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© 2012

Combinatorial Set Theory

With a Gentle Introduction to Forcing

  • Provides a comprehensive introduction to the sophisticated technique of forcing

  • Complete proofs of famous results are given, for instance, Robinson’s construction of doubling the unit ball using just five pieces

  • Offers an extensive list of references, historical remarks and related results

Book

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-XVI
  2. Topics in Combinatorial Set Theory

    1. Front Matter
      Pages 7-7
    2. Lorenz J. Halbeisen
      Pages 1-6
  3. Topics in Combinatorial Set Theory

    1. Front Matter
      Pages 7-7
    2. Lorenz J. Halbeisen
      Pages 9-24
    3. Lorenz J. Halbeisen
      Pages 25-70
    4. Lorenz J. Halbeisen
      Pages 71-100
    5. Lorenz J. Halbeisen
      Pages 101-141
    6. Lorenz J. Halbeisen
      Pages 143-155
    7. Lorenz J. Halbeisen
      Pages 157-177
    8. Lorenz J. Halbeisen
      Pages 179-199
    9. Lorenz J. Halbeisen
      Pages 201-213
    10. Lorenz J. Halbeisen
      Pages 215-233
    11. Lorenz J. Halbeisen
      Pages 235-255
  4. From Martin’s Axiom to Cohen’s Forcing

    1. Front Matter
      Pages 257-257
    2. Lorenz J. Halbeisen
      Pages 259-261
    3. Lorenz J. Halbeisen
      Pages 263-272
    4. Lorenz J. Halbeisen
      Pages 273-293
    5. Lorenz J. Halbeisen
      Pages 295-303
    6. Lorenz J. Halbeisen
      Pages 305-310

About this book

Introduction

This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing.

The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research.

This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.

Keywords

Axiom of Choice Cardinal Characteristics of the Continuum Combinatorics of Forcing Consistency and Independence Results Continuum Hypothesis Forcing Forcing Technique Infinite Combinatorics Ramsey Theory Set Theory

Authors and affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

Bibliographic information

  • Book Title Combinatorial Set Theory
  • Book Subtitle With a Gentle Introduction to Forcing
  • Authors Lorenz J. Halbeisen
  • Series Title Springer Monographs in Mathematics
  • Series Abbreviated Title Springer Monographs in Mathematics
  • DOI https://doi.org/10.1007/978-1-4471-2173-2
  • Copyright Information Springer-Verlag London Limited 2012
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4471-2172-5
  • Softcover ISBN 978-1-4471-6086-1
  • eBook ISBN 978-1-4471-2173-2
  • Series ISSN 1439-7382
  • Edition Number 1
  • Number of Pages XVI, 456
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Mathematical Logic and Foundations
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking

Reviews

From the reviews:

“The text is very well organised; each chapter ends with useful notes containing much historical background and the source of the main results, together with reports on related results and the references. The beautifully written book is intended for students on graduate courses in axiomatic set theory, and it is also excellent as a text for self-study.” (Peter Shiu, The Mathematical Gazette, Vol. 98 (541), March, 2014)

“The book under review provides a thorough and nicely written account of combinatorial set theory and infinite Ramsey theory together with a treatment of the underlying set theoretical axioms as well as of sophisticated methods which are involved in proving independence results. … I can recommend this book to all graduate students, PostDocs, and researchers who are interested in set theoretical combinatorics … . also mathematicians from other areas who are interested in the foundational aspects of their subject will enjoy this book.” (Ralf Schindler, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2013)

“This book provides a self-contained introduction to axiomatic set theory with main focus on infinitary combinatorics and the forcing technique. It is intended as a textbook in courses as well as for self-study. … The author gives the historical background and the sources of the main results in the Notes of each chapter. He also gives hints for further studies in his sections ‘Related results’.” (Martin Weese, Zentralblatt MATH, Vol. 1237, 2012)