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Author: Stevo Todorcevic
Publisher: World Scientific
ISBN: 9814571598
Category : Mathematics
Languages : en
Pages : 234
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Book Description
In the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths.
Author: Stevo Todorcevic
Publisher: World Scientific
ISBN: 9814571598
Category : Mathematics
Languages : en
Pages : 234
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Book Description
In the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths.
Author: Mohamed Bekkali
Publisher: Springer
ISBN: 9783540541219
Category : Mathematics
Languages : en
Pages : 128
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Book Description
During the Fall Semester of 1987, Stevo Todorcevic gave a series of lectures at the University of Colorado. These notes of the course, taken by the author, give a novel and fast exposition of four chapters of Set Theory. The first two chapters are about the connection between large cardinals and Lebesque measure. The third is on forcing axioms such as Martin's axiom or the Proper Forcing Axiom. The fourth chapter looks at the method of minimal walks and p-functions and their applications. The book is addressed to researchers and graduate students interested in Set Theory, Set-Theoretic Topology and Measure Theory.
Author: Lorenz J. Halbeisen
Publisher: Springer
ISBN: 3319602314
Category : Mathematics
Languages : en
Pages : 586
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Book Description
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: Thomas J. Jech
Publisher:
ISBN:
Category : Set theory
Languages : en
Pages : 137
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Book Description
Author: Kenneth Kunen
Publisher:
ISBN: 9781904987147
Category : Mathematics
Languages : en
Pages : 251
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Book Description
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: W. Hugh Woodin
Publisher: Walter de Gruyter
ISBN: 3110804735
Category : Mathematics
Languages : en
Pages : 944
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Book Description
The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.
Author: Thomas J. Jech
Publisher: Courier Corporation
ISBN: 0486466248
Category : Mathematics
Languages : en
Pages : 226
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Book Description
Comprehensive and self-contained text examines the axiom's relative strengths and consequences, including its consistency and independence, relation to permutation models, and examples and counterexamples of its use. 1973 edition.
Author: K. Kunen
Publisher: Elsevier
ISBN: 148329515X
Category : Mathematics
Languages : en
Pages : 1282
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Book Description
This Handbook is an introduction to set-theoretic topology for students in the field and for researchers in other areas for whom results in set-theoretic topology may be relevant. The aim of the editors has been to make it as self-contained as possible without repeating material which can easily be found in standard texts. The Handbook contains detailed proofs of core results, and references to the literature for peripheral results where space was insufficient. Included are many open problems of current interest.In general, the articles may be read in any order. In a few cases they occur in pairs, with the first one giving an elementary treatment of a subject and the second one more advanced results. These pairs are: Hodel and Juhász on cardinal functions; Roitman and Abraham-Todorčević on S- and L-spaces; Weiss and Baumgartner on versions of Martin's axiom; and Vaughan and Stephenson on compactness properties.
Author: Nik Weaver
Publisher: World Scientific
ISBN: 9814566020
Category : Mathematics
Languages : en
Pages : 153
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Book Description
Ever since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics.
Author: Saharon Shelah
Publisher: Cambridge University Press
ISBN: 1107168368
Category : Mathematics
Languages : en
Pages : 1069
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Book Description
This book presents the theory of proper forcing and its relatives from the beginning. No prior knowledge of forcing is required.
