The Foundations of Mathematics

This excellent book is designed to prepare students who have completed at least one calculus course for the abstraction of higher level mathematics courses. The primary goal of the text is to teach students how to read and do proofs but the author also introduces the student to far more. Chapter 1 introduces the logical language required for proofs and mathematics in general. Chapter 2 lays the groundwork for the remainder of the text and additional proof based courses by providing the reader with a comprehensive introduction to proofs including direct proofs, contrapositive and contradiction, existence and uniqueness, and mathematical induction. The author covers functions and relations in chapters 3 and 4. Chapter 5 provides the student with an excellent introduction to infinite sets to include the axiom of choice. A brief introduction to discrete mathematics is presented in chapter 6. Introduction to abstract algebra is covered in chapter 7 and the student is provided with a survey of groups, rings and fields, lattices and homomorphisms. Chapter 8 provides a top-notch overview of analysis including Zeno's paradoxes, limits of fuctions, continuous functions and counterexamples, sequences and series, and a nice section on discrete dynamical systems (elementary chaos theory). The author concludes the book with a well written and very informative chapter on metamathematics and philosophy of mathematics where he examines such topics as metalogic, Godel's Completeness and Incompleteness Theorems, philosophies of mathematics such as Platonism, Kant, Embodied Theories, etc. Best of all, the book is very affordable and should meet any student's budget requirements. In addition, each chapter directs the reader to highly informative references. My only complaint is that the author should have provided more solutions to the problems or a website offering additional solutions and/or hints. The book would also well serve any reader who wants to engage in self-study. In summary, the author has produced a well written, easy to follow book for the undergraduate student who is ready to transition from calculus to higher level proof based mathematics courses. And the publisher has not only provided an attractive, well laid-out text but has made it very affordable! Two years ago when I wrote this review, this book was selling for $55 but the price has nearly tripled so the last part of the previous sentence no longer applies. Still an excellent book!

This book was used for my mathematics proof class. It is very solid as a transitional book. I had originally bough Chartrand to help me learn the subject, but I would say this book has more opportunity for complete understanding through practice problems. Overall, it was a great book, BUT it did cause a little trouble when discussing images and pre-images. It does well to discuss images, but it does lack--particularly--an important note about the functional image of an empty set. As a result, I missed a question on that subject on my test. But again, it is very solid and it has quite a bit of content! It is more than just an intro to advanced math. It is also a great primer in to further math such as analysis, modern algebra, and discrete math.So, is this the best book on the subject? Probably. :) Great job Sibley!

nice explanation of true false logic scenarios

Easy to follow book. Good problems

This book is awesome. I used it for a course introducing students to advanced mathematics. Of course it has some flaws, but overall I loved the approach and material. But it's a shame that this book is so expensive and that so few students have the opportunity to take a course like this.