Set Theory

With an Introduction to Real Point Sets

  • Abhijit Dasgupta

Table of contents

  1. Front Matter
    Pages i-xv
  2. Abhijit Dasgupta
    Pages 1-23
  3. Dedekind: Numbers

    1. Front Matter
      Pages 25-27
    2. Abhijit Dasgupta
      Pages 29-46
    3. Abhijit Dasgupta
      Pages 47-65
  4. Cantor: Cardinals, Order, and Ordinals

    1. Front Matter
      Pages 73-75
    2. Abhijit Dasgupta
      Pages 77-107
    3. Abhijit Dasgupta
      Pages 109-129
    4. Abhijit Dasgupta
      Pages 131-147
    5. Abhijit Dasgupta
      Pages 149-174
    6. Abhijit Dasgupta
      Pages 175-198
    7. Abhijit Dasgupta
      Pages 199-219
    8. Abhijit Dasgupta
      Pages 221-243
    9. Abhijit Dasgupta
      Pages 245-250
  5. Real Point Sets

    1. Front Matter
      Pages 251-253
    2. Abhijit Dasgupta
      Pages 255-264
    3. Abhijit Dasgupta
      Pages 265-279
    4. Abhijit Dasgupta
      Pages 281-299

About this book

Introduction

What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner.

To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals.

Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.

Keywords

Cantor's Theorem Dedekind's Theorem Frege Zermelo-Fraenkel axiom system order, cardinals, and ordinals set theory, mathematical logic

Authors and affiliations

  • Abhijit Dasgupta
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Detroit MercyDetroitUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-8854-5
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Birkhäuser, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-8853-8
  • Online ISBN 978-1-4614-8854-5
  • About this book