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Set Theory for the Working Mathematician : London Mathematical Society Student Texts - Krzysztof Ciesielski

Set Theory for the Working Mathematician

London Mathematical Society Student Texts

Hardcover

Published: 28th August 1997
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This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.

' ... the author has produced a very valuable resource for the working mathematician. Postgraduates and established researchers in many (perhaps all) areas of mathematics will benefit from reading it.' Ian Tweddle, Proceedings of the Edinburgh Mathematical Society

Prefacep. ix
Basics of set theoryp. 1
Axiomatic set theoryp. 3
Why axiomatic set theory?p. 3
The language and the basic axiomsp. 6
Relations, functions, and Cartesian productp. 12
Relations and the axiom of choicep. 12
Functions and the replacement scheme axiomp. 16
Generalized union, intersection, and Cartesian productp. 19
Partial- and linear-order relationsp. 21
Natural numbers, integers, and real numbersp. 25
Natural numbersp. 25
Integers and rational numbersp. 30
Real numbersp. 31
Fundamental tools of set theoryp. 35
Well orderings and transfinite inductionp. 37
Well-ordered sets and the axiom of foundationp. 37
Ordinal numbersp. 44
Definitions by transfinite inductionp. 49
Zorn's lemma in algebra, analysis, and topologyp. 54
Cardinal numbersp. 61
Cardinal numbers and the continuum hypothesisp. 61
Cardinal arithmeticp. 68
Cofinalityp. 74
The power of recursive definitionsp. 77
Subsets of R[superscript n]p. 79
Strange subsets of R[superscript n] and the diagonalization argumentp. 79
Closed sets and Borel setsp. 89
Lebesgue-measurable sets and sets with the Baire propertyp. 98
Strange real functionsp. 104
Measurable and nonmeasurable functionsp. 104
Darboux functionsp. 106
Additive functions and Hamel basesp. 111
Symmetrically discontinuous functionsp. 118
When induction is too shortp. 127
Martin's axiomp. 129
Rasiowa-Sikorski lemmap. 129
Martin's axiomp. 139
Suslin hypothesis and diamond principlep. 154
Forcingp. 164
Elements of logic and other forcing preliminariesp. 164
Forcing method and a model for [not sign]CHp. 168
Model for CH and [diamonds suit symbol]p. 182
Product lemma and Cohen modelp. 189
Model for MA+[not sign]CHp. 196
Axioms of set theoryp. 211
Comments on the forcing methodp. 215
Notationp. 220
Referencesp. 225
Indexp. 229
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521594417
ISBN-10: 0521594413
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 252
Published: 28th August 1997
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 1.75
Weight (kg): 0.54