Publisher: Courier Corporation
  • ISBN: 0486276201
  • Category: Mathematics
  • Page: 352
  • View: 1316
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Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition.