Set Theory for the Working Mathematician

By: Krzysztof Ciesielski, None

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Paperback | 27 November 1997

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252 Pages

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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.

Industry Reviews

' ... 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

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Preface	p. ix
Basics of set theory	p. 1
Axiomatic set theory	p. 3
Why axiomatic set theory?	p. 3
The language and the basic axioms	p. 6
Relations, functions, and Cartesian product	p. 12
Relations and the axiom of choice	p. 12
Functions and the replacement scheme axiom	p. 16
Generalized union, intersection, and Cartesian product	p. 19
Partial- and linear-order relations	p. 21
Natural numbers, integers, and real numbers	p. 25
Natural numbers	p. 25
Integers and rational numbers	p. 30
Real numbers	p. 31
Fundamental tools of set theory	p. 35
Well orderings and transfinite induction	p. 37
Well-ordered sets and the axiom of foundation	p. 37
Ordinal numbers	p. 44
Definitions by transfinite induction	p. 49
Zorn's lemma in algebra, analysis, and topology	p. 54
Cardinal numbers	p. 61
Cardinal numbers and the continuum hypothesis	p. 61
Cardinal arithmetic	p. 68
Cofinality	p. 74
The power of recursive definitions	p. 77
Subsets of R[superscript n]	p. 79
Strange subsets of R[superscript n] and the diagonalization argument	p. 79
Closed sets and Borel sets	p. 89
Lebesgue-measurable sets and sets with the Baire property	p. 98
Strange real functions	p. 104
Measurable and nonmeasurable functions	p. 104
Darboux functions	p. 106
Additive functions and Hamel bases	p. 111
Symmetrically discontinuous functions	p. 118
When induction is too short	p. 127
Martin's axiom	p. 129
Rasiowa-Sikorski lemma	p. 129
Martin's axiom	p. 139
Suslin hypothesis and diamond principle	p. 154
Forcing	p. 164
Elements of logic and other forcing preliminaries	p. 164
Forcing method and a model for [not sign]CH	p. 168
Model for CH and [diamonds suit symbol]	p. 182
Product lemma and Cohen model	p. 189
Model for MA+[not sign]CH	p. 196
Axioms of set theory	p. 211
Comments on the forcing method	p. 215
Notation	p. 220
References	p. 225
Index	p. 229
Table of Contents provided by Syndetics. All Rights Reserved.

Set Theory for the Working Mathematician

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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