- Basics of set theory
- Axiomatic set theory
- Why axiomatic set theory?
- The language and the basic axioms
- Relations, functions, and Cartesian product
- Relations and the axiom of choice
- Functions and the replacement scheme axiom
- Generalized union, intersection, and Cartesian product
- Partial- and linear-order relations
- Natural numbers, integers, and real numbers
- Natural numbers
- Integers and rational numbers
- Real numbers
- Fundamental tools of set theory
- Well orderings and transfinite induction
- Well-ordered sets and the axiom of foundation
- Ordinal numbers
- Definitions by transfinite induction
- Zorn's lemma in algebra, analysis, and topology
- Cardinal numbers
- Cardinal numbers and the continuum hypothesis
- Cardinal arithmetic
- Cofinality
- The power of recursive definitions
- Subsets of R[superscript n]
- Strange subsets of R[superscript n] and the diagonalization argument
- Closed sets and Borel sets
- Lebesgue-measurable sets and sets with the Baire property
- Strange real functions
- Measurable and nonmeasurable functions
- Darboux functions
- Additive functions and Hamel bases
- Symmetrically discontinuous functions
- When induction is too short
- Martin's axiom
- Rasiowa-Sikorski lemma
- Martin's axiom
- Suslin hypothesis and diamond principle
- Forcing
- Elements of logic and other forcing preliminaries
- Forcing method and a model for [not sign]CH
- Model for CH and [diamonds suit symbol]
- Product lemma and Cohen model
- Model for MA+[not sign]CH.

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.

(source: Nielsen Book Data)