Set theory for the working mathematician / Krzysztof Ciesielski.

Format:

• Book

Author/Creator:

• Ciesielski, Krzysztof, 1957-

Series:

• London Mathematical Society student texts ; 39.
• London Mathematical Society student texts ; 39

Language:

English

Subjects (All):

• Set theory.

Physical Description:

xi, 236 pages ; 24 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, [1997]

Summary:

• 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 undergraduate students, this book should also interest researchers wanting to learn more about set-theoretic techniques applicable to their fields.

Contents:

• Part I Basics of set theory 1
• 1 Axiomatic set theory 3
• 1.1 Why axiomatic set theory? 3
• 1.2 The language and the basic axioms 6
• 2 Relations, functions, and Cartesian product 12
• 2.1 Relations and the axiom of choice 12
• 2.2 Functions and the replacement scheme axiom 16
• 2.3 Generalized union, intersection, and Cartesian product 19
• 2.4 Partial- and linear-order relations 21
• 3 Natural numbers, integers, and real numbers 25
• 3.1 Natural numbers 25
• 3.2 Integers and rational numbers 30
• 3.3 Real numbers 31
• Part II Fundamental tools of set theory 35
• 4 Well orderings and transfinite induction 37
• 4.1 Well-ordered sets and the axiom of foundation 37
• 4.2 Ordinal numbers 44
• 4.3 Definitions by transfinite induction 49
• 4.4 Zorn's lemma in algebra, analysis, and topology 54
• 5 Cardinal numbers 61
• 5.1 Cardinal numbers and the continuum hypothesis 61
• 5.2 Cardinal arithmetic 68
• 5.3 Cofinality 74
• Part III The power of recursive definitions 77
• 6 Subsets of R[superscript n] 79
• 6.1 Strange subsets of R[superscript n] and the diagonalization argument 79
• 6.2 Closed sets and Borel sets 89
• 6.3 Lebesgue-measurable sets and sets with the Baire property 98
• 7 Strange real functions 104
• 7.1 Measurable and nonmeasurable functions 104
• 7.2 Darboux functions 106
• 7.3 Additive functions and Hamel bases 111
• 7.4 Symmetrically discontinuous functions 118
• Part IV When induction is too short 127
• 8 Martin's axiom 129
• 8.1 Rasiowa-Sikorski lemma 129
• 8.2 Martin's axiom 139
• 8.3 Suslin hypothesis and diamond principle 154
• 9 Forcing 164
• 9.1 Elements of logic and other forcing preliminaries 164
• 9.2 Forcing method and a model for [not sign]CH 168
• 9.3 Model for CH and [diamonds suit symbol] 182
• 9.4 Product lemma and Cohen model 189
• 9.5 Model for MA+[not sign]CH 196
• A Axioms of set theory 211
• B Comments on the forcing method 215.

Notes:

Includes bibliographical references (pages 225-227) and index.

ISBN:

• 0521594413
• 0521594650

OCLC:

36621835