Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

Table of Contents

1 Introduction

2 Basic Set-Building Axioms and Operations

3 Relations and Functions

4 The Natural Numbers

5 On the Size of Sets

6 Transfinite Recursion

7 The Axiom of Choice (Revisited)

8 Ordinals

9 Cardinals

Review

"As far as I can see, Cunningham neglects no opportunity to make the subject as accessible as possible. The mathematical development is rigorous, as it should be, but not excessively so. ... Throughout the book, he offers many appropriate examples (or non-examples), and provides numerous and diverse exercises, which often prove results that are later used in the body of the text, drawing the reader into the subject. ... In short, this is an excellent book! You will know a good deal about basic set theory after working through it." Frederic Green, September 2017, ACM SIGACT News 48(3):7-9

"This introductory textbook on set theory clearly lays out the fundamental concepts of the topic. The overall format of this text appeals to a fairly `naive' reader with clearly exposited proofs that contain complete detail as well as a rigor that will be appreciated by a more sophisticated student. ... This book fulfills its stated goals: 'The textbook is suitable for a broad range of readers, from undergraduate to graduate students, who desire a better understanding of the fundamental topics in set theory that may have been, or will be, overlooked in their other mathematics courses.' An understanding of these foundational concepts will benefit any student of mathematics and this textbook would be perfectly suitable for use in an undergraduate course." Shoshana Friedman, MathSciNet

Book Description

Set theory can be considered a unifying theory for mathematics. This book covers the fundamentals of the subject.

About the Author

Daniel W. Cunningham is a Professor of Mathematics at State University of New York, Buffalo, specializing in set theory and mathematical logic. He is a member of the Association for Symbolic Logic, the American Mathematical Society, and the Mathematical Association of America. Cunningham's previous work includes A Logical Introduction to Proof, which was published in 2013.