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The rapid development of set theory in the last fifty years, mainly by obtaining plenty of independence results, strongly influenced an understanding of the structure of the real line. This book is devoted to the study of the real line and its subsets taking into account the recent results of set theory. Whenever possible the presentation is done without the full axiom of choice. Since the book is intended to be self-contained, all necessary results of set theory, topology, measure theory, and descriptive set theory are revisited with the purpose of eliminating superfluous use of an axiom of choice. The duality of measure and category is studied in a detailed manner. Several statements pertaining to properties of the real line are shown to be undecidable in set theory. The metamathematics behind set theory is shortly explained in the appendix. Each section contains a series of exercises with additional results.
List of contents
Preface.- 1 Introduction.- 2 The Real Line.- 3 Topology of Euclidean Spaces.- 4 Measure Theory.- 5 Useful Tools and Technologies.- 6 Descriptive Set Theory.- 7 Decline and Fall of the Duality.- 8 Special Sets of Reals.- 9 Additional Axioms.- 10 Undecidable Statements.- 11 Appendix.- Bibliography.- Index of Notation.- Index.
Summary
The rapid development of set theory in the last fifty years, mainly by obtaining plenty of independence results, strongly influenced an understanding of the structure of the real line. This book is devoted to the study of the real line and its subsets taking into account the recent results of set theory. Whenever possible the presentation is done without the full axiom of choice. Since the book is intended to be self-contained, all necessary results of set theory, topology, measure theory, and descriptive set theory are revisited with the purpose of eliminating superfluous use of an axiom of choice. The duality of measure and category is studied in a detailed manner. Several statements pertaining to properties of the real line are shown to be undecidable in set theory. The metamathematics behind set theory is shortly explained in the appendix. Each section contains a series of exercises with additional results.
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From the reviews:
“The history of investigations of real numbers is the starting point of this monograph. … Every chapter is supplied with a well-chosen series of exercises containing interesting complementary material. … This unique monograph presents beautiful scientific work prepared in a professional style and built upon a number of references to prominent mathematicians living in previous epochs … . A wide variety of researchers and students will find in it a large … attractive body of knowledge that should be highly inspiring in their work.” (Marek Balcerzak, Mathematical Reviews, Issue 2012 d)
“The book under review is addressed to experts and begins to address some of the questions about the complexities of the real line. … The author provides an appendix with background material on set theory, algebra, topology and metamathematics.” (William J. Satzer, The Mathematical Association of America, August, 2011)
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From the reviews:
"The history of investigations of real numbers is the starting point of this monograph. ... Every chapter is supplied with a well-chosen series of exercises containing interesting complementary material. ... This unique monograph presents beautiful scientific work prepared in a professional style and built upon a number of references to prominent mathematicians living in previous epochs ... . A wide variety of researchers and students will find in it a large ... attractive body of knowledge that should be highly inspiring in their work." (Marek Balcerzak, Mathematical Reviews, Issue 2012 d)
"The book under review is addressed to experts and begins to address some of the questions about the complexities of the real line. ... The author provides an appendix with background material on set theory, algebra, topology and metamathematics." (William J. Satzer, The Mathematical Association of America, August, 2011)
