Few mathematical results capture the imagination like Georg Cantor's groundbreaking work on infinity in the late nineteenth century. This opened the door to an intricate axiomatic theory of sets which was born in the decades that followed. Written for the motivated novice, this book provides an overview of key ideas in set theory, bridging the gap between technical accounts of mathematical foundations and popular accounts of logic. Readers will learn of the formal construction of the classical number systems, from the natural numbers to the real numbers and beyond, and see how set theory has evolved to analyse such deep questions as the status of the continuum hypothesis and the axiom of choice. Remarks and digressions introduce the reader to some of the philosophical aspects of the subject and to adjacent mathematical topics. The rich, annotated bibliography encourages the dedicated reader to delve into what is now a vast literature.
Barnaby Sheppard is a freelance writer. He has previously held positions at Lancaster University, the University of Durham and University College Dublin.
Preface; Synopsis; 1. Introduction; 2. Logical foundations; 3. Avoiding Russell's paradox; 4. Further axioms; 5. Relations and order; 6. Ordinal numbers and the axiom of infinity; 7. Infinite arithmetic; 8. Cardinal numbers; 9. The axiom of choice and the continuum hypothesis; 10. Models; 11. From Goedel to Cohen; Appendix A. Peano arithmetic; Appendix B. Zermelo-Fraenkel set theory; Appendix C. Goedel's incompleteness theorems; Bibliography; Index.