Axiomatic Set Theory

Author : G. Takeuti,W.M. Zaring
Publisher : Springer Science & Business Media
Page : 244 pages
File Size : 41,7 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781468487510

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Axiomatic Set Theory by G. Takeuti,W.M. Zaring Pdf

This text deals with three basic techniques for constructing models of Zermelo-Fraenkel set theory: relative constructibility, Cohen's forcing, and Scott-Solovay's method of Boolean valued models. Our main concern will be the development of a unified theory that encompasses these techniques in one comprehensive framework. Consequently we will focus on certain funda mental and intrinsic relations between these methods of model construction. Extensive applications will not be treated here. This text is a continuation of our book, "I ntroduction to Axiomatic Set Theory," Springer-Verlag, 1971; indeed the two texts were originally planned as a single volume. The content of this volume is essentially that of a course taught by the first author at the University of Illinois in the spring of 1969. From the first author's lectures, a first draft was prepared by Klaus Gloede with the assistance of Donald Pelletier and the second author. This draft was then rcvised by the first author assisted by Hisao Tanaka. The introductory material was prepared by the second author who was also responsible for the general style of exposition throughout the text. We have inc1uded in the introductory material al1 the results from Boolean algebra and topology that we need. When notation from our first volume is introduced, it is accompanied with a deflnition, usually in a footnote. Consequently a reader who is familiar with elementary set theory will find this text quite self-contained.

Author : Patrick Suppes
Publisher : Courier Corporation
Page : 265 pages
File Size : 54,5 Mb
Release : 2012-05-04
Category : Mathematics
ISBN : 9780486136875

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Axiomatic Set Theory by Patrick Suppes Pdf

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.

Author : G. Takeuti,W.M. Zaring
Publisher : Springer Science & Business Media
Page : 251 pages
File Size : 50,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461381686

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Introduction to Axiomatic Set Theory by G. Takeuti,W.M. Zaring Pdf

In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are high lighted, and second, the student who wishes to master the subject is com pelled to develop the detail on his own. However, an instructor using a "fast development" text must devote much class time to assisting his students in their efforts to bridge gaps in the text.

Author : J.L. Krivine
Publisher : Springer Science & Business Media
Page : 108 pages
File Size : 48,8 Mb
Release : 2012-12-06
Category : Philosophy
ISBN : 9789401031448

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Introduction to Axiomatic Set Theory by J.L. Krivine Pdf

This book presents the classic relative consistency proofs in set theory that are obtained by the device of 'inner models'. Three examples of such models are investigated in Chapters VI, VII, and VIII; the most important of these, the class of constructible sets, leads to G6del's result that the axiom of choice and the continuum hypothesis are consistent with the rest of set theory [1]I. The text thus constitutes an introduction to the results of P. Cohen concerning the independence of these axioms [2], and to many other relative consistency proofs obtained later by Cohen's methods. Chapters I and II introduce the axioms of set theory, and develop such parts of the theory as are indispensable for every relative consistency proof; the method of recursive definition on the ordinals being an import ant case in point. Although, more or less deliberately, no proofs have been omitted, the development here will be found to require of the reader a certain facility in naive set theory and in the axiomatic method, such e as should be achieved, for example, in first year graduate work (2 cycle de mathernatiques).

Author : Kenneth Kunen
Publisher : Unknown
Page : 251 pages
File Size : 48,5 Mb
Release : 2009
Category : Mathematics
ISBN : 1904987141

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The Foundations of Mathematics by Kenneth Kunen Pdf

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.

Author : Paul Richard Halmos
Publisher : Unknown
Page : 122 pages
File Size : 52,7 Mb
Release : 1960
Category : Mathematics
ISBN : UOM:39015006570702

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Naive Set Theory by Paul Richard Halmos Pdf

From the Reviews: .,."He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. ...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." Philosophy and Phenomenological Research

Author : Kenneth Kunen
Publisher : Unknown
Page : 0 pages
File Size : 54,6 Mb
Release : 1983
Category : Electronic
ISBN : 0720422000

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Set Theory by Kenneth Kunen Pdf

Author : Edward John Lemmon
Publisher : Unknown
Page : 156 pages
File Size : 51,6 Mb
Release : 1969
Category : Axiomatic set theory
ISBN : UCAL:B4248817

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Introduction to Axiomatic Set Theory by Edward John Lemmon Pdf

Author : Michael Potter
Publisher : Clarendon Press
Page : 362 pages
File Size : 55,7 Mb
Release : 2004-01-15
Category : Philosophy
ISBN : 9780191556432

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Set Theory and its Philosophy by Michael Potter Pdf

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Author : R.B. Chuaqui
Publisher : Newnes
Page : 385 pages
File Size : 47,5 Mb
Release : 2011-08-30
Category : Mathematics
ISBN : 0080871623

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Axiomatic Set Theory by R.B. Chuaqui Pdf

Axiomatic Set Theory

Author : Paul Bernays
Publisher : Courier Dover Publications
Page : 244 pages
File Size : 53,6 Mb
Release : 1958
Category : Axiomatic set theory
ISBN : UCAL:B4249731

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Axiomatic Set Theory by Paul Bernays Pdf

Author : Lorenz J. Halbeisen
Publisher : Springer
Page : 594 pages
File Size : 45,5 Mb
Release : 2017-12-20
Category : Mathematics
ISBN : 9783319602318

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Combinatorial Set Theory by Lorenz J. Halbeisen Pdf

This book, now in a thoroughly revised second edition, provides a comprehensive and accessible introduction to modern set theory. Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin’s problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah’s astonishing construction of a model with finitely many Ramsey ultrafilters. Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.

Author : Karel Hrbacek,Thomas J. Jech
Publisher : Unknown
Page : 272 pages
File Size : 53,5 Mb
Release : 1984
Category : Mathematics
ISBN : UOM:39076000787080

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Introduction to Set Theory by Karel Hrbacek,Thomas J. Jech Pdf

Author : Daniel W. Cunningham
Publisher : Cambridge University Press
Page : 265 pages
File Size : 43,7 Mb
Release : 2016-07-18
Category : Mathematics
ISBN : 9781107120327

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Set Theory by Daniel W. Cunningham Pdf

Set theory can be considered a unifying theory for mathematics. This book covers the fundamentals of the subject.

Author : Thomas J. Jech
Publisher : American Mathematical Soc.
Page : 232 pages
File Size : 53,8 Mb
Release : 1971
Category : Axiomatic set theory
ISBN : 9780821802465

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Axiomatic Set Theory, Part 2 by Thomas J. Jech Pdf