# Set Theory

## With an Introduction to Real Point Sets

• Abhijit Dasgupta
Book

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. Abhijit Dasgupta
Pages 67-72
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
5. Abhijit Dasgupta
Pages 301-311

### 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