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

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: 5330
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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.

The Foundations of Mathematics

The Foundations of Mathematics

  • Author: Kenneth Kunen
  • Publisher: N.A
  • ISBN: 9781904987147
  • Category: Mathematics
  • Page: 251
  • View: 765
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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: 5387
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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.

Problems in the Philosophy of Mathematics

Problems in the Philosophy of Mathematics

  • Author: Brouwer
  • Publisher: Elsevier
  • ISBN: 0080957668
  • Category: Computers
  • Page: 240
  • View: 3636
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Problems in the Philosophy of Mathematics

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: 6410
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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.

Grundzüge der Mengenlehre

Grundzüge der Mengenlehre

  • Author: Felix Hausdorff
  • Publisher: American Mathematical Soc.
  • ISBN: 9780828400619
  • Category: Mathematics
  • Page: 476
  • View: 947
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This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.

Principia Mathematica.

Principia Mathematica.

  • Author: Alfred North Whitehead,Bertrand Russell
  • Publisher: N.A
  • ISBN: N.A
  • Category: Logic, Symbolic and mathematical
  • Page: 167
  • View: 5023
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Harvey Friedman's Research on the Foundations of Mathematics

Harvey Friedman's Research on the Foundations of Mathematics

  • Author: L.A. Harrington,M.D. Morley,A. Šcedrov,S.G. Simpson
  • Publisher: Elsevier
  • ISBN: 9780080960401
  • Category: Mathematics
  • Page: 407
  • View: 7580
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This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.

Essays on the Foundations of Mathematics and Logic

Essays on the Foundations of Mathematics and Logic

  • Author: Giandomenico Sica
  • Publisher: Polimetrica s.a.s.
  • ISBN: 8876990143
  • Category: Mathematics
  • Page: 351
  • View: 4253
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Topics in Logic, Philosophy and Foundations of Mathematics, and Computer Science

Topics in Logic, Philosophy and Foundations of Mathematics, and Computer Science

In Recognition of Professor Andrzej Grzegorczyk

  • Author: Stanisław Krajewski
  • Publisher: IOS Press
  • ISBN: 9781586038144
  • Category: Computers
  • Page: 365
  • View: 3618
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Professor Andrzej Grzegorczyk has made fundamental contributions to logic and to philosophy. This volume honors Professor Grzegorczyk, the nestor of Polish logicians, on his 85th anniversary. It presents the work and life of Professor Grzegorczyk.

Abstract set theory

Abstract set theory

  • Author: Abraham Adolf Fraenkel
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: 479
  • View: 6934
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Kurt Gödel and the Foundations of Mathematics

Kurt Gödel and the Foundations of Mathematics

Horizons of Truth

  • Author: Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
  • Publisher: Cambridge University Press
  • ISBN: 1139498436
  • Category: Mathematics
  • Page: N.A
  • View: 4393
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This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Naive Mengenlehre

Naive Mengenlehre

  • Author: Paul R. Halmos
  • Publisher: Vandenhoeck & Ruprecht
  • ISBN: 9783525405277
  • Category: Arithmetic
  • Page: 132
  • View: 3961
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Axiomatic Set Theory

Axiomatic Set Theory

  • Author: Patrick Suppes
  • Publisher: Courier Corporation
  • ISBN: 0486136876
  • Category: Mathematics
  • Page: 265
  • View: 4190
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Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960 edition.

Foundational Studies

Foundational Studies

Selected Works

  • Author: Andrzej Mostowski,Kazimierz Kuratowski
  • Publisher: Elsevier
  • ISBN: 0444851038
  • Category: Electronic books
  • Page: 605
  • View: 9053
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Essays on the Foundations of Mathematics by Moritz Pasch

Essays on the Foundations of Mathematics by Moritz Pasch

  • Author: Stephen Pollard
  • Publisher: Springer Science & Business Media
  • ISBN: 9789048194162
  • Category: Mathematics
  • Page: 248
  • View: 5567
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Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert’s program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.

Introduction to the Foundations of Mathematics

Introduction to the Foundations of Mathematics

Second Edition

  • Author: Raymond L. Wilder,Mathematics
  • Publisher: Courier Corporation
  • ISBN: 0486488209
  • Category: Mathematics
  • Page: 327
  • View: 741
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This classic undergraduate text by an eminent educator acquaints students with the fundamental concepts and methods of mathematics. In addition to introducing many noteworthy historical figures from the eighteenth through the mid-twentieth centuries, the book examines the axiomatic method, set theory, infinite sets, the linear continuum and the real number system, and groups. Additional topics include the Frege-Russell thesis, intuitionism, formal systems, mathematical logic, and the cultural setting of mathematics. Students and teachers will find that this elegant treatment covers a vast amount of material in a single reasonably concise and readable volume. Each chapter concludes with a set of problems and a list of suggested readings. An extensive bibliography and helpful indexes conclude the text.

Logic, Foundations of Mathematics, and Computability Theory

Logic, Foundations of Mathematics, and Computability Theory

Part One of the Proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada-1975

  • Author: Robert E. Butts,Jaakko Hintikka
  • Publisher: Springer Science & Business Media
  • ISBN: 9401011389
  • Category: Science
  • Page: 416
  • View: 8067
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The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science.

Surveys in Set Theory

Surveys in Set Theory

  • Author: A. R. D. Mathias
  • Publisher: Cambridge University Press
  • ISBN: 0521277337
  • Category: Mathematics
  • Page: 247
  • View: 3729
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This book comprises five expository articles and two research papers on topics of current interest in set theory and the foundations of mathematics. Articles by Baumgartner and Devlin introduce the reader to proper forcing. This is a development by Saharon Shelah of Cohen's method which has led to solutions of problems that resisted attack by forcing methods as originally developed in the 1960s. The article by Guaspari is an introduction to descriptive set theory, a subject that has developed dramatically in the last few years. Articles by Kanamori and Stanley discuss one of the most difficult concepts in contemporary set theory, that of the morass, first created by Ronald Jensen in 1971 to solve the gap-two conjecture in model theory, assuming Gödel's axiom of constructibility. The papers by Prikry and Shelah complete the volume by giving the reader the flavour of contemporary research in set theory. This book will be of interest to graduate students and research workers in set theory and mathematical logic.

Practical Foundations of Mathematics

Practical Foundations of Mathematics

  • Author: Paul Taylor
  • Publisher: Cambridge University Press
  • ISBN: 9780521631075
  • Category: Mathematics
  • Page: 572
  • View: 9690
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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.