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Author : Robert Lee Moore
Publisher : American Mathematical Soc.
Page : 434 pages
File Size : 41,7 Mb
Release : 1932-12-31
Category : Mathematics
ISBN : 9780821810132
Author : Robert Lee Moore
Publisher : American Mathematical Soc.
Page : 434 pages
File Size : 40,7 Mb
Release : 1932
Category : Set theory
ISBN : 9780821874509
Author : Douglas Cenzer,Jean Larson,Christopher Porter,Jindrich Zapletal
Publisher : World Scientific
Page : 222 pages
File Size : 52,9 Mb
Release : 2020-04-04
Category : Mathematics
ISBN : 9789811201943
This book provides an introduction to axiomatic set theory and descriptive set theory. It is written for the upper level undergraduate or beginning graduate students to help them prepare for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra.The book is designed as a flexible and accessible text for a one-semester introductory course in set theory, where the existing alternatives may be more demanding or specialized. Readers will learn the universally accepted basis of the field, with several popular topics added as an option. Pointers to more advanced study are scattered throughout the text.
Author : A.A. Fraenkel,Y. Bar-Hillel,A. Levy
Publisher : Elsevier
Page : 415 pages
File Size : 52,6 Mb
Release : 1973-12-01
Category : Computers
ISBN : 9780080887050
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.
Author : Luca Incurvati
Publisher : Cambridge University Press
Page : 255 pages
File Size : 48,6 Mb
Release : 2020-01-23
Category : History
ISBN : 9781108497824
Presents a detailed and critical examination of the available conceptions of set and proposes a novel version.
Author : Abhijit Dasgupta
Publisher : Springer Science & Business Media
Page : 434 pages
File Size : 42,7 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461488545
What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.
Author : Jose Ferreiros
Publisher : Springer Science & Business Media
Page : 472 pages
File Size : 50,6 Mb
Release : 2001-11-01
Category : Mathematics
ISBN : 3764357495
"José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced, and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach in mathematics from the early nineteenth century. This takes up Part One of the book. Part Two analyzes the crucial developments in the last quarter of the nineteenth century, above all the work of Cantor, but also Dedekind and the interaction between the two. Lastly, Part Three details the development of set theory up to 1950, taking account of foundational questions and the emergence of the modern axiomatization." (Bulletin of Symbolic Logic)
Author : Sean Morris
Publisher : Cambridge University Press
Page : 221 pages
File Size : 46,6 Mb
Release : 2018-12-13
Category : History
ISBN : 9781107152502
Provides an accessible mathematical and philosophical account of Quine's set theory, New Foundations.
Author : Keith Devlin
Publisher : Springer Science & Business Media
Page : 204 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209034
This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naïve" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science.
Author : Felix Hausdorff
Publisher : American Mathematical Soc.
Page : 358 pages
File Size : 46,7 Mb
Release : 2005
Category : Mathematics
ISBN : 0821838350
In the early twentieth century, Hausdorff developed an axiomatic approach to topology, which continues to be the foundation of modern topology. The present book, the English translation of the third edition of Hausdorff's Mengenlehre, is a thorough introduction to his theory of point-set topology. The treatment begins with topics in the foundations of mathematics, including the basics of abstract set theory, sums and products of sets, cardinal and ordinal numbers, and Hausdorff's well-ordering theorem. The exposition then specializes to point sets, where major topics such as Borel systems, first and second category, and connectedness are considered in detail. Next, mappings between spaces are introduced. This leads naturally to a discussion of topological spaces and continuous mappings between them. Finally, the theory is applied to the study of real functions and their properties. The book does not presuppose any mathematical knowledge beyond calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first-year graduate students should have no difficulty in making the material their own.
Author : Kenneth Kunen
Publisher : Unknown
Page : 251 pages
File Size : 52,9 Mb
Release : 2009
Category : Mathematics
ISBN : 1904987141
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 : John D. Baum
Publisher : Courier Corporation
Page : 164 pages
File Size : 46,8 Mb
Release : 1991-01-01
Category : Mathematics
ISBN : 9780486668260
Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.
Author : Charles Arthur Coppin,W. Ted Mahavier,E. Lee May,Edgar Parker
Publisher : MAA
Page : 262 pages
File Size : 46,7 Mb
Release : 2009
Category : Education
ISBN : 0883851857
The Moore method is a type of instruction used in advanced mathematics courses that moves away from a teacher-oriented experience to a learner-centered one. This book gives an overview of the Moore Method as practiced by the four authors. The authors outline six principles they all have as goals : elevating students from recipients to creators of knowledge; letting students discover the power of their minds; believing every student can and will do mathematics; allowing students to discover, present and debate mathematics; carefully matching problems and materials to the students; and having the material cover a significant body of knowledge. Topics include establishing a classroom culture, grading methods, materials development and more. Appendices include sample tests, notes and diaries of individual courses.
Author : F. William Lawvere,Robert Rosebrugh
Publisher : Cambridge University Press
Page : 280 pages
File Size : 44,7 Mb
Release : 2003-01-27
Category : Mathematics
ISBN : 0521010608
In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
Author : Herbert B. Enderton
Publisher : Academic Press
Page : 279 pages
File Size : 46,9 Mb
Release : 1977-05-23
Category : Mathematics
ISBN : 9780080570426
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.
