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: 7614
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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: 3236
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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.

Handbook of Proof Theory

Handbook of Proof Theory

  • Author: S.R. Buss
  • Publisher: Elsevier
  • ISBN: 9780080533186
  • Category: Mathematics
  • Page: 810
  • View: 8180
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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.

Categorical Logic and Type Theory

Categorical Logic and Type Theory

  • Author: Bart Jacobs
  • Publisher: Gulf Professional Publishing
  • ISBN: 9780444508539
  • Category: Mathematics
  • Page: 760
  • View: 308
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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.

Set Theory

Set Theory

  • Author: Kenneth Kunen
  • Publisher: N.A
  • ISBN: 9781848900509
  • Category: Mathematics
  • Page: 412
  • View: 9465
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This book is designed for readers who know elementary mathematical logic and axiomatic set theory, and who want to learn more about set theory. The primary focus of the book is on the independence proofs. Most famous among these is the independence of the Continuum Hypothesis (CH); that is, there are models of the axioms of set theory (ZFC) in which CH is true, and other models in which CH is false. More generally, cardinal exponentiation on the regular cardinals can consistently be anything not contradicting the classical theorems of Cantor and Konig. The basic methods for the independence proofs are the notion of constructibility, introduced by Godel, and the method of forcing, introduced by Cohen. This book describes these methods in detail, verifi es the basic independence results for cardinal exponentiation, and also applies these methods to prove the independence of various mathematical questions in measure theory and general topology. Before the chapters on forcing, there is a fairly long chapter on "infi nitary combinatorics." This consists of just mathematical theorems (not independence results), but it stresses the areas of mathematics where set-theoretic topics (such as cardinal arithmetic) are relevant. There is, in fact, an interplay between infi nitary combinatorics and independence proofs. Infi nitary combinatorics suggests many set-theoretic questions that turn out to be independent of ZFC, but it also provides the basic tools used in forcing arguments. In particular, Martin's Axiom, which is one of the topics under infi nitary combinatorics, introduces many of the basic ingredients of forcing.

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: 6190
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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.

Undecidable Theories

Undecidable Theories

  • Author: Alfred Tarski,Andrzej Mostowski,Raphael Mitchel Robinson
  • Publisher: Elsevier
  • ISBN: 0444533788
  • Category: Decidability (Mathematical logic)
  • Page: 98
  • View: 6593
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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: 9926
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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.

Equivalents of the Axiom of Choice, II

Equivalents of the Axiom of Choice, II

  • Author: H. Rubin,J.E. Rubin
  • Publisher: Elsevier
  • ISBN: 9780080887654
  • Category: Mathematics
  • Page: 321
  • View: 9688
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This monograph contains a selection of over 250 propositions which are equivalent to AC. The first part on set forms has sections on the well-ordering theorem, variants of AC, the law of the trichotomy, maximal principles, statements related to the axiom of foundation, forms from algebra, cardinal number theory, and a final section of forms from topology, analysis and logic. The second part deals with the axiom of choice for classes - well-ordering theorem, choice and maximal principles.

The Foundations of Mathematics in the Theory of Sets

The Foundations of Mathematics in the Theory of Sets

  • Author: John P. Mayberry
  • Publisher: Cambridge University Press
  • ISBN: 9780521770347
  • Category: Mathematics
  • Page: 424
  • View: 9825
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This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

Descriptive Set Theory

Descriptive Set Theory

  • Author: Yiannis N. Moschovakis
  • Publisher: American Mathematical Soc.
  • ISBN: 0821848135
  • Category: Mathematics
  • Page: 502
  • View: 9922
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Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern ``effective'' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.

Problems and Theorems in Classical Set Theory

Problems and Theorems in Classical Set Theory

  • Author: Peter Komjath,Vilmos Totik
  • Publisher: Springer Science & Business Media
  • ISBN: 0387362193
  • Category: Mathematics
  • Page: 516
  • View: 1929
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This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

Discovering Modern Set Theory: The basics

Discovering Modern Set Theory: The basics

  • Author: Winfried Just,Martin Weese
  • Publisher: American Mathematical Soc.
  • ISBN: 0821802666
  • Category: Mathematics
  • Page: 210
  • View: 2054
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This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.

The Logical Foundations of Mathematics

The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series

  • Author: William S. Hatcher
  • Publisher: Elsevier
  • ISBN: 1483189635
  • Category: Mathematics
  • Page: 330
  • View: 3242
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The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

Constructivism in Mathematics

Constructivism in Mathematics

  • Author: A.S. Troelstra,D. van Dalen
  • Publisher: Elsevier
  • ISBN: 008095510X
  • Category: Mathematics
  • Page: 129
  • View: 5384
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Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.

Principia Mathematica

Principia Mathematica

  • Author: Alfred North Whitehead,Bertrand Russell
  • Publisher: N.A
  • ISBN: N.A
  • Category: Logic, Symbolic and mathematical
  • Page: N.A
  • View: 1905
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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: 2482
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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.

Studies in Logic and the Foundations of Mathematics - The Axiomatic Method with Special Reference to Geometry and Physics

Studies in Logic and the Foundations of Mathematics - The Axiomatic Method with Special Reference to Geometry and Physics

  • Author: L. Brouwer
  • Publisher: Brouwer Press
  • ISBN: 1443728128
  • Category: Mathematics
  • Page: 504
  • View: 7759
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Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

Practical Foundations of Mathematics

Practical Foundations of Mathematics

  • Author: Paul Taylor
  • Publisher: Cambridge University Press
  • ISBN: 9780521631075
  • Category: Mathematics
  • Page: 572
  • View: 8774
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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 Foundations of Mathematics

The Foundations of Mathematics

  • Author: Ian Stewart,David Orme Tall
  • Publisher: Oxford University Press on Demand
  • ISBN: 9780198531654
  • Category: Fiction
  • Page: 263
  • View: 2249
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"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books