Basic Set Theory - Joan Bagaria

This book provides a rigorous, step-by-step, and detailed introduction to axiomatic set theory within the standard framework of Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). It is suitable as a textbook for a one-semester course in set theory at the senior undergraduate or first-year graduate level.

Although the presentation is self-contained, the text is best suited for students in mathematics, philosophy, computer science, or linguistics who have some prior background in logic. The book covers all standard material, including a thorough construction of the natural numbers and the linear continuum, followed by the basic theory of transfinite ordinals and cardinals, and concluding with an overview of the general structure of the set-theoretic universe. Each chapter begins with an introduction and ends with a carefully designed set of exercises aimed at reinforcing the material and developing proof-writing skills.

Based on a course taught by the authors at the University of Barcelona over the past 12 years, the book has been refined over time to ensure clarity and accessibility. In addition to serving as a course textbook, it contains the core material in set theory typically required by working mathematicians, so can also be used as a reference for mathematicians and mathematically inclined philosophers.